Let $L$ be positive and $V$ a closed smooth hypersurface in a complex torus. The semipositive canonical bundle of a submanifold of a complex torus makes $P_a=(L\otimes \mathcal{O}(aV)){|}_V$ positive. Since $K_V=\mathcal{O}(V){|}_V$, the restriction of the $a$-th ambient twist is $K_V\otimes P_{a-1}$ when $a\ge 1$. Apply the Kodaira vanishing theorem on $V$, and use $0\to L((a-1)V)\to L(aV)\to L(aV){|}_V\to 0$ and its long exact sequence to induct from the positive bundle $L$ on the torus. No unproved ambient positivity of $\mathcal{O}(V)$ is needed.

With Hermitian angular-momentum generators $[J_a,J_b]=i{ϵ}_{abc}{J}_c$, the Killing form is $2{\delta }_{ab}$ and its inverse contraction is $C_{\kappa }=\frac{1}{2}{\sum }_a{J}_a^2$. Equivalently the anti-Hermitian basis $T_a=-i{J}_a$ has Killing form $-2{\delta }_{ab}$ and gives the same operator. In the complex highest-weight basis, $C_{\kappa }=H^2/8+(E^{+}{E}^{-}+E^{-}{E}^{+})/4$. It acts on spin $j$ by $j(j+1)/2$, so it differs in normalization from the usual angular-momentum quadratic Casimir operator.

Within the ideal-vapor Clausius-Clapeyron approximation, a local slope of the natural logarithm of saturation pressure against inverse temperature estimates the molar latent heat:In the stipulated vapor-pressure calculation, the slope is $-4811.16\mathrm{K}$, which recovers $40.0\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. For variable $L_m(T)$, a two-point estimate gives an average weighted by $T^{-2}$, rather than the latent heat at an arbitrary midpoint. Differentiating noisy pressure measurements amplifies their errors. Outside the ideal-vapor regime, use independently determined molar volumes in $L_m=T\Delta v_{m}dp/dT$; a straight-looking graph alone cannot establish that the gas is ideal.

Consider an unnamed model substance with molar latent heat $L_m=40.0\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and saturation pressure $p_1=10.0\mathrm{kPa}$ at $T_1=300\mathrm{K}$. Stipulate the ideal-vapor Clausius-Clapeyron approximation throughout $280\le T\le 340\mathrm{K}$, constant $L_m$, and $R=8.314\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. These parameters define an illustrative model, not a measured dataset or a fit to water. At $T_2=330\mathrm{K}$,At the reference state the gas molar volume is $R{T}_1/p_1=0.24942{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, and the local slope is $dp/dT\simeq 0.5346\mathrm{kPa}{\mathrm{K}}^{-1}$. The tangent estimate $p_2\approx 10.0+30(0.5346)=26.04\mathrm{kPa}$ is much lower than the integrated result: a local derivative should not be extrapolated over this whole interval. By contrast, over a small temperature increment, the fractional sensitivity is $d\ln (p/p_1)/dT=L_m/(R{T}_1^2)\simeq 0.05346{\mathrm{K}}^{-1}$.

For a closed holomorphic one-form $\sum b_{k}d{z}_k$ on a polydisc, the displayed coordinate-path integral is a holomorphic function. Differentiating in $z_j$ gives the $k=j$ endpoint term; for $k>j$ use ${\partial }_j{b}_k={\partial }_k{b}_j$ and integrate that derivative. The resulting sum telescopes to $b_j(z)$. Thus $\partial F=\sum b_{k}d{z}_k$, giving an explicit degree-one proof of the holomorphic Poincaré lemma.

If $f_i$ and $g_i$ are local meromorphic equations for two divisors on a complex manifold, then $f_i{g}_i$ defines their sum. With local frames satisfying $e_j=(f_i/f_j)e_i$, the transition functions of a vector bundle for the tensor product are the products of the two ratios, exactly those for $f_i{g}_i$. Multiplication of local frames therefore gives the displayed holomorphic line bundle isomorphism.

For reduced local equations $f_j=u_{ji}{f}_i$ with $u_{ji}$ a holomorphic unit, the holomorphic line bundle associated to a divisor $[-Y]$ has local generators $f_i$ and transition $u_{ji}$. Differentiation along $Y$ gives $d{f}_j{|}_Y=u_{ji}{|}_{Y}d{f}_i{|}_Y$. Sending the restricted local generator to $d{f}_i{|}_Y$ gives the conormal isomorphism. The term $f_{i}d{u}_{ji}$ disappears only after restricting to $Y$.

For a root reflection on the weight lattice, $s_{\alpha }\lambda =\lambda -m\alpha$ with $m=\lambda (H_{\alpha })\in \mathbb{Z}$. A finite anti-invariant element of the group ring of a weight lattice is a sum of terms $a_{\lambda }[e(\lambda )-e(s_{\alpha }\lambda )]$; fixed weights have zero coefficient. Each pair is divisible by $1-e(\alpha )$, because $1-t^m$ is divisible by $1-t$ in $\mathbb{Z}[t,t^{-1}]$ for every integer $m$. Thus the quotient has finite support.

The formal expression $({\sum }_{n\ge 0}e(n\alpha ))u$ is interpreted coefficientwise. Along each coset of $\mathbb{Z}\alpha$, only finitely many coefficients of $u$ are nonzero and their total is zero by reflection anti-invariance. Partial coefficient sums therefore vanish before and after that finite interval. The result has finite support, even though the individual summands $e(n\alpha )u$ do not eventually vanish.

// Target: lie-algebra.bigb

A good reduction in characteristic $p$ is ordinary when the geometric $p$-torsion has $p$ points, equivalently its formal group of an elliptic curve has height one. For a CM elliptic curve, a good prime split in its CM field is ordinary. One of the two roots of $X^2-a_{p}X+p$ is a $p$-adic unit, enabling the bounded ordinary p-adic L-function.

A good reduction in characteristic $p$ is supersingular when its geometric $p$-torsion is trivial, equivalently its formal group of an elliptic curve has height two. Good inert primes of a CM elliptic curve are supersingular. The ordinary bounded-measure and torsion-Selmer formulation must then be modified, for instance using signed local conditions in the cyclotomic tower.

An integral Weierstrass equation of an elliptic curve with unit elliptic-curve discriminant defines a smooth proper elliptic curve over the valuation ring. Properness identifies points over the local field with integral sections. The group law, constructed by adding degree-zero divisor classes, is a morphism over the valuation ring, and base change to the residue field preserves it. Hence reducing primitive projective coordinates is a group homomorphism, including exceptional affine addition cases.

Normalize the Fubini-Study form so that a complex projective line has symplectic area $\pi$. The displayed map takes the unit symplectic ball in ${\mathbb{C}}^n$ diffeomorphically onto the complement of the last-coordinate hyperplane. Its lift to the unit sphere has last coordinate positive real, so that coordinate contributes zero to the pulled-back standard symplectic form. The map is therefore a symplectomorphism.

On a compact Hermitian manifold without boundary, this operator is the inverse of the Dolbeault Laplacian on the orthogonal complement of its harmonic forms and is zero on the harmonic space. Elliptic theory gives ${\Delta }_{\bar{\partial }}G=G{\Delta }_{\bar{\partial }}=I-H$, with $H$ the harmonic projection, and ensures that $G$ sends smooth forms to smooth forms. It commutes with $\bar{\partial }$ and ${\bar{\partial }}^{\ast }$; their commutation with the Laplacian and orthogonality to its kernel give this by uniqueness of the inverse. If $\eta$ is $\bar{\partial }$-exact, then $\eta =\bar{\partial }{\bar{\partial }}^{\ast }G\eta$. On a Kähler manifold, the same operator serves the holomorphic Laplacian because ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$.

In frames with $e^{(\beta )}=g_{\alpha \beta }{e}^{(\alpha )}$ and input-index connection matrices, the Chern connection has ${\theta }_{\beta }-g_{\alpha \beta }{\theta }_{\alpha }{g}_{\alpha \beta }^{-1}={\sigma }_{\alpha \beta }$. These holomorphic one-forms, interpreted in the beta frame, satisfy the transported Čech cocycle condition because $g_{\alpha \gamma }=g_{\beta \gamma }{g}_{\alpha \beta }$. Local primitives of curvature differ by this cocycle. A holomorphic frame change $e_{\alpha }^{\prime }=u_{\alpha }{e}_{\alpha }$ adds the Čech coboundary of $u_{\alpha }^{-1}\partial u_{\alpha }$, so the resulting curvature Dolbeault class is intrinsic.

For odd nonzero $D$, the intrinsic bad primes of $y^2=x^3-D^{2}x$ are exactly the displayed set. If $v_p(D)=2k$, the change $x=p^{2k}X$, $y=p^{3k}Y$ removes that even valuation from the parameter and gives unit discriminant at odd $p$. If the valuation is odd, after the same removal the discriminant valuation is six. Discriminant valuations of isomorphic Weierstrass models differ by multiples of twelve, so no integral model has unit discriminant. At two the original discriminant valuation is also six. Thus primes dividing a nonminimal displayed discriminant need not be intrinsic bad primes: $E_9$ is isomorphic to $E_1$ and has good reduction at three.

Discriminants of isomorphic Weierstrass equations of an elliptic curve differ by a twelfth power of the scale parameter, so their valuations differ by a multiple of twelve. An integral model with good reduction has discriminant valuation zero. Thus a displayed integral model with discriminant valuation not divisible by twelve proves intrinsic bad reduction. More generally, a displayed discriminant valuation between zero and eleven is already minimal, since any smaller integral-model valuation would be nonnegative and differ by twelve. Positive valuations divisible by twelve alone do not establish bad reduction; rescaling may remove them.

Let Roche-lobe overflow transfer mass conservatively within a circular binary while a separate fast wind escapes with the donor's specific orbital angular momentum. Then ${\dot{M}}_d=\dot{M}-{\dot{M}}_a$ and $\dot{a}/a=-\dot{M}/M+2{\dot{M}}_a(1/M_d-1/M_a)$. Differentiating the donor and Roche lobe radii and enforcing contact gives the displayed rate. It is positive and stabilizing when the wind drives overfill and $6q<5-3n$. When the latter inequality is reversed, transfer increases the overfill, producing runaway rather than a steady negative-transfer solution.

A weight basis of a Lie algebra representation is a basis consisting of weight vectors for a chosen Cartan subalgebra. When the Cartan action is diagonalizable, choose a basis separately in every weight space and combine them. This makes weight multiplicities directly visible and gives diagonal matrices for Cartan elements.

Multiply the Weyl character formula by the generating series of the Kostant partition function. In each numerator summand, the coefficient of $e^{\mu }$ occurs when $\nu =w(\lambda +\rho )-(\mu +\rho )$. Summing those coefficients with the Weyl group signs proves the displayed weight multiplicity formula. Although $\rho$ need not itself be an integral group weight, $w\rho -\rho$ is in the root lattice, so each partition-function argument is an integral weight.

The group ring of a weight lattice $P$ consists of finite sums ${\sum }_{\lambda }{a}_{\lambda }e(\lambda )$ with integer coefficients, where $e(\lambda )e(\mu )=e(\lambda +\mu )$. A finite-dimensional Lie algebra representation has formal character ${\sum }_{\lambda }(\dim V_{\lambda })e(\lambda )$ in this ring. The Weyl group acts by $e(\lambda )\mapsto e(w\lambda )$. The ambient ring need not consist of genuine characters; the representation ring of a semisimple Lie algebra of finite-dimensional modules for a complex semisimple Lie algebra identifies with the invariant subring $\mathbb{Z}[P{]}^W$.

// Target: semisimple-lie-algebra.bigb