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The determinant-one matrices form the normal subgroup $S{L}_n({\mathbb{F}}_q)=\ker (\det )$.

The group $PG{L}_2(F)=G{L}_2(F)/Z$ acts faithfully onby Möbius transformations. For $|F|=q$, this embeds $PG{L}_2(F)$ into the symmetric group $S_{q+1}$.

The upper unitriangular group consists of upper triangular matrices with every diagonal entry equal to one. Over ${\mathbb{F}}_p$ it has order $p^{n(n-1)/2}$ and is a Sylow subgroup of $G{L}_n({\mathbb{F}}_p)$.

Its elements form the upper triangular, diagonal-one subset of triangular matrices.

Choosing linearly independent columns successively gives

An Ornstein-Uhlenbeck process combines mean reversion with continuing random fluctuations. In the scalar model, its stochastic differential equation iswhere $W$ is a standard Brownian motion. The drift points towards $\mu$: it is negative above $\mu$ and positive below it. The coefficient $\theta$ sets the speed of mean reversion, while $\sigma$ sets the strength of the random forcing. We consider a fixed or normal initial value independent of future Brownian increments, giving a Gaussian process and a Markov process.

For a stipulated example, use $\theta =1$, $\mu =2$, $\sigma =\sqrt{2}$ and the fixed initial value $X_0=6$, in arbitrary consistent state and time units. Then the expected value relaxes as $2+4e^{-t}$ and the variance grows as $1-e^{-2t}$. At time one they are about $3.4715$ and $0.8647$; the limiting variance is one. The sample paths keep fluctuating even after the expected value is close to two. The Ornstein-Uhlenbeck transition distribution describes this evolving uncertainty; Ornstein-Uhlenbeck stationary initialization instead starts with that limiting normal distribution already in place.

A physical motivation is the velocity of a particle undergoing Brownian motion: linear drag removes a persistent initial velocity, while random forcing continually renews it. The original calculation gives a normal distribution for the velocity after subtracting its exponentially decaying initial contribution, with decay rate equal to the friction coefficient divided by the particle's mass.^[ref] A position coordinate in an overdamped harmonic optical trap gives another interpretation of the same scalar equation, with $\theta =k/\zeta$ for spring constant $k$ and drag coefficient $\zeta$. Its physical assumptions and connection to measurements are developed in thermal calibration of an optical trap.

A Gaussian measure on a real topological vector space is a Borel probability measure whose image under every continuous linear functional is a one-dimensional normal distribution.

Every linear subspace that is measurable for the cylinder sigma-algebra has probability zero or one under a centered Gaussian process.

A Gaussian random field is a spatially indexed Gaussian process; its law is determined by its mean and covariance kernel.

The binomial series gives $F(A,C;C;u)=(1-u{)}^{-A}$ wherever the defining series converges, followed by analytic continuation.

Away from resonant parameter cases, two independent local solutions at infinity areand the expression obtained by exchanging $a$ and $b$. Every analytic continuation of a hypergeometric solution into their common domain is a constant linear combination of this basis.

When $C$ is not an integer, a second local solution at zero is $u^{1-C}F(A-C+1,B-C+1;2-C;u)$.

With compatible branches and initially near $z=0$,It follows by applying a Möbius transformation and a dependent-variable rescaling to the same Papperitz symbol, then matching the normalized analytic solution at zero.

BRST symmetry is a nilpotent fermionic symmetry of a gauge-fixed action. Its differential $s$ replaces an infinitesimal gauge parameter by the ghost field and satisfies $s^2=0$.

An axial gauge imposes $n^{\mu }{A}_{\mu }=0$ for a fixed vector $n^{\mu }$. Its Faddeev-Popov operator is $n^{\mu }{D}_{\mu }$; on the strict gauge slice its gauge-field-dependent part vanishes, so its ghosts decouple.

The Faddeev-Popov determinant is the functional Jacobianthat compensates for the change from integration along a gauge orbit to a gauge-fixing condition $G(A)=0$. It can be represented by a path integral over a Faddeev-Popov ghost field pair.

Adding $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\alpha )$ gives a family of Lorentz-covariant gauges; $\alpha =1$ is Feynman gauge.

The beta--gamma identity gives $B(z,z)=\Gamma (z{)}^2/\Gamma (2z)\sim 2/z$ as $z\to 0$ through the right half-plane.

For a symmetric random walk started at $i\in \{-a,\dots ,b\}$ and stopped on first reaching $-a$ or $b$, the mean duration isIn particular, ${\mathbb{E}}_{0}T=ab$. This follows either from the first-step recurrence or by stopping the martingale $S_n^2-n$.

The Galactic scale height measures the vertical thickness of a tracer. Gravity, velocity dispersion, thermal pressure, magnetic and cosmic-ray support, particle transport, and source lifetime determine its value.

Diffuse Galactic gamma rays arise mainly from neutral-pion decay after cosmic-ray nuclear collisions, electron bremsstrahlung in gas, and inverse-Compton scattering from radiation fields.