deuterium proton capture is much faster than the weak reaction that produces deuterium. Its abundance therefore adjusts so that ${\lambda }_{21}{n}_2{n}_1\simeq {\lambda }_{11}{n}_1^2/2$. Substituting this balance eliminates the fast intermediate from the slower proton and helium-3 evolution equations. This is a quasi-steady approximation, not an assumption that hydrogen itself is in nuclear equilibrium.

With $A={\lambda }_{33}$, $B={\lambda }_{34}{n}_4$ and $C={\lambda }_{11}{n}_1^2/2$, helium-3 evolves as ${\dot{n}}_3=C-A{n}_3^2-B{n}_3$ after deuterium elimination. The positive root is its unique attracting equilibrium for a fixed background. In the pp-I limit it reduces to $n_1\sqrt{{\lambda }_{11}/(2{\lambda }_{33})}$, making equilibrium abundance strongly temperature dependent.

The normalized universal line bundle restricts to the class $\xi$ on $A\times \{\xi \}$ and is trivialized on both zero sections. For a line bundle $L$ on an abelian variety, its homomorphism associated to a line bundle on an abelian variety satisfiesas Picard group classes. A choice of trivialization of $L_0$ normalizes the right side; uniqueness then follows from the Seesaw theorem.

For the Levi-Civita connection of a Riemannian metric, the divergence of $X$ is ${\mathrm{div}}_{g}X=\mathrm{tr}(Y\mapsto {\nabla }_{Y}X)$. In coordinates it equals $(\det g{)}^{-1/2}{\partial }_i((\det g{)}^{1/2}{X}^i)$. On an oriented manifold it satisfies ${\mathcal{L}}_X{\omega }_g=({\mathrm{div}}_{g}X){\omega }_g$. The codifferential on a one-form satisfies $\delta \theta =-{\mathrm{div}}_g({\theta }^{♯})$, where $♯$ is the musical isomorphism; this follows by integration by parts or the Hodge star formula.

For the dipole similarity solution of a draining porous current, $V=\varphi \int hdx=\varphi ad{t}^{-1/4}/18$. Thus $\dot{V}/V=-1/(4t)$. A shifted similarity origin replaces $t$ by $t+t_0$. Neither a positive rate nor a rate $-1/(2t)$ is compatible with this two-dimensional first-moment-conserving profile.

For each index, collect the ground subsets which some condition forces to be the corresponding diamond guess. Distinct forced values have incompatible witnessing conditions, so the countable chain condition for forcing makes this a countable family. Every ground subset is guessed by these families on a ground stationary set, by testing each ground club set. The countable-family diamond equivalence then yields diamond in the ground model. A generic guess need not itself be a ground subset; only values forced equal to one are collected.

A periodic complex field permits $\varphi (\beta ,\mathbf{r})-\varphi (0,\mathbf{r})=2\pi w(\mathbf{r})$. Its uniform-density first-order term is therefore a winding-dependent imaginary action, not generally a removable global constant. It vanishes in the smooth zero-winding sector used for Gaussian phonons. Density defects and vortices require more than that smooth-sector approximation.

Expand the density-phase action of a Bose gas around ${\rho }_0=\mu /g>0$ and a constant phase. Density and phase are coupled by $i\delta \rho {\partial }_{\tau }\varphi$, and retaining density gradients gives the full quadratic Bogoliubov spectrum. Dropping those gradients is a long-wavelength approximation. A small-fluctuation Gaussian extension around ${\rho }_0$ does not exactly remove the original nonnegative-density constraint.

The complex of smooth differential forms resolves the constant complex sheaf by the Poincaré lemma and consists of fine sheaves. Its global cohomology therefore agrees with sheaf cohomology of that constant sheaf. A good-cover Čech-de Rham double complex gives an explicit comparison, used in Čech-de Rham curvature descent.

For a connection on a vector bundle written $D=d+A\wedge$ in a local frame, its curvature form of a connection is represented by $\Theta =dA+A\wedge A$. The products combine matrix multiplication with the exterior product. The formula follows by squaring $D$ and using the graded Leibniz rule.

For normalized columns and $1\le s\le N$, the restricted isometry constant is bounded by the cumulative coherence at $s-1$. A restricted Gram matrix has diagonal one and every off-diagonal row sum at most ${\mu }_1(s-1)$. The Gershgorin circle theorem and the finite-dimensional spectral theorem bound its eigenvalues between $1-{\mu }_1(s-1)$ and $1+{\mu }_1(s-1)$. At $s=1$ the distortion is zero, and at $s=2$ it is exactly the mutual coherence.

For a cyclic extension $E/K$ of degree $m$ with generator $\sigma$ and $a\in K^{\times }$, the cyclic algebra is generated by $E$ and $u$ with $ub=\sigma (b)u$, $u^m=a$. It is a central simple algebra of degree $m$. Its Brauer class can also be written $(\chi ,a)$, where $\chi (\sigma )=1/m$.

For independent unit-rate exponential random variables, the cumulant-generating function is $-\log (1-\theta )$ for $\theta <1$. Its Legendre transform of a cumulant-generating function is $I(x)=x-1-\log x$ for $x>0$ and infinite otherwise. The Cramér theorem gives the upper-tail logarithmic rate $-I(a)$ for $a>1$, and zero for $0\le a\le 1$. At the mean threshold the infima on both the open and closed upper half-lines are zero; below the mean the strong law of large numbers applies.

When the adjustment coefficient lies inside the finite domain of the claim moment-generating function, the tilted kernel has finite mean $(\lambda M^{\prime }(R)-c)/(cR)$. The forcing integral is $(c-\lambda \mu )/(cR)$. Their ratio is the constant $C$ in $\psi (u)\sim C{e}^{-Ru}$, by the key renewal theorem. An extra exponential moment controls the forcing tails and proves direct Riemann integrability.

For a CIR model started at a positive rate, this condition makes the zero boundary inaccessible. Below the threshold, zero can be reached, but the usual nonnegative solution does not cross into negative rates. Thus nonnegativity and strict positivity are different properties.

For the CIR model under its money-market risk-neutral measure, the Feynman-Kac formula gives $B^{\prime }=1-bB-{\sigma }^2{B}^2/2$ and $A^{\prime }/A=-aB$, with $B(0)=0$, $A(0)=1$. Solving this Riccati equation gives explicit exponential-affine zero-coupon bond prices. The positive root parameter is $\sqrt{b^2+2{\sigma }^2}$; the squared diffusion coefficient, not the volatility itself, enters this expression.

Exterior product of differential forms combined with endomorphism composition is associative and generally noncommutative. For an endomorphism-valued one-form $a$, $(a\wedge a)(u,v)=[a(u),a(v)]$. The endomorphism bundle connection extends as a graded derivation of this algebra.

A compound-symmetry covariance matrix has common diagonal and off-diagonal entries:For group size $k\ge 2$, contrasts orthogonal to $1$ have eigenvalue $a$, with multiplicity $k-1$, while the constant direction has eigenvalue $a+kc$. Thus positive semidefiniteness requires exactly $a\ge 0$ and $a+kc\ge 0$. Negative $c$ is allowed down to $-a/k$.

When $c\ge 0$, a shared random intercept of variance $c$ plus independent individual errors of variance $a$ realizes the matrix. The group sample mean has variance $c+a/k$. The difference between means of two disjoint subgroups of sizes $m,\ell$ has variance $a(1/m+1/\ell )$: their common covariance contribution $c$ cancels.

For $k=1$, there is no contrast subspace. The sole eigenvalue is $a+c$, so only $a+c\ge 0$ is required. For example, $a=-1,c=2$ still gives the valid one-dimensional covariance matrix $[1]$, showing why the two-inequality statement needs $k\ge 2$.

For uniform independent draws of $n$ coupon types, the waiting times with $r$ unseen types remaining are independent positive-support geometric distributions of parameter $r/n$. Summing their means gives $n{\sum }_{r=1}^{n}1/r$, and summing their variances gives the displayed formula. Independence follows from fresh draws after each discovery and the fact that the next stage's conditional distribution depends only on the count of remaining types.

A differential one-form $\theta$ is a smooth section of a vector bundle $T^{\ast }X\to X$. Its graph is its image as a section, and the projection restricts to a diffeomorphism onto $X$. Pullback of the Liouville one-form by that section equals $\theta$, making the twisted Lagrangian graph criterion immediate.