A nondivergence-form elliptic operator applies its principal coefficients directly to the Hessian matrix: $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u+cu$. Only the symmetric part of $a$ contributes to the second-derivative term. Uniform ellipticity bounds its quadratic form below by $\lambda |\xi {|}^2$ with a constant $\lambda >0$ independent of position.

On a connected locally finite recurrent graph, root Wilson's algorithm at a fixed graph vertex $r$ and process a countable enumeration of the graph vertices. Every walk hits the existing tree almost surely, since it contains $r$. Each attached graph path is finite, and the union is a connected acyclic spanning subgraph. Its law is the infinite uniform spanning tree.

It is also the common free and wired finite-volume limit of a sequence along every graph exhaustion. For a finite edge set, run a finite initial segment of the chosen enumeration containing all its endpoints. Its membership is then permanently decided, because later attachments have new internal graph vertices. Only finitely many finite random-walk trajectories were used, and their graph vertices and graph neighbours fit inside all sufficiently large exhaustion sets. Coupling then gives convergence of those finite-dimensional edge laws for both boundary conventions. Finite Wilson's algorithm is root/order independent, so the limiting law is too. There is no uniform counting measure on all infinite spanning trees; the definition is this limiting probability law.

The complement of an $\ell$-uniform hypergraph on $V$ has all $\ell$-subsets of $V$ that are not hyperedges of the original hypergraph. An edge in every $r$-set of one hypergraph is equivalent to the absence of a complete $\ell$-uniform $r$-vertex hypergraph in its complement.

The complete $\ell$-uniform hypergraph on $r$ vertices contains every $\ell$-subset of its vertex set. A hypergraph clique here means a restriction to a vertex subset of this form, rather than merely a clique in the pairwise shadow. For $\ell =2$, it is an ordinary complete graph.

With the convention $\zeta =\Phi -\delta \rho /[3(\bar{\rho }+\bar{P})]$, energy conservation gives ${\zeta }^{\prime }=\mathcal{H}\delta P_{\mathrm{nad}}/(\bar{\rho }+\bar{P})+(\nabla \cdot \mathbf{v})/3$. The non-adiabatic pressure perturbation vanishes for adiabatic cosmological perturbations, and regular velocity gradients are suppressed on superhorizon scales. Curvature is then conserved to leading gradient order, allowing primordial initial conditions to be propagated between cosmological eras.

For spatial metric $a^2(1-2\Phi ){\delta }_{ij}$, the combination $\zeta =-\Phi +\delta \rho /[3(\bar{\rho }+\bar{P})]$ is invariant under a time displacement: $\Phi$ shifts by $\bar{N}HT$ and $\delta \rho$ by $3\bar{N}H(\bar{\rho }+\bar{P})T$. The opposite overall sign is also valid. The sum $\Phi +\delta \rho /[3(\bar{\rho }+\bar{P})]$ is not invariant. This is a useful convention check before applying superhorizon conservation of uniform-density curvature.

If a uniformly continuous function $u:{\mathbb{R}}^d\to \mathbb{R}$ belongs to Lp space for some $1\le p<\infty$, then $u(x)\to 0$ as $|x|\to \infty$. Otherwise choose points escaping to infinity with $|u(x_j)|\ge 2\theta >0$. Uniform continuity gives one radius $\rho >0$ on which $|u|\ge \theta$ around every $x_j$. Extract disjoint such balls. Their contributions to $\int |u{|}^p$ are each at least ${\theta }^p|B_{\rho }|$, a contradiction. The finite-$p$ condition is essential: the constant function one lies in $W^{1,\infty }$ and does not vanish at infinity.

A product of two uniformly continuous functions need not have uniform continuity on an unbounded domain. The functions $f(x)=g(x)=x$ give the counterexample $x^2$, since inputs $n$ and $n+1/n$ approach each other while their squared values remain separated. For bounded factors the estimate $|f(x)g(x)-f(y)g(y)|\le \Vert f{\Vert }_{\infty }|g(x)-g(y)|+\Vert g{\Vert }_{\infty }|f(x)-f(y)|$ restores uniform continuity.

Choose the uniform continuity input tolerance corresponding to output tolerance one. A totally bounded domain is covered by finitely many balls of smaller radius centred in the domain. Every function value differs by less than one from the value at one of their finitely many centres. Taking the largest centre value bounds the whole function. Compactness or closedness of the domain is unnecessary.

If $B$ is a countably infinite set and $A$ is a finite set with at least two elements, the set of surjective functions $B\to A$ is uncountable. Fix finitely many values to cover $A$, then encode an arbitrary infinite binary sequence using two values of $A$ on the remaining inputs.

The bijections of a countably infinite set form an uncountable set. Partition an enumeration into disjoint pairs and encode an infinite binary sequence by independently fixing or swapping each pair.

A vertex potential in the dual of an uncapacitated minimum-cost flow yields the edge inequalities ${\pi }_i-{\pi }_j\le c_{ij}$. Its objective is ${\sum }_i{\pi }_i{b}_i$. Adding a common constant leaves all network reduced costs and the objective unchanged when total net supply is zero. Potentials tight on a spanning tree are fixed up to this common constant.

An estimator is conditionally unbiased for a parameter if its expectation, conditional on the specified observation or selection event, equals that parameter. Unconditional unbiasedness does not imply this property after selection. Independent new-stage data and the Rao-Blackwell theorem can provide useful constructions.

In the orthonormal history basis $|e_t⟩=|{\psi }_t⟩|c_t⟩$, circuit propagation restricts to $E=\frac{1}{2}{\sum }_{t=1}^T(|e_t⟩-|e_{t-1}⟩)(⟨e_t|-⟨e_{t-1}|)$. This is a graph Laplacian on a path. Its uniform zero-energy vector is the computational history state, while initialization restricts to $D=I-|e_0⟩⟨e_0|$.

If $U$ is a kappa-complete nonprincipal ultrafilter on $\kappa$ and $|M|\ge \kappa$, then $|M^{\kappa }/U|>\kappa$. Otherwise enumerate representatives $f_{\alpha }$ for its classes and choose $g(i)\notin \{f_{\alpha }(i):\alpha <i\}$. Every tail of $\kappa$ belongs to $U$, so $g$ differs modulo $U$ from every representative. The small set is absent from a complete nonprincipal ultrafilter lemma supplies the tail property.

For an ultrafilter $U$ on $X$ and a function $g:X\to Y$, define $g_{\ast }U=\{B\subseteq Y:g^{-1}[B]\in U\}$. This is an ultrafilter and inherits countable completeness from a countably complete ultrafilter. It is nonprincipal exactly when no single fibre of $g$ belongs to $U$. Restricting $U$ to a member of $U$ before pushing forward preserves these properties.

The displayed coordinates identify determinant-one real symmetric matrices of size two with the two-sheeted hyperboloid. The special linear congruence action on symmetric matrices is transitive on each sheet, with stabilizer $SO(2)$. In particular the positive sheet is the homogeneous space $SL(2,\mathbb{R})/SO(2)$, a model of the hyperbolic plane.

The stretching contribution to upper-layer potential vorticity is $-F({\psi }_1-{\psi }_2)$, with the opposite sign below. An interface displacement thickens one layer and thins the other. In a parcel's potential-vorticity equation, its material change balances changes of relative and planetary vorticity and any forcing. The two layer derivatives cannot in general be replaced by one common advection operator.

Every nontrivial quotient group of a group generated by two involutions is generated by two involutions, allowing repeated generators. The two generator images have squares equal to the identity. If one image becomes trivial, the other generates $C_2$; if both become trivial, the quotient is trivial and is excluded. Thus the two-involution characterization of a dihedral group is preserved by nontrivial quotients.

The coupling $q{\int }_{\Sigma }{X}^{\ast }B$ integrates the pullback of the two-form gauge field over the string worldsheet. Its gauge variation is $q{\int }_{\partial \Sigma }{X}^{\ast }\alpha$, by Stokes theorem. It therefore vanishes for a closed worldsheet, while an open surface requires suitable boundary fields or boundary-state transformations.