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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 $(2,0),(0,1),(-2,2),(2,-1),(0,0),(-2,1),(2,-2),(0,-1),(-2,0)$.

Getzler rescaling combines the change $x\mapsto \sqrt{t}x$ in geodesic normal coordinates with the Clifford algebra filtration, assigning degree one to a derivative and to a Clifford generator and degree minus one to a coordinate. The associated graded Clifford algebra is the exterior algebra. This is useful because the spinor supertrace annihilates every Clifford monomial except the top-degree one. Applied to the Lichnerowicz formula for a twisted Dirac operator, it retains a quadratic curvature model, whose heat kernel is computed by a Gaussian ansatz; all terms of smaller rescaling degree disappear from the local index limit. Thus the cancellations in a supersymmetric heat trace can be computed before solving the complete curved heat equation.

On a closed oriented smooth four-manifold, an anti-self-dual connection on a line bundle has $c_1(L{)}^2[X]=-\Vert i{F}_A/(2\pi ){\Vert }_2^2\le 0$. A line bundle with positive square therefore admits no such connection for any metric of the chosen orientation. The splitting $L\oplus L^{-1}$ has $c_2=-c_1(L{)}^2$, giving the equivalent nonnegative-instanton-charge obstruction.

// Destination: geometry-and-topology.bigb

A smooth anti-self-dual connection with compact structure group on a punctured four-ball and finite $L^2$ curvature extends smoothly over the puncture after changing gauge and extending the bundle. The gauge on the punctured ball can carry nontrivial clutching degree; the extended bundle need not agree globally with the original one in a bubbling sequence.

// Destination: geometry-and-topology.bigb

On a closed oriented Riemannian four-manifold, a sequence of $SU(2)$ anti-self-dual connections of second Chern number one has, modulo gauge and subsequence, either smooth convergence with charge one or smooth convergence away from one point to a flat charge-zero connection. In the second case the curvature-energy measures converge to $8{\pi }^2{\delta }_x$. Small-energy gauges give convergence away from finitely many points; removable singularities extend the limit; Chern-Simons clutching integers quantize each positive defect. Total charge one leaves only the two stated cases. A flat limit need not have trivial holonomy on a nonsimply connected base.

// Destination: geometry-and-topology.bigb

The infinitesimal gauge and curvature complex is $0\to {\Omega }^0(\mathrm{ad}E)\stackrel{d_A}{\to }{\Omega }^1(\mathrm{ad}E)\stackrel{d_A^{+}}{\to }{\Omega }^{2,+}(\mathrm{ad}E)\to 0$. Its composite on $\xi$ is $[F_A^{+},\xi ]=0$. Its cohomology describes infinitesimal stabilizers, deformations and obstructions. The gauge-fixed operator is $D_A=d_A^{\ast }\oplus d_A^{+}$.

// Destination: geometry-and-topology.bigb

Work in an open subset of ${\mathbb{C}}^2$, with complex coordinates $w,z$, the Euclidean metric and its complex orientation. Use an imaginary-valued local potential for a U(1) connection. If its curvature form has no $(2,0)$ or $(0,2)$ part, the Dolbeault-Poincaré lemma gives $A=\partial u+\bar{\partial }v$. Unitarity permits $v=-\bar{u}$.

Write $u=p+iq$. A unitary bundle gauge transformation removes $idq$, leaving $A=\partial p-\bar{\partial }p$. With $f=-2p$ this isFor the chosen flat Riemannian metric, anti-self-duality of gauge curvature is exactlythe scalar Laplace equation. For example, $f=|w{|}^2-|z{|}^2$ gives the constant primitive curvature form $dw\wedge d\bar{w}-dz\wedge d\bar{z}$.

Alternatively, the complex gauge transformation $e^{-u}$ gives $A=\bar{\partial }(v-u)$. This gauge need not be unitary. The scalar harmonic equation above uses the specified flat metric; a general Riemannian metric changes the differential equation.

// Destination: complex-geometry.bigb

For constant isotropic transport coefficients and zero mean flow, a divergence-free Fourier mode has two circular polarizations that diagonalize curl with eigenvalues $\pm k$. The mean induction equation therefore has the displayed growth rates. Growth requires an admissible $k$ with $|\alpha |>(\eta +\beta )k$. Freely variable wavenumber gives $k_{\ast }=|\alpha |/[2(\eta +\beta )]$, provided it respects mean-field scale separation. A uniform field has zero curl and cannot grow by constant alpha alone; boundaries fix the longest available spatial mode.

For nonzero coupling ${ϵ}_1$, an alpha-quenching rotating wave can be made real in its toroidal amplitudes by a common phase rotation. Set $a={\alpha }_0-2\omega /\Omega$. The amplitude equations are $(a-p{B}_1^2)B_1+{ϵ}_1{B}_2=0$ and $(a-p{B}_2^2)B_2+{ϵ}_1{B}_1=0$. Subtracting after cross multiplication gives $(B_1^2-B_2^2)(p{B}_1{B}_2+{ϵ}_1)=0$. Unequal magnitudes therefore require $B_1{B}_2=-{ϵ}_1/p$ and $B_1^2+B_2^2=a/p$. They exist exactly when $a>2|{ϵ}_1|$ and have the displayed squared amplitudes. Exchanging hemispheres yields the companion solution. This breaks equatorial reflection symmetry and combines both dipole and quadrupole parity in a mean-field dynamo. Existence does not alone prove nonlinear stability.

Boundary-connected-summing Seifert surfaces gives a block-diagonal Seifert matrix, because basis curves in the separated summands have zero linking. The Seifert-matrix presentation of the Alexander module is block diagonal too. Thus the Alexander modules form a direct sum and their Alexander polynomials of a knot multiply up to a unit.