Suppose $a^{\prime }$ is bounded and let $d{Z}_t=Z_t{a}^{\prime }(S_t)d{W}_t$, $Z_0=1$. The Novikov condition makes $Z$ a strictly positive true martingale. For ${\pi }_t=U(t,S_t)$ solving the call delta equation for a driftless local volatility diffusion, the product formula givesThe two drift terms cancel through quadratic covariation. Thus $Z\pi$ is a local martingale. If it is a true martingale on the closed horizon, its terminal value is $Z_T{1}_{\{S_T\ge K\}}$, soThe upper bound uses $\mathbb{E}[Z_T\mid {\mathcal{F}}_t]=Z_t$. Equivalently, the option delta is the terminal-exceedance probability under the measure with density $Z_T$.

Solutions of one fixed bounded-coefficient uniformly elliptic equation $Lu=f$ that are bounded in global $L^2$ have a subsequence converging strongly in $H^1$ on every relatively compact subdomain. Local energy bounds and the Rellich-Kondrachov compactness theorem first give local strong $L^2$ and weak $H^1$ convergence on a smooth exhaustion. Pass to the weak equation to identify the limit. Differences solve $L(u_k-u)=0$, so the interior Caccioppoli inequality upgrades the local strong $L^2$ convergence to strong gradient convergence. Global weak $L^2$ convergence preserves the original global bound.

Suppose a compact parameter set carries a locally finite countable multiset of positive quantities, and a fixed injective transform of each labelled quantity is a polynomial in finitely many ambient coordinates. Assume only finitely many labels can have bounded value anywhere in the compact set. Then a uniform finite initial multiset determines the entire multiset. For each source label there are finitely many possible matching labels. Equality for finitely many labels in both directions is a finite union of algebraic sets, obtained from injective matchings and polynomial equalities. These algebraic sets decrease as more labels are included, so they stabilize by the Hilbert basis theorem. A compatible infinite matching in each direction follows by the König infinity lemma, and local finiteness makes multiplicities equal. Squared traces of hyperbolic holonomy representations give the application to the Buser finite length spectrum theorem.

Let the initial dust mass be $m_d$, let its differential number distribution be $K{D}^{-\alpha }$ with $\alpha <4$ between $D_{\min }$ and $D_{\max }$, and put $\eta =D_{\min }/D_{\max }$. Under burst transport under inverse-size radial drag, complete absorption at a fixed radius occurs at $t=TD/D_{\max }$, with $T=t_r(D_{\max })$. The arriving mass is proportional to $D^{3-\alpha }dD$. Substitute $D=D_{\max }t/T$ and $dD/dt=D_{\max }/T$ and use mass normalization of a power-law size distribution to prove the displayed rate for $\eta T\le t\le T$, with zero rate elsewhere. Its integral is exactly $m_d$. Resonant delays, scattering losses, a finite release duration or grain destruction change this ideal pulse.

For initial dust mass $m_d$ in a ring at $r_0$ with a fixed-diameter power-law size distribution of index $\alpha <4$, burst transport under inverse-size radial drag maps radius to diameter $D_r=2\gamma t/(r_0^2-r^2)$. Set $T_r=t_r(D_{\max })$ and $\eta =D_{\min }/D_{\max }$. The surface density is the azimuthal average: $2\pi r\Sigma dr=(\pi \rho K/6)D_r^{3-\alpha }d{D}_r$. Since $d{D}_r/dr=2r{D}_r/(r_0^2-r^2)$, mass normalization of a power-law size distribution gives the displayed expression. It is supported where $\eta \le t/T_r\le 1$. It applies at $t>0$ before removal at the radius in question; the initial state is a delta-function ring. A local angular density needs additional phase information.

The work served in one slot is capped at the service rate of a queue. The remainder is lost, so output is input minus its positive part above capacity. Every downstream threshold below the cap has exactly the same overflow event for input and output; thresholds at or above the cap cannot be exceeded by the output.

For fixed Poisson mean $\lambda >0$ and capacity $c\to \infty$, set $m=\lfloor c\rfloor +1$. The optimized Chernoff bound gives $\log \mathbb{P}(X\ge m)\le -I_P(m)$. The lower bound by $\mathbb{P}(X=m)$ and Stirling's formula differs by only $O(\log m)$. Rounding $m$ to $c$ changes the cost by $O(\log c)$, proving the displayed formula and the leading scale $-c\log c$.

For $u^{\prime }=1-(b+1)u+a{u}^{2}v$, $v^{\prime }=bu-a{u}^{2}v$, the equilibrium $(1,b/a)$ has a Hopf bifurcation at $b=1+a$. Coordinates $x=u-1$, $y=-\sqrt{a}[(u-1)+(v-b/a)]$ give the threshold equations $x^{\prime }=-\sqrt{a}y+(1-a)x^2-2\sqrt{a}xy-a{x}^3-\sqrt{a}{x}^{2}y$, $y^{\prime }=\sqrt{a}x$. In the usual planar radial normalization the cubic coefficient is $-(a+2)/8<0$, proving a nondegenerate supercritical bifurcation. The leading small-amplitude angular frequency is $\sqrt{a}$.

For a deterministic $h>0$, the processes $B_{h-t}-B_h$, $0\le t\le h$, and $B_{h+t}-B_h$, $t\ge 0$, are independent standard Brownian motions. The first uses reversed, sign-changed past increments; the second uses future increments independent of the entire past. The first process is not generally independent of $B_h$, but translating a range by $B_h$ preserves its Lebesgue measure pathwise.

For Brownian motion with drift $dX=dB+\mu dt$ in $(-b,b)$, the displayed probability of exiting at $-b$ follows by stopping the harmonic function $e^{-2\mu X}$. A unit-time increment larger than $2b$ has a uniform positive probability and forces exit, so the exit time has a geometric tail bound from a uniform escape probability and all polynomial moments. At zero drift the limiting probability is $(b-x)/(2b)$.

The process $B_t^k-\mathbb{E}{B}_t^k$ is a martingale only for $k=1,2$. For higher powers its finite-variation drift is $k(k-1)(B_t^{k-2}-\mathbb{E}{B}_t^{k-2})/2$, which is not identically zero. Subtracting the random compensator $k(k-1){\int }_0^t{B}_s^{k-2}ds/2$ instead produces a true martingale on each finite horizon.

// Target: probability-and-statistics.bigb

For standard Brownian motion started at zero, choose an independent radius $R$ with $\mathbb{P}(R\le s)=(s/r{)}^n$, $0\le s\le r$, and stop on first reaching that radius. Rotational invariance of Brownian motion makes the conditional exit position uniform on each sphere. Mixing with radius density $n{s}^{n-1}/r^n$ gives the uniform probability distribution on the ball. In dimension one, the conditional sign has probabilities $1/2,1/2$. The stopped point samples volume measure even though each conditional exit law is supported on a boundary.

For independent coordinates of standard Brownian motion and $x_i>0$, the Brownian reflection principle gives ${\mathbb{P}}_x(T>t)={\prod }_i[2\Phi (x_i/\sqrt{t})-1]$. This is asymptotic to $(2/\pi {)}^{n/2}{\prod }_i{x}_i{t}^{-n/2}$. The tail integral formula for expectation therefore proves the stated criterion, with logarithmic divergence in dimension two.

For $d\ge 3$ and $|x|>\epsilon$, stop the harmonic function $u(x)=|x{|}^{2-d}$ at the two boundaries of $\{\epsilon <|x|<R\}$. Bounded optional stopping gives the probability of hitting the inner sphere first as $(u(x)-u(R))/(u(\epsilon )-u(R))$. Letting $R\to \infty$ gives the displayed formula; a finite hitting path is contained in some finite-radius ball.

For a cumulative distribution function $F$ and a standard Brownian bridge $\mathbb{G}$, the F-Brownian bridge is the centred Gaussian process ${\mathbb{G}}_F=\mathbb{G}\circ F$. Its covariance is $F(\min (s,t))-F(s)F(t)$, by substituting $F(s),F(t)$ into the bridge covariance. It can jump when $F$ jumps. Its law is tight in the supremum norm, since composition with $F$ is a continuous map from the separable space of continuous bridge paths.

For $t<1$ this stochastic integral solves $d{X}_t=d{B}_t-X_t(1-t{)}^{-1}dt$. The Itô product rule also gives $X_t=B_t-(1-t){\int }_0^t{B}_s(1-s{)}^{-2}ds$. The process is centered Gaussian, with covariance $\min (s,t)-st$, and has a continuous extension $X_1=0$. Time reversal of a Brownian bridge proves endpoint continuity without assuming it at the start.

The Brownian bridge covariance kernel is invariant under reversing both times, and its mean is zero. The two centered Gaussian processes therefore have equal finite-dimensional laws. Equality on countably many rational coordinates transfers the event of continuity at zero to the reversed endpoint. Interior continuity then extends the rational-time limit to all times. This proves the terminal continuity of the stochastic integral representation of a Brownian bridge.

The cubic crystal system has cubic point symmetry. Its three cubic Bravais lattices are primitive, body-centered, and face-centered. The body-centered cubic lattice is one member of that classification.

A tetragonal conventional cell with lattice points at its vertices and face centers describes a Bravais lattice. It is equivalent, after a rotation and a change of conventional cell, to a body-centered tetragonal description. The reciprocal lattice of a body-centered tetragonal lattice with conventional sides $a,a,b$ consists of $2\pi (h/a,k/a,l/b)$ with $h+k+l$ even; these points form a face-centered tetragonal conventional cell with sides $4\pi /a,4\pi /a,4\pi /b$.

A region whose translates tile a lattice's ambient space, together with the lattice points represented in it. A primitive unit cell represents one lattice point; a conventional unit cell may represent several to display symmetry.