Two identical radius-$a$ spheres moving together through a viscous fluid at separation $b\gg a$ entrain each other. The leading far velocity of one sphere at the other is $3aU/(4b)$ for separation perpendicular to the velocity, and $3aU/(2b)$ for parallel separation. Replacing $U$ by its velocity relative to this induced flow in Stokes law gives the displayed drag reductions relative to $D_0=6\pi \mu aU$, with errors of order $(a/b{)}^2$. Reflection symmetry and linearity imply equal drag on the two spheres in either geometry.

With body-outward normal $n$, put $F=-\int \sigma ndS$, ${\Sigma }_{ij}=\int x_i(\sigma n{)}_{j}dS$, $U_{ij}=\int u_i{n}_{j}dS$, and $M=\Sigma -2\mu U$. Taylor expansion of the exterior boundary integral representation of Stokes flow gives a Stokeslet from $F$, a rotlet from $G_j=-{ϵ}_{jkl}{\Sigma }_{kl}$, a source from $Q=\mathrm{Tr}U$, and a stresslet from $S=(M+M^T)/2-I\mathrm{Tr}M/3$. Thus $F$ and $G$ are the force and torque exerted on the fluid, while $Q$ is the instantaneous displaced-volume rate. A volume-preserving deforming body has $Q=0$.

For a two-dimensional decaying Stokes flow in $y>0$, prescribe boundary Fourier mode $(U,V)e^{ikx}$. The boundary shear stress and normal stress are the displayed diagonal map, with the common exponential understood. For $k>0$ and $V=0$, the Unscaled Papkovich–Neuber representation gives $u=U(1-ky)e^{ikx-ky}$, $v=-ikUy{e}^{ikx-ky}$ and $p=-2i\mu kU{e}^{ikx-ky}$. Differentiation proves the tangential traction and vanishing normal stress. The normal-velocity calculation supplies the second diagonal entry. The map converts a viscous exterior into a nonlocal boundary resistance proportional to $|k|$.

A layer of half-thickness $h_0$, dynamic viscosity $\mu$ and density $\rho$, between semi-infinite fluids of viscosity $\lambda \mu$ and densities $\rho \mp \Delta \rho$, relaxes a symmetric thickness Fourier mode. In the plug-like extensional viscosity regime, mass conservation and $4\mu (h{u}_x{)}_x=h\Delta \rho g{h}_x-{\sigma }_{xz}^{+}$ together with the Fourier traction map for a viscous half-space give the displayed rate, where $\kappa =|k|h_0$. Inner longitudinal extension dominates for $\lambda \ll \kappa$; outer tangential resistance dominates for $\kappa \ll \lambda$ while the plug approximation remains valid.

For positive stocks satisfying $d{S}_t^i/S_t^i={\sum }_j{\sigma }_{ij}d{W}_t^j+{\mu }_{i}dt$, let $q=n^{-1}1$ and $V=\sigma {\sigma }^T$. The Itô formula givesThus its volatility is ${\sigma }^{T}q$. This geometric average is generally not the value of the constant equal-weight self-financing portfolio: its instantaneous drift is $q^T\mu -\mathrm{tr}(V)/(2n)+q^{T}Vq/2$, whereas the fully invested equal-weight portfolio has drift $q^T\mu$.

A proportional continuous dividend payment at rate $\delta$ per unit time relative to the ex-dividend stock price. The total gain of one share is $d{S}_t+\delta S_{t}dt$. Under a risk-neutral measure with short rate $r$, a geometric stock diffusion therefore has ex-dividend drift $(r-\delta )S_t$. A fixed cash dividend at a specified date is a different convention and generally requires an ex-dividend price jump.

An increment of a real- or vector-valued stochastic process is the difference $X_t-X_s$ between its values at two times $s<t$. A random walk accumulates its one-step increments; a Poisson process has nonnegative integer increments counting arrivals in $(s,t]$. A process can have independent increments even though its values at different times are dependent.

A process is locally bounded if there are stopping times ${\tau }_n\uparrow \infty$ and finite deterministic constants $K_n$ bounding it on the stopped intervals. A locally bounded predictable process is integrable against a semimartingale. Continuous adapted processes are locally bounded by stopping their magnitudes and time.

// Target: probability-and-statistics.bigb

A random field is a family of random variables indexed by points of a spatial domain, or more generally by a multi-dimensional parameter set. Its componentwise covariance function and spectral tensor describe second-order dependence. A Gaussian random field has jointly Gaussian finite-dimensional distributions; a general random field need not be Gaussian.

A process $Z$ is self-similar with exponent $H$ if $Z(a\cdot )$ and $a^{H}Z$ have the same finite-dimensional distributions for every $a>0$, or the same path law when the process is considered as a random path. Fractional Brownian motion has this property with its Hurst exponent. It converts dilation of time to deterministic scaling of amplitude: $Z(N\cdot )/N\stackrel{d}{=}{N}^{H-1}Z$. Consequently a small-noise large deviation principle with inverse-variance speed produces the large-deviation speed $N^{2(1-H)}$ for those rescaled paths.

Find a constrained path and a boundary regulator which together modify a given driving path. In the half-line case, write $x=g+k\ge 0$, require $k(0)=0$ and $k$ nondecreasing, and require its increases to occur only where $x=0$. For multidimensional orthants the regulator acts through a specified reflection matrix. The Skorokhod reflection map on the half-line solves the one-dimensional continuous-path case explicitly.

A Bernoulli process is a sequence of independent and identically distributed random variables with a Bernoulli distribution: $\mathbb{P}(B_n=1)=p$ and $\mathbb{P}(B_n=0)=1-p$. It models repeated Bernoulli trials with a fixed success probability and no dependence between trials. The count $K_n={\sum }_{j=1}^n{B}_j$ has a binomial distribution, expected value $np$, and variance $np(1-p)$; the first success time has a geometric distribution on $\{1,2,\dots \}$. The observations $B_n$ and the accumulated count $K_n$ are different stochastic processes: the observations are independent, while consecutive counts share all earlier observations. ^[ref]

Mean reversion is a tendency for a stochastic process to move back towards a reference level. For an Ornstein-Uhlenbeck process, the conditional expected value of a future deviation is the present deviation multiplied by $e^{-\theta h}$ after time $h$. Individual sample paths can move away from that level: mean reversion describes a conditional tendency, not monotone motion or a guaranteed return by a deadline. It also does not by itself imply a stationary process; the initial probability distribution matters.

A controlled Markov process has transition probabilities or rates depending on its current state and the chosen control action. A Markov policy selects an action from the current state and possibly the time. Once such a policy is fixed, the induced state evolution is a Markov process. For general history-dependent controls, the state alone need not remain Markov. A dynamic programming operator optimizes over the available actions at each state.

At stationarity, the mean number ${\mu }_i$ divided by the molecular removal flux $v_i$ is the mean time a molecule remains in the system. Birth or death event rates must be multiplied by the numbers of molecules changed in each event before computing this lifetime.

// Target: mathematical-biology.bigb

This noise-relevant event size weights absolute stoichiometric jump sizes by their molecular turnover flux. At stationarity the denominator is twice the balanced production or removal flux $v_i$, so the normalized diffusion diagonal is $D_{ii}=2⟨s_i⟩/({\tau }_i{\mu }_i)$. Fixed birth bursts of size $b$ and unit deaths give $⟨s⟩=(b+1)/2$. This quantity is not the event-count-weighted average absolute jump size.

// Target: mathematical-biology.bigb

The Bell number $B_n$ counts all set partitions of an $n$-element finite set, with no restriction on the number of blocks of a set partition. Summing the Stirling numbers of the second kind gives $B_n={\sum }_{k=0}^{n}S(n,k)$. The first five values, for $n=0,1,2,3,4$, are $1,1,2,5,15$. In particular, $B_4=1+7+6+1=15$. ^[ref]

A population inversion puts more emitters in the upper than in the lower quantum state of a transition in the nondegenerate two-level model. Let $N_2,N_1$ be their number densities. Equal stimulated-transition strengths make the emission rate proportional to $N_2$ and the competing photon absorption rate proportional to $N_1$, so net optical gain is proportional to $N_2-N_1$. For $N_2=6\times 10^{20}{\mathrm{m}}^{-3}$ and $N_1=4\times 10^{20}{\mathrm{m}}^{-3}$, the difference is positive; swapping them makes the medium absorb. Equal populations give zero net stimulated contribution. With degenerate levels, the relevant comparison uses populations per statistical weight of an atomic level, rather than simply $N_2>N_1$.

Why pump through extra levels? Ordinary steady resonant pumping of a nondegenerate two-level system drives both photon absorption and stimulated emission. With equal induced rate $W$ and upper-state decay rate $\Gamma$, its elementary population equation isat a stationary population. Increasing this pump saturates towards $N_2=N_1$; it cannot sustain population inversion in this model. This is a steady rate-equation conclusion, not a universal prohibition on all driven two-level dynamics or all possible laser gain mechanisms. ^[ref].

In a familiar three-level path, pumping excites a higher level which relaxes into the upper lasing level; the lower lasing level is the ground state, so substantial ground-state depletion is needed. A four-level path also provides rapid emptying of the lower lasing level into a separate ground state. Suitable lifetimes then allow accumulation above and depletion below the lasing transition. Pumping still supplies the energy: extra levels provide a route to population inversion, not free energy. ^[ref].

Uniform solid-body rotation adds the centrifugal potential $-{\Omega }^2{\varpi }^2/2$ to the gravitational potential. In the corotating frame, hydrostatic equilibrium becomes $\nabla P=-\rho \nabla \varphi$. The Poisson equation gives ${\nabla }^2\varphi =4\pi G\rho -2{\Omega }^2$. On regular connected equipotential surfaces, $P$ is constant; taking the curl shows that $\rho$ is constant too. Uniform composition and an invertible equation of state then make $T$ constant on each such surface. These statements do not require $|\nabla \varphi |$ to be constant there.