For independently driven nonnegative squared Bessel processes $X,Y$, put $Z=X+Y$. Where $Z>0$, use Brownian weights $\sqrt{X/Z}$ and $\sqrt{Y/Z}$. Where $Z=0$, fill the weights with $(1,0)$. Their squared norm is always one and their cross-variation is zero, so their stochastic integral is Brownian motion by the Lévy characterization of Brownian motion. Multiplying by $2\sqrt{Z}$ recovers the sum's martingale part even at zero, and the drifts add to $\alpha +\beta$.

For bounded predictable $\sigma$ on $[0,T]$, the stochastic exponential $Z_s=\exp (\frac{1}{2}{\int }_0^s{\sigma }_{u}d{W}_u-\frac{1}{8}{\int }_0^s{\sigma }_u^{2}du)$ is a positive martingale by the Novikov condition. Under $dQ/dP=Z_T$, the Bayes formula for conditional expectation givesIndeed, the square-root stock ratio is $(Z_T/Z_t)\exp (-\frac{1}{8}{\int }_t^T{\sigma }_u^{2}du)$. This exact identity gives the bounds on square-root stock implied volatility by bounding integrated variance pathwise. The Girsanov theorem gives $W^Q=W-\frac{1}{2}\int \sigma du$. The measure is an auxiliary pricing identity for this payoff, not a claim that it is an equivalent martingale measure for the original stock.

For Positive elements of a C-star algebra $x\le y$, inversion reverses the order of strictly positive elements, so $(t1+x{)}^{-1}\ge (t1+y{)}^{-1}$ for $t>0$. Subtracting their square-root resolvent integral in a C-star algebra formulas givesThe integrand is positive, and the positive cone is closed, so the norm limit of these integrals is positive. No commutativity of $x$ and $y$ is required. In particular, $b^2\ge a^2$ for positive $a,b$ implies $b\ge a$, even though squaring itself is not generally order preserving.

After $y={\epsilon }^{2}Y$ and $x=\epsilon \xi$ or a translated version, the layer equation is $Y^{\prime \prime }-\sqrt{Y}+k=0$. Multiplying by $Y^{\prime }$ and matching $Y=k^2$, $Y^{\prime }=0$ fixes the energy constant to $k^3/3$ and gives the displayed factorization. The increasing branch below $k^2$ approaches it at positive infinity; the increasing branch above it leaves it at negative infinity and grows like ${\xi }^4/144$. This supplies both interfaces of the square-root reaction problem with nested boundary layers.

For solutions with independent Brownian drivers in a common filtration, combine the drivers with weights $\sqrt{Z/(Z+Z^{\prime })}$ and $\sqrt{Z^{\prime }/(Z+Z^{\prime })}$ where the sum is positive. At zero use one original driver. The resulting continuous local martingale has bracket $t$ and is Brownian motion by the Lévy characterization of Brownian motion. Multiplying its differential by $\sqrt{Z+Z^{\prime }}$ gives exactly the sum of the original stochastic differentials.

Replace the square-root coefficient near zero by smooth globally Lipschitz cutoffs that vanish below half the cutoff level. With a common Brownian driver, local pathwise uniqueness makes the solutions agree until the larger cutoff level. The exit times increase as the levels decrease. The compatible integrands $1_{\{s\le T_n\}}\sqrt{Z_s}$ have increasing squares and expected time-integrals at most $zt$ by the nonnegative local martingale bound. Their limit is square-integrable on every finite horizon. The Itô isometry gives a continuous global limiting stochastic integral; at a finite limiting exit time its value is zero, and it remains absorbed afterward.

The three generators $-1,2,5$ of the square-class group of the 2-adic numbers give these seven distinct extensions. In characteristic different from two, completing the square shows every quadratic extension is of this form, and comparing coefficients in $\sqrt{e}=a+b\sqrt{d}$ proves that two such fields coincide precisely when $e/d$ is a square. The only unramified member is ${\mathbb{Q}}_2(\sqrt{5})$: $(1+\sqrt{5})/2$ is integral with irreducible reduction $X^2+X+1$. Conversely any unramified quadratic extension has that residue field, and Hensel lemma lifts a residue root of $X^2-X-1$, recovering the same field.

For $G:\mathcal{D}\to \mathcal{C}$, a parallel pair in $\mathcal{D}$ is G-split if its image admits a split coequalizer in $\mathcal{C}$. Reflection of such coequalizers means that an existing equalizing arrow in $\mathcal{D}$ is a coequalizer whenever its image is the corresponding split coequalizer. This asserts a property of existing arrows; it does not assert that a lift exists for every base quotient.

For unit-spaced order-three B-splines sampled at support midpoints $i+3/2$, the B-spline collocation matrix has diagonal $3/4$ and neighboring entries $1/8$. Conjugating by alternating diagonal signs reduces its inverse absolute values to the nonnegative inverse of $\mathrm{tridiag}(-1,6,-1)/8$. With $\tau =3-2\sqrt{2}$, the absolute row sums are $r_i=2[1-({\tau }^i+{\tau }^{n+1-i})/(1+{\tau }^{n+1})]$. This follows by solving $6r_i-r_{i-1}-r_{i+1}=8$ with $r_0=r_{n+1}=0$. The maximum occurs at the middle row or rows and tends to two as $n$ grows. Nonnegativity and subpartition of unity of the B-splines transfer this bound to the spline interpolation operator.

The spinor covariant derivative has curvature $d\Omega +\Omega \wedge \Omega =\frac{1}{4}{R}_{AB}{\gamma }^{AB}$, since the spin representation preserves the Lorentz algebra bracket. In a coordinate basis with the torsion-free connection, antisymmetric second covariant derivatives therefore give the displayed commutator. With unit-weight antisymmetrization, ${\nabla }_{[\mu }{\nabla }_{\nu ]}\psi =\frac{1}{8}{R}_{\mu \nu AB}{\gamma }^{AB}\psi =\frac{1}{16}{R}_{\mu \nu AB}[{\gamma }^A,{\gamma }^B]\psi$. A noncoordinate frame requires its frame-commutator correction when using the operator commutator; the full second covariant derivative includes that correction automatically.

A parallel spinor field is annihilated by the spinor covariant derivative. Its value at one point determines it on a connected domain by parallel transport. The spinor curvature identity implies $R_{\mu \nu AB}{\gamma }^{AB}ϵ=0$, but this local condition alone need not settle global existence: the spin structure and holonomy also matter. Lorentzian parallel spinors need not force the whole Ricci tensor to vanish, unlike the positive-definite case.

The complexified tangent space of a four-dimensional Lorentzian manifold factors locally into an unprimed and a primed two-dimensional spinor representation. Their alternating forms ${ϵ}_{AB}$ and ${ϵ}_{A^{\prime }{B}^{\prime }}$ encode the metric tensor and raise or lower spinor indices. A convention must specify the order of contraction with these alternating forms. An antisymmetric spacetime tensor decomposes into symmetric unprimed and primed spinors, $F_{A{A}^{\prime }B{B}^{\prime }}={\varphi }_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}+{\tilde{\varphi }}_{A^{\prime }{B}^{\prime }}{ϵ}_{AB}$. This splits the six-dimensional space of differential two-forms into the two three-dimensional eigenspaces of the complexified Hodge star operator.

A twistor spinor has vanishing gamma-trace-free part of its spinor covariant derivative. In $n$ dimensions the displayed equation uses gamma matrices satisfying $\{{\gamma }_a,{\gamma }_b\}=2g_{ab}$. It is weaker than being a parallel spinor. In four dimensions it implies $[{\gamma }_a{\nabla }_b+{\gamma }_b{\nabla }_a-g_{ab}{\gamma }^c{\nabla }_c/2]ϵ=0$.

For the Landau free energy $V(M)=r{M}^2/2+u{M}^4/4-hM$ with $r<0$ and $u>0$, stationary branches satisfy $h=rM+u{M}^3$. Their local stability requires $r+3u{M}^2>0$. At a spinodal point, $M_{\mathrm{sp}}=\pm \sqrt{|r|/(3u)}$ and $h_{\mathrm{sp}}=(2r/3)M_{\mathrm{sp}}$, giving the displayed field magnitude. Following local minima without nucleation produces a hysteresis loop switching at these limits. Equilibrium instead switches between the two global minima at $h=0$.

An atomic read-modify-write that returns a flag's old value while setting it. In a sequential consistency model with the flag initially $0$, exchange(flag, 1) permits entry only when it returns $0$; an owner resets the flag to $0$ on release. Between releases only one exchange can observe $0$. By contrast, separate read(flag) and write(flag, 1) steps allow both contenders to read $0$ before either writes $1$. The indivisibility of the exchange is an assumed primitive, not something proved by writing its steps as ordinary assignments.^[ref]

A Spin-c structure on an oriented rank-$n$ real vector bundle is a lift of its oriented frame bundle to ${\mathrm{Spin}}^c(n)=(\mathrm{Spin}(n)\times U(1))/\{(1,1),(-1,-1)\}$. Its determinant line bundle is associated with $[s,z]\mapsto z^2$; its First Chern class reduces modulo two to the second Stiefel–Whitney class. When such structures exist, their isomorphism classes form an affine space over $H^2$; tensoring by a complex line bundle of first Chern class $h$ changes the determinant class by $2h$. On a closed oriented three-manifold they always exist. On a four-manifold, the determinant class is characteristic: $⟨c_1(\mathfrak{s}),v⟩\equiv v\cdot v(\mathrm{mod}2)$.

Well-separated shrinking spheres respond independently at leading order, so their polarization contributions add according to volume fraction. Clusters kept at fixed relative internal separation can have a different leading response; the dilute limit alone does not guarantee independent-sphere polarizability.

For a spherical, nonstreaming stellar population with isotropic velocity dispersion, each component of the random pressure tensor equals $p=n⟨v_r^2⟩$. The observed second stellar velocity moment satisfies $N(R){\sigma }_{\mathrm{los}}^2(R)=2{\int }_R^{\infty }p(r)r/\sqrt{r^2-R^2}dr$. Composing this Abel transform with itself gives the displayed inverse under sufficient decay and regularity. The Spherical Jeans equation then gives ${\psi }^{\prime }=p^{\prime }/n$, with gravitational acceleration $\nabla \psi$. The Poisson equation for Newtonian gravity recovers total gravitating mass density, not just the tracer density. Without velocity isotropy the same projected moment also contains anisotropic stresses.

For $A,b>0$ and $\alpha >1/2$, the projected profile $F(R)=A(R^2+b^2{)}^{-\alpha }$ has the displayed spherical Abel transform inverse. Indeed its composed transform is $A(r^2+b^2{)}^{1/2-\alpha }{I}_{\alpha }$, with $I_{\alpha }={\int }_1^{\infty }{s}^{-\alpha }(s-1{)}^{-1/2}ds=B(1/2,\alpha -1/2)$. Differentiate in $r$, divide by $-2\pi r$, and use the Gamma function recurrence to obtain the coefficient. The same kernel reconstructs tracer number density or isotropic random pressure from their respective projected profiles.

Contraction sends the zero prime ideal to zero. A complex closed point $(x-z)$ contracts to $(x-z)$ when $z$ is real, and to $((x-\mathrm{Re}z{)}^2+(\mathrm{Im}z{)}^2)$ otherwise. Thus the map on the spectrum of a commutative ring is surjective and identifies each nonreal conjugate pair. Both spectra have only finite proper Zariski-closed sets, so this is also the topological quotient by complex conjugation.