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For a stationary cylinder of radius $a$ in the far-field shear $(\Gamma y,0)$, the stream function is $\psi =(\Gamma /4)[r^2-2a^2\log (r/a)-a^2-(r^2-2a^2+a^4/r^2)\cos 2\phi ]$. It solves the biharmonic equation and the no-slip boundary condition. The fluid exerts torque $-2\pi \mu \Gamma a^2$ per unit length, because the angular part of the surface traction averages to zero and the mean tangential traction is $-\mu \Gamma$.

The Oseen approximation retains advection of a small disturbance by a uniform incident velocity $\mathbf{U}$, but neglects advection of the disturbance by itself. Its steady equations areA Stokes disturbance of a sphere decays as $Ua/r$, so the ratio of background advection to viscous diffusion grows as $Ur/\nu$. Thus the approximation resolves the outer region $r\sim \nu /U$, where the Stokes flow expansion is nonuniform, while self-convection remains smaller by $a/r$.

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.

Calcium chloride is $\mathrm{CaC}{\mathrm{l}}_2$. One doubly charged calcium ion balances two singly charged chloride ions, a direct example of the stoichiometry required by electroneutrality.

It forms several hydrates and interacts strongly with water. Its hydration state affects dissolution energetics and the mass-to-amount conversion; dissolved-ion counts must also be distinguished from counts of solid formula units.

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 locally bounded process can be made bounded by stopping it on an increasing sequence of random time intervals. For stochastic integration on positive times, one useful convention is that there exist stopping times ${\tau }_n\uparrow \infty$ almost surely and finite deterministic constants $K_n$ such thatThe interval convention matters at time zero: including zero additionally requires a deterministic bound on $H_0$, whereas the positive-time Stochastic integral does not use that value.

Every continuous adapted process with finite initial value has this positive-time localization. Set ${\tau }_n=\inf \{t\ge 0:|H_t|\ge n\}\wedge n$. If its initial magnitude already exceeds $n$, this Stopping time is zero and the indicated interval is empty. Otherwise continuity prevents overshoot at the first exit, so $K_n=n$ works. continuity on each compact time interval makes ${\tau }_n\to \infty$ pathwise.

A predictable locally bounded process is integrable against a semimartingale: the bounded stopped integrands define compatible integrals, which are joined by localization. Predictability supplies the required information condition; a magnitude bound alone does not replace it. For example, Brownian motion is not bounded on the whole infinite time axis but its continuous adapted paths give the stated localization.

// 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. When second moments are finite, 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: $Z(N\cdot )/N\stackrel{d}{=}{N}^{H-1}Z$. For $H<1$, so that $N^{H-1}\to 0$, 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 a mean-reverting Ornstein-Uhlenbeck process with $\theta >0$, 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]