For barotropic equilibrium $dp/d\Psi =-\rho$, this coefficient equals $-{\rho }^{-1}(dp/d\Psi )[{\gamma }^{-1}d\ln p/d\Psi -d\ln \rho /d\Psi ]$. It weights displacement across effective-potential surfaces in the Cowling energy principle for a rotating barotropic star. Its sign distinguishes stabilizing from destabilizing buoyancy there. It is not itself a squared frequency: the physical local buoyancy-frequency square contains an additional factor $|\nabla \Psi {|}^2$.

For fixed $\xi \ne 0$, the Lorentzian quadratic density $-F^2/4+(\partial \cdot A{)}^2/(2\xi )$ has momentum kernel $K^{ab}=-p^2{\eta }^{ab}+(1+1/\xi )p^a{p}^b$. At $p^2\ne 0$ its inverse, defined by $K^{ac}{G}_{cb}={\delta }_b^a$, is $G_{ab}=-{\eta }_{ab}/p^2+(1+\xi )p_a{p}_b/(p^2{)}^2$. A vacuum photon propagator is $iG$ with the appropriate Feynman i-epsilon prescription. The usual parameter in the negative-sign covariant gauge term is $\alpha =-\xi$, so Feynman gauge is $\xi =-1$.

Gauge fixing removes the noninvertible longitudinal direction of the Maxwell quadratic action. In a gauge with parameter $\xi$, the Fourier kernel is $-k^2{\eta }^{\mu \nu }+(1-{\xi }^{-1})k^{\mu }{k}^{\nu }$. Multiplying its inverse by $i$ gives the photon propagator, with the Feynman i-epsilon prescription. At $\xi =1$ the answer is $-i{\eta }_{\mu \nu }/(k^2+i0)$.

For square-integrable complex random variables, the Hermitian covariance is ${\mathrm{Cov}}_{\mathbb{C}}(Z,W)=\mathbb{E}[(Z-\mathbb{E}Z)\overline{(W-\mathbb{E}W)}]$. Its diagonal is nonnegative and equals $\mathbb{E}|Z-\mathbb{E}Z{|}^2$. The conjugation distinguishes it from the pseudo-covariance.

For paired observations $(x_i,y_i)$, the sample correlation coefficient isIt is the normalized inner product of the centered data vectors, or the sample covariance divided by the product of the corresponding sample standard deviations. The choice of divisor $n$ or $n-1$ cancels when used consistently in all three quantities. It is undefined if either centered data vector is zero, which occurs when that variable is constant in the observed statistical sample.

For $x=(1,2,3,4)$ and $y=(2,1,4,3)$, both sample means are $2.5$. The sum of centered products is $3$, and each sum of centered squares is $5$, hence the sample correlation coefficient is $3/5=0.6$. This is a summary of these four observed pairs; it is not automatically the population Pearson correlation coefficient. Philip Stark, Computing the correlation coefficient explains the centered and standardized calculation.

For square-integrable random variables, $\mathrm{Cov}(X,Z\mid \mathcal{G})=\mathbb{E}[XZ\mid \mathcal{G}]-\mathbb{E}[X\mid \mathcal{G}]\mathbb{E}[Z\mid \mathcal{G}]$. It is a $\mathcal{G}$-measurable random variable. Replacing $X$ by its conditional expectation given a larger sigma-field leaves this covariance unchanged when $Z$ is measurable with respect to that larger sigma-field. This observation underlies recovery of a martingale-transform integrand by conditional covariance.

Let $Z$ be a nondegenerate real random variable and $g$ a strictly increasing function. Assume $Z$, $g(Z)$ and $Zg(Z)$ are integrable. For an independent copy $Z^{\prime }$,The product is positive whenever $Z\ne Z^{\prime }$ and zero otherwise. Nondegeneracy gives positive probability of unequal copies, proving strict positivity. The same argument applies to a conditional distribution when that conditional law is nondegenerate and has the required integrability.

For a market with one trading period, suppose an increasing terminal payoff is replicated by a risk-free asset holding and a stock holding: $g(Z)=a+bZ$. Under current information $\mathcal{G}$, let $a,b$ be $\mathcal{G}$-measurable and $0<\mathrm{Var}(Z\mid \mathcal{G})<\infty$. ThenThis is a conditional one-period statement. General dynamic attainability alone does not specify the affine representation used in this calculation.

Countable partial functions from ${\omega }_1$ to a nonempty ground-model set $A$, ordered by extension, add a surjection ${\omega }_1\to A$. Countable closure preserves ${\omega }_1$ and adds no reals.

Conditions are countable successor-length sequences $⟨A_{\xi }\subseteq \xi ⟩$, ordered by end extension. A countable fusion construction decides a named subset below a fresh limit index and writes that trace at the index inside any named club. The generic sequence satisfies the diamond principle.

A Riemannian metric induces a dual inner product on each cotangent space. On decomposable covectors define $⟨{\xi }_1\wedge \cdots \wedge {\xi }_p,{\zeta }_1\wedge \cdots \wedge {\zeta }_p⟩=\det (⟨{\xi }_i,{\zeta }_j⟩)$ and extend bilinearly. Increasing wedges of an orthonormal coframe form an orthonormal basis. This positive-definite inner product is the one used to define the Hodge star operator.

For pressureless matter in Newtonian gauge in cosmology with a common metric potential $\Phi$, let $\theta =\nabla \cdot \mathbf{v}$. The linear equations ${\delta }_m^{\prime }=-\theta +3{\Phi }^{\prime }$ and ${\theta }^{\prime }=-\mathcal{H}\theta -{\nabla }^2\Phi$, where primes denote conformal time, yieldThis follows by differentiating the first equation and eliminating $\theta$. On deeply subhorizon scales with a slowly varying matter potential, the metric time derivatives are small compared with its spatial Laplacian.

In simple hierarchical models, increasing system mass raises the virial temperature and can carry primordial gas beyond its efficient atomic line-cooling interval. At assembly densities, the cooling-time constraint then distinguishes galaxy-sized condensations from larger hot groups or clusters. The resulting characteristic mass is approximate, depends on composition and structure, and does not prohibit subsequent mergers of already formed stellar systems.

Primordial atomic line cooling becomes effective near the temperature where electronic excitations are accessible, of order ten thousand kelvin. Molecular hydrogen has lower-energy transitions and can cool colder gas if it forms and survives. The corresponding virial temperature thresholds translate into epoch-dependent halo mass thresholds. Radiation, molecular chemistry, metal-line cooling and gas density qualify a single universal minimum galaxy mass.

Increasing marginal contributions imply $m_i^{\pi }\ge v((S\cap P_i)\cup \{i\})-v(S\cap P_i)$ for $i\in S$. Summing telescopes to ${\sum }_{i\in S}{m}_i^{\pi }\ge v(S)$, while efficiency follows by telescoping over the full ordering. Hence every marginal contribution vector is in the core of a cooperative game. The core is a convex set, so their average, the Shapley value, is in it too. This gives an elementary proof of nonemptiness and stability for a convex cooperative game.

If a bounded sequence $(y_n)$ in $X$ converges weak-star to $\varphi \in X^{\ast \ast }$ under the canonical embedding into the bidual, then any convex blocks $(u_n)$ have the same scalar limit on every $f\in X^{\ast }$. Thus $y_n-u_n$ is a weakly null sequence, even when $\varphi$ does not belong to the embedded space $X$.

The displayed condition implies convergence in density of a sequence to $\ell$, since the fraction of indices with $|a_n-\ell |\ge \epsilon$ is at most ${\epsilon }^{-2}$ times the averaged squared error. A signed Cesaro limit alone is insufficient: $a_n=(-1{)}^n$ has zero signed average but no density limit.

The global Eddington closure $L=(1-\beta )4\pi cGM/\kappa$, combined with a representative stellar gas-pressure fraction, makes the exterior stellar radiative temperature gradient approximately $M/(4m)$ when the luminosity is constant outside the core. Matching it to the adiabatic temperature gradient estimates the displayed boundary mass. This is a closure estimate: imposing exactly uniform $\beta$ throughout a finite-mass radiative envelope also conflicts with constant luminosity, as described by the constant-beta radiative-envelope obstruction.

Linearize around $A=R{e}^{iqX}$ and write $r=R^2>0$. For sideband wavenumber $l$, the sum and difference of the two amplitude sidebands, together with the field amplitude $b$, evolve underExpanding $\det (\lambda I-M_l)$ givesAt $l=0$, the eigenvalues are $-2r,0,0$; the field's constant mode must be removed if its mean is prescribed. Nonzero long-wave field modes are still dynamically active.

For a real steady amplitude and zero-mean field, twice integrating the field equation yields $B=\kappa (⟨A^2⟩-A^2)$, $\kappa =\mu /\sigma$. The amplitude equation becomes $A^{\prime \prime }+(1-\kappa ⟨A^2⟩)A+(\kappa -1)A^3=0$. Using $(R\mathrm{sech}cX{)}^{\prime \prime }=c^{2}A-2c^2{A}^3/R^2$ gives $\kappa ⟨A^2⟩=1+c^2$ and $R^2=2c^2/(\kappa -1)$, requiring $\kappa >1$. Its finite-window mean is $2R^2\tanh (cL/2)/(cL)$. For $cL\gg 1$, consistency gives $c+c^{-1}=4\kappa /[L(\kappa -1)]$. The leading roots coalesce at $c=1$; for $L>2$ the leading upper parameter limit is $\kappa \le L/(L-2)$, while retaining the finite-window factor makes the necessary inequality strict. The isolated sech profile has an exponentially small derivative mismatch on a periodic interval, so exact periodic tails need correction. For $L\le 2$ the algebra supplies no finite upper bound of this form.

For an entire function of polynomial growth and exponential type $H_K$, multiply it by $\hat{\rho }(\epsilon z)$ with a unit-mass mollifier. The product becomes rapidly decreasing in real frequency, so its inverse Fourier transform has an absolutely convergent contour integral. Shifting in a separating direction forces that smooth inverse to vanish outside $K+\epsilon {\overline{B}}_1$. Letting $\epsilon$ tend to zero recovers a distribution supported in $K$.