The normalized Probabilists' Hermite polynomials diagonalize the Gaussian number operator with eigenvalues $n\ge 0$. Its self-adjoint closure has domain $\sum n^2|c_n{|}^2<\infty$. Polynomial truncations are dense for its graph norm. The semigroup $e^{-tN}$ is the Ornstein-Uhlenbeck semigroup and its energy form is the Gaussian Dirichlet energy.

The normalization of an Abelian gauge coupling matters in testing gauge coupling unification. If $\mu d{g}_i/d\mu =b_i{g}_i^3$, then ${\alpha }_i^{-1}(\mu )={\alpha }_i^{-1}({\mu }_0)-8\pi b_i\log (\mu /{\mu }_0)$. For normalized hypercharge ${\alpha }_Y^{\mathrm{GUT}}=(5/3){\alpha }_Y$, the inverse slope is $(3/5)b_Y$. Subtracting pairs of inverse couplings at the common scale and eliminating its logarithm gives a low-energy consistency relation. Division by the difference of two slopes requires that difference to be nonzero; a scale above ${\mu }_0$ additionally requires a positive inferred logarithm.

With the gamma scale inverse-gamma conjugacy model and prior shape $k>2$, the Bühlmann model has $v/a=(k-1)/\alpha$. Its Bühlmann credibility premium equals $\alpha (\lambda +{\sum }_j{X}_j)/(k+n\alpha -1)$, exactly the Bayes estimator under squared error loss of $\alpha \Theta$. Equality holds because the posterior mean is already affine in the observed sample mean.

Relative to a family $Y$, an infinite reservoir accepts a finite stem $s$ when its entire Ellentuck topology neighbourhood $[s,A]$ lies in $Y$. It rejects $s$ when no infinite refinement accepts it. Every reservoir has a refinement deciding a prescribed stem, and both outcomes are hereditary under further infinite thinning. These definitions turn open-set homogeneity into a fusion construction.

For a smooth left Lie group action, define the fundamental vector field by $X^{\#}(p)={\frac{d}{ds}|}_0\exp (sX)\cdot p$. With the Lie bracket of vector fields $[U,V]f=U(Vf)-V(Uf)$, this map is an anti-homomorphism. The map $X\mapsto v_X=-X^{\#}$ is a Lie algebra representation. A right action with the positive exponential has the opposite homomorphism convention, so specifying action side and bracket convention prevents a sign ambiguity.

The distributional derivative of this indicator function is $e_1{\mathcal{H}}^1$ on the vertical segment $x_1=1/2$, so its total variation seminorm on a domain is one. This is singular with respect to planar Lebesgue measure, whereas Sobolev space weak derivatives in $W^{1,1}$ have integrable densities. Its bounded-variation space norm is $1/2+1=3/2$.

In the $L^2$ formulation of total variation denoising on the entire plane, one may allow finite distributional total variation seminorm on a domain without requiring global $L^1$ membership. Its extended penalty is a supremum of continuous test-function pairings and hence is sequentially lower semicontinuous. This domain can be larger than $BV({\mathbb{R}}^2)\cap L^2$, since the usual bounded-variation space includes an $L^1$ condition.

On a bounded Lipschitz domain, a sequence with bounded $L^1$ norms and bounded total variation seminorm has a subsequence converging strongly in $L^1$. The limit is in the BV space, and its variation does not exceed the limit inferior. This supplies the compactness step of the direct method in the calculus of variations for linearly growing gradient energies.

For a BV space function with $Du=\nabla udx+D^{s}u$, relaxed graph area is $A(u)=\int \sqrt{1+|\nabla u{|}^2}dx+|D^{s}u|(\Omega )$. Its dual supremum uses the coupled constraint ${\phi }_0^2+|\phi {|}^2\le 1$ in the pairing $\int ({\phi }_0+u\mathrm{div}\phi )$. It is convex and lower semicontinuous in $L^1$. Independent bounds on the two test fields define a different functional.

A BV space function is special when its derivative has no Cantor part of a bounded-variation derivative: $Du=\nabla udx+[u]{\nu }_u{\mathcal{H}}^{n-1}\lfloor J_u$. The remaining singular part is supported on the jump set of a bounded-variation function. This space accommodates sharp interfaces and is the natural setting for the Mumford–Shah functional.

The jump set consists of points where a BV space function has distinct one-sided approximate values across an approximate normal. In the plane its derivative contribution is $[u]{\nu }_u{\mathcal{H}}^1\lfloor J_u$. The Mumford–Shah functional penalizes its length rather than the jump amplitude.

The derivative measure of a BV space function decomposes into its absolutely continuous, jump and Cantor parts. The Cantor part of a bounded-variation derivative is singular but not a jump measure on a rectifiable hypersurface. The special bounded-variation space excludes it; finite-length image signal interfaces alone cannot describe it.

A measurable set $E$ has finite perimeter in an open domain $\Omega$ when the distributional derivative of ${\chi }_E$ is a finite vector Radon measure there:The indicator is always locally integrable. Equivalently it belongs locally to BV space and has finite total derivative variation on the whole domain. If $|E\cap \Omega |<\infty$, it also belongs to the global BV space with its L1 norm. Finite perimeter itself does not require that finite-volume condition.

For example, the half-line $(0,\infty )$ has perimeter one in $\mathbb{R}$, from its single endpoint Dirac measure, despite infinite length. The entire real line has perimeter zero, although its indicator is not in $L^1(\mathbb{R})$. On a bounded domain, finite volume is automatic, so the simpler condition ${\chi }_E\in BV(\Omega )$ is equivalent.

Relative perimeter counts the interface inside $\Omega$ and excludes boundary outside that open domain. In a Caccioppoli partition, each shared interior interface appears in the perimeters of both adjacent regions; a segmentation cost must account for that double counting.

With constant volume flux per unit width $Hu=q_0>0$, define $\beta =l/\lambda$, $\mathcal{H}=H/\beta$, and $\mathcal{Q}=\nu q_0/(g{\beta }^3)$. The friction closure gives $H_x=\alpha -q_0\nu /(\beta g{H}^2)$. Differentiating constant flux and applying the unbuttressed Newtonian grounding-line stress condition gives $q_0{H}_x/H+g^{\prime }{H}^2/(8\nu )=0$. Substitution proves the displayed relation. For $\alpha \ge 0$, $g^{\prime }>0$ and $\mathcal{Q}>0$, the polynomial $(g^{\prime }/8g){\mathcal{H}}^5+\alpha \mathcal{Q}{\mathcal{H}}^2-{\mathcal{Q}}^2$ increases strictly on the positive half-line, begins negative and tends to infinity; hence it has exactly one positive root. Its root satisfies $\mathcal{Q}>\alpha {\mathcal{H}}^2$, so the thickness decreases seaward at that steady grounding line.

In ZFC, the Freiling axiom of symmetry is equivalent to failure of the Continuum hypothesis. Under the hypothesis, countable initial segments of a well-order of the reals violate symmetry. When the continuum exceeds ${\aleph }_1$, the union of the countable values on an ${\aleph }_1$-sized set leaves a real outside; avoiding that real's countable value gives the two witnesses.

A free product $A\ast B$ is nontrivial as a splitting when both factors differ from the trivial group. Alternating words $(ab{)}^r$ with $a,b\ne 1$ in different factors show that every such splitting gives an infinite group. The distinct normal forms $ab$ and $ba$ also show that the group is not an abelian group.

Given group homomorphisms $f_A:A\to H$ and $f_B:B\to H$, there is exactly one group homomorphism $A\ast B\to H$ extending both. Its value on an alternating normal form is the product of the factor-map values. Thus the free product is the coproduct in the category of groups.

Let $D_h$ be the scaled Dirichlet discrete Laplacian and $V_h$ a bounded diagonal reaction matrix. If $0\le k/h_x^2\le 1/2$, the heat update $H_h=I+k{D}_h$ is contractive in both the maximum norm and the mesh-weighted L2 norm. For $\Vert V_h\Vert \le A$,This proves finite-time mesh-uniform stability without requiring the full update to have nonnegative entries or to be contractive. Negative reaction terms can destroy nonnegativity at the endpoint $k/h_x^2=1/2$, so it is the heat part alone that is treated as a nonnegative contraction.

For each formula, its forcing relation on names and conditions is uniformly first-order definable inside the ground model. This permits ground-model separation and replacement using forcing predicates.

A formula holds in a generic extension exactly when some condition in the generic filter forces it for names of its parameters.