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For Brownian motion with generator $\frac{1}{2}\Delta$, its local time is the occupation density normalized by ${\int }_0^{t}h(B_s)ds={\int }_{\mathbb{R}}h(a)L_t^{a}da$ for nonnegative Borel $h$. In the Tanaka formula it satisfies $|B_t-a|=|B_0-a|+{\int }_0^t\mathrm{sgn}(B_s-a)d{B}_s+L_t^a$. The symmetric-window approximation is $(2\epsilon {)}^{-1}{\int }_0^t{1}_{\{|B_s-a|<\epsilon \}}ds$. The factor two is essential. Occupation approximations for Brownian local time give a direct derivation at level zero from smooth approximations and maximal martingale estimates.

For a finite-type morphism of schemes, local fibre dimension is upper semicontinuous on the source, in the displayed sense. If the morphism is proper, the maximal fibre dimension is also upper semicontinuous on the target. Without properness a component of a large fibre can disappear from the source over a special point, so the target statement is not valid in that generality. The Stacks Project: dimension of fibres distinguishes these statements.

A bipartite graph has equal list-chromatic index, chromatic index and maximum degree of a graph $\Delta$. Start with a proper $\Delta$-edge coloring. Orient its line graph toward smaller colors at one part and toward larger colors at the other. An edge of color $c$ has at most $(c-1)+(\Delta -c)=\Delta -1$ outgoing neighbours. Every induced orientation has a digraph kernel obtained from a stable matching in a bipartite graph with those preferences. Apply the kernel list-coloring lemma.

A group admits a two-sided invariant total order exactly when every finite list of nonidentity elements can be signed so that the conjugation-invariant semigroup generated by the signed elements excludes the identity. A compactness theorem argument supplies a compatible global sign assignment. Omitting the conjugates changes the criterion to a different orderability problem.

For a torsion-free abelian group of finite rational rank $r$, each convex subgroup of an ordered group is pure: if $ng\in C$ for a positive integer $n$, then $g$ lies between zero and $ng$ or their negatives and hence belongs to $C$. A proper inclusion of convex subgroups therefore increases rational rank. They form a chain, so there are at most $r+1$, including zero and the whole group. A lexicographically ordered copy of ${\mathbb{Z}}^r$ attains the bound.

For ordered Abelian groups, compare tuples at their first differing coordinate. This gives a two-sided invariant total order on their direct product. The additive group ${\mathbb{Z}}^2$ with its first coordinate dominant is not an Archimedean ordered group, since $n(0,1)<(1,0)$ for every positive integer $n$.

An ordered group is Archimedean when $a\ge 1$ and $a^n\le b$ for all positive integers force $a=1$. In an additive Abelian group this excludes a positive element whose every integer multiple stays below one fixed bound. This generalizes the Archimedean property of the real numbers.

For independent $Y_i\sim N(\beta x_i,{\sigma }^2)$ with $n\ge 2$ and $\sum x_i^2>0$, the least-squares estimator is $\hat{\beta }=\sum x_i{Y}_i/\sum x_i^2$. The projection of the isotropic Gaussian vector onto the design direction is independent of its orthogonal residual; hence $\hat{\beta }\sqrt{\sum x_i^2}/\sigma$ is standard normal under $\beta =0$ and $\mathrm{SSE}/{\sigma }^2$ is independent chi-squared with $n-1$ degrees of freedom. The displayed statistic therefore has Student's t-distribution with $n-1$ degrees of freedom. A two-sided test rejects at the corresponding upper/lower quantiles. The zero design has no information about the slope; with one observation there is no residual variance degree of freedom.

The parameter $u$ in the linear limb-darkening law specifies its contrast between disk centre and the limiting edge. Its interpretation requires the chosen law and interval of frequency or wavelength. A limb-darkening coefficient fitted to an exoplanet transit may trade off against the transit impact parameter and planet radius; it is not a direct measurement of one layer's temperature.

On $S^2=SU(2)/U(1)$ a degree-$q$ equivariant line bundle corresponds to isotropy character $\mathrm{diag}(e^{it},e^{-it})\mapsto e^{-iqt}$. The Peter-Weyl theorem computes its section multiplicities by the occurrence of weight $-q$ in the SU(2) representation ${\mathrm{Sym}}^n{\mathbb{C}}^2$. Its weights $n,n-2,\dots ,-n$ give the displayed multiplicity-one decomposition. The spinor lines have degrees $-1,+1$ and thus contain all odd $n$, once each.

The orientation line bundle of a smooth $n$-manifold is the real line bundle ${\bigwedge }^{n}TM$. Its transition functions have the signs of the tangent-coordinate transition determinants. A nonzero continuous section specifies an orientation of the manifold. Its orientation double cover records the two possible local orientations.

A half-density transforms by the square root of the absolute Jacobian under a coordinate change. The product of a half-density with its complex conjugate is an integrable density, so its norm is coordinate-independent without choosing a separate volume form. Half-densities provide an intrinsic configuration-space Hilbert space for a real cotangent polarization.

On a smooth projective curve, a nonzero rational section represents a line bundle by its divisor on an algebraic curve. The degree is the sum of its vanishing orders minus its pole orders, including residue-field degrees over a nonclosed field. Principal divisors on an algebraic curve have degree zero, so this number is independent of the section. A nonzero regular section gives an effective divisor and therefore a nonnegative degree. Degrees add under tensor products and change sign under duality.

Consider two work-conserving slotted queues in tandem, with upstream service $c$ and downstream service $d<c$, where upstream departures can be served downstream in the same slot. Put $D_t=\min \{c,Q_{t-1}+a_t\}$. Then $Q_t=Q_{t-1}+a_t-D_t$ and $R_t=(R_{t-1}+D_t-d{)}^{+}$. If $R_{t-1}+D_t\ge d$, addition gives $(Q_{t-1}+R_{t-1}+a_t-d{)}^{+}$. Otherwise $D_t<d<c$, so the upstream queue empties and both sides of that identity are zero. Thus total workload obeys the Lindley recursion with service $d$. Starting empty in the remote past yields the displayed identity whenever the upstream workload is finite. Both workloads, and hence their difference, are continuous in the weighted cumulative-input topology for a slotted queue under strictly subcritical mean input $\lambda <d<c$.

A quantum-trajectory unravelling represents a Lindblad equation as an ensemble of stochastic pure-state histories. In one jump unravelling, a state evolves between jumps under the non-Hermitian effective Hamiltonian $H_{\mathrm{eff}}=H-\frac{i}{2}{\sum }_{\alpha }{L}_{\alpha }^{†}{L}_{\alpha }$, while a jump of type $\alpha$ sends it to a normalized multiple of $L_{\alpha }|\psi ⟩$.

A Lindblad dissipator is the non-Hamiltonian generator associated with a Lindblad operator $V$. It preserves the trace because cyclicity gives $\mathrm{Tr}(V\rho V^{†})=\mathrm{Tr}(V^{†}V\rho )$, cancelling the two anticommutator terms. It preserves Hermiticity and satisfies $\mathcal{D}[V](I)=[V,V^{†}]$; it need not preserve the identity or the purity of a density operator.

Every nonempty closed boundary subset invariant under a non-elementary Kleinian group contains the Kleinian limit set. Such a subset cannot be a singleton. Its hyperbolic convex hull contains an interior point, whose orbit lies in the hull; in the Klein ball model the boundary of this hull is contained in the original closed boundary subset.

An equivalence-invariant lifting property permits the underlying cone of the lift to be identified with the chosen cone by an isomorphism, rather than requiring literal equality. Lifts are unique up to a unique isomorphism compatible with that identification. A monadic adjunction defined by equivalence of its comparison functor has this property by monad algebra forgetful functor creates limits. Literal creation follows if the comparison is an isomorphism over the base, or suitable strict transport of objects is part of the convention. An equivalence alone does not guarantee a literally prescribed underlying object lies in the image.

A Bartlett correction rescales a regular likelihood-ratio test statistic so that its null expected value matches its limiting chi-squared degrees of freedom to a higher asymptotic order. If ${\mathbb{E}}_{0}W=d(1+b/n+O(n^{-2}))$, divide $W$ by $1+b/n$. The operation uses a model-specific Bartlett correction coefficient; matching the mean alone is not a general proof of any claimed distributional error rate.

A log-likelihood ratio compares two likelihoods on an additive scale. For independent observations its increments add. Under distributions $P_0,P_1$ with suitable integrability and common support, $E_0[\log (d{P}_1/d{P}_0)]=-D(P_0\Vert P_1)\le 0$ and $E_1[\log (d{P}_1/d{P}_0)]=D(P_1\Vert P_0)\ge 0$, where $D$ is Kullback-Leibler divergence. This sign change motivates likelihood-based sequential monitoring.