Choose ordered minimal topological generators $a_1,\dots ,a_d$ of a uniform pro-p group for odd $p$, put $b_i=a_i-1$, and choose $1/(p-1)<s<1$. Ordered monomials $b^{\alpha }=b_1^{{\alpha }_1}\cdots b_d^{{\alpha }_d}$ give power-series coordinates in the integral completed group algebra. The displayed Non-Archimedean absolute value filtration gives a complete normed ${\mathbb{Q}}_p$-algebra after completion of the rational group algebra. The power relation raises valuation by one, and interchanging $b_i,b_j$ adds terms of valuation at least $1+s>2s$, so multiplication does not decrease total valuation. The p-adic logarithm and p-adic exponential are inverse on the relevant small balls. In particular $\log G$ is a faithful realization of the Lie algebra of a uniform pro-p group, useful for verifying its intrinsic limit operations and the Baker--Campbell--Hausdorff formula without assuming the correspondence in advance.

On ${\mathbb{R}}^2$, the vector fields $X=y{\partial }_x$ and $Y=(x^2/2){\partial }_y$ are complete, since their local flows are $(x+ty,y)$ and $(x,y+t{x}^2/2)$. Their Lie bracket of vector fields is $-(x^2/2){\partial }_x+xy{\partial }_y$. Its integral curve of a vector field starting at $(2,0)$ is $(2/(1+t),0)$, which cannot continue through $t=-1$. The sum has an integral curve of a vector fieldthat cannot continue through $t=2\sqrt{3}$. Thus neither operation preserves complete vector fields on a general noncompact smooth manifold. This contrasts with a compact manifold without boundary, where every smooth vector field is complete.

An $r$-uniform hypergraph is strongly saturated with respect to a complete uniform hypergraph of order $r+t$ when every missing edge immediately completes such a clique. Existing cliques are allowed. All $r$-sets meeting a fixed $t$-set give an example: the witness for a missing edge is its union with that fixed set. It has $\left(\genfrac{}{}{0ex}{}{n}{r}\right)-\left(\genfrac{}{}{0ex}{}{n-t}{r}\right)$ edges, and the exterior-power quotient bound for strong hypergraph saturation proves optimality.

A partially ordered set is directed-complete if every directed set in it has a least upper bound. A complete lattice is directed-complete. Conversely, directed completeness together with all finite joins, including the empty join, gives arbitrary joins: take the directed set of joins of finite subsets.

For a Banach space $E$ satisfying $E\cong ({⨁}_{n\ge 1}E{)}_p$, a complemented subspace $W$ is absorbed by $E$: write $E\cong W\oplus V$, expand the countable sum, and absorb one additional copy of $W$ into the countably many $W$ summands. Hence $E\oplus W\cong E$. If $Z$ is complemented in $E$ and contains a complemented copy of $E$, write $Z\cong E\oplus W$ with $W$ complemented in $E$; the preceding argument gives $Z\cong E$. The decompositions are Banach space isomorphisms, not merely algebraic decompositions.

If a counting process $N$ has predictable conditional intensity $\lambda$, subtracting its compensator of a counting process gives the associated martingale $M$. Conditional expected event increments and compensator increments agree, so their difference is conditionally centered. With the necessary integrability, $M$ is a mean-zero martingale; for a simple no-ties process its predictable quadratic variation is ${\int }_0^t\lambda (u)du$. A predictable weighted martingale integral has mean zero and variance determined by the square of its weights. In survival analysis, $\lambda =Yh$ produces the event-minus-expected representation used to derive the Nelson–Aalen estimator.

The metric construction of a compatible almost complex structure is continuous in the auxiliary Riemannian metric, and recovers $J$ when applied to $g_J=\omega (\cdot ,J\cdot )$. Interpolating $g_J$ linearly to a fixed auxiliary metric therefore contracts the space of compatible almost complex structures. In particular it is nonempty and path connected.

An explicit objective isotropic example is $\mathcal{W}=\mu (S-1)/2+h(S-1{)}^2/4+(\lambda +\mu )(J-1{)}^2/2$, with $J=\det A$, $S=\mathrm{tr}(A^{\mathsf{T}}A)-2J$, and $\lambda ,\mu >0$, $h\ge 0$. On $A=[(a,q,0{)}^{\mathsf{T}},e_2,e_3]$, $S-1=(a-1{)}^2+q^2$ and $J=a$, so $P_{11}=0$ at $a=1$ while the shear nominal stress is the displayed cubic. The natural-state quadratic energy has Lamé parameters $\lambda ,\mu$, and the shear speed is $\sqrt{(\mu +3h{q}^2)/{\rho }_0}$.

If $x_n>0$ and $c>0$, clipping every term at $c$ preserves divergence. If infinitely many terms are at least $c$, the clipped series has infinitely many terms equal to $c$ and diverges. Otherwise its tail agrees with the original series. This lemma is useful in proving convergence from minimum-decrement inequalities.

For positive sequences $(a_n),(b_n)$, suppose $a_{n+1}/a_n\le b_{n+1}/b_n$ eventually. Then $a_n/b_n$ is eventually nonincreasing, so $a_n\le C{b}_n$ beyond a fixed index. The comparison test for series proves convergence of $\sum a_n$ whenever $\sum b_n$ converges. Reversing the ratio inequality gives an eventual lower bound $a_n\ge c{b}_n$ and hence divergence when the positive comparison series diverges. The initial finite set of terms does not affect either conclusion.

For the $a=1$ compact self-similar disk with density-linear viscosity, radial drift is $\dot{r}=r/(2t)-R/(10t)$. An integrating factor gives the displayed trajectories. From $0<r_i<R(t_i)$, the arrival time at the origin is $t_i[1-r_i/R(t_i){]}^{-10}$, finite for every interior point. The outer-edge trajectory has zero surface density of a disk and carries no mass.

The $a=5/4$ compact self-similar disk with density-linear viscosity has outward drift $r/(2t)$ but zero limiting mass flux at the origin. Its viscous torque in an accretion disk has a positive finite inner limit for $t>0$, equal to $dJ/dt$. Thus disk mass conservation coexists with growth of angular momentum; an external central torque supplies that growth.

An arbitrary product of compact locales is compact. Finite products correspond to preframe tensor products, whose compactness follows by tensoring the top tests. An arbitrary product is obtained by the filtered colimit of finite frame coproducts; the eventual-top test proves that filtered colimits preserve compactness. No choice of points is required.

Let $G$ be a connected compact Lie group, $T$ a maximal torus, and $W$ its Weyl group. With normalized Haar measures and $D(t)={\prod }_{\alpha >0}(1-e^{-\alpha }(t))$, every continuous class function satisfies ${\int }_{G}f(g)dg=|W{|}^{-1}{\int }_{T}f(t)|D(t){|}^{2}dt$. For a general continuous function, replace $f(t)$ by its average ${\int }_{G/T}f(gt{g}^{-1})d(gT)$. Conjugation from $G/T\times T$ covers the regular elements $|W|$ times, and its real Jacobian determinant is $|D(t){|}^2$.

A real torus is the compact Lie group obtained by identifying real vectors differing by integer vectors, with addition induced from Euclidean addition. It is a finite product of circle groups and has normalized Haar measure. Dimension two is the familiar surface torus, but toral dynamics and Fourier characters naturally use arbitrary finite dimension.

For a nonnegative self-adjoint operator with compact resolvent, this function counts eigenvalues up to a given level with their multiplicities. For the positive Laplace-Beltrami operator on a closed $n$-dimensional Riemannian manifold, $N(\Lambda )\le C(1+\Lambda {)}^{n/2}$. To prove this bound, multiply each vector in the low-eigenvalue subspace by a finite set of chart cutoffs and record its local Fourier coefficients of frequency at most $R$. The total number of retained coefficients is $O(R^n)$, while the sum of discarded squared norms is at most $C{R}^{-2}(1+\Lambda )\Vert u{\Vert }_2^2$ by Parseval identity and the energy bound. For $R$ a sufficiently large multiple of $\sqrt{1+\Lambda }$, vanishing of all recorded coefficients forces $u=0$. The recording map is injective and proves the dimension bound. Inverting this estimate gives ${\lambda }_j\ge c(j+1{)}^{2/n}-1$.

For proper number density $n(t)$ in an FLRW metric $d{s}^2=d{t}^2-a^2[d{r}^2/(1-k{r}^2)+r^{2}d{\Omega }^2]$, a radial shell has proper volume $4\pi a^3{r}^2|dr|/\sqrt{1-k{r}^2}$. A radial null geodesic obeys $|dr|=\sqrt{1-k{r}^2}dt/a$. Multiplication gives the displayed shell count on a local radial branch of the past light cone. Spatially uniform number density need not be time independent or have conserved comoving number density.

A unital subalgebra $A\subseteq \mathrm{End}(V)$ closed under the adjoint operator, on a finite-dimensional real or complex inner product space, equals its double commutant of an operator algebra. Here is a proof that does not require a classification of algebras. For $N=\dim V$, choose a basis $(v_1,\dots ,v_N)$ and let $W=\{(a{v}_1,\dots ,a{v}_N):a\in A\}\subseteq V^N$. The diagonal action of $A$ preserves $W$ and $W^{\perp }$, so the orthogonal projection onto $W$ commutes with that action. Each matrix entry of this projection belongs to $A^{\prime }$. Hence the diagonal action of $b\in A^{\prime \prime }$ commutes with the projection and preserves $W$. Since $1\in A$, the tuple $(v_1,\dots ,v_N)$ belongs to $W$, and therefore $(b{v}_1,\dots ,b{v}_N)=(a{v}_1,\dots ,a{v}_N)$ for one $a\in A$. This implies $b=a$. Without the unital hypothesis the conclusion is $A^{\prime \prime }=A+\mathbb{F}1$.

For a unital star-subalgebra $A$ of the bounded operators on a complex Hilbert space, its strong operator topology and weak operator topology closures equal its double commutant of an operator algebra. To approximate $T\in A^{\prime \prime }$ on finitely many vectors, take their tuple in $H^m$. The projection onto its diagonal-$A$ cyclic subspace has matrix entries in $A^{\prime }$, so it commutes with the diagonal action of $T$. Therefore the image tuple under $T$ lies in that cyclic subspace and can be approximated by a single $a\in A$.

For a finite, time-homogeneous Discrete-time Markov chain, state $j$ is accessible from $i$ when some finite path of positive transition probabilities leads from $i$ to $j$. Equivalently, $(P^n{)}_{ij}>0$ for some $n\ge 0$, allowing a zero-step path when $i=j$. Accessibility is reflexive and transitive; communicating states require accessibility in both directions. See Ward Whitt's lecture notes of September 24, 2013, classification of states. In the closed-cycle example in communicating class, $s\to c_2$ despite $P_{s,c_2}=0$, because $P_{s,c_1}{P}_{c_1,c_2}=1/4>0$; $c_2\to s$ fails at every step count.