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For $H=p^2/(2m)-1/|q|=-|E|$, collision continuation between the two turning points gives $I=\sqrt{2m/|E|}$.

When a Hamiltonian parameter varies slowly compared with the orbital period and no separatrix is crossed, the action changes only at higher adiabatic order.

The action is an adiabatic invariant under these assumptions; other slowly varying systems can have different adiabatic invariants.

Equating the absolute reflected and transmitted displacement amplitudes, then using acoustic Snell's law, gives an algebraic condition on incidence angle and the two moduli-over-speed impedances.

For a P-wave incident at angle $\theta$ from a liquid $(\rho ,c,\lambda )$ onto $({\rho }^{\prime },c^{\prime },{\lambda }^{\prime })$, with transmitted angle ${\theta }^{\prime }$ obeying Snell law for elastic and acoustic waves,No reflection is equivalent to ${\rho }^{\prime }{c}^{\prime }\cos \theta =\rho c\cos {\theta }^{\prime }$.

For a line bundle $L$ on an abelian variety and points $x,y$,

Translation identifies the cotangent sheaf of a group scheme with the constant bundle determined by its cotangent space at the identity:

One form of Mumford's rigidity lemma says that if $X$ is complete, $Y$ is connected, and a morphism $f:X\times Y\to Z$ maps one fiber $X\times \{y_0\}$ to a point, then $f$ factors through the projection to $Y$ under the usual pointed hypotheses.

The root condition is the polynomial criterion equivalent to zero-stability for a linear multistep method.

The seven consecutive zeros followed by one force degree at least eight. Degree eight is attained by $s_{n+8}=s_n$, with initial state $10000000$ and feedback polynomial $x^8+1$.

Shell crossing occurs when the Lagrangian-to-Eulerian map ceases to be one-to-one. Its Jacobian determinant vanishes, the pressureless single-stream density formally diverges, and several streams subsequently occupy the same position.

For an irrotational displacement $\mathbf{s}=-{\nabla }_q\Phi$, the deformation tensor is the symmetric Hessian $d_{ij}=-{\partial }_i{\partial }_j\Phi$. Its real eigenvalues determine the principal collapse directions.

In four dimensions the Yang-Mills theta term is proportional to $\theta \mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$. Its density is a total derivative, but nontrivial gauge-field topology can make its spacetime integral physically relevant in the quantum theory.

The term weights topological sectors and supplies the angle labeling a theta vacuum.

For the traceless diffeomorphism operatorthe ghost action is

Static gauge identifies selected target-space coordinates with worldvolume coordinates, such as $X^0=\tau$ for a string.

If a continuously differentiable $L$-periodic function $f$ has mean zero, thenEquality holds exactly for $f(s)=A\cos (2\pi s/L)+B\sin (2\pi s/L)$.

An ocean gyre is a basin-scale rotating circulation driven principally by wind stress and shaped by planetary rotation and boundaries.

Two large cliques joined by a bipartite cross graph with edge count $\ell$ and vertex-cover number $k$ realize edge connectivity $\ell$ and vertex connectivity $k$ while preserving a chosen minimum degree.

Given a choice function on all nonempty subsets of $X$, recursively choose the next point from the complement of all earlier choices. Hartogs theorem forces the recursion to exhaust $X$ before it defines an injection from $h(X)$ into $X$, producing a bijection from an ordinal to $X$.

Generalized least squares minimizes $(Y-X\beta {)}^T{\Omega }^{-1}(Y-X\beta )$ for a known covariance shape $\Omega$.

Writing $G_{2r}={\sum }_{\omega \ne 0}{\omega }^{-2r}$ for the lattice Eisenstein sums, expansion about zero givesThus the constant coefficient vanishes and, for $k\ge 1$, the coefficient of $z^{2k}$ is $(2k+1)G_{2k+2}$.