Let $P$ project onto a degree-one equivariant line bundle on the complex-oriented round two-sphere, and let $V$ be its positive-to-negative chiral Dirac operator phase. The compressed map has parametrix $P{V}^{\ast }P$ modulo compacts. Its domain bundle has degree zero and its target degree two. Their homogeneous line-bundle representations on the two-sphere are respectively $E_0\oplus E_2\oplus E_4\oplus \cdots$ and $E_2\oplus E_4\oplus \cdots$, so the equivariant Fredholm index is the trivial representation and the ordinary index is one. Reversing chirality negates the index. Compressing the full self-adjoint phase instead gives ordinary index zero.

For a Hermitian metric on a complex manifold, the Chern connection on its holomorphic tangent bundle corresponds to a real metric-compatible connection preserving the almost complex structure. Its mixed torsion vanishes, and in holomorphic coordinates its remaining coefficients are ${\Gamma }_{ij}^k=h^{k\bar{l}}{\partial }_i{h}_{j\bar{l}}$. Symmetry in $i,j$ is equivalent to $\partial \omega =0$; its conjugate gives $d\omega =0$. Thus zero torsion is exactly the Kähler metric condition, and then uniqueness identifies this connection with the Levi-Civita connection.

If a connected closed manifold $M^3$ has a noncontractible universal cover, then $M\times \mathbb{R}$ has no complete metric making it a Ricci-flat Riemannian manifold. The product has two ends and hence contains a line in a Riemannian manifold for any complete metric. The Cheeger-Gromoll splitting theorem would split it isometrically as $N^3\times \mathbb{R}$. Its three-dimensional factor is Ricci-flat and therefore flat by three-dimensional curvature from the Ricci tensor. The complete simply connected cover of the product would be Euclidean four-space. But that cover is $\tilde{M}\times \mathbb{R}$, which retracts onto $\tilde{M}$, contradicting noncontractibility.

Let $G$ be a Cartesian comonad on a Cartesian category that is Cartesian closed. Its forgetful functor creates finite limits. For coalgebras $(A,a)$ and $(B,b)$, set $D=B^A$, $W=(GB{)}^A$. Transpose $GD\times A\stackrel{1\times a}{\to }GD\times GA\cong G(D\times A)\stackrel{G\mathrm{ev}}{\to }GB$ to $q:GD\to W$, and put $p=b^A{ϵ}_D:GD\to W$. Their cofree coalgebra transposes $Gp{\delta }_D,Gq{\delta }_D:R(D)\rightrightarrows R(W)$ have a coalgebra equalizer. A map into this equalizer is exactly a map $C\times A\to B$ respecting coalgebra structures, so it is the required exponential object. No subobject classifier on the base category is needed.

For a family of transition graph paths, Path congestion is the largest, over transitions, ratio of routed pair weight times path length to stationary transition capacity. The canonical paths Poincare bound makes this an upper bound on the relaxation time.

Let $Y$ be a smooth complex analytic hypersurface through the blowup centre with $[-Y]\cong K_X$. Its multiplicity at that point is one, so ${\sigma }^{\ast }Y=\tilde{Y}+E$. Combining this with the canonical bundle formula for a point blowup gives $[-\tilde{Y}]\cong K_{\tilde{X}}\otimes [E{]}^{\otimes (2-n)}$. The nontrivial powers of the exceptional line bundle of a point blowup prove the equivalence for $n\ge 2$. On a compact complex curve the premise cannot hold for an irreducible hypersurface: it is one point, so its negative divisor has degree $-1$, whereas the canonical bundle has even degree $2g-2$.

A nonunit element of a commutative monoid is irreducible if each factorization has a unit factor. In a submonoid of the positive integers under multiplication, the only unit is one. Irreducibility is relative to the allowed factors: a composite integer can be irreducible in a smaller monoid even though it is not a prime number.

For an integer $m\ge 2$, the positive integers congruent to one modulo $m$ form a cancellative commutative monoid under multiplication. The identity is one, and it is the only unit. Proper factors outside this congruence class do not witness reducibility inside the monoid. Hence irreducible elements of a commutative monoid need not coincide with ordinary prime numbers.

Two distinct closed lambda terms in normal form for both beta reduction and eta conversion admit a separating context that maps them respectively to the two Church Booleans. Finite Böhm trees and the Böhm-out lemma isolate their first genuine head disagreement. The qualification about eta conversion matters: distinct beta-normal forms that are eta-equivalent cannot be separated this way.

The Bishop-Gromov inequality forces every volume ratio to equal one if the asymptotic ratio is one. Equality in the radial Jacobian comparison forces the radial operator $A(X)={\nabla }_X{\partial }_r$ of each distance sphere to be $r^{-1}I$ (it is the negative of the outward shape operator in the convention $S=-\nabla \nu$), by the traced radial Riccati equation for distance spheres and equality in Cauchy-Schwarz inequality. Any finite cut time would remove a positive cone of radial volume at a later radius, contradicting equality. Thus the Riemannian exponential map is global and one-to-one, and the polar metric is $d{r}^2+r^2{g}_{S^{n-1}}$, the Euclidean metric.

For a complete connected $n$-dimensional Riemannian manifold with $\mathrm{Ric}\ge (n-1)kg$, $k>0$, put $D=\pi /\sqrt{k}$ andHere ${\omega }_{n-1}$ is the area of the unit $(n-1)$-sphere, and $v_k(r)$ is the Riemannian volume of a radius-$r$ ball in the round curvature-$k$ sphere. The Bishop-Gromov inequality asserts that $V_p(r)/v_k(r)$ is nonincreasing for $0<r<D$, tends to one at zero, and is at most one. In geodesic polar coordinates, the radial volume density before the cut time is at most $s_k(t{)}^{n-1}$. The bound uses trace Ricci curvature; with normalized Ricci curvature it reads $\overline{\mathrm{Ric}}\ge kg$.

A binary quartic is a homogeneous degree-four polynomial in two variables. Over the complex numbers it determines four points of the projective line, counted with multiplicity. Its discriminant is nonzero exactly when the four points are distinct; this is also its geometric invariant theory stability condition under $S{L}_2$.

For a smooth three-dimensional biharmonic function regular throughout a ball, its surface average has exactly this form. Taylor expansion and rotationally invariant spherical moments turn each even term into a multiple of an iterated Laplacian; terms from the fourth derivative onward vanish when ${\Delta }^{2}f=0$. This is the mean-value identity that converts the reciprocal surface average into Faxén's first law.

For a nearest-neighbour random walk with upward probability $0<p<q=1-p$, start at $S_0=0$ and put $M={\sup }_{n\ge 0}{S}_n$. Its probability distribution is geometric on the nonnegative integers, with the displayed mass and expected value $p/(q-p)$. To prove this, the exponential martingale of a biased simple random walk $Z_n=(q/p{)}^{S_n}$ has expectation one. Stop at the first hit $H_k$ of $k$. Before that hit, $Z_{n\wedge H_k}\le (q/p{)}^k$; the strong law of large numbers gives $S_n\to -\infty$ on paths without a hit. Dominated convergence theorem therefore gives $1=(q/p{)}^k\mathbb{P}(H_k<\infty )$. Taking differences of $\mathbb{P}(M\ge k)=(p/q{)}^k$ yields the mass. If time zero is excluded, the maximum can equal $-1$ with probability $q-p$ and does not have this nonnegative geometric distribution.

For independent lattice steps of length $a$ at intervals $\tau$, with probabilities $\alpha +\beta =1$, the diffusive scaling $a^2/\tau \to 2D$ and $(\alpha -\beta )a/\tau \to V$ requires a bias of order $a$. Expanding the master equation gives the advection-diffusion equation $p_t+V{p}_x=D{p}_{xx}$. The finite-step variance coefficient is $2a^2\alpha \beta /\tau$; it has the same limit because the squared bias vanishes. Fixed nonzero bias does not give finite drift under this diffusive scaling.

On $\{0,\dots ,N\}$, stop at zero, move left with probability $q>1/2$ and right with probability $p=1-q$ at interior states, and move deterministically from $N$ to $N-1$. For $m_i={\mathbb{E}}_i{T}_0$, the first-step analysis gives $m_0=0$, $m_i=1+q{m}_{i-1}+p{m}_{i+1}$ and $m_N=1+m_{N-1}$. Setting $D_i=m_i-m_{i-1}$ gives $q{D}_i-p{D}_{i+1}=1$, $D_N=1$, hence $D_i=[1-2p(p/q{)}^{N-i}]/(q-p)$. Summing these differences proves the displayed formula, and more generally $m_i=i/(q-p)-2pq(p/q{)}^{N-i}[1-(p/q{)}^i]/(q-p{)}^2$. A downward block has a uniformly positive probability of reaching zero, which supplies a geometric tail bound and justifies finiteness of these expectations.

For independent p-values, with $m_0$ true-null values independent and uniform on $[0,1]$, let $R^{(i)}$ denote the Benjamini-Hochberg procedure rejection count after replacing one true-null value by zero. The Benjamini-Hochberg leave-one-out identity givesFor distinct true-null indices, the Benjamini-Hochberg leave-two-out identity givesIndeed, conditional on the remaining values and $R^{(ij)}=r$, the two independent uniform values both meet their threshold with probability $(\alpha r/m{)}^2$, cancelling the denominator $r^2$. Expanding the square of the false rejection count into its diagonal terms and ordered distinct pairs, and using symmetry among the true-null values, provesIf no null is true, both sides are zero; if exactly one is true, the pair contribution is absent. This formula complements the exact false discovery rate $\alpha m_0/m$ by describing second-moment variability.

Flatness of a projective family does not by itself force global sections to commute with base change. For a discrete valuation ring with uniformizer $\pi$, glue $\mathrm{Spec}R[\pi t,t^2,t^3]$ to $\mathrm{Spec}R[t^{-1}]$ along $\mathrm{Spec}R[t,t^{-1}]$. In the projective model with coordinates $(1,\pi t,t^2,t^3)$, the global ring is $R$. The special-fibre first chart has an additional square-zero element $a$ annihilated by $t^2,t^3$; it vanishes on the overlap and therefore extends by zero to a global section. The special-fibre global ring is $k[a]/(a^2)$, and base change from $R$ reaches only its constants.

For a sample of size $n\ge 2$ from a normal distribution with unknown mean, the null variance likelihood-ratio test statistic is $W=n[-\log (U/n)+U/n-1]$, where $U\sim {\chi }_{n-1}^2$. With $V=(U-n)/n$, its expected Taylor terms start with $n[\mathbb{E}{V}^2/2-\mathbb{E}{V}^3/3+\mathbb{E}{V}^4/4]$. Here $\mathbb{E}{V}^2=(2n-1)/n^2$, $\mathbb{E}{V}^3=(2n-3)/n^3$, and $\mathbb{E}{V}^4=(12n^2+4n-15)/n^4$. Higher fixed-order terms contribute only $O(n^{-2})$ under valid termwise asymptotic integration. Thus ${\mathbb{E}}_{0}W=1+11/(6n)+O(n^{-2})$ and $W_B=W/[1+11/(6n)]$.

For simple roots $(1,0),(-1,1)$, Dynkin labels of a Euclidean weight $(x,y)$ are $(2x,-x+y)$. The highest root $(1,1)$ gives the Adjoint representation, of dimension ten. Its eight nonzero weights are the eight roots of $B_2$; the zero weight has multiplicity two. The nine distinct label pairs are $(2,0),(0,1),(-2,2),(2,-1),(0,0),(-2,1),(2,-2),(0,-1),(-2,0)$.