For $L(v)=(p{v}^{\prime }{)}^{\prime }+av$, direct differentiation gives $uL(v)-vL(u)=[p(u{v}^{\prime }-v{u}^{\prime }){]}^{\prime }$. The potential terms cancel. Integrating with common homogeneous Dirichlet boundary conditions eliminates the endpoint term. If $L(u)=0$ and $L(v)=f$, this gives $\int uf=0$, a necessary solvability condition for the inhomogeneous equation.

For $p\in C([0,1])$ with $p\ge p_{\ast }>0$ and bounded real $q$, the form $a(u,v)=\int (p{u}^{\prime }{v}^{\prime }+quv)$ on the zero-trace Sobolev space defines the weak operator $L=-(p{u}^{\prime }{)}^{\prime }+qu$. Its literal $L^2$ operator domain is the displayed flux domain. If a weak solution has right-hand side $f\in L^2$, then $(p{u}^{\prime }{)}^{\prime }=qu-f\in L^2$, so it belongs to that domain. This avoids assuming that a merely differentiable coefficient has bounded derivative. With $p\in C^1$ bounded away from zero, the domain reduces to $H^2\cap H_0^1$.

Write $\mathcal{A}y=-(p{y}^{\prime }{)}^{\prime }+qy$ and $\mathcal{A}{y}_n={\lambda }_{n}r{y}_n$, where $r$ is the positive weight. Normalize the real eigenfunctions using $\int r{y}_m{y}_n={\delta }_{mn}$. For $\mu$ outside the spectrum, applying $\mathcal{A}-\mu r$ to the displayed Green function expansion gives $r(x){\sum }_n{y}_n(x)y_n(\xi )=\delta (x-\xi )$ by weighted completeness. This is the kernel with respect to ordinary source integration $d\xi$; using the weighted measure would change its normalization.

A gradient integrand is uniformly convex in this elliptic variational sense when its Hessian matrix in the gradient variables has a common positive lower bound. Then $(F_p(x,p)-F_p(x,q))\cdot (p-q)\ge \lambda |p-q{|}^2$. Averaging $F_{pp}$ along a line segment in gradient space preserves the same lower bound.

Every strongly continuous unitary group has a unique possibly unbounded self-adjoint operator $A$ as its generator. Its domain consists of the vectors for which the norm limit of $(U_t\xi -\xi )/(it)$ exists as $t\to 0$. Conversely, the Borel functional calculus for a normal operator constructs a strongly continuous unitary group from every self-adjoint $A$. Smooth averaging supplies a dense generator domain; Laplace integrals of $U_t$ and $U_{-t}$ prove that $A\pm i$ are surjective, which establishes self-adjointness.

An iterated strong graph product has vertices that are words of length $m$ in the graph's vertex set. Distinct words are nonadjacent exactly when some coordinate contains distinct nonadjacent entries. This differs from the distance-based graph power.

A finite nonempty connected graph whose edges are undirected admits a strong orientation if and only if it has no bridge in a graph. This statement holds for finite multigraphs as well as finite simple graphs; self-loops do not help connect distinct vertices. The single-vertex case is allowed, with length-zero directed paths.^[ref]

Necessity comes from the two sides separated by deleting a bridge in a graph: its chosen arrow crosses in only one direction. For sufficiency, the cited constructive proof uses depth-first search to find a cycle in a graph, orients it cyclically, contracts it, and repeats. Undoing the contractions replaces junctions by directed cycles, retaining two-way reachability. Contraction can produce multiple edges, so these must remain distinct. This is a construction outline; the input graph is finite and bridgeless, and remaining edges within contracted pieces can receive arbitrary directions.^[ref]

This topology on bounded operators is generated by the seminorms $(\Vert T\xi {\Vert }^2+\Vert T^{\ast }\xi {\Vert }^2{)}^{1/2}$. A net converges exactly when both it and its adjoint net converge in the strong operator topology. It is the topology used in the full-unit-ball version of the Kaplansky density theorem.

In conformal gauge and light-cone gauge in string theory $X^{+}=\kappa \tau$, the transverse part of the string sigma model in a spacetime metric reduces toThe omitted longitudinal term is a total time derivative. Each transverse coordinate obeys a massive wave equation. If the spatial period is $L$, the longitudinal momentum is $p^{+}=T\kappa L$ in this normalization. Keeping the gauge slope explicit prevents missing a factor of $p^{+}$ when changing conventions.

For an integrable finite-dimensional random vector $Y$, let $L=\{h:h\cdot Y=0\text{ almost surely}{\}}^{\perp }$. The displayed convex cone is open relative to $L$. Given a bounded strictly positive $Z$, the perturbation $Z_{\epsilon }=Z(1+\epsilon \cdot Y/(1+\Vert Y\Vert ))$, $\Vert \epsilon \Vert <1$, changes its weighted expectation by $A_Z\epsilon$, whereFor $0\ne v\in L$, $v^T{A}_{Z}v>0$: otherwise $v\cdot Y=0$ almost surely. Thus $A_Z$ is invertible on $L$, and these perturbations give a neighborhood of every point of the cone. The hyperplane separation theorem consequently gives the alternative: either zero is a strictly positively weighted expectation, or a nonzero direction in $L$ has a nonnegative dot product with $Y$ almost surely and a positive dot product with positive probability. To obtain the latter assertion, test the separating direction with $Z=1_{\{h\cdot Y<0\}}+\epsilon$ and let $\epsilon \downarrow 0$.

If two distinct elements $u,v$ of a linear subspace both minimize distance to $f$, normalize their residuals by the common positive minimum $d$. Strictly convex normed space geometry gives $\Vert f-(u+v)/2\Vert <d$, a contradiction. If $d=0$, both elements equal $f$. This proves uniqueness when a best approximant exists, without asserting existence for a nonclosed subspace.

Here $v,e,a$ are the numbers of vertices, edges and graph automorphisms of $H$. Vertex-disjoint ordered $r$-tuples contribute $(n{)}_{rv}{p}^{re}/a^r\to (c^e/a{)}^r$ to the factorial moment. Adding a copy overlapping the previous union in $f$ vertices and $g$ edges changes the power of $n$ by $-f+(v/e)g\le 0$, strictly when the intersection is a proper nonempty subgraph. The first overlap is strict because $H$ is connected and the previous copies are disjoint. Finitely many overlap types therefore contribute $o(1)$. The factorial-moment criterion for Poisson convergence gives the result.

The left stretch tensor is $V=(F{F}^{\mathsf{T}}{)}^{1/2}$, the positive matrix square root of the left Cauchy-Green deformation tensor. In the polar decomposition in continuum mechanics $F=VR$, it applies stretches along directions in the current configuration after the rotation $R$.

It is related to the right stretch tensor by $V=RU{R}^{\mathsf{T}}$. The two tensors therefore have the same eigenvalues, the principal stretches, but their eigenvectors belong to differently oriented configurations. For $F=R\mathrm{diag}(2,1,1/2)$, $V=R\mathrm{diag}(2,1,1/2)R^{\mathsf{T}}$. This distinction is useful when comparing a material-direction measurement before loading with an observed direction after the specimen has both stretched and turned. For a pure rotation, $V=I$, correctly reporting no local change in length.

The right stretch tensor is $U=(F^{\mathsf{T}}F{)}^{1/2}$, the positive matrix square root of the right Cauchy-Green deformation tensor. In the polar decomposition in continuum mechanics $F=RU$, it stretches material directions in the reference configuration before the rotation $R$ carries them to the final orientation.

For a unit reference direction $\mathbf{N}$, the squared length ratio is $|F\mathbf{N}{|}^2=\mathbf{N}\cdot U^2\mathbf{N}$. Along an eigenvector of $U$, that ratio is the square of its principal stretch. For $F=R\mathrm{diag}(2,1,1/2)$, one has $U=\mathrm{diag}(2,1,1/2)$ regardless of $R$: the local lengths double, stay fixed and halve along the three reference axes. Since the product is one, this example preserves volume. The Biot strain tensor $U-I$ expresses these extension ratios relative to unity.

The principal stretches are the eigenvalues of a stretch tensor, or equivalently the singular values of the deformation gradient. They measure local length ratios along the principal reference directions.

An $N$-dimensional conically stratified space has a closed filtration $\varnothing =X_{-1}\subseteq X_0\subseteq \cdots \subseteq X_N=X$, with each nonempty $X_j\setminus X_{j-1}$ a $j$-manifold, $X_{N-1}=X_{N-2}$, and dense top stratum. At an $s$-stratum the conical link is a compact $(N-s-1)$-dimensional stratified pseudomanifold. In dimension zero a compact link is a finite set. Connected links are an additional normality condition, not part of this definition.

A stratum is one manifold piece of a stratification, with locally constant dimension and prescribed conical normal neighbourhoods. Its codimension in a pure $N$-dimensional space is $N$ minus its dimension.

Suppose the variance is ${\sum }_i{v}_i/x_i$ and the sampling cost is ${\sum }_i{a}_i{x}_i\le b$, with positive costs, variance coefficients and budget. The Cauchy-Schwarz inequality gives $A^2\le ({\sum }_i{v}_i/x_i)({\sum }_i{a}_i{x}_i)\le b{\sum }_i{v}_i/x_i$. Equality forces $x_i$ proportional to $\sqrt{v_i/a_i}$ and a tight budget, proving the displayed unique optimum. The minimal variance is $A^2/b$. For the optimization Lagrangian $f-\lambda ({\sum }_i{a}_i{x}_i-b)$, the multiplier and budget derivative both equal $-A^2/b^2$. Therefore a small resource change $\delta b$ changes the optimum by $\lambda \delta b+O((\delta b{)}^2)$. This is a continuous allocation; integer sample-size restrictions define a separate optimization problem.

Instantaneous mixing efficiency is the fraction of instantaneous irreversible mechanical-energy loss that raises background potential energy, after subtracting internal-to-potential exchange. The ratio $\Gamma =M/ϵ$ is a different quantity, related by $\eta =\Gamma /(1+\Gamma )$. Replacing $M$ by buoyancy exchange $B$ additionally assumes negligible change of available potential energy.

For fixed wavevector with $k_z\ne 0$, fixed nonzero Alfvén frequency $a$, and $|\Omega |\to \infty$, the two roots in $X={\omega }^2$ of the stratified magneto-Coriolis wave polynomial have $X_f\sim 4{\Omega }^2{k}_z^2/K^2$ and the displayed slow root. This follows from the sum of roots being $4{\Omega }^2{k}_z^2/K^2+O(1)$ and their product being $a^2(K^2{a}^2-H{h}^2)/K^2$. The slow branch is oscillatory for $K^2{a}^2>H{h}^2$ and growing or decaying for the reversed inequality. At equality its frequency vanishes and the original stationary equations must be used.