Let a fixed-mass, fully mixed stellar model evolve by hydrogen burning, with $L=C{M}^6{Z}^{-7/6}H(X)$ and $H(X)=\mu (X{)}^{26/3}{X}^{-1/3}(1+X{)}^{-7/6}$. Energy conservation gives $\dot{X}=-L/(M{E}_H)$. The time to reach $X=f{X}_0$ is therefore $t=(E_H/C)M^{-5}{Z}^{7/6}{\int }_{f{X}_0}^{X_0}H(X{)}^{-1}dX$. If $X_0,f$ and the function $\mu (X)$ are common to the population, the integral is constant and $M_{\mathrm{to}}\propto Z^{7/30}{t}^{-1/5}$. This includes the change of luminosity during fuel depletion rather than replacing the luminosity by its initial value.

At a spherical interface with an insulating exterior, equal magnetic permeability on both sides and no surface current, all components of the magnetic field are continuous. The solenoidal magnetic-field constraint then makes $P=\mathbf{B}\cdot \mathbf{x}$ and ${\partial }_{r}P$ continuous. A current-free exterior has $\Delta P=0$. For an isolated field decaying at infinity, its degree-$l$ spherical harmonic satisfies ${\partial }_r{P}_{lm}=-(l+1)P_{lm}/a$ at a sphere of radius $a$. This boundary condition incorporates the exterior field without setting the radial field to zero.

// Target: fluid-mechanics.bigb

For independent Rademacher random variables, this series converges absolutely on every sign sequence and has the uniform distribution on $[-1,1]$. Its first $n$ terms are uniform on the $2^n$ midpoints of equal subintervals of $[-1,1]$, giving weak convergence of probability measures by Riemann sums. The remaining tail has absolute value at most $2^{-n}$, and its mean square is $4^{-n}/3$ by independence. Thus convergence is also almost surely and in $L^2$.

Suppose an $s$-stage Runge-Kutta method has order $2s-1$, an invertible matrix $A$, and $b^T{A}^{-1}1=1$. Its stability function is $R(z)=1+z{b}^T(I-zA{)}^{-1}1$. The adjugate matrix formula gives a rational function with denominator degree $s$ and numerator degree at most $s$; its zero limit at infinity lowers the latter degree to at most $s-1$. The order of a numerical method gives $R(z)-e^z=O(z^{2s})$, identifying the $[s-1/s]$ Padé approximant. Uniqueness follows because the cross-multiplied difference of two candidates has degree at most $2s-1$ but a zero of order at least $2s$, so is identically zero. The A-stability of near-diagonal exponential Padé approximants then proves A-stability, and the zero limit proves L-stability. These conditions identify the stability function; they do not alone claim uniqueness of all entries of the Runge-Kutta method tableau.

Over ${\mathrm{Gr}}_k({\mathbb{R}}^N)$, quotient the trivial bundle by the tautological bundle. The quotient is a vector bundle because it identifies continuously with the orthogonal-complement bundle via $[v]\mapsto P_{W^{\perp }}v$. Its rank is $N-k$.

Two elements of a commutative ring are congruent modulo an ideal if their difference lies in that ideal, equivalently if their images in the quotient ring agree. This is an equivalence relation compatible with addition and multiplication. For $R={\mathbb{Z}}_p[[T]]$ and $I=p^{k}R$, it means every coefficient difference is divisible by $p^k$.

Let $f$ be a chain map between degreewise finitely generated free abelian groups. If its reduction modulo every prime number is a quasi-isomorphism, the mapping cone acyclicity criterion makes $M(f)\otimes {\mathbb{F}}_p$ acyclic. The short exact sequence of chain complexes $0\to M\stackrel{p}{\to }M\to M\otimes {\mathbb{F}}_p\to 0$ implies that multiplication by $p$ is an isomorphism on every $H_i(M)$. These are finitely generated abelian groups. Their free summands would make multiplication by $p$ fail to be surjective, and any nonzero finite summand would make it fail to be injective for a prime dividing its order. Thus $H_i(M)=0$ and $f$ is a quasi-isomorphism. Degreewise finite generation is essential to this argument.

For a cubic-quintic Gross–Pitaevskii equation with normalized bulk amplitude one, a straight quantum vortex of integer winding number $\mathcal{N}$ has radial equation $R^{\prime \prime }+R^{\prime }/r-{\mathcal{N}}^{2}R/r^2+(1-\alpha R^2-\beta R^4)R=0$. With $\alpha +\beta =1$, the derivative of the nonlinear term at $R=1$ is $-2(1+\beta )$. Balancing the order-$r^{-2}$ centrifugal term with the bulk amplitude restoring term proves the displayed tail for a stable nondegenerate bulk state, $1+\beta >0$. A vortex's amplitude tends to one; its full complex field tends to $e^{i\mathcal{N}\theta }$, not a direction-independent constant. At $\beta =-1$ the stated inverse-square expansion cannot cancel the centrifugal term.

For a centered quantum vortex in a two-dimensional Thomas–Fermi condensate, use an intermediate radius ${\xi }_{\mathrm{core}}\ll L\ll R$. The approximately uniform core contributes $\pi n_c{\hslash }^2{\mathcal{N}}^2[\log (L/\xi )+L_{0|\mathcal{N}|}]/m$. The outer superfluid velocity is $\hslash \mathcal{N}/(mr)$, so its kinetic contribution is $(\pi {\hslash }^2{\mathcal{N}}^2/m){\int }_L^R{n}_c(1-r^2/R^2)dr/r$. Integration cancels $L$ and produces the constant $-1/2$. The displayed result is the matched small-core approximation at fixed particle number, not a formula for arbitrarily large winding at fixed cloud size. The angular momentum is exactly $L_z=N\hslash \mathcal{N}$ for a normalized axisymmetric single-winding state.

In healing-length units a straight quantum vortex obeys $f^{\prime \prime }+f^{\prime }/r-{\mathcal{N}}^{2}f/r^2+(1-f^2)f=0$ and $f\to 1$. Inserting $f=1-a/r^2+\cdots$ makes its order-$r^{-2}$ balance $-{\mathcal{N}}^2+2a=0$, proving the tail. The physical healing length convention must be stated; the deficit of density $1-f^2$ has twice the leading coefficient of the amplitude deficit $1-f$.

For a continuous local martingale and a locally bounded predictable process, stop at a localizing time, an integrand bound, a time bound, and level bounds for the centered martingale and its bracket. The stopped martingale is bounded and its bracket and integrand are deterministically bounded. The Itô isometry, applied to $1_F{1}_{(s,t]}H$ for $F\in {\mathcal{F}}_s$, gives the conditional second-moment identity. Expanding the square proves that $(H\cdot M{)}^2-H^2\cdot [M]$ is a martingale after stopping. Uniqueness of an increasing square compensator identifies the bracket, and increasing stopping times patch the identities into the displayed local result.

Let $c=\varphi /{\varphi }_{\max }$ be particle volume fraction normalized by the packed value, and use the quadratic hindered-settling flux $j(c)=-u_{s}c(1-c)$. Takewith clear fluid above $h$. To isolate breaking of this smooth profile, do not impose an additional packed-bed discontinuity. A characteristic curve labelled by its initial height $\xi$ carries constant concentration and followsThe mapping first loses invertibility whenThe earliest time occurs at $\xi =h$, giving $t_c=3h/(4u_s)$ and $z_c=h/4$. This explicitly shows why the time uses normalized concentration and the isolated settling-speed scale $u_s$; the physical flux for $\varphi$ is $-u_s\varphi (1-\varphi /{\varphi }_{\max })$.

The new sedimentation shock joins concentrated fluid below to clear fluid above. Its signed speed is $u_s(c_L-1)$, initially $-u_s$ as $c_L\to 0$. As the state behind it becomes more concentrated, its downward speed decreases until a boundary interaction changes the problem. This is the onset of characteristic breaking, not the onset of a separately imposed deposition front.

For zero-starting L2-bounded continuous martingales, this condition is equivalent to $\mathbb{E}[M_t{N}_t]=0$ at equal times, by conditioning the later factor. The covariance identity for continuous square-integrable martingales also makes it equivalent to $\mathbb{E}[M,N{]}_t=0$. It is weaker than the vanishing bracket condition for orthogonal continuous local martingales: $B_{t\wedge 1}$ and $B_{t\wedge 1}^2-(t\wedge 1)$ have zero equal-time covariance but nonzero bracket $2{\int }_0^{t\wedge 1}{B}_{s}ds$.

For two zero-starting continuous martingales square-integrable at time $t$, the Itô isometry gives $\mathbb{E}(M_t\pm N_t{)}^2=\mathbb{E}[M\pm N{]}_t$. The polarization identity yields the displayed identity. Integrability of the cross-bracket follows from the Kunita-Watanabe inequality and the Cauchy-Schwarz inequality.

For an analytic function $f=u+iv$ and planar Brownian motion $Z=X+iY$ with independent standard coordinates, the Cauchy-Riemann equations make $u,v$ harmonic. The Itô formula gives $dU=u_{x}dX+u_{y}dY$ and $dV=-u_{y}dX+u_{x}dY$. Localizing the derivatives proves both coordinates are local martingales. Squaring their diffusion coefficients and multiplying the two rows gives the displayed quadratic variations and zero quadratic covariation.

If stars are neither created nor lost and all follow quadratic accretion in a central Keplerian potential, conservation of number gives $F(M,t)dM=F_0(M_0)d{M}_0$. Inverting the trajectory gives $M_0=M/(1+\lambda tM)$ and its Jacobian determinant is $(1+\lambda tM{)}^{-2}$. This proves the displayed mass-density-in-mass-space transformation. In a broad high-mass range with $\lambda tM\gg 1$, whose inverse images remain in a narrow interval where $F_0$ varies little, the power law is $F\propto M^{-2}$. A strictly monochromatic initial population remains monochromatic. Finite initial support imposes a finite upper cutoff before the earliest formal blow-up, so the power law is an intermediate asymptotic subject to the accretion model's limits.

For a morphism of ringed spaces, there is a natural morphismIt is an isomorphism when $\mathcal{E}$ is a locally free sheaf of finite rank, because the claim is local and then reduces to distributivity over a finite direct sum.

The same formula holds for higher direct images: $R^i{f}_{\ast }\mathcal{F}\otimes \mathcal{E}\cong R^i{f}_{\ast }(\mathcal{F}\otimes f^{\ast }\mathcal{E})$ when $\mathcal{E}$ is locally free of finite rank.

For a smooth map $\varphi :M^{\prime }\to M$ and a connection on a vector bundle $E\to M$, the curvature form of a connection on the pullback vector bundle satisfiesas an endomorphism of $E_{\varphi (p)}$. In a frame of a vector bundle written as a row, with section components written as columns, the connection one-form is $\Omega$ and its curvature form of a connection is $d\Omega +\Omega \wedge \Omega$. The pullback connection has matrix ${\varphi }^{\ast }\Omega$; commutation of the pullback of a differential form with the exterior derivative and wedge product of differential forms proves the identity. No pushforward to an ambient vector field is required: $d\varphi (X)$ is a vector field along a map.

For a commutative ring $R$, this locale is presented by basic opens $U_r$ recording membership in a multiplicatively closed prime filter. Products give intersections. A ring map induces a locale map in the opposite direction. The basic opens satisfy $U_r\vee U_s=1$ exactly when $(r,s)=R$, making unit-ideal calculations a substitute for choosing prime ideals.

For this smooth projective bundle, put $L={\pi }^{\ast }{\mathcal{O}}_{{\mathbb{P}}^1}(1)$ and $M={\mathcal{O}}_{\mathbb{F}}(1)$, using projectivization by quotients. If every $a_i>0$, the complete system $|M|$ embeds it as a rational normal scroll. If every $a_i\ge 0$, the same system defines a scroll image which may be singular. Allowing arbitrary integers gives an abstract scroll; a common twist makes an embedding available. Its intersection numbers are $L^2=0$, $L{M}^{n-1}=1$, $M^n={\sum }_i{a}_i$, and its canonical divisor is $-nM+({\sum }_i{a}_i-2)L$.