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Let $a={\mathbb{P}}_i(T_j<T_i)$ and $b={\mathbb{P}}_j(T_i<T_j)$. Before returning to $i$, the chain visits $j$ zero times with probability $1-a$ and, for $k\ge 1$, exactly $k$ times with probabilityWhen $b>0$, the expected number of visits is $a/b$.

If $H>0$ at one time, the inequality $d(H^{-1})/dt\ge 1$ forces $H\to +\infty$ and $a\to 0$ within finite time to the past. If $H<0$, it forces $H\to -\infty$ and $a\to 0$ within finite time to the future.

A machine computes the characteristic function of a language $X$ when it halts on every input with the one-letter output $1$ for inputs in $X$ and $0$ for inputs outside $X$.

The induced worldsheet metric is the pullback of the target-space metric:

A strictly convex function has at most one minimizer: if distinct points attained the same minimum, every strict convex combination of them would have a smaller value.

On the positive orthant,is strictly concave. Its unique maximizer $x^{\ast }$ on a compact convex feasible set satisfies the first-order inequalityequivalently

A stratum is a subgroup assigned its own nuisance baseline.

For bounded stopping times $\sigma \le \tau$, a supermartingale satisfies $\mathbb{E}[V_{\tau }\mid {\mathcal{F}}_{\sigma }]\le V_{\sigma }$; equality holds for a martingale.

For a discrete martingale and stopping time $T$,The indicator is measurable at time $n$, so the stopped process is a martingale. If it is uniformly bounded and $T<\infty$ almost surely, bounded convergence gives $\mathbb{E}{M}_T=\mathbb{E}{M}_0$ directly.

A stochastic differential equation specifies infinitesimal drift and random diffusion through a stochastic integral equation.

The stochastic integral $\int HdX$ integrates a predictable process against a semimartingale. It extends pathwise integration against finite-variation processes and the Itô integral against local martingales.

The quadratic variation of a continuous semimartingale is the limit in probabilityFinite-variation processes have zero quadratic variation, while a Brownian motion satisfies $[B{]}_t=t$.

A semimartingale is the sum of a local martingale and an adapted finite-variation process. This is the broad class of integrators for which the Itô stochastic integral is defined.

Two-body relaxation is the cumulative randomization of stellar velocities by many weak gravitational encounters. Its timescale in a system of $N$ comparable stars is of order $N/\log N$ crossing times.

A body is in radiative equilibrium when absorbed radiative power equals emitted radiative power. A surface absorbing flux $F$ and emitting as a grey body has equilibrium temperature $T=(F/(ϵ\sigma ){)}^{1/4}$.

Emissivity compares thermal emission from a surface with emission from an ideal blackbody at the same absolute temperature, using the same wavelength and angular averaging convention. For ordinary passive surfaces in the far field at local thermal equilibrium, the ratio lies between zero and one. A blackbody has emissivity one. Unless a spectral or directional qualification is given, the $ϵ$ in $M=ϵ\sigma T^4$ is total hemispherical emissivity: a ratio of emitted radiant exitances, not of all outgoing radiant flux including reflected radiation.^[ref]

Emissivity belongs to a specified surface state, not just a bulk material name. Coating, oxidation and surface finish can change it; wavelength, viewing direction and temperature also matter. A surface that looks white in visible light need not have low infrared emissivity. This distinction is useful when choosing a radiator coating or interpreting a radiation-based temperature measurement: a narrow-band detector needs the appropriate spectral emissivity, and its signal can include reflected surrounding radiation. The grey body and diffuse thermal emitter assumptions are separate simplifications, addressing wavelength and direction respectively.

Writing $A=D+L+U$, where $D$ is diagonal and $L,U$ are strictly triangular, the Jacobi iteration is

A splitting $A=M-N$ with invertible $M$ produces

If $B=U⋊T$ is the upper-triangular subgroup of $S{L}_2({\mathbb{F}}_p)$ and $\theta$ is one-dimensional on $T$, thenis the direct sum of three pairwise nonisomorphic irreducibles. Their restrictions to $U$ have character supports given by the trivial character, the square orbit, and the nonsquare orbit, so their dimensions are $1,(p-1)/2,(p-1)/2$.

For $A{s}^4+s^2-1=0$, the physical speed approaches $c_0(1-A/2)$ as $A$ tends to zero and scales as $c_0{A}^{-1/4}$ as $A$ tends to infinity.