On a Stein manifold $X$, every coherent analytic sheaf $\mathcal{F}$ has $H^q(X,\mathcal{F})=0$ for $q>0$. In particular this applies to the sheaf of holomorphic functions. If every nonempty finite intersection in an open cover is Stein, these vanishing results and the acyclic cover theorem compute the corresponding sheaf cohomology from its Čech cochain complex.

For a complex Lie group, holomorphic principal bundles over a Stein manifold are classified up to holomorphic isomorphism by their topological bundle class. In particular, a topologically trivial holomorphic principal bundle over a Stein base has a global holomorphic trivialization. This supplies global complex gauges for a flat partial connection on contractible complex affine space; it does not impose prescribed behaviour at infinity.

Choose the degree-four generator with pullback $u\mapsto x^2$ under the complex inclusion into quaternionic projective space. On infinite complex projective space, the Cartan formula gives $P^i(x^{2k})=\left(\genfrac{}{}{0ex}{}{2k}{i}\right)x^{2k+i(p-1)}$. Injectivity of the infinite-space pullback proves the displayed formula for quaternionic projective space. Restriction to ${\mathbb{HP}}^n$ sets powers above $n$ to zero. Coefficients are reduced modulo the odd prime $p$; the exponent is an integer because $p-1$ is even.

At a correctly guessing limit stage of a normal tree construction, make every new level node extend a member of the guessed maximal tree antichain. It follows that the antichain has no later member. This produces a Suslin tree from the stationary diamond principle.

The Null Raychaudhuri equation and Einstein field equations give ${\partial }_U(f^{-1}{r}_{,U})=-r{R}_{UU}/(2f)=-4\pi r{T}_{UU}/f$, with an exchanged-coordinate identity in $V$. Under the null energy condition, each rescaled radial derivative is nonincreasing along its corresponding future null direction.

The maximum antichains in the Boolean lattice are exactly its full middle levels. For even $n$ there is one such level, and for odd $n$ either middle level works. The LYM inequality first confines a maximum antichain to those levels; connectedness of the regular inclusion bipartite graph rules out a mixture of the two odd middle levels.

For a Hermitian matrix with unitary diagonalization $A=U\mathrm{diag}({\lambda }_i)U^{\ast }$, define $f(A)=U\mathrm{diag}(f({\lambda }_i))U^{\ast }$ for a function on its finite spectrum. This acts on eigenvalues, unlike an entrywise matrix function $f[A]$. For example, the spectral square of a matrix is its ordinary matrix product with itself, whereas its entrywise square is a Hadamard power.

The Davis-Kahan theorem bounds the change of an invariant eigenspace of a symmetric matrix in terms of a matrix perturbation and an appropriate spectral gap. The rank-one eigenprojector perturbation bound below depends only on the unperturbed gap. Yu, Wang and Samworth's formulation gives population-gap bounds in greater generality.

Suppose $H\ge 0$ has ground energy zero and positive spectral gap at least $g$, and $0\le V\le I$. Let $\nu$ be the minimum of $V$ restricted to the ground space. Splitting a normalized vector into ground and excited components bounds its energy below by $\delta \nu +(g-\delta )y^2-2\delta y$, with $y$ the excited norm. Completing the square yields the displayed bound for $0<\delta <g$. The error is controlled by the gap and cannot be treated as a uniform $O({\delta }^2)$ when $g$ closes.

For the special linear Lie algebra $\mathfrak{sl}(3)$, with Dynkin labels indexing irreducible highest-weight representations,The summands have dimensions $60,24,15,15,6$, summing to $120=15\cdot 16/2$. The identity follows from the formal character of a weight module formula ${\chi }_{S^{2}W}(z)=({\chi }_W(z{)}^2+{\chi }_W(z^2))/2$ and the Weyl character formula.

For $n\ge 2$ and $m\ge 0$, the representation ${\mathrm{Sym}}^m{\mathbb{C}}^n$ of ${\mathfrak{sl}}_n(\mathbb{C})$ is irreducible with highest weight $m{\omega }_1$. Its monomial basis supplies weights by adding $m$ defining-representation weights, with repeated weights counted separately.

A standard sufficient form uses a bounded Lipschitz domain $\Omega$, scalar functions $u_k\in SBV(\Omega )$, a uniform essential supremum bound, and some $p>1$ such thatThere is a subsequence converging in L1 norm to a function $u\in SBV(\Omega )$. Its approximate gradients converge weakly in $L^p$, and the jump-set measure is bounded by the lower limit of the approximants' jump-set measures. The theorem keeps the Cantor part of a BV derivative out of the limit. ^[ref]

The hypotheses do distinct jobs. The value bound and jump-size bound control jump variation; $p>1$ gives uniform integrability of the absolutely continuous derivatives. Ordinary bounded-variation compactness alone permits a diffuse singular part in the limit. Domain or tightness control also matters for global convergence: translating a fixed interval indicator arbitrarily far along $\mathbb{R}$ keeps all these local size bounds but gives no strongly convergent global $L^1$ subsequence.

Over an algebraically closed field, a smooth four-variable projective quadratic equation can be written $x_0{x}_3-x_1{x}_2=0$, including in characteristic two. Its quadratic form splits into two hyperbolic planes: choose an isotropic vector and its paired vector, adjust the latter to be isotropic, and repeat on the orthogonal complement. In characteristic two, smoothness forces the alternating polar form in dimension four to have full rank, since a positive-dimensional radical has a nonzero isotropic vector over the algebraically closed field and would give a singular point. The Segre embedding $([s_0:s_1],[t_0:t_1])\mapsto [s_0{t}_0:s_0{t}_1:s_1{t}_0:s_1{t}_1]$ then gives the isomorphism. Each matrix of coordinates has rank one, which supplies the inverse pair of projective factors.

Over an algebraically closed field, remove a smooth hyperplane conic $C$ from a smooth quadric surface $Q$. The rulings of a smooth quadric surface generate $\mathrm{Pic}(Q)={\mathbb{Z}}^2$, and $C$ has class $(1,1)$. Picard-group localization on a smooth variety gives the quotient by $\mathbb{Z}(1,1)$. The restricted ruling bundle ${\mathcal{O}}_Q(1,0)$ is a generator, exhibiting a nontrivial line bundle on an affine surface.

For $X=\{x_0{x}_1=x_2{x}_3\}$ and $P=[1:0:0:1]$, the displayed map is defined on $X\setminus \{P\}$. Its fibre over $[u:v:w]$ has equation $(u-v)\lambda +uw=0$ on an affine line. Away from $u=v$ the fibre is one reduced point. On that line it is empty except at $[0:0:1]$ and $[1:1:0]$, where it is a punctured ruling line, hence an affine line. The two exceptional lines are the rulings of a smooth quadric surface through $P$.

Let $A,B$ be positive operators on a finite-dimensional Hilbert space, with every nonzero eigenvalue at least $\gamma >0$. Let $P,Q$ be the orthogonal projections onto $\ker A,\ker B$ and define $\cos \vartheta =\Vert PQ\Vert$. ThenWhen the kernels have no common nonzero vector, this gives a positive lower bound. The projection formula also covers a zero kernel without taking a supremum over an empty unit sphere.

Indeed, $A\ge \gamma (I-P)$ and $B\ge \gamma (I-Q)$. For $p\in \mathrm{ran}P$, $q\in \mathrm{ran}Q$, $|⟨p,q⟩|\le \Vert PQ\Vert \Vert p\Vert \Vert q\Vert$. The map $(p,q)\mapsto p+q$ consequently has squared operator norm at most $1+\Vert PQ\Vert$, so $P+Q\le (1+\Vert PQ\Vert )I$. Combining these inequalities proves the result. With common kernel $N$, restrict to $N^{\perp }$ and use the kernel projections there to bound the positive spectral gap.

In infinite dimension, replace the eigenvalue condition by the actual operator inequalities $A\ge \gamma (I-P)$ and $B\ge \gamma (I-Q)$. An assertion only about eigenvalues can be vacuous in the presence of continuous spectrum, and does not ensure this uniform gap.

For sine collocation of square modified Helmholtz global relations, each opposite-side mode has a scaled matrixThe assembled system has strict diagonal dominance, and its spectral condition number of a positive-definite matrix is below $(1+e^{-\pi })/(1-e^{-\pi })<1.091$. The conclusion concerns the explicit paired sine rows, not arbitrary uncombined complex rows.

Let $R_{jn}={\int }_{\partial \Omega }f{\partial }_n{v}_{jn}ds$ for a square sine adjoint test, and put ${\rho }_{jn}=e^{-{\omega }_n}{R}_{jn}$, ${\delta }_n=e^{-2{\omega }_n}$. The opposite-side normal-trace coefficients obeywith the same formulas for $L,R$. When the known Dirichlet boundary condition is integrated exactly, these are exact coefficients, since orthogonality eliminates all other unknown modes.

In ${\epsilon }^2{\Psi }^{\prime \prime }+2{\epsilon }^2(c^{\prime }/c){\Psi }^{\prime }+Q(Y)\Psi =0$, with $Q\sim -q(Y-Y_s)$ and $q>0$, set $Y-Y_s=({\epsilon }^2/q{)}^{1/3}\eta$. The leading equation is ${\Psi }_{\eta \eta }-\eta \Psi =0$, the Airy ordinary differential equation. The first derivative and nonlinear coefficient corrections are smaller by $O({\epsilon }^{2/3})$.

For the quadratic hindered-settling flux, let the initial normalized concentration be $b$ below $z_m$ and $a>b$ above it, within a column $0<z<H$. The internal settling rarefaction fan hasBefore entering the fan, the upper clearing front follows $H-u_s(1-a)t$ and the lower deposition front follows $u_{s}bt$. They reach the fan at respective times $t_u=(H-z_m)/(u_{s}a)$ and $t_d=z_m/[u_s(1-b)]$.

Inside the fan, the Rankine-Hugoniot condition givesIntegrating these linear ordinary differential equations and matching to the entry events yieldswhere $A=a(H-z_m)$ and $D=(1-b)z_m$. Each curve applies only after that front enters the fan and while its adjacent concentration lies between $b$ and $a$. If a front leaves the fan, its subsequent speed is set by the uniform state it encounters, rather than by extrapolating these curves.