For two density operators $\rho ,\sigma$, let $P_{+}$ project onto the positive spectral subspace of $\rho -\sigma$. The binary POVM $\{P_{+},I-P_{+}\}$ has output probability difference $(D,-D)$, where $D=D(\rho ,\sigma )$. Its measurement channel therefore preserves this pair's trace distance exactly. The optimizing quantum measurement depends on the pair; a single fixed measurement need not preserve every pair's distance.

Vacuum energy is the energy assigned to a theory's vacuum state. In a free bosonic field description it includes the sum of oscillator zero-point energies, formally $\frac{1}{2}{\sum }_{\mathbf{k}}\hslash {\omega }_{\mathbf{k}}$. The continuum limit of that expression is generally divergent and requires a prescription, rather than direct identification with a finite measured energy density.

Adding a constant to a nongravitational Hamiltonian changes an overall evolution phase but not transition probabilities. Energy differences between boundary configurations can nevertheless have physical effects, as in the Casimir effect. When gravity is included, a vacuum energy density contributes to the gravitational source and is related to the cosmological constant. Normal ordering removes a reference-theory constant by convention; it does not solve every interacting or gravitational vacuum-energy problem.

A lambda term has head normal form when it beta-reduces to a term $\lambda x_1\cdots x_k.y{M}_1\cdots M_r$ with a variable $y$ at the head. The argument terms need not be in beta-normal form. A term with no head normal form cannot be beta-equivalent to a Church numeral.

For a non-Archimedean local field $K$, every finite extension $\ell /k_K$ of its residue field is realized by an unramified extension of $K$, unique up to $K$-isomorphism. Lift the irreducible polynomial of a residue generator to a monic integral polynomial and adjoin a root. Its residue degree equals its field degree, forcing ramification index one. In any other such extension the Hensel lemma lifts the same simple residue root, giving an isomorphism.

For a nonzero ideal $I$ in a Dedekind domain, $v_P(I)$ is the exponent of the nonzero prime ideal $P$ in the unique factorization of ideals in a Dedekind domain. Then $I\subseteq J$ if and only if $v_P(I)\ge v_P(J)$ for every $P$, and $v_P(I+J)=\min (v_P(I),v_P(J))$.

Let $B[u,v]={\int }_U{a}^{ij}{\partial }_{i}u{\partial }_{j}v+cuv$ on $H_0^1(U)$ for a bounded smooth domain, smooth symmetric uniformly positive coefficients, and bounded real $c$. If $B[u,u]>0$ for every nonzero $u$, then $B[u,u]\ge \alpha \Vert u{\Vert }_{H^1}^2$ for some $\alpha >0$. If this failed, choose $\Vert u_j{\Vert }_{H^1}=1$ with $B[u_j,u_j]\to 0$. The Rellich-Kondrachov compactness theorem gives a subsequence weakly converging in $H_0^1$ and strongly in $L^2$. The principal energy has weak lower semicontinuity, and the potential term converges, so strict positivity forces the limit to be zero. Uniform ellipticity then gives $\Vert D{u}_j{\Vert }_2\to 0$, contradicting normalization. Merely being a positive semidefinite operator is insufficient: $-\Delta -{\lambda }_1$ has a nonzero Dirichlet eigenfunction of zero energy.

A divergence-form elliptic operator is $Lu=D_i(a^{ij}{D}_{j}u)$, with bounded measurable coefficients whose symmetric part has a uniform positive lower bound. Its weak formulation involves only first weak derivatives, so differentiability of the coefficients is unnecessary.

A uniform spanning forest is an infinite-volume limit of uniform spanning trees on finite graph approximations. Free and wired boundary identifications give the free uniform spanning forest and wired uniform spanning forest, respectively.

The uniform layer consists of the characteristic vectors of sets of size $r$ in $[n]$. It has size $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$, a binomial coefficient. Restricting polynomials to a uniform layer introduces identities absent on the entire Boolean lattice, since the coordinate sum is fixed. In particular, homogenisation on a uniform layer spans all restricted multilinear polynomials of degree at most $s\le r$ using just the degree-$s$ monomials.

For one integrable random variable $Y$, the family $Z_{\mathcal{G}}=\mathbb{E}[Y\mid \mathcal{G}]$, over arbitrary sigma-algebras $\mathcal{G}$, has uniform integrability. On $A=\{|Z_{\mathcal{G}}|>K\}$, the defining property of conditional expectation gives $\mathbb{E}[|Z_{\mathcal{G}}|1_A]\le \mathbb{E}[|Y|1_A]$, while $\mathbb{P}(A)\le \mathbb{E}|Y|/K$ by the Markov inequality. The integrable random variable $Y$ has uniformly small absolute integrals over events of sufficiently small Probability.

For integrable random variables $Z,Y$,Apply this with $Z=X_n$ and $Y=X$ to prove that convergence in L1 implies uniform integrability; control finitely many early terms separately.

For $p>1$, ${\sup }_n\mathbb{E}|X_n{|}^p=C<\infty$ gives ${\sup }_n\mathbb{E}[|X_n|1_{|X_n|>M}]\le C{M}^{1-p}\to 0$. Thus an $L^p$ bound implies uniform integrability; a mere $L^1$ bound does not.

If $Q\ll P$, uniform convergence on compacts in probability under $P$ implies the same convergence under $Q$. Indeed, let $Z=dQ/dP$ be the Radon-Nikodym derivative. For every event $E$,Integrability makes the second term arbitrarily small for large $K$. Thus $P(E_n)\to 0$ implies $Q(E_n)\to 0$; apply this to each event where the supremum on $[0,T]$ exceeds a fixed tolerance. Only absolute continuity of measures is needed, not equivalence.

Every continuous $f:\mathbb{R}\to \mathbb{R}$ can be approximated with pointwise convergence by uniformly continuous functions. Set $c_n(x)=\max (-n,\min (x,n))$ and $f_n=f\circ c_n$. The map $c_n$ is Lipschitz continuous with constant one and $f{|}_{[-n,n]}$ is uniformly continuous by the Heine-Cantor theorem, so $f_n$ is uniformly continuous on the whole line. On every fixed bounded interval, $f_n=f$ eventually.

Even when most of an unconfined aquifer with depth-dependent permeability is deep, its height vanishes at an absorbing outlet. Near that outlet, finite outward volume flux per unit width $Q$ gives $u_{b}h{h}_x\simeq Q$ and hence $h\simeq (2Qx/u_b{)}^{1/2}$. The high-depth outer approximation instead has $h\simeq [6Qx/(u_b\beta ){]}^{1/3}$. Their crossover lies at height of order $1/\beta$ and distance of order $u_b/({\beta }^{2}Q)$; the thin inner boundary layer transmits the same leading discharge.

For ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{V}$ on $J$, putThis is an ultrafilter on $I\times J$: upward closure and finite intersection follow at both levels, and the ultrafilter dichotomy follows by complementing at both levels. Iterating gives ordered finite products. Projection onto a subsequence of coordinates pushes this product to the product on that subsequence, since omitted coordinates are absent from the tested first-order formula. The order of coordinates matters; these products need not be symmetric.

For a typical set of probability at least $1-\delta$,The upper bound sums the lower probability bound for each word; the lower bound sums the upper probability bound and uses the set's total probability. These bounds turn the asymptotic equipartition property into exponential estimates of source-coding dimension.

For an IID random variable source with probability mass function $p$, a word $x^n$ is weakly typical when $|-(1/n){\log }_{2}p(x^n)-H(X)|\le \epsilon$. It has probability on the scale $2^{-nH(X)}$. The weak law of large numbers makes the typical set probable, but does not make every word in it representative: for a uniform source every possible word is weakly typical.

The use of a convergent oracle computation is one more than its largest queried position, or zero if there are no queries. Preserving the oracle below this number preserves the computation. This finite dependence supplies the restraints in a finite-injury priority construction.

Post problem asks whether a computably enumerable set can have a Turing degree strictly between the computable degree and the degree of the halting problem. The Friedberg–Muchnik theorem answers yes by constructing two incomparable computably enumerable degrees.