A donor star with stellar radius response exponent ${\zeta }_{\ast }$ can maintain exact contact in the donor-wind Roche-lobe response model for some retained fraction $0\le f\le 1$ precisely when ${\zeta }_{\ast }$ lies between ${\zeta }_0$ and ${\zeta }_1$. This follows because the attainable Roche-lobe radius response exponents form that closed interval. If its endpoints coincide, all fractions work when ${\zeta }_{\ast }$ equals the common value and none work otherwise. With negative donor mass-loss rate, $d\log (R_d/R_L)/dt=({\zeta }_{\ast }-{\zeta }_L){\dot{M}}_d/M_d$ distinguishes detachment from increasing overfill. Dynamical stability of binary mass transfer requires the adiabatic donor response.

For a complex torus $X=V/\Lambda$ of complex dimension $g$, constant forms give a natural isomorphism $H_{\bar{\partial }}^{p,q}(X)\cong {\bigwedge }^p{V}^{\ast }\otimes {\bigwedge }^q\overline{V^{\ast }}$. A flat Kähler metric and the Dolbeault Hodge decomposition prove that these constant forms are precisely the harmonic representatives. Thus $h^{p,q}=\left(\genfrac{}{}{0ex}{}{g}{p}\right)\left(\genfrac{}{}{0ex}{}{g}{q}\right)$ for $0\le p,q\le g$, and zero otherwise.

This is the cohomology of ${\mathcal{A}}^{p,\bullet }(X,E)$ with the bundle Dolbeault operator. The Dolbeault theorem identifies it with $H^q(X,{\Omega }_X^p\otimes \mathcal{O}(E))$. For compact Hermitian $X$, Dolbeault Hodge decomposition gives unique harmonic representatives.

For $X={\mathbb{P}}^1\times \mathbb{C}$, the two standard affine charts and their intersection form an acyclic cover for the sheaves of holomorphic forms. Their Čech cohomology gives $H_{\bar{\partial }}^{0,0}=\mathcal{O}(\mathbb{C})$ and $H_{\bar{\partial }}^{2,1}=\mathcal{O}(\mathbb{C})$, with all other groups for $p\in \{0,2\}$ zero. For the top-form sheaf, the transition $d(1/z)=-z^{-2}dz$ removes every Laurent series power except $z^{-1}$ from the first cohomology quotient; its coefficient may be an arbitrary entire function on the affine factor.

On a Stein manifold, every smooth $\bar{\partial }$-closed $(0,q)$-form with $q>0$ is globally $\bar{\partial }$-exact. The same holds componentwise for forms valued in a finite-dimensional constant vector space. Complex conjugation yields the corresponding global $\partial$ primitive for a closed $(1,0)$-form, as used in the complex potential reduction of anti-self-dual Yang-Mills.

If a symmetric matrix $H$ has largest eigenvalue ${\lambda }_1$ separated from ${\lambda }_2$ by $\gamma >0$, and $P=v{v}^{\top }$ is its leading rank-one orthogonal projection matrix, then every unit vector $u$, with $Q=u{u}^{\top }$, satisfies the displayed inequality. Expanding $u$ in an orthonormal eigenbasis gives ${\lambda }_1-u^{\top }Hu\ge \gamma (1-|u^{\top }v{|}^2)$, while $\Vert P-Q{\Vert }_F^2=2(1-|u^{\top }v{|}^2)$. The analogous statement for a smallest eigenvalue follows by replacing $H$ by $-H$.

Suppose both curved settling fronts within a rarefaction fan meet while still inside the fan. Write $A=a(H-z_m)$ and $D=(1-b)z_m$, where $0<b<a<1$ are the initial normalized concentrations below and above $z_m$. Setting $z_u=z_d$ givesAt that point the fan concentration isThe final surviving characteristic curve runs from the initial internal interface to the meeting point. Its concentration $c_{\ast }$ must satisfy $b\le c_{\ast }\le a$. Failure of either bound means that a front leaves the fan before final deposition, requiring another branch of the entropy solution.

The meeting height reduces to $z_m+A-D=a(H-z_m)+b{z}_m$, agreeing independently with particle-volume mass conservation for a unit-normalized packed deposit.

Ignoring the last-scattering angular source and later scattering, the tensor-induced temperature anisotropy is $\Theta =-\frac{1}{2}{\int }_{{\eta }_{\ast }}^{{\eta }_0}{\dot{h}}_{ij}(\eta ,{\mathbf{x}}_0-({\eta }_0-\eta )\mathbf{e})e^i{e}^{j}d\eta$. An instantaneous cosmological visibility function alone leaves a scattering source at last scattering; leading tensor tight coupling supplies the additional reason to neglect it.

For simple shear flow $u=(\dot{\gamma }y,0,0)$, put $s=\dot{\gamma }$ and $\ell =\lambda s$. The steady deviatoric stress isThis follows by setting ${\omega }_{xy}=-s$, ${\omega }_{yx}=s$ in the corotational Maxwell fluid equation: ${\sigma }_{xx}=\lambda s{\sigma }_{xy}$, ${\sigma }_{yy}=-\lambda s{\sigma }_{xy}$, and ${\sigma }_{xy}+\lambda s({\sigma }_{xx}-{\sigma }_{yy})/2=\mu s$. Hence the shear viscosity is $\mu /(1+{\lambda }^2{s}^2)$, which exhibits shear thinning, while $N_1=2\mu \lambda s^2/(1+{\lambda }^2{s}^2)$ and $N_2=-N_1/2$. These differences are independent of the isotropic pressure.

Update one link blocking probability at a time using the reduced load accepted by all other links, then cycle through the links. In logarithmic acceptance coordinates this is exact coordinate descent for the convex potential for the Erlang fixed point. Potential values decrease within a compact sublevel set. Continuity of a whole-sweep update forces every limit point to minimize every coordinate, hence to equal the unique global minimizer. This proves convergence without requiring a simultaneous substitution scheme.

For a closed immersion, this quasi-coherent sheaf measures the first-order equations of the subvariety. Its local module is $I/I^2$, and its map to the restricted Kähler differential sheaf sends the class of $f$ to $df$. This is the definition in Stacks Project, Section 29.32.

For $D>0$, a nontorsion rational point on the congruent number elliptic curve $y^2=x^3-D^{2}x$ gives infinitely many positive rational right triangles of area $D$. Apply the formulas $a=|(x^2-D^2)/y|$, $b=|2Dx/y|$, $c=|(x^2+D^2)/y|$ to its multiples. The map is finite-to-one: if $r=c/b$ is fixed, then $|x|$ satisfies $X^2-2DrX+D^2=0$, leaving only finitely many coordinate signs. Thus infinitely many points give infinitely many distinct triangles.

For conjugated Loewner chains, ${\varphi }_t={\tilde{g}}_t\circ \varphi \circ g_t^{-1}$ and ${\tilde{\xi }}_t={\varphi }_t({\xi }_t)$. The image capacity speed is $2{\varphi }_t^{\prime }({\xi }_t{)}^2$. Differentiating this identity gives ${\dot{\varphi }}_t(z)=2{\varphi }_t^{\prime }({\xi }_t{)}^2/({\varphi }_t(z)-{\varphi }_t({\xi }_t))-2{\varphi }_t^{\prime }(z)/(z-{\xi }_t)$. The removable singularity at the driver has value $-3{\varphi }_t^{\prime \prime }({\xi }_t)$.

In its block form, a matrix-unit basis of the unitary Lie algebra spans the anti-Hermitian block $A$, with the corresponding conjugate block below. Independent real and imaginary symmetric matrix units span $B$, with lower block $-\overline{B}$. These give $n^2+n(n+1)$ independent real generators, including diagonal symmetric entries.

Let $X$ be a Noetherian scheme and suppose $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf $\mathcal{I}$. For a coherent sheaf $\mathcal{F}\subseteq {\mathcal{O}}_X^r$, projection onto the last coordinate has image an ideal sheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$ and kernel ${\mathcal{F}}^{\prime }\subseteq {\mathcal{O}}_X^{r-1}$. Both are coherent sheaves, sois a short exact sequence of sheaves. Thus $\mathcal{F}$ is a sheaf extension of the quotient ideal $\mathcal{I}$ by the lower-rank submodule ${\mathcal{F}}^{\prime }$, with this order of the two terms.

For $r=1$, the assertion is exactly the assumed vanishing for coherent ideal sheaves. For larger $r$, mathematical induction gives $H^1(X,{\mathcal{F}}^{\prime })=0$, and the long exact sequence in sheaf cohomology places $H^1(X,\mathcal{F})$ between this zero group and $H^1(X,\mathcal{I})=0$. Hence $H^1(X,\mathcal{F})=0$.

Let $X$ be an algebraic variety over an algebraically closed field $k$, and assume $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf $\mathcal{I}$. Given a closed point $P$, choose an affine open subscheme $U$ containing it and give $Y=X\setminus U$ its reduced induced subscheme structure. Since $P\notin Y$, evaluation at $P$ is a surjective morphism of sheaveswith kernel ${\mathcal{I}}_Y\cap {\mathcal{I}}_P$. Here $k_P$ is the skyscraper sheaf at the closed rational point, and the kernel is again a coherent ideal sheaf. The long exact sequence in sheaf cohomology therefore lifts $1\in k_P$ to a global section $f$ of ${\mathcal{I}}_Y$.

Viewed as a global regular function, $f$ vanishes on $Y$ and has $f(P)=1$. Thus $P\in X_f\subseteq U$, and $X_f=U_{f{|}_U}$ is a principal open subscheme of an affine scheme, hence affine. Such neighborhoods cover $X$: every nonempty closed subset of an algebraic variety contains a closed point. This is the local step in the cohomological criterion for affineness; the closed-point hypothesis makes the coherent skyscraper argument valid.

If finitely many $X_{f_i}$ cover $X$, the morphism ${\mathcal{O}}_X^r\to {\mathcal{O}}_X$ with coefficients $f_i$ is onto. If its kernel has vanishing first sheaf cohomology, the long exact sequence in sheaf cohomology lifts the global section $1$ to coefficients $g_i$. This turns a geometric cover into a unit ideal in the ring of global sections.

If $S$ is a holomorphic subbundle of a Hermitian vector bundle $E$, its induced Chern connection is $D_S={\pi }_S{D}_E$ on sections of $S$. Orthogonal projection preserves metric compatibility, and holomorphicity of $S$ preserves the required $(0,1)$ operator. Uniqueness of the Chern connection therefore identifies the projected connection.

For a comonad, the cofree coalgebra on $X$ is $(GX,{\delta }_X)$. A map $UA\to X$ transposes to the coalgebra morphism $Gfa:A\to GX$. This gives the adjunction $U⊣R$ with the forgetful functor.

For a finite-limit-preserving comonad on a topos, start with the cofree coalgebra on the ambient exponential $B^A$. Its underlying evaluation is $e=\mathrm{ev}(ϵ\times 1)$. Transpose the two expressions $be$ and $Ge(\delta \times a)$ first using the ambient exponential and then the cofree adjunction. Their coalgebra equalizer represents exactly the maps whose evaluations respect coalgebra structure. Thus coalgebra exponentials need not be the underlying ambient exponentials.