For $q=(q_0,q_1,q_2,q_3)\in S^3$, the vectors $(-q_1,q_0,q_3,-q_2)$, $(-q_2,-q_3,q_0,q_1)$ and $(-q_3,q_2,-q_1,q_0)$ are tangent and orthonormal. Under SU(2) as the three-sphere they are the left-invariant vector fields obtained by translating $-i$ times the Pauli matrices. This explicitly realizes the parallelization of a Lie group by left translations.

For a mixed-characteristic local field with residue characteristic $\ell$, the p-adic logarithm and p-adic exponential function are mutually inverse continuous group homomorphisms on these domains. Bounds on the valuations of the denominators show that higher terms have greater valuation than the linear term. Dividing the logarithm by ${\pi }^r$ identifies the multiplicative higher principal-unit group with the additive integer ring; in particular it is torsion-free.

For $a>0$, polar-coordinate reduction and integration by parts give $m\sin (m||x{|}^2-a|)=2\delta (|x{|}^2-a)-\pi \cos (ma){\delta }_0+o_{{\mathcal{D}}^{\prime }}(1)$ in two dimensions. The circle term comes from the cusp in the folded phase, while the origin term is a radial endpoint contribution. A punctured-domain limit therefore need not extend across the quadratic critical point.

Here $Z$ has the standard normal distribution. Applied to the generating function $e^{sx-s^2/2}$, the Gaussian moment-generating function replaces $s$ by $e^{-t}s$, proving the Hermite eigenvalue identity. Positivity and preservation of Gaussian measure then give the $L^2$ contraction and identify the full Ornstein-Uhlenbeck semigroup by polynomial density.

On polynomials in $L^2$ of standard Gaussian measure, Gaussian integration by parts makes $D$ and $x-D$ adjoints, with commutator $[a^{-},a^{+}]=I$. The Probabilists' Hermite polynomials are obtained by applying $a^{+}$ repeatedly to $1$, and $a^{-}{\mathrm{He}}_n=n{\mathrm{He}}_{n-1}$. These give a Gaussian-space realization of creation and annihilation operators.

Differentiating the Mehler formula for the Ornstein-Uhlenbeck semigroup proves the identity for continuously differentiable functions with bounded derivative. Density extends it to the Gaussian Sobolev form domain. Together with a weighted Cauchy-Schwarz inequality, it yields the Ornstein-Uhlenbeck entropy dissipation identity proof of the Gaussian logarithmic Sobolev inequality.

Partial regression first projects a response and a selected predictor orthogonally off the remaining design columns. Regressing the residualized response on the residualized predictor yields its coefficient in the full ordinary least squares fit.

For a correctly specified full-rank fixed-design Gaussian linear regression with $p<n$, dividing the residual sum of squares by the residual degrees of freedom $n-p$ gives an unbiased estimator of the noise variance. The projection removes the whole deterministic mean, so the numerator's expectation is $(n-p){\sigma }^2$. A misspecified subset model need not have this property.

The Taylor expansion of the exact-solution residual for a linear multistep method has the same coefficient sequence asVanishing through degree $p$ is therefore equivalent to order at least $p$. Exact order $p$ additionally requires a nonzero coefficient at degree $p+1$. This symbol conveniently collects all polynomial consistency and order conditions.

The completion of a total order by its nonprincipal proper cuts, supplying least upper bounds. For a dense order it contains the original order densely, and preserves the interval countable chain condition and separability or nonseparability.

An order automorphism of an order-dense subset $D$ of the real line extends uniquely by $\tilde{\phi }(x)=\sup \{\phi (d):d\in D,d<x\}$. Order completeness supplies the extension, and the lower cuts in $D$ give uniqueness.

An order-dense subset with no nonidentity order automorphism. A transfinite diagonal construction marks an included point and an excluded image for every nonidentity real-line order automorphism, yielding such a subset of size $2^{{\aleph }_0}$.

In a right-handed reference frame, the rotation matrix from an orbital plane frame whose first axis points along the ascending node is $R_z(\Omega )R_x(I)$. Its columns are the orbital frame's orthonormal basis expressed in reference coordinates. The unit vector normal is $(\sin \Omega \sin I,-\cos \Omega \sin I,\cos I)$; its dot product with another such normal gives the cosine of their mutual inclination.

For a Markov process with reward $f$, the value starting at time $t$ in state $s$ is the supremum of expected rewards over admissible stopping times. In finite discrete time it obeys $V(T,s)=f(s)$ and $V(t,s)=\max \{f(s),PV(t+1,\cdot )(s)\}$ for the transition operator $P$, wherever expectations are well-defined. Statewise absolute integrability of every remaining-horizon reward makes these values finite.

At a regular exercise boundary for a diffusion optimal stopping problem, value matching is commonly accompanied by matching first derivatives. This makes the value $C^1$ across the boundary and eliminates a derivative-jump local time of a semimartingale term in the generalized Itô formula. It does not assert that second derivatives agree, or that smooth fit holds for every possible stopping problem.

Equal local resistance magnitudes and opposite coupling signs give ${\Omega }_1=\omega /2$, ${\Omega }_2=-\omega /2$ and $U=-B\omega /(2A)$. Translation-induced torques cancel, while propulsive forces add because handedness and rotation both reverse.

If the second rotor has zero drag, it cannot supply reaction torque. The single remaining helix must be both force-free and torque-free; a positive-definite resistance matrix forces its translation and rotation to vanish. The relative motor motion is absorbed by the zero-resistance rotor.

As the length ratio of an opposite-handed reaction helix tends to infinity, its translation and rotation tend to zero as $1/n$, while the finite first helix rotates at almost the full motor rate. Small motions of the long rotor balance finite force and torque. The common propulsion vanishes in this limit of the additive model.

With row vectors, external arrival rates $\nu$ and transient routing matrix $P$, the total arrival rates satisfy $\alpha =\nu +\alpha P=\nu (I-P{)}^{-1}$. These include all internal migrations and do not depend on colony service speeds.

For an open migration process, colony $j$ with population $k$ has total departure rate ${\varphi }_j(k)$. Its stationary weight is $b_j(k)={\alpha }_j^k/{\prod }_{h=1}^k{\varphi }_j(h)$. The colony weights factor independently and normalize when ${\sum }_{k\ge 0}{b}_j(k)<\infty$ for every colony. Global balance for a continuous-time Markov chain proves the formula even when routing is not reversible.