For pricing drift ${\mu }_Q$, a unit zero-coupon bond price $P(t,r;T)$ satisfies $P_t+{\mu }_Q{P}_r+{\eta }^2{P}_{rr}/2-rP=0$, with $P(T,r;T)=1$. Itô formula makes its bank-account-discounted value a local martingale; suitable integrability gives $P(t,r;T)={\mathbb{E}}^Q[\exp (-{\int }_t^T{r}_{s}ds)\mid r_t=r]$. The rate kills the diffusion semigroup, as in the Feynman-Kac formula.

If all traded zero-coupon bonds depend on one scalar rate driven by one Brownian motion, their diffusion exposures form one vector $v_i=\eta P_{i,r}$. Their instantaneous covariance matrix is $v{v}^T$, of rank at most one. Nondegenerate maturity changes are therefore instantaneously perfectly correlated, with correlation $+1$ or $-1$, though finite-horizon bond-return correlations need not be one.

For smooth derivative prices depending on a Markov short rate, Itô formula gives physical drift $m=V_t+b{V}_r+{\sigma }^2{V}_{rr}/2$ and exposure $s=\sigma V_r$. The common short-rate market price of risk gives $m-rV=\theta s$, yielding this equation. Under the risk-neutral measure, the rate drift is $b-\sigma \theta$ and the bank-discounted price is a martingale under suitable integrability. A unit zero-coupon bond has terminal value one and price ${\mathbb{E}}^Q[\exp (-{\int }_t^T{r}_{s}ds)\mid r_t=r]$.

If $p$ divides the primitive level $M$ of a trivial-character newform and $N=M{p}^r$, put $e_j=p^{jk/2}f(p^{j}z)$. Then $U_p{e}_0=a_p{e}_0$ and $U_p{e}_j=p^{k/2}{e}_{j-1}$ for $j\ge 1$. The normalized Fricke involution acts by $e_j\mapsto \epsilon e_{r-j}$, where $W_{M}f=\epsilon f$. These two operators act irreducibly: a nonzero invariant subspace contains an eigenvector of $U_p$; its eigenvalue is $a_p$ or zero. Either it contains $e_0$ immediately or applying the reversal and a power of $U_p$ produces $e_0$, using $a_p^2\ne p^k$. Reversal then produces $e_r$, and its successive images produce every basis vector.

Let $M\mid N$ and $D\mid N/M$. For a weight-$k$ cusp form $f$ on ${\Gamma }_0(M)$, the function $\tau \mapsto f(D\tau )$ is a weight-$k$ cusp form on ${\Gamma }_0(N)$. Indeed, conjugating a matrix $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\Gamma }_0(N)$ by $\mathrm{diag}(D,1)$ givesbecause $MD\mid N$. The same argument applies to ${\Gamma }_1(M)$ and ${\Gamma }_1(N)$: the congruences $a,d\equiv 1(\mathrm{mod}N)$ imply the required congruences modulo $M$ as well. This supplies both common versions of the degeneracy maps used in the old subspace of cusp forms.

At a rational modular cusp, factor the rational change of variable into an integral cusp representative followed by an upper-triangular rational matrix of positive slope. The original Fourier expansion of a modular form then remains holomorphic and has zero constant term. Thus the cusp form condition, as well as the weight transformation, is preserved.

Author of the 1827 treatise developing the electrical transport relation now called Ohm's law. ^[ref].

For a finite-state homogeneous continuous-time Markov chain with transition intensity matrix $Q$, integrate the indicator of occupying state $j$ over $[0,T]$. Interchanging the bounded time integral and expected value gives the displayed integral of transition probabilities. It counts all visits, not only the first holding time. Summing over states gives $T$. A zero eigenvalue of $Q$ does not cause a problem for this integral, although it prevents casually writing it with an inverse matrix $Q^{-1}$.

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.