Differentiate $\log \Vert u{\Vert }_q$ with $u=P_{t}f$ and a changing exponent. The derivative is $q^{\prime }\mathrm{Ent}(u^q)/(q^2\int u^q)-\mathcal{E}(u^{q-1},u)/\int u^q$. Apply the logarithmic Sobolev inequality to $u^{q/2}$ and the power inequality for symmetric Markov energies. Choosing $q^{\prime }=4(q-1)/c_{\mathrm{LS}}$ makes the upper bound zero.

// Target: probability-and-statistics.bigb

For planar Brownian motion started at $1$, continuous winding angles sampled at the displayed radii have independent stationary increments. At $T_r$, apply the Strong Markov property, rotate by the current angle, and rescale space by $e^r$ and time by $e^{-2r}$. The future radial-level winding increment has the original law and is independent of the past. The logarithmic radius and winding angle are independent Brownian coordinates on the common clock $\int |B_s{|}^{-2}ds$. At radial passage they are therefore $U_{{\eta }_r}=-r$ and $V_{{\eta }_r}$, where ${\eta }_r$ is a Brownian first-passage time. Its Laplace transform gives the displayed characteristic function. The level process is a symmetric Cauchy process; replacing the passage clock by its right-continuous version supplies a càdlàg modification without changing any fixed-level law.

An increasing family of compact H-hulls of half-plane capacity $2t$ with the Loewner local growth property corresponds uniquely to a continuous real Loewner driving function. Its mapping-out functions solve the Chordal Loewner equation up to the maximal lifetime of each point. Conversely each continuous driver generates that hull family. A continuous Loewner trace is an additional property, not a consequence for every continuous driver. For SLE it is supplied by the trace-existence theorem.

Over a field of characteristic zero, a locally nilpotent operator has an algebraically defined exponential, because the sum is finite for each vector. Its inverse is $\exp (-T)$. The nilpotence bound may depend on the vector. In an sl2 Lie algebra Verma module, the raising operator is locally nilpotent but the lowering operator is not, so the latter's exponential does not define an endomorphism of the ordinary algebraic module.

// Target: lie-algebra.bigb

The space $C_c^{\infty }(X)$ consists of locally constant complex functions with compact support. On a locally profinite group, finite compact-open coset covers of the support show that every such function is fixed by one common compact-open subgroup under right translation. The symbol infinity describes local constancy in this context, not real differentiability.

A scale-invariant law of unparameterized chords from zero to infinity in the complex upper half-plane is local when a boundary-preserving conformal map between neighborhoods of zero, fixing zero, transports its curve stopped on leaving the first neighborhood to the same-law curve stopped on leaving the second. Equality concerns the stopped curves up to increasing reparameterization. For SLE with parameter six, the transformed SLE driving function is a continuous local martingale with bracket six times the image capacity clock. The Lévy characterization of Brownian motion makes that driver Brownian in the new clock, proving the local stopped-curve invariance.

This normalization uses the left derivative of absolute value in the Tanaka formula. Its increasing correction is the right-sided spatial local time; equivalently $(X_t-a{)}^{+}=(X_0-a{)}^{+}+{\int }_0^t{1}_{\{X_s>a\}}d{X}_s+\frac{1}{2}{L}_t^{a,+}$. It need not equal symmetric local time for a general continuous semimartingale with finite-variation mass at the level. In the smooth convex approximation proof of the Tanaka formula, the second derivatives are supported on the positive side of the level, agreeing with this convention.

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.

On a smooth manifold, a half-density has local expression $\sigma =\psi (q)|d{q}_1\cdots d{q}_n{|}^{1/2}$. Under a change from coordinates $q$ to $Q$,Consequently $\overline{\sigma }\sigma$ is a nonnegative density on a manifold. It can be integrated without choosing an orientation or a volume form, but the integral need not be finite. Square-integrable half-densities are precisely those satisfyingTheir inner product is $⟨\sigma ,\tau ⟩={\int }_M\overline{\sigma }\tau$. Completing the smooth compactly supported half-densities gives an intrinsic Hilbert space; the Jacobian transformation makes its norm independent of coordinates.^[ref]

For instance, $|dq{|}^{1/2}$ on the real line is a smooth half-density with infinite squared norm, whereas $e^{-q^2}|dq{|}^{1/2}$ has finite squared norm. In geometric quantization, half-densities let configuration-space states carry a coordinate-independent inner product.

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.