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.

For the period matrix of a genus-$g$ compact Riemann surface, there is a vector $\kappa$ depending on the marking and base point such that the Riemann theta function vanishes exactly on this translate of the image of the Abelian sum map. Pairing sides of the cut surface proves that a nonzero pullback $\theta (u(p)-z)$ has a degree-$g$ zero divisor whose Abel sum is $z+\kappa$. At a theta zero that pullback includes the base point. Irreducibility and dimension of the effective-divisor locus give equality of supports; the primitive First Chern class of the theta line bundle makes the divisor reduced.

For a finite stationary axisymmetric self-gravitating system with similar oblate density surfaces, let $q=b/a$, $s=\sqrt{1-q^2}$, $A_1=q\arccos (q)/s^3-q^2/s^2$, and $A_3=2/s^2-2q\arccos (q)/s^3$. Its stellar potential-energy tensor has $Q=A_1/(q^2{A}_3)$; the common radial integral cancels. The spherical limit has $A_1=A_3=2/3$ and $Q=1$. With mass-weighted in-plane velocity dispersion ${\sigma }_x$ and constant azimuthal streaming $v$, the tensor virial theorem gives $v^2/{\sigma }_x^2=2[(1-\beta )Q-1]$, where ${\sigma }_z^2=(1-\beta ){\sigma }_x^2$. Neither $Q=1/q$ nor the omission of the Cartesian streaming factor is exact.

Use the position second-moment tensor $I_{ij}=\int \rho x_i{x}_j{d}^{3}x$, not the engineering moment-of-inertia convention. If a density pattern rotates steadily about its third principal axis, $I(t)=R(\Omega t)I(0)R(\Omega t{)}^T$. Differentiating twice at a principal-axis instant gives the displayed tensor. The tensor virial theorem then determines the pattern speed from the difference of its first two diagonal components. This density-pattern speed is distinct from stellar mean streaming; in the axisymmetric limit $I_{11}=I_{22}$, this difference cannot determine it.

For $V={\mathbb{C}}^n$, $n\ge 3$, the tensor product of Lie algebra representations for the special linear Lie algebra splits as displayed. The exterior product map $v\otimes (u\wedge w)\mapsto v\wedge u\wedge w$ has an equivariant right inverse obtained by dividing its three-term alternating insertion by three. Its kernel is the Schur functor of shape $(2,1)$, with highest weight ${\omega }_1+{\omega }_2$ and dimension $n(n^2-1)/3$. The other summand has highest weight ${\omega }_3$ for $n\ge 4$ and is trivial for $n=3$.

The weight space of $2{\epsilon }_i+{\epsilon }_j$ has dimension one when $i\ne j$. The weight space of ${\epsilon }_i+{\epsilon }_j+{\epsilon }_k$ has dimension three for distinct indices. Restricting these functionals to the traceless diagonal Cartan subalgebra creates no additional coincidences; for $n=3$ the latter weight is zero.

Give $\mathfrak{g}$ its Adjoint representation. The representation identity gives $a(y\cdot (x\otimes v))=[y,x]v+xyv=y(xv)$, so the action map is a Lie algebra representation homomorphism. For a nontrivial simple module it is surjective. If the acting algebra is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and the module is finite dimensional, Weyl complete reducibility theorem splits the surjection and makes $V$ a direct summand.

For the defining two-dimensional sl2 Lie algebra module with coordinates $x,y$, put $u_{ij}=x^{3-i}{y}^i\otimes x^{2-j}{y}^j$. Its three highest-weight vectors are $u_{00}$, $u_{10}-u_{01}$ and $u_{20}-2u_{11}+u_{02}$, of weights five, three and one. Lowering the last once gives $u_{30}-2u_{21}+u_{12}$, an explicit second vector of its two-dimensional irreducible summand.

The summands have Dynkin labels $(2,1)$, $(0,2)$ and $(1,0)$. Convolving the triplet and octet weight diagrams and applying highest-weight character subtraction removes these three characters in that order. Their dimensions $15+6+3=24$ match the product.

// Target: quantum-field-theory.bigb

A closed symplectic manifold of real dimension four cannot be a connected sum of two oriented smooth four-manifolds each with positive positive index of the intersection form. The total $b_2^{+}$ exceeds one; Taubes nonvanishing theorem and connected-sum vanishing of the Seiberg–Witten invariant would force the same invariant to be both nonzero and zero.

Choose positions $z_a$ and positive integer multiplicities $m_a$. Solve the Taubes equation with $h\to 0$ at infinity and $h=2m_a\log |z-z_a|+O(1)$ near each position. The planar existence theorem supplies such a solution for every finite multiset of positions; see the original existence theorem. Set $\chi ={\sum }_a{m}_a\arg (z-z_a)$, $\Phi =e^{h/2+i\chi }$ and $A=\nabla \chi -J\nabla h/2$, with $J(v_1,v_2)=(-v_2,v_1)$. The singular terms cancel in $A$, while $\Phi$ has precisely the selected zeros. These fields satisfy the Bogomolny vortex equations, with $B=(1-|\Phi {|}^2)/2$. Their vortex number is ${\sum }_a{m}_a$. Complex conjugation of $\Phi$ together with $A\mapsto -A$ gives antivortices.

At parameter six, changing a boundary target preserves the initial unparameterized SLE law until the growing hull separates the two possible targets. After separation, each curve continues in the complementary component whose boundary contains its own target. A curve targeted at infinity follows the unbounded component, whereas a finite separated target lies on a different component. The target-change locality of SLE6 proof no longer applies: its conformal automorphism of the upper half-plane conjugates corresponding surviving domains only before this separation. A zero drift before contact does not give equality of complete curve laws afterwards.

For $n\ge 2$, the defining Lie algebra representation $V$ of ${\mathfrak{sp}}_{2n}$ has the displayed tensor product decomposition. The crystal of the defining symplectic representation and the crystal tensor-product rule give exactly these three highest vertices, each occurring once. The symmetric square is $L(2{\omega }_1)$ of dimension $n(2n+1)$. The exterior square splits into its invariant inverse-form line and the primitive exterior square $L({\omega }_2)$ of dimension $n(2n-1)-1$. Their dimensions sum to $4n^2$. At $n=1$, the exterior square is only the invariant line, giving $L(2{\omega }_1)\oplus L(0)$.