The second variation of the Sine-Gordon theory action about its static kink gives ${\Delta }_x=-{\partial }_x^2+1-2{\mathrm{sech}}^{2}x$. Let $A={\partial }_x+\tanh x$. Then ${\Delta }_x=A^{†}A$ and $A{A}^{†}=-{\partial }_x^2+1$. Thus ${\Delta }_x$ is nonnegative, with its translational zero mode of a sine-Gordon kink and a continuum at ${\omega }^2=k^2+1$. The construction is the supersymmetric factorization of the one-soliton potential.

For $0<v<1$ and $\Gamma =(1-v^2{)}^{-1/2}$, this real two-soliton solution has angular winding $4\pi$. Its separated kink velocities are $\pm v$. At large times the centers satisfy $|m\Gamma x|=|m\Gamma vt|-\log v+o(1)$, giving a right-moving shift $-2\log v/(m\Gamma )$. It follows from Bianchi permutability for sine-Gordon Bäcklund transformations with oppositely signed seed parameters.

This localized two-soliton field has zero net winding and describes an elastic kink-antikink collision for $0<v<1$. Two reciprocal positive Bäcklund parameters produce it from the vacuum. Analytic continuation of $v$ to an imaginary value gives a real Sine-Gordon breather. The overall field sign and spacetime translations change conventions, not the field equation.

The second breather is a bound pair of first breathers with constituent shifts $\pm ia$. The bound-state fusion of factorized S-matrices gives $S_{2,1}=S_{1,1}(\vartheta +ia)S_{1,1}(\vartheta -ia)$, which factors as displayed. The nearest physical-strip pole at $\vartheta =ia$ is a crossed-channel exchange of the first breather. At $n=3$ the more distant center pole is double, without changing that nearest-pole interpretation.

Write the transformed angular field as $v=u+2d$. The Sine-Gordon Bäcklund transformation gives $\sin d=a(u_{\xi }+d_{\xi })$, determining $d$ as a formal local derivative expansion in $a$. The displayed exact current identity yields a conservation law at each order. Formal convergence is unnecessary because each coefficient obeys an exact identity on solutions.

Two compatible Bäcklund steps commute after integration constants are matched. The displayed superposition relation constructs their common output algebraically from the seed and the two one-step outputs, in the reciprocal-parameter convention of the Sine-Gordon Bäcklund transformation. It uses the angular field $u=\beta \varphi$. Smooth inverse-tangent branch continuation is required to retain the correct vacuum labels.

For the Euclidean kinetic normalization $(4\pi {\alpha }^{\prime }{)}^{-1}\int g_{ab}(X)\partial X^a\partial X^b$, vanishing two-form and constant dilaton, the leading beta function is ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}$. The covariant Gaussian fluctuation operator contains a curvature vertex whose one-loop trace contracts to the Ricci tensor. Its logarithmic divergence renormalizes the metric. The scale convention is $\beta =\mu dg/d\mu$; Weyl consistency requires vanishing of this beta function as well as the other anomaly coefficients.

For a scalar equation with strictly convex flux and states $u_l>u_r$, the Rankine-Hugoniot condition determines a discontinuity speed strictly between the right and left characteristic speeds. Characteristic curves enter this compressive discontinuity from both sides. Its entropy flux for a scalar conservation law inequalities make it an entropy solution. It is the limit of a phase-normalized decreasing travelling wave in the vanishing viscosity approximation.

For constant isothermal sound speed, time-independent shearing-sheet tidal potential ${\Phi }_t$, and inviscid flow, define $E=\Sigma u^2/2+c_s^2\Sigma \ln (\Sigma /{\Sigma }_{\mathrm{ref}})+\Sigma {\Phi }_t$. The barotropic energy density and kinetic energy balances combine to give ${\partial }_{t}E+\nabla \cdot [(E+P)\mathbf{u}]=0$. Coriolis acceleration does no work. Integral conservation requires vanishing net boundary energy flux.

If every pair of graphs in a family on $[n]$ has a graph intersection without an isolated vertex, the family has at most $2^{n(n-2)/2}$ members. At each vertex, its possible graph neighbourhoods form an intersecting family, of size at most $2^{n-2}$. Every edge occurs in two such neighbourhood projections. Shearer inequality bounds twice the full information entropy by the sum of their information entropies.

For a presheaf, take families of elements of its stalks which near each point are the germs of one presheaf section. Pointwise operations and restriction make these families into a sheaf. The map sending a section to all its germs is its sheafification map.

Every presheaf morphism from $\mathcal{F}$ to a sheaf $\mathcal{H}$ factors uniquely through the canonical sheafification map. Local representatives give local images in $\mathcal{H}$, and the sheaf gluing axiom makes their union unique.

The direct sum of sheaves of abelian groups is the sheafification of the presheaf $U\mapsto {⨁}_i{\mathcal{F}}_i(U)$. It need not equal that presheaf: sections are locally represented using finitely many summands, while globally infinitely many can occur. For point-supported skyscraper sheaves at closed points, allowable point supports are a locally finite family of subsets.

For $X={\mathbb{C}}^2\setminus \{0\}$, the cover $\mathbb{C}\times {\mathbb{C}}^{\ast }$ and ${\mathbb{C}}^{\ast }\times \mathbb{C}$ is acyclic for ${\mathcal{O}}_X$. The first Čech cohomology quotient is represented uniquely by normally convergent doubly negative Laurent series. Under $w_i=1/z_i$ it is $w_1{w}_2\mathcal{O}({\mathbb{C}}_w^2)$, the entire functions divisible by both coordinates. This is an infinite-dimensional analytic vector space; restricting representatives to finite Laurent polynomials loses classes. The class of $1/(z_1{z}_2)$ is nonzero, so $X$ is not a Stein manifold.

A positive-degree sheaf cohomology class restricts to zero on some neighbourhood of every point, because a representing cocycle in a flasque resolution is locally a boundary. If a basis closed under finite intersections has vanishing local cohomology in degrees $1,\dots ,i-1$, the neighbourhoods can be chosen in that basis so that the image of the class in $H^i(X,{}_U\mathcal{F})$ is zero. Here ${}_U\mathcal{F}$ is the direct image from an open restriction. Lower local vanishing makes the edge map $H^i(X,{}_U\mathcal{F})\to H^i(U,\mathcal{F}{|}_U)$ injective. Local vanishing of a class alone does not imply global vanishing.

If $0\to \mathcal{F}\to {\mathcal{I}}^0\to {\mathcal{I}}^1\to \cdots$ is an exact resolution of sheaves and each ${\mathcal{I}}^j$ has vanishing positive sheaf cohomology, then the cohomology of its global-section complex computes $H^q(X,\mathcal{F})$. Fine sheaves provide acyclic terms on a paracompact manifold, giving the analytic Dolbeault resolution of holomorphic differential forms.

For a player ordering $\pi$, $P_i$ is the set before player $i$, and the displayed entries form its marginal contribution vector. Their sum is $v(N)-v(\varnothing )$. The Shapley value is the average of these vectors. In a convex cooperative game, increasing marginal contributions make each such vector satisfy every coalition constraint of the core of a cooperative game.

With the unnormalized Fourier transform, the scaling function with transform $1_{[-\pi ,\pi )}$ is $\varphi (x)=\sin (\pi x)/(\pi x)$. Its integer translates are orthonormal by orthonormal translates and Fourier periodization. Its refinement mask is the periodic indicator of $[-\pi /2,\pi /2)$, with $a_0=1$ and $a_n=2\sin (n\pi /2)/(\pi n)$. It belongs to $L^2$ but not $L^1$, so its transform is naturally interpreted through the Plancherel theorem.

The wavevectors of a nonzero-frequency Rossby wave in a resting shallow layer satisfyThis follows by completing the square in the shallow-water Rossby-wave dispersion relation. A real circle requires $|\omega |\le |\beta |/(2|a|)$. Its two intersections with a given tangential wavenumber provide the incident and reflected normal wavenumbers at a meridional wall.

A stationary straight meridional wall preserves frequency and tangential wavenumber $l$. The two normal wavenumbers obeyIncidence and reflection are distinguished by the normal group velocity. In the leading quasi-geostrophic approximation, a nonzero $l$ and the impermeability condition give reflected height amplitude $A_r=-A_i$. The $l=0$ no-normal-flow condition is degenerate in that reduced model; exact ageostrophic transport generally changes the amplitude ratio.