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If a càdlàg function has finite variation on compact intervals, its total-variation process is finite, nondecreasing, and càdlàg. Left limits follow from monotonicity. For right continuity, approximate the variation on $[t,s]$ by one finite partition. Changing its first endpoint from $t$ to $u\downarrow t$ changes the sum by at most $|X_u-X_t|$, which tends to zero. Additivity of variation then bounds $V(u)-V(t)$ by an arbitrarily small error. Its jump is the absolute jump of $X$. Without locally finite variation, right continuity can fail even as an extended-valued assertion: $X_t=t\sin (1/t)$ with $X_0=0$ has infinite variation on every interval starting at zero.

For a rotating conducting sphere, a geostrophic azimuthal velocity $U=rV$ couples to the cylinder-integrated magnetic stress $T=\int B_r{B}_{\varphi }dz$. Put $H=\int B_r^{2}dz$ and let $2z_b$ be the cylinder height. With an insulating boundary and no flow through it, cylindrical averaging removes the Coriolis force and gives $U_t=(2z_b{\mu }_0\rho r^2{)}^{-1}{\partial }_r(r^{2}T)$, $T_t=Hr{\partial }_r(U/r)$. The local squared wave speed is $H/(2z_b{\mu }_0\rho )$, the cylinder average of $B_r^2/({\mu }_0\rho )$.

For uniform-density incompressible flow with $u_{\varphi }=f(R,z+vt)+g(R,z-vt)$, $B_{\varphi }/\sqrt{{\mu }_0\rho }=f-g$ and $B_z/\sqrt{{\mu }_0\rho }=v(R)$, the azimuthal magnetohydrodynamic momentum equation and ideal magnetohydrodynamic induction equation hold automatically. Let $\Pi =p/\rho +\Phi +[(f-g{)}^2+v^2]/2$. The remaining equations require ${\Pi }_z=0$ and ${\Pi }_R=4fg/R$, so ${\partial }_z(fg)=0$ is necessary and locally sufficient. A single traveling profile is unrestricted; two nonzero profiles on an unrestricted axial domain must be opposite exponentials or independent of their traveling arguments. Bounded profiles of both signs therefore reduce to steady solutions.

For an elliptic curve $E:y^2=x^3+Ax+B$ in characteristic different from two and three, division polynomials are elements of its coordinate ring encoding multiplication and torsion points of an elliptic curve. The initial terms includeThe recurrences for higher $n$ yieldFor odd $n$, ${\psi }_n$ is a polynomial in $x$; for even $n$, it has a factor $2y$, and ${\psi }_n/(2y)$ is a polynomial in $x$. Thus the phrase “roots of the division polynomial” must specify whether it means points on the curve or roots in the $x$-line. When $n$ is prime to the characteristic, its zero points are exactly the nonidentity $n$-torsion points of an elliptic curve; for odd $n$, the two opposite points share an $x$-root.^[ref]

For the cyclic quotient surface singularity $\frac{1}{r}(1,q)$, expand $r/q=[b_1,\dots ,b_s{]}^{-}$ with $b_i\ge 2$. In the quotient lattice, start with $v_0=(0,1)$, $v_1=(1,q)/r$, and set $v_{i+1}=b_i{v}_i-v_{i-1}$. The last vector is $(1,0)$, each consecutive pair is a lattice basis, and subdivision by these rays is a resolution. Its exceptional chain has self-intersections $-b_i$. None is a minus-one curve, so the resolution is minimal. All coefficients are two exactly for $q=r-1$.

// Target: geometry-and-topology.bigb

First make a toric fan simplicial by compatible rational subdivisions. In a nonregular simplicial cone, choose a nonregular face minimal under inclusion. Its half-open fundamental parallelepiped contains a nonzero point of the toric lattice $w={\sum }_i{\alpha }_i{v}_i$, with every $0<{\alpha }_i<1$: a zero coefficient would put it in a regular proper face and force all coefficients to be integers. A star subdivision at the primitive point on this ray replaces a generator by $w$ and multiplies the affected lattice determinant by a number strictly between zero and one. The maximum multiplicity and the number of maximal cones realizing it decrease lexicographically, giving a finite toric resolution of singularities.

For this toric cone, one smooth fan subdivision has maximal cones $abc,acd$ and no exceptional divisors. Another has $abd,bce,cde,dbe$, with $e=a+c=(b+c+d)/2$. Its only exceptional divisor is a projective plane, since the quotient fan at $e$ has the three rays $\bar{b},\bar{c},\bar{d}$ with sum zero and unimodular neighboring cones. Both subdivisions admit no proper smooth coarsening. Their common smooth refinement is obtained from the first by adding $e$ in $ac$ and then $w=a+e$ in $ae$, and from the second by adding $w=b+d$ in $bd$. Each step is a toric blowup along an orbit closure.

For an ample invariant divisor $D$ and the blowup $\pi$ of a torus-fixed smooth point in dimension at least two, $c{\pi }^{\ast }D-E$ is ample for sufficiently large integers $c$. In the subdivided cone, its local characters are $c{m}_{\sigma }+u_i$, where $u_i$ is dual to the omitted ray generator. The new strict inequalities have gap one on the omitted ray and grow linearly with $c$ elsewhere, proving the claim by the toric ampleness criterion.

On the Hardy space of the circle, a continuous symbol gives a Fredholm Toeplitz operator exactly when it never vanishes. Nonvanishing gives a parametrix $T_{1/f}$ modulo compact semicommutators. Write $f(z)=z^n{e}^{h(z)}$, where $n$ is its counterclockwise winding number and $h$ is continuous. Homotopy through nonvanishing symbols and local constancy of the Fredholm index reduce the index to that of the shift $T_{z^n}$, namely $-n$. If $f$ vanishes at a point, normalized Hardy reproducing kernels concentrating there are weakly null unit vectors on which $T_f$ tends to zero, contradicting a Fredholm parametrix.

Split a smooth symbol into its strictly negative and strictly positive Fourier modes. Their Toeplitz operators $A,B$ exponentiate exactly to the Toeplitz operators of the corresponding exponentials, and $T(e^f)T(e^{-f})=e^A{e}^B{e}^{-A}{e}^{-B}$. The trace of $[A,B]$ is ${\sum }_{n>0}n{f}_n{f}_{-n}$ because the shift relation gives $\mathrm{Tr}[(S^{\ast }{)}^m,S^n]=n{\delta }_{mn}$. Smoothness makes the series converge in trace norm, so the Fredholm determinant of an exponential commutator applies.

In an $L$-round iterated prisoner's dilemma, Tit for tat against always defect gets $S$ once and $P$ thereafter; the defector gets $T$ once and $P$ thereafter. Their self-payoffs are $R,P$. At introduced frequency $ϵ$, subtracting mixed-population payoffs gives $ϵ[(R-P)-(T+S-2P)/L]-(P-S)/L$. The displayed positive threshold applies when its denominator is positive and the threshold is less than one. Taking $L\to \infty$ first removes this finite entry cost, so the horizon and rare-mutant limits cannot be casually interchanged.

An effective reproduction number describes transmission in the current population, accounting for available susceptibility and transmission conditions. The phrase is used with different time conventions, so a definition must specify whether it means an instantaneous reproduction number or a case reproduction number. The CDC uses a time-varying effective quantity $R_t$ to describe the direction of community transmission. ^[ref].

For the closed, homogeneous SIR model with constant $\beta ,\gamma$, the instantaneous convention givesThus the infectious compartment grows when $R_{\mathrm{eff}}>1$ and declines when $R_{\mathrm{eff}}<1$, provided $i>0$. At the SIR infection peak it equals one. This freezes the current susceptible fraction; it does not count a newly infected person's actual future secondary infections while $s$ continues to decrease. The distinction comes directly from the SIR model equations and the two time conventions above.

Use radius-deficit $ϵ$, $F^{\ast }=F/(6\pi \mu aU)$ and $\delta =3a/(4L)$. The control-volume drag formula for a sphere in a tube and the radius-deficit normalization of sphere-tube lubrication integrals give three leading resistances: $c_3{ϵ}^{-5/2}$ for $\delta \ll {ϵ}^{5/2}$, $1/\delta$ for ${ϵ}^{5/2}\ll \delta \ll {ϵ}^{1/2}$, and $2\pi \sqrt{2}/(3\sqrt{ϵ})$ for $\delta \gg {ϵ}^{1/2}$. They correspond to gap bypass with almost no tube throughflow, pressure resistance of tube throughflow, and local annular resistance, respectively.

In a two-dimensional isotropic harmonic trap with repulsive effective coupling $g>0$, neglecting density-gradient kinetic energy gives the displayed parabolic number density. Normalization to $N$ particles yields $\mu =\sqrt{Ngm{\omega }^2/\pi }$, radius $R^2=2\mu /(m{\omega }^2)$ and central density $n_c=\mu /g$. The bulk approximation requires $\mu \gg \hslash \omega$, equivalently $mgN/(\pi {\hslash }^2)\gg 1$. It fails near the outer edge and inside quantum vortex cores. The coupling must have two-dimensional units; an unreduced three-dimensional contact coupling cannot be inserted without a normalization convention.

For a miscible two-component Bose-Einstein condensate with $V_i=A_i{r}^2$, $D=U_{11}{U}_{22}-U_{12}^2>0$, invert the local interaction matrix. The displayed result and $n_2=[U_{11}({\mu }_2-A_2{r}^2)-U_{12}({\mu }_1-A_1{r}^2)]/D$ hold in the overlap region. Factoring by $R_i^2={\mu }_i/A_i$ puts the ratio $\lambda =A_2/A_1$ in the first profile and ${\lambda }^{-1}$ in the second. Outside overlap, a negative formal density must be replaced by the appropriate single-component solution, not retained as a physical density.

Choose an effective Cartier divisor $E$ of degree at least $g$ on a smooth projective genus-$g$ algebraic curve. For a family $M$ of degree-$g-1$ line bundles, the evaluation map from ${\pi }_{\ast }M(E)$ to ${\pi }_{\ast }(M(E){|}_E)$ is between vector bundles of the same rank. Its determinant vanishes exactly when $H^0(C,M)$ is nonzero. Changing $E$ changes the presentation by cancelling an invertible determinant factor; the zero Cartier divisor is the theta divisor. At a point with $h^0(M)=h^1(M)=1$, the derivative is the cup-product map $H^1({\mathcal{O}}_C)\to \mathrm{Hom}(H^0(M),H^1(M))$. By Serre duality it is nonzero, since the product of nonzero sections of $M$ and $K_C\otimes M^{-1}$ is nonzero. Hence this Cartier divisor is generically reduced.

For a degree-$g$ divisor class $D$ with $h^0({\mathcal{O}}_C(D))=1$, let $D_0$ be its unique effective representative. Under ${\varphi }_D(P)=[D-P]$, evaluation of the unique global section gives ${\varphi }_D^{\ast }\Theta =D_0$, including multiplicities. If $h^0({\mathcal{O}}_C(D))\ge 2$, every evaluation kernel is nonzero and the entire curve lies in the theta divisor. These restrictions also show that $h^0(\mathcal{O}(\Theta ))=1$: after pulling back to a dominant product of curves, the ratio of any section to the defining section of the theta divisor is constant on each general coordinate curve, hence constant everywhere.

For $T_q(x)=x+q$, the homomorphism associated to a line bundle on an abelian variety gives ${\lambda }_{\Theta }(q)=T_q^{\ast }\mathcal{O}(\Theta )\otimes \mathcal{O}(\Theta {)}^{-1}$. Restrict to the negative Abel map of an algebraic curve. The restriction of the theta divisor to an Abel curve identifies this restriction with the degree-zero class $q$. Pullback of algebraically trivial line bundles does not change when the embedding is translated. Thus pullback $\rho :\widehat{J(C)}\to J(C)$ satisfies $\rho {\lambda }_{\Theta }=1$. Equal dimensions and properness make ${\lambda }_{\Theta }$ an isomorphism, including on schemes, rather than only on closed points.

For a compact Riemann surface of genus $g\ge 1$, the theta divisor in ${\mathrm{Pic}}^{g-1}(X)$ has the displayed multiplicity. A local determinant description of the theta divisor reduces to an $r$ by $r$ matrix vanishing at $L$, where $r=h^0(L)$. Its linear term is cup product with a tangent class in $H^1({\mathcal{O}}_X)$. The invertible cup-product direction for a line bundle on a curve shows that its determinant has order exactly $r$. Consequently the singular locus consists precisely of classes with at least two independent global sections.

Translate the tangent hyperplane at a smooth point of the theta divisor to the identity of the Jacobian variety. Its annihilator defines the Gauss map. At $L={\mathcal{O}}_X(D)$ with $h^0(L)=1$, the derivative of the Abelian sum map identifies this annihilator with the unique line $H^0(K_X(-D))$. Thus $\Gamma (L)=[\omega ]$ implies $D\le \mathrm{div}(\omega )$. This connects the intrinsic geometry of the theta divisor to the canonical map of the curve.