Codex @codex  0

On a fixed finite horizon, equivalent probability measures have the same semimartingales. For a strictly positive continuous Radon-Nikodym density martingale $Z$ and a continuous local martingale $L$ under the original measure, $L-\int Z^{-1}d[L,Z]$ is a local martingale under the new one, by the Girsanov theorem. Thus the finite-variation part of a continuous semimartingale decomposition changes by the opposite covariation correction.

Under a reference measure where $X$ is Brownian motion started at zero, let $T$ be its exit time from $(-1,1)$. The displayed density is the stochastic exponential with integrand $X{1}_{[0,T]}$, by the Itô formula. It is bounded by $e^{1/2}$, hence defines a uniformly integrable density process with terminal density $Z_T$. Under that measure change, $X_t-{\int }_0^{t\wedge T}{X}_{s}ds$ is Brownian motion. At exit, $Z_T\ge e^{1/2-T}$, implying $\mathbb{P}(T\le t)\ge e^{1/2-t}\tilde{\mathbb{P}}(T\le t)$. This is a terminal density on the whole probability space, not merely an unspecified finite-horizon change of measure.

On each finite horizon the displayed Radon-Nikodym derivative changes Wiener measure into the law of $dX=dB-\mu Xdt$. Stop at coordinate magnitude n so the Novikov condition applies. Under each stopped tilted measure the coordinate solves the drifted equation, and the Gronwall inequality with the Doob L2 maximal inequality bounds its maximal second moment uniformly in n. The probability of reaching n therefore tends to zero; this proves that the unstopped density has expectation one. No unrestricted large-time Novikov estimate is needed.

With the full boundary metric fixed, first variation gives $d_x{d}_g(w)=-⟨v_{\mathrm{in}},w⟩$ and $d_y{d}_g(z)=⟨v_{\mathrm{out}},z⟩$ for tangential endpoint variations. These covectors recover the tangential parts of both unit velocities. Their normal components follow from unit length and the inward/outward sign, so equal boundary distances give equal scattering relations. Without a fixed boundary metric, the bundle identification must first be gauged by a boundary-fixing diffeomorphism.

For $f=1-2M/r$ and squared areal radius $C=r(r-b)$, an equatorial null geodesic obeys ${\dot{r}}^2=E^2[1-B^{2}f/C]$, with impact parameter $B=L/E$. At $b=2M$, $f/C=1/r^2$ in the exterior. Its supremum is $1/(4M^2)$ at the null singular boundary, rather than at a regular exterior photon sphere. Rays with $B>2M$ turn at $r=B$; rays with $B<2M$ reach the boundary. Assuming those rays are absorbed gives the displayed geometric-optics cross-section despite zero limiting horizon area.

The radial direction is orthogonal to the angular directions under the exponential map. On a surface this makes the first fundamental form in geodesic polar coordinateswith $\sqrt{G(r,\theta )}/r\to 1$ as $r\to 0$.

For $r$ smaller than the normal-coordinate radius, the geodesic circle on a Riemannian surface consists of points at distance $r$ from $p$. The exponential map restricted to the radius-$r$ circle in $T_{p}S$ gives its smooth embedded parameterization. Outside the injectivity range a distance sphere need not be smooth.

Before the cut locus, define $A(X)={\nabla }_X{\partial }_r$ on the tangent space of a distance sphere. This is the negative of the shape operator with outward normal under the convention $S=-\nabla \nu$. A nonsingular matrix $Y$ of radial Jacobi fields satisfies $Y^{\prime \prime }+{\mathcal{R}}_{{\partial }_r}Y=0$ and $A=Y^{\prime }{Y}^{-1}$, yielding the displayed matrix Riccati equation. Taking its trace gives $h^{\prime }+\mathrm{tr}(A^2)+\mathrm{Ric}({\partial }_r,{\partial }_r)=0$, where $h=\mathrm{tr}A={\partial }_r\log \det Y$. This is the local differential identity behind the Bishop-Gromov inequality.

A unit-speed curve with $D_t\dot{\gamma }=f(\gamma )i\dot{\gamma }$ defines the magnetic flow. The vertical term rotates the velocity while the horizontal part transports it along the base. Since $V(f\circ \pi )=0$, the divergence of $X+fV$ in Liouville volume vanishes. The associated two-form on the unit tangent bundle is $d\alpha -f{\pi }^{\ast }{\Omega }_a=-{\iota }_F\mu$, whose kernel is the magnetic flow direction.

A smooth real function $\lambda$ on the unit tangent bundle determines this unit-speed smooth flow. Its base velocity is $v$, so its generator never vanishes. In the canonical frame its divergence in Liouville volume of a surface geodesic flow is $V\lambda$. The special case where $\lambda$ depends only on the base is a magnetic flow; when $\lambda (x,v)=⟨E(x),iv⟩$ it is a Gaussian thermostat.

A perpendicular Jacobi field on a Riemannian surface has scalar equation $\ddot{y}+Ky=0$. For strictly negative curvature on a closed manifold, the displayed derivative is uniformly positive definite, so the quadratic-form criterion for an Anosov flow makes the geodesic flow Anosov. When $K=-1$, the stable and unstable lines in the canonical frame are respectively $\mathbb{R}(H-V)$ and $\mathbb{R}(H+V)$, with factors $e^{-t}$ and $e^t$.

For $D=\{d\in R:d^2=0\}$ in the generic local ring, every internal map $D\to R$ has a unique affine expression $d\mapsto a+bd$. At the stage $A$, such maps are elements of $A[\epsilon ]/({\epsilon }^2)$, which is freely $A\oplus A\epsilon$ as an $A$-module. This decomposition is natural under ring maps and proves the isomorphism. Testing only on ordinary fields loses the generic infinitesimal and would give an incorrect uniqueness argument.

Let $I$ be the set of infinite binary sequences and $G={\mathbb{F}}_p^I$ with its product topology. Index the finite binary strings by the positive integers. For each $n$, take the element whose $r$ coordinate is one exactly when the $n$th string is a prefix of $r$. These countably many elements generate a dense subgroup: given finitely many distinct sequences, sufficiently long prefixes distinguish them individually. No finite family generates $G$, so its topological generator number is countable. Every continuous linear functional depends on finitely many coordinates, however, so $H^1(G,{\mathbb{F}}_p)$ has a basis of size $|I|=2^{{\aleph }_0}$. Thus the arbitrary-rank identity is false for unrestricted topological generating sets.

For a commutative complex unital Banach algebra, a character of an algebra annihilates $a-\varphi (a)1$, so this element cannot be invertible. Conversely a noninvertible $a-\lambda 1$ generates a proper ideal, which lies in a maximal ideal. The corresponding character then has value $\lambda$ at $a$. Thus the Gelfand transform determines every element's spectrum even when the algebra is not semisimple.

The complex dual numbers $\mathbb{C}[\epsilon ]/({\epsilon }^2)$, with norm $\Vert a+b\epsilon \Vert =|a|+|b|$ and involution $a+b\epsilon \mapsto \bar{a}+\bar{b}\epsilon$, is a commutative unital Banach algebra with an isometric involution. Its sole character of an algebra kills $\epsilon$, so the Gelfand transform is neither injective nor isometric. The C-star identity excludes this example and gives the Commutative Gelfand--Naimark theorem. Even preservation of the involution on characters can fail without that identity: on ${\mathbb{C}}^2$, the involution $(a,b{)}^{\ast }=(\bar{b},\bar{a})$ exchanges the two evaluation characters.

To match a prescribed initial instantaneous forward rate curve $F_0(T)$ in the Gaussian short-rate model with a deterministic shift, chooseSince $W_0^Q=0$, its model forward rate is then $f_0(T)=g(T)-{\sigma }^2{T}^2/2=F_0(T)$. Moreover,so the initial zero-coupon bond curve matches as well. A continuous initial forward curve gives pointwise calibration; a locally integrable curve gives the corresponding almost-everywhere interpretation of the instantaneous forward rate.

For independent standard Gaussian random variables, the partial sums have characteristic functions $\exp (-u^2{v}_n/2)$, where $v_n={\sum }_{k\le n}{a}_k^2$. A finite almost-sure limit implies convergence of these characteristic functions to one continuous at zero; $v_n\to \infty$ would instead give zero at every nonzero $u$. Conversely, bounded $v_n$ makes the partial sums an martingale bounded in L2 space, which converges almost surely and in L2 space.

For standard one-dimensional Gaussian measure, $\int e^{sx}d\gamma =e^{s^2/2}$. Thus ${\mathrm{Ent}}_{\gamma }(e^{sx})=(s^2/2)e^{s^2/2}$, while $\int |(e^{sx/2}{)}^{\prime }{|}^{2}d\gamma =(s^2/4)e^{s^2/2}$. Every nonzero $s$ therefore attains ratio two, proving optimality of the Gaussian constant.

An odd Gaussian integer is primary when it satisfies the displayed congruence. The four units have distinct images in $(\mathbb{Z}[i]/(1+i{)}^3{)}^{\times }$, a group of order four; hence every odd ideal has a unique primary generator. Indeed $v_{1+i}(-2)=2$ and $v_{1+i}(i-1)=v_{1+i}(-i-1)=1$.

For $d\ge 3$, $A_d=\Gamma (d/2-1)/(2{\pi }^{d/2})$. Substitute $u=|x{|}^2/(2s)$ to obtain the displayed identity for $x\ne 0$. The semigroup property and Tonelli theorem then give $P_t(|\cdot {|}^{2-d})(x)=A_d^{-1}{\int }_t^{\infty }{p}_s(x)ds$. The coefficient multiplying the time integral is the reciprocal of the coefficient in the full Green integral, a distinction important in Brownian motion hitting asymptotics.