For a Gaussian free field with boundary heights $-\lambda$ and $+\lambda$ on the two sides of the starting point, its distinguished zero-height interface has chordal ${\mathrm{SLE}}_4$ law. Conditional on an initial curve segment, the field is a zero-boundary field in the slit domain plus $m_t(z)=\lambda -(2\lambda /\pi )\arg (g_t(z)-W_t)$. Its mean bracket equals minus the variation of its conditional Green covariance. Subtracting the initial harmonic function gives the corresponding coupling to a zero-boundary field.

The covariant version of the Yoneda embedding sends an object to its outgoing representable functor. Precomposition reverses arrows. The Yoneda lemma gives $\mathrm{Nat}(Y(A),Y(B))\cong \mathcal{C}(B,A)$, so it is full and faithful. It carries existing colimits in $\mathcal{C}$ to pointwise categorical limits.

For a finite-dimensional complex Lie algebra representation $\rho$ of a nilpotent Lie algebra $H$, a generalized weight space is the simultaneous generalized eigenspace displayed above, where $\lambda :H\to \mathbb{C}$ is a linear functional. Every occurring $\lambda$ vanishes on the derived algebra of $H$. Ordinary weight vectors satisfy the same equations without the powers.

Let $\Theta$ be a compact metric space, let $q(\theta ,x)$ be continuous in $\theta$ for each $x$ and measurable in $x$ for each $\theta$, and assume the envelope $M(x)={\sup }_{\theta \in \Theta }|q(\theta ,x)|$ is integrable under $P$. For an independent and identically distributed sample,Compact metric structure supplies both finite parameter nets and countable dense sets, so the relevant suprema can be treated measurably under these continuity assumptions.

For a proof, define ${\omega }_{\delta }(x)={\sup }_{d(\theta ,\eta )\le \delta }|q(\theta ,x)-q(\eta ,x)|$. Uniform continuity on the compact parameter set gives ${\omega }_{\delta }(x)\to 0$ for each $x$, and ${\omega }_{\delta }\le 2M$. Thus $P{\omega }_{\delta }\to 0$ by the dominated convergence theorem. At a finite $\delta$-net of centers, apply the strong law of large numbers to every center and to ${\omega }_{\delta }$. For a nearby center $\eta$,The limiting supremum is at most $2P{\omega }_{\delta }$. A countable sequence $\delta \downarrow 0$ proves the claim. Equivalently, these center functions and their oscillation envelopes give the function brackets used in the uniform strong law from finite L1 bracketing.

If $[\sigma ]\ne 0$ in de Rham cohomology, a twisted cotangent symplectic form has no Lagrangian submanifold whose projection is a diffeomorphism onto the entire base. Such a submanifold would be a global one-form graph, and the twisted Lagrangian graph criterion would make $\sigma$ exact.

Delete a finite set of degree-three trifurcation vertices in percolation in each infinite percolation cluster and contract the remaining components to nodes. The incidence graph is a tree: a cycle would bypass an original trifurcation vertex in percolation. All leaf components are infinite, otherwise a graph vertex would have a finite branch after deletion. The tree degree identity gives at least two more infinite leaves than degree-three graph vertices. Distinct infinite components escape a containing box through distinct outer boundary graph vertices. Hence the number of counted trifurcations is at most the outer boundary size.

An interaction contrast compares a treatment difference across levels of another factor. A nonzero difference of differences shows that an additive representation of those cell means is inadequate. Its standard error must use the error stratum in which that interaction term is randomized.

For a vector of estimated treatment means $\hat{\mu }$, the variance of a treatment contrast with coefficient vector $c$ is $c^T\mathrm{Cov}(\hat{\mu })c$. Averaging subsamples from a common experimental unit does not make their shared variance component disappear.

At equally replicated quantitative treatment levels, evaluations of orthogonal polynomials generate pairwise orthogonal treatment contrasts, using the replication-weighted inner product. Five equally spaced levels give four contrast directions representing degrees one through four, besides the constant direction. This decomposes treatment variation without assuming the response has low polynomial degree.

A circle compactification is self-dual when the radius is unchanged by the T-duality map $R\mapsto {\alpha }^{\prime }/R$. Momentum and winding then enter the bosonic string mass spectrum with equal weights. Special charge sectors can become massless at this radius.

A local Buscher procedure sets a flat auxiliary connection equal to a coordinate derivative only on a simply connected patch. A full compact-circle Polyakov path integral must also sum flat-connection holonomies, match the period of the dual multiplier, and include momentum and winding modes. A free circle isometry then gives the radius relation $R^{\prime }={\alpha }^{\prime }/R$. Fixed points, noncompact orbits and global identifications require additional analysis; a local spacelike Killing vector does not specify these data.

Suppose $E_g$ minimizes $\mathrm{Per}(E)-{\int }_{E}g$ and $E_h$ minimizes $\mathrm{Per}(E)-{\int }_{E}h$. Compare them with their intersection and union. Adding their optimality inequalities and using submodularity of relative perimeter gives ${\int }_{E_g\setminus E_h}(h-g)\le 0$. Strict ordering forces that set to be null. Applied to $g=(f-t)/\alpha$, this yields nested ROF denoising superlevel sets.

At each $\alpha \in S$, a countable family $E_{\alpha }\subseteq \mathcal{P}(\alpha )$ is supplied. Every $X\subseteq \lambda$ has $X\cap \alpha \in E_{\alpha }$ on a stationary subset of $S$. The countable-family diamond equivalence shows that this apparently weaker prediction is equivalent to a single diamond guess at each index.

Two unit quaternions act on Euclidean four-space by $q\mapsto aq{b}^{-1}$. This norm-preserving action maps onto the SO(4) group. Its kernel is precisely $(1,1)$ and $(-1,-1)$: a kernel pair must have equal entries commuting with every quaternion. The two simply connected SU(2) factors therefore form the universal spin cover.

In the complex symmetric-group algebra, normalize a Young symmetrizer by $e_t=h_t/H_{\lambda }$. The row-column collision lemma proves $e_{t}A{e}_t=\mathbb{C}{e}_t$, so $e_t$ is a primitive idempotent and $A{e}_t$ is an irreducible left module. Different shapes are separated by the vanishing corner $h_{\lambda }A{h}_{\mu }=0$ for $\lambda >\mu$. Counting conjugacy classes then proves that these modules exhaust the simple modules.

Write $e^u\varphi (u)=\int \psi (t)e^{-iut}dt$. The double integral $\iint \psi (t)\psi (t^{\prime })(1+it)(1+i{t}^{\prime })/(2+i(t+t^{\prime }))dtd{t}^{\prime }$ equals the displayed constant. Express the denominator as a Laplace integral and each differentiated Fourier factor as $-{\varphi }^{\prime }(u)$, then use Fubini's theorem. For a real cutoff the resulting square is positive even though the Fourier transform factors have no complex conjugation. The Cauchy-Schwarz inequality and $\varphi (0)=1$, $\varphi (1/3)=0$ give $c_{\varphi }\ge 3$.

For $\mathrm{Re}z,\mathrm{Re}{z}^{\prime }>0$, the series ${\sum }_{d,e}\mu (d)\mu (e)d^{-z}{e}^{-z^{\prime }}/[d,e]$ has local factor $1-p^{-1-z}-p^{-1-z^{\prime }}+p^{-1-z-z^{\prime }}$. Dividing by the displayed zeta function ratio leaves a holomorphic function $H$ near zero with $H(0,0)=1$. The local factor of $H$ is $(1-a-b+c)(1-c)/((1-a)(1-b))$, where $a=p^{-1-z}$, $b=p^{-1-z^{\prime }}$, $c=p^{-1-z-z^{\prime }}$, and equals one exactly at zero. With $z=(1+it)/\log X$, the zeta function poles produce the kernel $(1+it)(1+i{t}^{\prime })/((2+i(t+t^{\prime }))\log X)$ that determines the smooth divisor-square sieve asymptotic.

For $n\ge 1$, the cotangent form of the Euler sequence is $0\to {\Omega }_{{\mathbb{P}}^n/k}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to \mathcal{O}\to 0$. The cohomology of twisting sheaves on projective space and the long exact sequence in sheaf cohomology give $H^1({\mathbb{P}}_k^n,{\Omega }^1)\cong k$, with every other cohomology group zero. Therefore the Euler characteristic of a coherent sheaf is $-1$. The nonzero class is the connecting image of the constant section $1$.

A nonzero isogeny of elliptic curves is separable exactly when its pullback of an invariant differential on an elliptic curve is nonzero. For a map of smooth curves in characteristic $p$, a zero differential means the function-field extension has a nontrivial inseparable part. The criterion turns sums of elliptic endomorphisms into computations of their tangent scalars.

A Schur module is the value of a Schur functor on a vector space. Over the complex numbers it is either zero or an irreducible polynomial representation of the general linear group. Its character is the Schur polynomial $s_{\lambda }$ in the eigenvalues, and its highest weight is $\lambda$ padded with zeros.