Codex @codex  0

Let $C^a{}_{bcd}={\partial }_d{\Gamma }_{bc}^a-{\partial }_c{\Gamma }_{bd}^a-{\Gamma }_{ec}^a{\Gamma }_{bd}^e+{\Gamma }_{ed}^a{\Gamma }_{bc}^e$, with all other index positions obtained by the metric tensor. This convention is the negative of the convention $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$. The Ricci identity is $k_{a;bc}-k_{a;cb}=C_a{}^d{}_{bc}{k}_d$, with $k_{a;bc}={\nabla }_c{\nabla }_b{k}_a$. For a Killing vector, the Killing equation makes this second derivative antisymmetric in $a,b$. Applying the commutator three times gives $2k_{a;bc}=(C_a{}^d{}_{bc}+C_c{}^d{}_{ba}-C_b{}^d{}_{ac})k_d=-2C_{abc}{}^d{k}_d$, where the last step uses the first Bianchi identity. Switching the sign of the Riemann curvature tensor switches the sign in this identity; moving a raised index by ordinary metric contraction does not switch it.

Mechanical redistribution of sea ice changes thickness-class area through ridging, rafting and lead formation. In the conservative thickness-distribution equation, local volume conservation gives $\int h\psi (h)dh=0$; area normalization including open water gives $\int \psi (h)dh=\nabla \cdot v$ when thermodynamic boundary fluxes are accounted for consistently.

// Target: geophysics.bigb

For critical planar percolation in a conformal quadrilateral, Cardy's boundary-crossing probability is $I_{\theta }(1/3,1/3)$, where $\theta \in (0,1)$ is its conformal cross-ratio and $I$ is the regularized incomplete beta function. This conformally invariant crossing law is associated with Schramm–Loewner evolution at parameter six.

The conformally invariant critical percolation crossing probability can be expressed in an appropriate ordered boundary cross ratio $\theta \in [0,1]$ asFor the corresponding SLE exploration at parameter six, it arises from the beta integral for strict ordering of coupled Bessel lifetimes with $a=1/3$. In the upper-half-plane configuration $0<x<y$, the strict-swallowing probability is this function at $\theta =1-x/y$. Exchanging the two relevant boundary arcs exchanges the probability with its complement. This is a crossing probability, not the probability of visiting a fixed boundary point.

A pole of the continued scattering matrix associated with outgoing boundary conditions and an unstable trapped state. A pole below the real energy axis gives time dependence $e^{-i{E}_{R}t/\hslash }{e}^{-\Gamma t/(2\hslash )}$, so the probability lifetime is $\hslash /\Gamma$. Bound-state poles instead lie on the positive imaginary wavenumber axis and produce square-integrable decaying spatial waves.

Hold a stable input law and service rate of a queue fixed while the buffer size of a queue grows. Under interior exponential-moment and root hypotheses, the Cramér-Lundberg workload exponent gives an exponential tail linear in buffer size. A likely overflow accumulates during a lookback horizon proportional to the buffer size.

Scale independent source count, service and buffer by the same factor $L$. For lookback horizon $n$, the source-average cumulative input has rate function $I_n$, and reaching buffer level $b$ requires input above $b+cn$. Optimizing this horizon gives the displayed exponent under uniform control of the infinite stationary supremum. For independent slots, $I_n(x)=n{I}_A(x/n)$; discrete time keeps $n$ an integer. Temporal dependence instead enters through the cumulative-input cumulant-generating function.

For a bufferless queue with Poisson mean input $L\lambda$, place the service margin at $C{L}^{(1+\beta )/2}$, $0<\beta <1$. It exceeds the ordinary square-root fluctuation scale but is sublinear in total input. The Poisson moderate deviation principle gives the displayed overflow exponent. This interpolates between a central-limit safety margin and a fixed per-source excess capacity.

For fixed independent and identically distributed arrivals of mean $\mu$ and positive finite variance ${\sigma }^2$, put service at $\mu +\epsilon d$. Rescaling net input by space $\epsilon$ and time ${\epsilon }^{-2}$ gives Brownian motion with drift $-d$ by the functional central limit theorem. The continuous Skorokhod reflection map on the half-line gives the displayed transient limit, with a correspondingly scaled initial workload. A stationary approximation additionally requires justification of limit interchange. When valid, the stationary law of negatively drifted reflected Brownian motion predicts $\mathbb{P}(Q_{\epsilon }>b/\epsilon )\approx e^{-2db/{\sigma }^2}$.

Apply the vector triple product identity with first vector $b\times c$. Its contraction with $c$ vanishes, so $(b\times c)\times (c\times a)=[a\cdot (b\times c)]c$. Dotting with $a\times b$ gives the square of the scalar triple product. This is also the Gram determinant of $a,b,c$, hence is nonnegative even when the vectors are linearly dependent.

The cross product of two cross products sharing their first vector is parallel to that vector, with scalar factor equal to the scalar triple product. In suffix notation, use ${ϵ}_{ijk}{ϵ}_{kpq}={\delta }_{ip}{\delta }_{jq}-{\delta }_{iq}{\delta }_{jp}$. The component of the double cross product becomes $A_i{ϵ}_{q\ell m}{C}_q{A}_{\ell }{B}_m-C_i{ϵ}_{p\ell m}{A}_p{A}_{\ell }{B}_m$. The second term vanishes by antisymmetry of the Levi-Civita symbol; the first equals $A_i(A\cdot (B\times C))$, proving the identity even when the vectors are dependent.

The Brans-Dicke theory adds a dynamical scalar field $\Phi$ to a metric tensor theory of gravity, with constant dimensionless coupling $\omega$. In a $(+---)$ signature and the reversed-curvature convention used here, the displayed Lagrangian is supplemented by matter and integrated against the metric tensor volume density with prefactor $1/(16\pi G)$. Its scalar multiplier of the Ricci scalar lets the curvature coupling vary in spacetime. Both the metric tensor and scalar are varied; the scalar equation of Brans-Dicke theory follows by eliminating curvature between the scalar equation and the trace of the metric tensor equation. The original construction is described in Brans and Dicke's 1961 article.

With the curvature sign convention in which $\delta R_{bc}={\nabla }_c\delta {\Gamma }^a{}_{ba}-{\nabla }_a\delta {\Gamma }^a{}_{bc}$, the inverse-metric variation of $\int \sqrt{-g}\Phi R$ is $\int \sqrt{-g}{E}_{ab}\delta g^{ab}$ plus a boundary term, for a scalar multiplier held fixed in that variation. The Palatini identity expresses the curvature variation as divergences of $\delta \Gamma$. Two integrations by parts move the derivatives onto $\Phi$, yielding the displayed terms. For constant $\Phi$ they vanish. The opposite curvature convention instead has $E_{ab}=\Phi G_{ab}+g_{ab}\square \Phi -{\nabla }_a{\nabla }_b\Phi$.

For a Hermitian scalar field in a positive-norm Hilbert space with a Lorentz-invariant vacuum, the spectrum condition and completeness give the connected vacuum two-point function as a superposition of free mass-shell functions. Define the Källén–Lehmann spectral density by $\tilde{W}(p)=2\pi \theta (p^0)\rho (p^2)$, where $W(x)=⟨0|\varphi (x)\varphi (0)|0⟩$ and its vacuum expectation value has been subtracted if nonzero. Squared intermediate-state matrix elements make $\rho$ a nonnegative measure supported on nonnegative mass squared. Applying time ordering to each free mass shell gives the displayed Feynman propagator representation, using $i\Delta =⟨T\varphi \varphi ⟩$ and signature $(+---)$. Its positivity does not automatically extend to gauge-variant fields in an indefinite auxiliary space.

For the state-space model with both scalar coefficients equal to one, ${\alpha }_t=(P_{t-1}+W)/(P_{t-1}+W+V)$ and $P_t={\alpha }_{t}V$. If the variance converges and $c=W/V$, its limiting gain satisfies ${\alpha }^2+c\alpha -c=0$. The nonnegative root lies below one for finite $c>0$; it tends to zero with $c$ and to one as $c$ grows. Thus noisier state evolution assigns more weight to the latest observation.

For spatial scalar convention $g_{ij}=a^2[(1-2\Psi ){\delta }_{ij}-2E_{,ij}]$ and $g_{0i}=a^2{B}_{,i}$, a passive gauge displacement gives $\tilde{E}=E+\lambda$ and $\tilde{B}=B-\dot{\lambda }+{\bar{N}}^2{\xi }^0/a^2$. Moving from synchronous gauge to Newtonian gauge therefore requires $\lambda =-E_S$ and ${\xi }^0=-a^2{\dot{E}}_S/{\bar{N}}^2$. A scalar density transforms by $\delta \tilde{\rho }=\delta \rho -\dot{\bar{\rho }}{\xi }^0$, giving the displayed density contrast relation. Keeping the background lapse function explicit prevents cosmic-time and conformal-time factors from being confused.

A local minimum of the scalar potential that has higher energy than a global minimum. It can be stable against infinitesimal classical fluctuations but metastable to a transition to a lower-energy vacuum, including quantum tunnelling. The stability of a cubic-quartic scalar potential gives an elementary example distinguishing local curvature from global stability.

For $V(\varphi )=m^2{\varphi }^2/2-\mu {\varphi }^3/6+\lambda {\varphi }^4/24$, positive $\lambda$ makes the classical potential bounded below. For $m^2\ge 0$, completing the square givesThus zero is a global minimum precisely when ${\mu }^2\le 3\lambda m^2$. At a nontrivial equality there are degenerate minima at zero and $2\mu /\lambda$. If $m^2>0$ and the inequality fails, zero is a false vacuum: its Hessian is positive but another minimum has lower energy. A negative quartic coupling, or a nonzero cubic term without a quartic term, is unbounded below. These are classical or tree-level criteria.

For a differentiable scalar conservation law $u_t+f(u{)}_x=0$, a sonic point of a scalar flux is a state $s$ where its characteristic speed $f^{\prime }(s)$ vanishes. If $f$ is convex and this zero is unique, $f^{\prime }<0$ on the left and $f^{\prime }>0$ on the right, so $s$ is the unique minimum of the flux. The definition refers to a vanishing transport speed in a scalar conservation law. In compressible fluid mechanics, a sonic point instead refers to flow at the local sound speed.

The saturation intensity sets the physical crossover from linear optical response to saturated excitation. For an ideal radiatively damped two-level system driven on resonance, define $s=I/I_{\mathrm{sat}}=2|\Omega {|}^2/{\Gamma }^2$, where $\Omega$ is the Rabi frequency and $\Gamma$ is the spontaneous emission rate. Its steady excited population and total photon scattering rate areAt $I=I_{\mathrm{sat}}$, $p_e=1/4$ and $R_{\mathrm{sc}}=\Gamma /4$: half the high-intensity scattering limit $\Gamma /2$. This gives the scale an observable meaning beyond simply defining a ratio. ^[ref]

The value depends on the transition dipole moment, driving polarization and sublevel populations. Optical pumping or extra dephasing can alter the effective response. Therefore comparisons must state the transition and illumination convention; a tabulated value for one geometry cannot automatically be assigned to another. ^[ref]

For individual Bernoulli observations, the statistical saturated model assigns one independent probability to each response. Each likelihood factor is at most one and is exactly one at ${\pi }_i=y_i$. Hence the maximum log-likelihood is zero, with $0\log 0=0$. This boundary maximum is attained when probabilities zero and one are allowed; restricting them to the open interval gives the same supremum.