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For a continuous-time Markov chain with absorbing endpoints $0,N$ and rates $q_{j,j\pm 1}={\lambda }_j/2$, the jump chain is a simple symmetric random walk. On leaving $i$, absorption before returning has probability $p_i=\frac{1}{2}(1/i+1/(N-i))$, by gambler's ruin on each side. The geometric return count before absorption and independent rate-${\lambda }_i$ holding times give ${\mathbb{E}}_{i}I=1/(p_i{\lambda }_i)$. The rates at other states affect elapsed excursion times but not their return probabilities.

Using the component-first velocity gradient $K_{ij}=\partial v_i/\partial x_j$, the Rivlin-Ericksen tensors are defined by the displayed recurrence. In particular $A_1$ is twice the rate-of-strain tensor. These tensors transform objectively under time-dependent rigid observer changes. With this convention the upper-convected derivative of $A_1$ is $A_2-2A_1^2$, an identity useful in the slow-flow expansion of a second-order fluid.

For an affine null geodesic congruence, $B_{ab}={\nabla }_b{U}_a$ annihilates $U$ in both slots. Relative to a null partner, its antisymmetric part is its screen projection plus a term of the form $U_{[a}{q}_{b]}$. Hence zero null twist implies $U\wedge dU=0$, and Frobenius theorem makes $U_a$ locally proportional to a hypersurface gradient. Conversely, if $U_a=h{\nabla }_{a}S$, its antisymmetric derivative has a factor of $U$, so the screen projection vanishes. Geodesicity is essential to the converse from vanishing screen twist to the full Frobenius condition.

For an affine null congruence in four spacetime dimensions with a parallel-transported screen, the displayed equation follows by taking the symmetric trace-free part of the Sachs optical equations. The Ricci contribution to the optical tidal tensor is $P_{ab}{R}_{cd}{U}^c{U}^d/2$, so it changes the expansion but contributes no screen-trace-free source. The projected Weyl tensor supplies the tidal source of shear.

A unit nowhere-zero section $\sigma$ of a real vector bundle with a fiber metric defines a rank-one trivial vector bundle. Its fiberwise orthogonal complement is a subbundle: locally complete $\sigma$ to an orthonormal frame. The vector bundle isomorphism $(v,t)\mapsto v+t\sigma$ has inverse $w\mapsto (w-⟨w,\sigma ⟩\sigma ,⟨w,\sigma ⟩)$. For an oriented bundle, orient the complement so that its positive frame followed by $\sigma$ is positive.

Identify ${\mathbb{R}}^{2m+2}$ with ${\mathbb{C}}^{m+1}$ and let $J$ be multiplication by $i$. On the unit sphere $S^{2m+1}$, the vector field $V(x)=Jx$ has norm one and satisfies $⟨x,Jx⟩=0$. It is therefore a continuous tangent nowhere-zero section, including on $S^1$.

Put ${\mathcal{N}}_m=\{f:f(t)\equiv t(\mathrm{mod}{t}^{m+1})\}$ for $m\ge 1$. These are open normal subgroups and ${\mathcal{N}}_m/{\mathcal{N}}_{m+1}\cong ({\mathbb{F}}_p,+)$ via the coefficient of $t^{m+1}$. Thus $[\mathcal{N}:{\mathcal{N}}_m]=p^{m-1}$. The depth of a nonidentity series is ${\mathrm{ord}}_t(f(t)-t)-1$. If $f=t+a{t}^{r+1}+\cdots$ and $g=t+b{t}^{s+1}+\cdots$, expansion of substitution givesHere $[f,g]=f^{-1}\circ g^{-1}\circ f\circ g$. This formula controls commutator subgroups and normal subgroups.

Realize a finite p-group $P$ as the Galois group of a totally ramified extension $E/{\mathbb{F}}_p((t))$, using Artin-Schreier embedding problems over a characteristic-p field. A uniformizer $u$ of $E$ gives $E\cong {\mathbb{F}}_p((u))$: every element has a unique convergent Laurent expansion with coefficients in the residue field. An automorphism has $u\mapsto a_{1}u+a_2{u}^2+\cdots$, and $a_1$ defines a homomorphism $P\to {\mathbb{F}}_p^{\times }$. Its image is trivial because $|P|$ is a $p$-power and $p\nmid |{\mathbb{F}}_p^{\times }|$. Thus the faithful Galois action lies in the Nottingham group, embedding $P$.

Every algebraically irreducible representation of a Banach algebra that is normed is continuous. The continuous orbit vectors form an invariant subspace. If all nonzero orbit maps were discontinuous, apply the countable annihilation sequence in an irreducible Banach module and the gliding-hump continuity principle to right multiplication on the algebra and evaluation of the operator-valued representation. Off-diagonal maps vanish, whereas diagonal maps are discontinuous, a contradiction. Once every orbit map is continuous, the Uniform boundedness principle on the Banach algebra gives the common operator-norm bound. Completing the original normed representation space supplies the Banach operator space needed in the argument.

For discrete random variables, normalized subset entropy decreases with subset size. For finite entropies, the chain rule for information entropy and conditioning reduces entropy give ${\sum }_{i\in S}h(X_{S\setminus \{i\}})\ge (|S|-1)h(X_S)$. Sum this over all $k$-subsets $S$. Every $(k-1)$-subset occurs $n-k+1$ times, and $k\left(\genfrac{}{}{0ex}{}{n}{k}\right)=(n-k+1)\left(\genfrac{}{}{0ex}{}{n}{k-1}\right)$ cancels the normalization factors. This proves $h_k^{(n)}\le h_{k-1}^{(n)}$. If a marginal entropy is infinite, every one of these subset averages is infinite; otherwise all entropies used in the proof are finite.

A lattice-ordered group is normal-valued precisely when the displayed inequality holds for all its elements. The theorem is proved without using it as a definition. In the forward direction a normal value has a linearly ordered quotient of its cover with no proper nontrivial convex subgroup. That quotient is Archimedean, hence Abelian by Archimedean linearly ordered groups are Abelian. Apply squared bound from comparison at values to $p=b,q=a,x=ab$. For the reverse direction, power cofinality in the cover of a value excludes a conjugate $u=a^{-1}va$ outside the value: its right cosets $V{u}^n$ are all bounded by $Va$, contradicting cofinality above $V{a}^2$. Left-coset cofinality treats $av{a}^{-1}$. Every element is a quotient of nonnegative lattice parts, proving normality in the cover. In a linearly ordered group, comparison of $a,b\ge 1$ immediately gives $ab\le \max (a,b{)}^2\le b^2{a}^2$, so every such group is normal-valued.

A fundamental normal surface has a nonzero admissible coordinate vector which is not the sum of two nonzero admissible vectors. Fixing the allowed quadrilateral type in every tetrahedron reduces the problem to the Hilbert basis of a rational cone. There are finitely many choices of quadrilateral types and finitely many indecomposable vectors for each choice.

The Haken sum of compatible normal surfaces is obtained by resolving their intersection curves in the way that preserves the normal disk types. It adds normal surface coordinates, Euler characteristic and mod-two homology classes. The alternative local resolution can create returning face arcs and need not be normal.

The weight of a normal surface counts its intersections with the edges of the triangulation. It is a positive linear function of its coordinates: each corner of a normal disk contributes the reciprocal of the number of tetrahedral sectors around its edge. Thus a nontrivial decomposition into compatible coordinate vectors gives both summands strictly smaller weight.

The canonical commutation relation gives $[P^{\mu },a_p^{†}]=p^{\mu }{a}_p^{†}$ for a free real scalar field. Differentiate the exponential conjugation: its derivative is a scalar multiple of the same operator, yielding the displayed phase. The annihilation operator receives the opposite phase, so at equal time $e^{-i\mathbf{P}\cdot \mathbf{y}}\varphi (\mathbf{x})e^{i\mathbf{P}\cdot \mathbf{y}}=\varphi (\mathbf{x}+\mathbf{y})$. The spatial translation operator translates a state's position wavefunction in the opposite argument direction.

A finite beta reduction can be rearranged into standard order, contracting earlier residual redexes before later ones. In particular, a lambda term with a head normal form reaches one by head reduction, and a term with a beta-normal form reaches it by normal-order beta reduction. Head reduction contracts the outermost redex on the left application spine, including under initial lambda abstractions.

The spectral theorem represents a normal operator by multiplication by its spectral parameter. In a finite-dimensional inner product space, this is diagonalization in an orthonormal basis; for a self-adjoint operator on a Hilbert space, it uses a projection-valued measure. Unbounded operators additionally require their natural square-integrability domain.

In a normal neighbourhood $U={\exp }_p(V)$ with $V$ star-shaped, define $r(q)=|{\exp }_p^{-1}(q)|$. Away from $p$, the Gauss lemma gives $\nabla r=(d{\exp }_p{)}_v(v/|v|)$ for $v={\exp }_p^{-1}(q)$. Thus $|\nabla r|=1$ and this unit vector field is tangent to unit-speed radial geodesics. The function $r$ is not smooth at $p$, but $r^2/2$ is smooth and $\nabla (r^2/2)=(d{\exp }_p{)}_{v}v$ extends by zero. On a sufficiently small normal ball, $r$ equals the Riemannian distance from $p$.

// Target: riemannian-geometry.bigb

For the Dirac equation with two equal quark masses $m$, the equations for $q$ and $\bar{q}$ give the divergence of the nonsinglet axial current as $im\bar{q}{\gamma }_5{\tau }_{a}q=m{P}_a$. The color chiral anomaly is absent in this nonsinglet identity because the quark flavor generator is traceless. Equivalently, the mass term in the Hamiltonian is $m\int S{d}^{3}x$, and the nonsinglet axial charge algebra of quark densities gives ${\dot{Q}}_{5,a}=m\int P_a{d}^{3}x$.

A first-order theory locally omits a family $\Gamma (\mathbf{x})$ if every formula $\theta (\mathbf{x})$ consistent with it has a consistent refinement $\theta \wedge \neg \gamma$ for some $\gamma \in \Gamma$. Equivalently no consistent formula implies all of $\Gamma$. In a countable language the omitting types theorem converts this local condition into a countable model with no realization of the family.