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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 $(2k+1)G_{2k+2}$.

The Weierstrass zeta function satisfies ${\zeta }^{\prime }=-\wp$ and is quasi-periodic: for every period $\omega$, the difference $\zeta (z+\omega )-\zeta (z)$ is constant. It has a simple pole of residue one at each lattice point. Consequently, if ${\sum }_j{c}_j=0$, thenis an elliptic function.

It is the quasi-periodic zeta member of the Weierstrass functions.

For a function $f:[0,1]\to \mathbb{R}$ and an integer $n\ge 1$, its Bernstein polynomial isThe Bernstein basis supplies nonnegative weights whose sum is one. Equivalently, the weights are the probabilities of a binomial distribution, and the result is an expected value. At a given $x$, this averages the sampled values $f(k/n)$ with weights concentrated near $k/n=x$.^[ref]

The endpoints are exact: $B_{n}f(0)=f(0)$ and $B_{n}f(1)=f(1)$. Interior sampled points need not be interpolated. For example, binomial distribution moments give $B_n(x)=x$ and $B_n(x^2)=x^2+x(1-x)/n$. Thus every affine function is reproduced exactly, whereas $x^2$ is generally changed. Bernstein polynomial degree preservation is a statement about degree of a polynomial, not equality of the entire polynomial.

For a continuous $f$, the uniform continuity of $f$ and the variance $\mathrm{Var}(K/n)=x(1-x)/n\le 1/(4n)$ explain uniform convergence. Split the expected value of the error into $|K/n-x|<\delta$, where uniform continuity makes the change small, and its complement, whose probability is at most $1/(4n{\delta }^2)$ by Chebyshev's inequality. The bounded function $f$ bounds the remaining error. This proves convergence without assuming differentiability.

For $f\in C([0,1{]}^d)$, the tensor-product Bernstein polynomial is the expected value of $f(X_1/n,\dots ,X_d/n)$ for independent random variables $X_j\sim \mathrm{Bin}(n,x_j)$. The same uniform continuity and coordinatewise variance estimates give uniform convergence for each fixed finite $d$.