A Cartesian second-rank tensor invariant under half-turns about all three coordinate axes is diagonal. Each half-turn has diagonal signs, and each off-diagonal entry changes sign under at least one of them. Invariance therefore kills all off-diagonal entries. It does not make the diagonal entries equal: that requires additional rotational invariance, as for an isotropic second-rank tensor.

For a hollow slender viscous thread, $A=R_e^2-R_i^2$ and both surfaces contribute axial surface tension. The liquid cut force relative to exterior pressure, including these surface terms, is $\pi [3\mu A{w}_z+(p_i-p_e)R_i^2+\gamma (R_e+R_i)]$. The axial traction on the sloping inner gas surface contributes $-\pi (p_i-p_e)(R_i^2{)}_z$ to force balance. For constant gas pressures the extra derivative cancels, giving $3\mu (A{w}_z{)}_z+\rho gA+\gamma (R_e+R_i{)}_z=0$. The displayed $T$ is this effective tension after accounting for gas-pressure work, not the unadjusted liquid cut force.

For $p>1$, the zero-interest curve $C(K)=(1+K^p{)}^{1/p}-K$ has positive second derivative $f_p$. Its mass and first moment are both one, and $\int (u-K{)}^{+}{f}_p(u)du=C(K)$. It prices an integrable payoff by $\int g(u)f_p(u)du$ when this law is equivalent to the physical terminal stock law, or on a canonical model with this pricing law. Finite-strike consistency alone is insufficient for that equivalence.

Finitely many call prices impose finitely many payoff-moment constraints. They do not require a continuous terminal law or determine prices of arbitrary new claims. For example, $S_0=1$, terminal values $1/2$ and $2+\sqrt{2}$, and upper-state probability $3-2\sqrt{2}$ give $C(1)=\sqrt{2}-1$, matching the $p=2$ power curve. A payoff vanishing on these two states must cost zero, whereas integration against the power curve's strictly positive density can assign it a positive cost. A pricing density intended for arbitrary claims must be compatible with an equivalent law on the actual state space.

The experience weight is $Z=Wa/(Wa+v)$, where $W$ is total observed exposure, $a$ the variance of hypothetical means, and $v$ the expected process variance. With positive $a$, it is $W/(W+v/a)$. More exposure raises the weight; larger process noise lowers it. If $a=0$, the unknown conditional expectation is constant and one can set $Z=0$.

With a Poisson-gamma conjugacy with unequal exposures model, $m_0=v=\alpha /\beta$ and $a=\alpha /{\beta }^2$, so the Bühlmann–Straub credibility factor is $W/(W+\beta )$. The credibility estimate simplifies to $(\alpha +{\sum }_j{Y}_j)/(\beta +W)$, exactly the posterior mean and hence the Bayes estimator under squared error loss. The agreement is exact because that posterior mean is already affine in the exposure-weighted data.

Let ${\rho }_0>0$ be a density threshold and let $a,\mu$ be the radius and enclosed mass where a monotonically decreasing stellar density reaches that threshold. Then $\mu \ge 4\pi {\rho }_0{a}^3/3$. A local compactness bound $\mu /a<4/9$ implies $a<(3\pi {\rho }_0{)}^{-1/2}$ and $\mu <4/(9\sqrt{3\pi {\rho }_0})$, independently of the equation of state above the threshold. A total-mass bound also requires suitable control of the known low-density envelope.

Writing a transformation as $\delta \Phi =ϵs\Phi$ with the odd parameter on the left defines this graded derivation convention. For Yang-Mills fields, $sA=Dc$, $sc=-[c,c]/2$, $sb=h$, and $sh=0$. The Jacobi identity supplies nilpotence; the placement of the odd parameter controls signs in a Noether current.

A BRST charge is nilpotent when $Q^2=0$, making BRST cohomology well-defined. For the critical bosonic string theory, the matter and ghost central charges cancel, $D-26=0$, and the intercept is one. In a ghost convention with a zero-mode linear shift, the shifted total constraint generators, rather than the unshifted ones, obey the centerless Witt algebra.

For a matter state tensored with the ghost oscillator vacuum, the BRST charge produces the independent ghost states $c_0|0⟩$ and $c_{-m}|0⟩$ with coefficients $(L_0-1)|\Psi ⟩$ and $L_m|\Psi ⟩$. Thus BRST closure is equivalent to $(L_0-1)|\Psi ⟩=0$ and $L_{m>0}|\Psi ⟩=0$ in this sector. It reproduces old covariant string quantization and the open bosonic string mass spectrum.

A brightness temperature equates a measured intensity with a Planck function; it does not determine its contribution pressure. In hydrostatic equilibrium, ${\tau }_{\lambda }(P)={\int }_0^P{\kappa }_{\lambda }d{P}^{\prime }/g$. Estimating $P\sim g/{\kappa }_{\lambda }$ therefore needs gravity and opacity, as well as a model of the contribution function. Different pressures can produce the same band brightness.

For $p_u\in \partial J(u)$ and $p_v\in \partial J(v)$, add the two Bregman distances to get $D_J^{p_v}(u,v)+D_J^{p_u}(v,u)=⟨p_u-p_v,u-v⟩\ge 0$. The choice of subgradients matters. This is generally not a metric: it need not separate points or satisfy a triangle inequality.

Starting with $u^0=0$ and $p^0=0\in \partial J(0)$, update $p^{k+1}=p^k+K^{\ast }(f-K{u}^{k+1})/\alpha$. The optimality condition makes $p^{k+1}\in \partial J(u^{k+1})$. The iteration replaces a fixed penalty by its Bregman distance from the preceding iterate and accumulates residual information in the dual variable. The first update uses $k=0$.

Let $W_i$ count observed wins and $n_{ij}$ count comparisons of players $i,j$. Differentiating the log-likelihood with respect to $\log {\theta }_i$ gives observed wins minus model-expected wins. Vanishing derivatives therefore give the displayed equations. One equation is redundant because abilities are identifiable only up to common scaling.

The notation $Z_n=O_{\mathbb{P}}(a_n)$ means that $Z_n/a_n$ is bounded in probability asymptotically: for every $\epsilon >0$ some $C$ gives ${\mathrm{limsup}}_n\mathbb{P}(|Z_n|>C{a}_n)\le \epsilon$. A tail bound $\mathbb{P}(|Z_n|>z{a}_n)\le C_0/z^2$ is sufficient. In statements involving bandwidths, specify whether the constants must be uniform over bandwidth sequences.

For the free Schrodinger equation with compatible smooth initial and boundary data, subtract $g(t)e^{-x}$ before taking a Fourier sine transform. The lifted initial datum vanishes at the endpoint, making its sine transform $O(k^{-2})$ if its second derivative is integrable. The transformed forcing is $k(ig-g^{\prime })/(1+k^2)$; time integration followed by integration by parts makes its solution contribution uniformly $O(k^{-3})$. An integrable spectral majorant proves uniform convergence even at the compatible initial-boundary corner.

For a positive Borel measure on a topological space, the support consists of the points whose every open neighborhood has positive measure. For a finite regular complex measure use its variation measure. Regularity makes the complement of the support null: compact subsets of that open complement have finite covers by null neighborhoods. A regular probability measure on a compact Hausdorff space whose support is a single point is therefore a Dirac measure.

A basic bootstrap interval reflects quantiles of the bootstrap estimation error about the observed estimate. For the sample mean it uses $[\bar{X}-q_{1-\alpha /2}/\sqrt{n},\bar{X}-q_{\alpha /2}/\sqrt{n}]$, where $q$ is a conditional quantile of $\sqrt{n}({\bar{X}}^b-\bar{X})$.

In a cold pressureless stream of asymptotic speed $v$, conservation of specific angular momentum and the eccentricity vector makes a streamline at impact parameter $d$ cross the downstream axis at $b=d^2{v}^2/(2GM)$. Its axial velocity there remains $v$, while opposing transverse velocities collide in a shock.

For $\lambda \ge 0$ and a linear map $A$ between finite-dimensional real inner product spaces,Thus the penalty is the support function of the compact convex set $A^{\ast }\{p:{\max }_i\Vert p_i{\Vert }_{q^{\prime }}\le \lambda \}$. The Moreau decomposition makes its proximal operator the identity minus the Euclidean projection onto a convex set.