For a compact H-hull, $v(z)=\mathrm{Im}{g}_K(z)$ is harmonic and satisfies $0<v(z)\le \mathrm{Im}z$. Local boundedness and the interior continuity of $g_K^{-1}$ show that $v$ tends to zero at every finite boundary point. The maximum principle for harmonic functions applied to $v-\mathrm{Im}z$, with an exhaustion controlling infinity through the Laurent expansion, proves the inequality without boundary smoothness.

Let $D=\mathbb{H}\setminus K$ and stop planar Brownian motion from $z\in D$ on reaching height $R$ or leaving $D$. Then $\mathrm{Im}{g}_K(z)={\lim }_{R\to \infty }R{\mathbb{P}}_z(\mathrm{Im}{B}_{{\tau }_R}=R)$. The stopped harmonic height is a bounded martingale. The bounded difference $g_K(z)-z$ makes its top-boundary value differ from $R$ by a uniform constant, while optional stopping of the imaginary coordinate bounds the top exit probability by $\mathrm{Im}z/R$. Killing on the hull is essential.

For propagation in a direction with field component $v_{an}$ of the Alfvén velocity, the squared fast magnetosonic wave and slow magnetosonic wave speeds are the roots of $c^4-(c_s^2+v_a^2)c^2+c_s^2{v}_{an}^2=0$. They are real and nonnegative, and satisfy $c_{\mathrm{s}}^2\le v_{an}^2\le c_{\mathrm{f}}^2$. A steady acceleration equation can become singular when the flow equals either of these speeds.

Every fixed $z\in \mathbb{H}$ has infinite interior-point swallowing time for a Loewner chain under Schramm–Loewner evolution at $\kappa =4$. The centered flow is bounded on any finite horizon, making $\log |Z_t|$ bounded above. The one-sided bound criterion for a martingale clock gives a finite logarithmic limit at any putative finite swallowing time. Thus $|Z_t|$ stays away from zero and the imaginary coordinate remains positive, so the differential equation extends, a contradiction. In particular the trace misses each fixed interior point almost surely.

The linearized relative-flux matrix is $J=[(-q,q/2,0);(m,-2m,m);(f,0,-f)]$. Its characteristic polynomial has constant term $qmf<0$ for physical positive-buoyancy similarity, so at least one real eigenvalue is positive. The growing mode behaves as $z^{\lambda }$ and makes the similarity trajectory unstable with increasing height, rather than with physical time.

Changing the target boundary point by a conformal automorphism of the upper half-plane preserves the local chordal SLE law at parameter six, up to a capacity time change and until the relevant boundary point is swallowed. The transformed SLE driving function has zero drift at six; the Dambis-Dubins-Schwarz theorem makes it Brownian in image capacity time.

Combine a local projector cone in a frustration-free chain with the contraction bound from the detectability lemma. If the localized operator misses a second observable's support, their commutation and the exact ground-state bra identity convert the connected correlation function into an approximation error. The two-projection power estimate yields spatial decay rate $\alpha =\frac{1}{3}\ln (1+\Delta /2)$ for the stated detectability contraction.

When $N\le P$,An arc of length $1/P$ contains at most three points of each grid of denominator $p\le 2P$. The Chebyshev estimate gives $O(P/\log P)$ such primes. Apply the local-multiplicity large sieve. Using only the separation of distinct reduced fractions gives the weaker $O(P^2)\sum |a_n{|}^2$.

For a frozen real parallel profile and modes proportional to $e^{i\alpha (x-ct)}$, continuity eliminates tangential velocity and gives this third-order equation. Wall conditions are $v=v^{\prime }=0$; outer tangential matching gives $v^{\prime }\to 0$, with bounded $v$. Real coefficients give the spectral symmetry $(\alpha ,c)\mapsto (-\alpha ,c^{\ast })$. For positive real wavenumber, growth means $\mathrm{Im}c>0$.

A two-resource linear flow network has one local route on each of two resources and one through route using both. With unit capacities and proportional fairness, its exponential-document flow-level network model has the stationary law of a two-resource linear flow network. The local counts are independent in equilibrium despite their dependence on the shared through-route count.

A line bundle trivial on each open set is determined by unit-valued transition functions on pairwise intersections. Their cocycle identities make the local trivial bundles glue; changing frames changes the cocycle by a coboundary. The tensor product of sheaves multiplies the transition functions, giving the displayed group isomorphism with Čech cohomology.

A conditional likelihood function conditions on an initial observation or another specified statistic. In an autoregressive model, conditioning on the first observed state avoids modeling its initial density. For Gaussian transitions, maximizing this likelihood is a regression problem; treating an autoregressive coefficient as known changes the exact distribution of an estimated mean.

A Bayesian deviance is minus twice the log-likelihood, with any chosen additive data-only constant held consistent across the models being compared. Its Bayesian posterior expectation measures average fit. In the deviance information criterion, evaluating it at a parameter's posterior mean also enters the effective complexity penalty. This likelihood-based convention differs by a data-only constant from a saturated-model exponential-family deviance when a common saturated model exists.

An observed-data likelihood integrates the full-data statistical probability density over unobserved values, rather than inserting arbitrary values or discarding incomplete records. When the observation pattern is random, its mechanism belongs to the joint observation model unless the conditions for an ignorable missingness mechanism permit its factor to be omitted.

The tensor product of Lie algebra representations has action $x(v\otimes w)=xv\otimes w+v\otimes xw$. The Lie algebra representation identity follows because operators on the two different tensor factors commute. This action descends to exterior powers and symmetric powers.

The abelian Lie algebra $kI\subseteq \mathrm{End}(V)$, for $V\ne 0$, is a nilpotent Lie algebra but contains the nonnilpotent identity map. Thus nilpotence of the abstract Lie algebra does not imply that a particular Lie algebra representation consists of nilpotent endomorphisms. Engel theorem requires nilpotence of every represented endomorphism to obtain strict upper triangularity.

A Lie algebra representation on $V$ induces one on each exterior power by acting on every factor and summing. The tensor product of Lie algebra representations preserves the defining alternating relations, so this descends to the quotient in every characteristic of a field. In characteristic two define the exterior algebra by the relations $v\wedge v=0$, rather than by dividing a tensor antisymmetrizer by $2$.

A linear map between Lie algebra representations is a representation homomorphism when it commutes with the action of every element. Its kernel and image of a linear map are invariant subspaces. This is a map of modules, distinct from a Lie algebra homomorphism between the acting algebras themselves.

For a unitary derived representation $\rho$, its generators are anti-Hermitian. Define Hermitian observables $H_X=i\rho (X)$; then their ordinary operator commutator is $[H_X,H_Y]=i{H}_{[X,Y]}$. Thus real Lie algebra structure constants and Hermitian-operator commutation relations differ by an $i$. Products in a universal enveloping algebra require one conversion factor for every generator. In particular, the abstract bilinear Pauli-Lubanski element has representation image minus the pseudovector formed from the Hermitian physical generators.

The determinant transformation formula for the Levi-Civita symbol shows that it is invariant under every proper rotation matrix, for which $\det R=1$. Under an orthogonal reflection its sign reverses. This distinguishes proper-rotation isotropy from the pseudotensor transformation convention when both orientations are included.