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For $1<p<\infty$, local Radon-Nikodym derivatives $g_E$ representing an $L^p$ functional on finite-measure sets have uniformly bounded $L^q$ mass. Choose a sequence of finite-measure sets approaching the supremum of that mass, and glue their compatible densities on their union $S$. Any finite-measure set outside $S$ must have zero local density, since otherwise it would increase the supremum. Every Lp space function is approximable by simple functions of finite-measure support, so the glued density represents the functional on the whole space.

For two outgoing particles with centre-of-mass momentum magnitude $k$ and total energy $\sqrt{s}$, integrating the four-momentum delta function gives $d{\Phi }_2/d\Omega =k/(16{\pi }^2\sqrt{s})$. A factor $1/2!$ for identical outgoing particles is inserted when counting events over the full phase space, not into this labelled kinematic measure itself.

The covariant force on a charged particle is its charge times the electromagnetic field-strength tensor contracted with its four-velocity. Antisymmetry keeps the four-force orthogonal to that velocity and preserves the rest mass.

In uniform perpendicular electric and magnetic fields, the guiding centre of a charged particle drifts with ${\mathbf{v}}_D=\mathbf{E}\times \mathbf{B}/|\mathbf{B}{|}^2$. This velocity is independent of the charge and mass; cyclotron motion is superposed on it.

For smooth stationary fields on ${\mathbb{R}}^3$, set ${\Gamma }^i{}_{tt}=-E^i$ and ${\Gamma }^i{}_{tj}={\Gamma }^i{}_{jt}={ϵ}^i{}_{jk}{B}^k$, with all other Christoffel symbols zero. This torsion-free connection makes the equation $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$ a geodesic equation with time as affine parameter. Arbitrary smooth fields suffice; a metric realization imposes additional potential conditions.

When the long-run variance of a stationary process is positive, matching the variance of its average to an average of independent observations gives $T_{\mathrm{eff}}\simeq T\gamma (0)/{\sum }_h\gamma (h)$. Positive aggregate correlation reduces this size; negative aggregate correlation can increase it beyond the observation count. This extends the variance interpretation of effective sample size of a Markov chain.

A path graph state simulates a sequence of Clifford gates $J({\alpha }_j)$, ${\alpha }_j\in \{0,\pi /2\}$, using fixed equatorial qubit measurement bases on successive path vertices. Let $s_j$ be raw outcomes, $t_j=2{\alpha }_j/\pi \in \{0,1\}$, and start the Pauli frame at $a_0=b_0=0$. Matrix commutation and $J(-\pi /2)=XJ(\pi /2)$ giveInduction shows the final unmeasured state differs from the ideal circuit output by $X^{a_m}{Z}^{b_m}$, up to global phase. A final computational basis result $d$ is corrected to $d\oplus a_m$. All bases are fixed, and projectors on distinct vertices commute, so all measurements, including the final one, can occur in a single layer. The frame recurrence is classical outcome processing, not quantum feed-forward.

A scale-invariant curvature seed gives wavenumber-independent dimensionless matter power at radiation-era horizon entry. Horizon entry at ${\tau }_h\sim 1/k$, followed by logarithmic growth to equality, gives a density transfer proportional to $\ln (k{\tau }_{\mathrm{eq}})$. Subsequent matter-era growth is proportional to ${\tau }^2$, so the dimensionless density power is proportional to $(\tau /{\tau }_{\mathrm{eq}}{)}^4{\ln }^2(k{\tau }_{\mathrm{eq}})$. This is a high-wavenumber asymptote, not an expression at the transition point.

The Cauchy principal value integral $\mathrm{li}(x)=\mathrm{PV}{\int }_0^{x}dt/\log t$ is the logarithmic integral. The offset version $\mathrm{Li}(x)={\int }_2^{x}dt/\log t$ differs by a constant and is often used with the prime-counting function. It has $\mathrm{Li}(x)\sim x/\log x$.

If $\log S=m+sZ$ with $s>0$, $K>0$ and $Z$ having the standard normal distribution, then for every real $p$,Here $\Phi$ is the standard normal distribution function. Multiplying the normal distribution density by $e^{psZ}$ and completing the square shifts its mean by $ps$ and supplies the factor $e^{p^2{s}^2/2}$; this is a Gaussian exponential tilt. Integrating above or below the shifted threshold proves both formulas. They apply to noninteger and negative powers, including the inverse-price moment $p=-1$. The lower-tail formula gives the European power put formula after a binomial theorem expansion.

For a Loewner chain with half-plane capacity $2t$, scale invariance means $({\lambda }^{-1}{K}_{{\lambda }^{2}t}{)}_{t\ge 0}$ has the original law for every $\lambda >0$. Its mapping-out maps are ${\lambda }^{-1}{g}_{{\lambda }^{2}t}(\lambda z)$ and its Loewner driving function is ${\lambda }^{-1}{U}_{{\lambda }^{2}t}$. This follows by differentiating the Chordal Loewner equation. Brownian scaling gives this property for Schramm–Loewner evolution.

Conditionally on the entire past up to $t$, the future domains mapped and centered by $g_t-U_t$ have the original chain law and are independent of that past. For Schramm–Loewner evolution this follows from independent stationary Brownian motion increments and the composition rule for chordal Loewner driving functions. The Strong Markov property gives the version at almost surely finite stopping times.

In a short exact sequence of sheaves $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$, if $\mathcal{H}$ is locally free of finite rank, the sequence splits near every point. Trivialize $\mathcal{H}$, lift its finitely many basis sections locally, and shrink to one neighbourhood on which all lifts exist. The lifts define a splitting. If $\mathcal{F}$ is also locally free, then so is $\mathcal{G}$. Conversely locally free kernel and middle term need not give a locally free quotient: multiplication by $t$ on ${\mathcal{O}}_{\mathrm{Spec}k[t]}$ has quotient supported at the origin.

Local linear regression fits $a+b(X_i-x)$ to responses near each target $x$ by weighted least squares, then returns the fitted intercept. With weights $w_i=K((X_i-x)/h)$, put $S_k={\sum }_i{w}_i(X_i-x{)}^k$ and $T_k={\sum }_i{w}_i(X_i-x{)}^k{Y}_i$. If $S_0{S}_2-S_1^2>0$, then $\hat{f}(x)=(S_2{T}_0-S_1{T}_1)/(S_0{S}_2-S_1^2)$. Its polynomial reproduction property of local polynomial regression reproduces affine functions, reducing boundary bias relative to a local constant fit.

A uniformly bounded local martingale is a true uniformly integrable martingale: a localizing sequence gives uniformly dominated stopped values, so their conditional identities pass to the limit. A deterministic bound on each compact time interval gives a true martingale on each such interval, even when the bound grows with time.

For a circular cone with intrinsic metric $d{\rho }^2+{\lambda }^2{\rho }^{2}d{\theta }^2$, $0<\lambda \le 1$, local development uses plane angle $\varphi =\lambda \theta$. A plane straight segment that avoids the origin lifts to a geodesic. Choosing an angular difference at most $\pi$ gives plane difference at most $\pi \lambda$, so for $\lambda <1$ this constructs a joining geodesic between any two distinct points. Different permitted angle lifts can yield distinct joining geodesics; the missing apex cannot be used to join straight segments through the plane origin.

For a statistical scalar theory with gradient coefficient $K/2>0$, this rescaling gives a canonical $(\nabla \psi {)}^2/2$ term. A mass coefficient $r$ becomes $r/K$, a quartic coefficient $g$ becomes $g/K^2$, and a source $h$ becomes $h/\sqrt{K}$. Thus the simple propagator denominator $p^2+R$ and loop-coupling formulas assume these normalized coefficients. The associated measure Jacobian is field independent and belongs to the free-energy normalization.

The stabilizer of a timelike four-momentum in $D$-dimensional Minkowski spacetime has connected group $SO(D-1)$. Massive particle polarizations form representations of this group.

The stabilizer of a nonzero null four-momentum has connected group $ISO(D-2)$. For ordinary finite-polarization particles its translation part acts trivially, leaving a representation of $SO(D-2)$.

A particle polarization specifies a physical spin state at fixed four-momentum. Massive spin states transform under the rotation little group $SO(D-1)$; finite-spin massless states transform under its transverse rotation part $SO(D-2)$. A massive two-form field has $\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$ such states, while a two-form gauge field has $\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)$. Polarization components must be counted after field equations and any gauge equivalence are imposed.