The local ring of a complex manifold is the unique factorization domain of convergent power series in local coordinates. A reduced local defining function of a complex analytic hypersurface has each local irreducible factor once. The analytic zero-set ideal is the radical of its principal ideal, and reducedness makes that principal ideal radical. Thus two reduced equations for the same germ generate the same ideal and differ by a unit. A globally irreducible hypersurface may have several local branches; reducedness includes each once. Without reducedness, $f$ and $f^2$ have the same zero set and need not differ by a unit.

In positive characteristic $p$, $(S^{\lambda }{)}^{S_n}$ is nonzero exactly when ${\lambda }_i\equiv -1(\mathrm{mod}{p}^{t_i})$ for every adjacent pair, where $t_i$ is least with ${\lambda }_{i+1}<p^{t_i}$. The only invariant line in $M^{\lambda }$ is generated by the sum $z_{\lambda }$ of all tabloids. The map ${\psi }_{i,v}$ sends it to $\left(\genfrac{}{}{0ex}{}{{\lambda }_i+{\lambda }_{i+1}-v}{{\lambda }_i}\right)z_{\lambda (i,v)}$. Thus the kernel intersection theorem for Specht modules gives vanishing of $\left(\genfrac{}{}{0ex}{}{{\lambda }_i+j}{j}\right)$ modulo $p$ for $1\le j\le {\lambda }_{i+1}$. By Lucas's theorem, all these binomial coefficients vanish exactly when the last $t_i$ base-$p$ digits of ${\lambda }_i$ are $p-1$.

On a nonempty compact Kähler manifold of complex dimension $n$, the Kähler form is closed and ${\int }_X{\omega }^n>0$. If ${\omega }^k=d\eta$ for $1\le k\le n$, then ${\omega }^n=d(\eta \wedge {\omega }^{n-k})$; Stokes theorem contradicts the positive integral. Thus the de Rham cohomology is nonzero in every even degree through $2n$.

Coverings surviving all local solubility tests need not have rational points. The quotient of an isogeny Selmer group by the rational-point image is a kernel in the Tate–Shafarevich group. A nonzero class therefore passes every local test but contributes no rational point to a search. This explains why two-isogeny descent can leave a gap between a certified upper bound and found lower bound for the rank of an elliptic curve. Higher descents address finer obstructions; see Tom Fisher's paper on higher descents with rational two-torsion.

A simple island is an island of a meromorphic function of mapping degree one. It is mapped biholomorphically onto the target Jordan domain. This is stronger than the source island being simply connected.

For an irreducible root system spanning $V$, the span $U$ of a nonzero root's Weyl group orbit is invariant, as is $U^{\perp }$. A root reflection preserving both subspaces must have its normal entirely in one of them: otherwise its rank-one correction mixes the subspaces. Thus a proper nonzero $U$ would partition all roots into mutually orthogonal subsets, contradicting irreducibility. Consequently some orbit root is nonorthogonal to any given root. Since Weyl reflections preserve lengths, the pairwise integrality restrictions for nonorthogonal roots also bound length ratios of orthogonal roots. All squared-length ratios lie in $\{1,2,1/2,3,1/3\}$; there cannot be three distinct lengths, since ratios two and three relative to a shortest root would yield the forbidden ratio $3/2$.

For binary predictors $A,B$, a logistic regression with linear predictor $a+bA+cB+\delta AB$ has conditional odds ratios $e^b$ at $B=0$ and $e^{b+\delta }$ at $B=1$. Their ratio is $e^{\delta }$. The same interaction can be computed from the four cell logit link values as $\delta ={\eta }_{11}-{\eta }_{10}-{\eta }_{01}+{\eta }_{00}$. Independent positive event and nonevent counts in each cell give an estimated variance for this contrast equal to the sum of all eight reciprocal counts. An interaction on the odds ratio scale is not an interaction on the absolute-risk scale.

For $f_a(x)=af(ax)$ with a positive, differentiation and change of variables give $R(f_a^{\prime \prime })=a^{5}R(f^{\prime \prime })$, while its standard deviation is $\sigma (f)/a$. Their product D is therefore invariant under scale changes, and translation also leaves it unchanged. Raw integrated squared curvature can be arbitrarily small under dilation even though this dimensionless shape measure is unchanged.

Positive association of random variables and translation invariance give $C_{m+n}\ge C_m{C}_n$ for axis connection probabilities. The Fekete lemma applied to $-\log C_n$ proves the displayed limit of a sequence and the bound $C_n\le e^{-n\alpha }$. The lower single-edge conditional bound gives $C_n\ge [p/(p+q(1-p)){]}^n$, so the decay rate is finite and nonnegative. It may be zero.

If the molar latent heat $L_m$ is approximately constant on the chosen temperature interval, the ideal-vapor Clausius-Clapeyron approximation integrates toFor $L_m>0$, higher absolute temperature gives higher saturation pressure. A plot of $\ln (p/p_1)$ against inverse temperature $1/T$ is a straight line of slope $-L_m/R$. One reference pair $(T_1,p_1)$ fixes its intercept. On a per-unit-mass basis replace $L_m/R$ by $L/R_s$, with $R_s=R/M$ and $M$ the molar mass; using $L$ in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$ with the molar $R$ would give incompatible units.

If only the constant-latent heat approximation is relaxed, the same ideal-vapor model instead givesConsequently an apparent straight-line fit across a broad temperature range need not represent a single material constant. The simple exponential also approaches a finite value as $T\to \infty$; this is a property of the stipulated formula, not a prediction that liquid–gas phase coexistence extends beyond a thermodynamic critical point.

The endpoint Laplace method for a Legendre polynomial ceases to localize when $n\sqrt{x-1}=O(1)$. On the scale $x=1+\nu /n^2$, its real integral representation givesThe Modified Bessel function of the first kind equals one at $\nu =0$ and has large-$\nu$ behavior $e^{\sqrt{2\nu }}/\sqrt{2\pi \sqrt{2\nu }}$, matching the Hyperbolic Debye asymptotic for Legendre polynomials. This boundary layer resolves an increasingly steep function at an endpoint where its value remains fixed.

For $n\ge 3$, the least ordinary irreducible dimension exceeding one is $n-1$, except at $n=4$, where $S^{(2,2)}$ has dimension two. The dimension-$n-1$ shapes are $(n-1,1)$ and its conjugate partition, with the additional shapes $(3,3)$ and $(2,2,2)$ at $n=6$. For $n=2$ there is no non-linear irreducible. A proof uses the restriction branching rule for a symmetric group: a nonstandard shape with at least two corners has two non-linear predecessors, and a rectangular shape has one predecessor which in turn has two non-linear predecessors. This bounds its degree strictly above $n-1$ once $n\ge 9$; the smaller cases follow directly from the hook-length formula.

Closed holomorphic one-forms inject into de Rham cohomology: an exact one would have a global holomorphic primitive and hence vanish. If a holomorphic class equals an antiholomorphic class, their difference is $df$, and $\partial \bar{\partial }f=0$; compact pluriharmonic rigidity makes both forms vanish. Their direct sum gives the lower bound. The sheaf of closed holomorphic one-forms exact sequence gives $0\to H^0({\Omega }^1)\to H^1(\mathbb{C})\to H^1(\mathcal{O})$, yielding the upper bound and $h^{1,0}\le h^{0,1}$. These inequalities do not require a Kähler manifold.

For a complex submanifold $Y\subset X$, the holomorphic conormal bundle is the annihilator of $T^{1,0}Y$ inside ${\Omega }_X^1{|}_Y$, or equivalently the dual of the holomorphic normal bundle. For a smooth analytic hypersurface with local equation $f$, it is generated by $df{|}_Y$. It is also the locally free analytic sheaf ${\mathcal{I}}_Y/{\mathcal{I}}_Y^2$.

For $\Phi (\rho )=p\rho +(1-p)I/2$, every output qubit has Von Neumann entropy at least $h((1-p)/2)$. A product input therefore has output entropy at least $nh((1-p)/2)$. The average output has entropy at most $n$, so its Holevo quantity is bounded by the displayed expression. This allows correlated classical message labels and joint output measurements, but requires product quantum inputs. The Holevo bound and Fano's inequality turn it into the corresponding asymptotic rate bound.

For a nonsingular Short Weierstrass form $y^2=x^3+ax+b$ over a number field, the unordered pair of coordinates $x(P+Q),x(P-Q)$ depends only on $x(P),x(Q)$. As a binary quadratic's roots, that pair is represented in ${\mathbb{P}}^2$ by the coefficientswith polynomial $C{T}^2-BT+A$. Homogenizing gives three forms of bidegree $(2,2)$ on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. They have no common zero. Off the diagonal $C\ne 0$; on the finite diagonal $x_1=x_2=t$, $B=4f(t)$ and $A=f^{\prime }(t{)}^2-8tf(t)$ for $f(t)=t^3+at+b$, so simultaneous vanishing would make a multiple root of $f$. At $(\infty ,\infty )$, the homogeneous $A$ is nonzero. Therefore they define a global morphism, also when a sum or difference is the identity.

The biquadratic morphism for elliptic sums and differences has no common zero, so the local coefficient norms giveuniformly in $P,Q$. Upper bounds use the fixed coefficients. Lower bounds follow from the absence of common zeros over every local field; the constants are $1$ outside finitely many places, where the integral morphism still has no base point after reduction.

For the binary quadratic with roots $u,v\in {\mathbb{P}}^1(K)$, the projective height of its coefficient vector is $h(u)+h(v)+O(1)$. At nonarchimedean places the coefficient maximum norm of a product of linear polynomials is the product of their maximum norms by the multiplicativity of the nonarchimedean polynomial norm. At archimedean places the same norms are comparable up to a fixed factor for degree two. This also includes roots at infinity. Combining the comparisons yields $h_x(P+Q)+h_x(P-Q)=2h_x(P)+2h_x(Q)+O(1)$. After dividing the identity for $2^{n}P,2^{n}Q$ by $4^n$ and taking limits, it gives the exact height parallelogram identity for the canonical height of an elliptic curve.

At most two values can have counting order smaller than a positive finite function order. Otherwise the Nevanlinna second main theorem with three such values bounds the characteristic by a smaller power outside a finite-length exceptional set. Monotonicity extends this bound to all large radii by choosing a good radius in $[r,r+1]$, contradicting the original order. Order zero is immediate from nonnegativity and the first theorem.

The first component is a proper partition marking an initial segment of each row of the second component. The second component may be a composition rather than a decreasing partition. This convention specifies a generalized Specht module and differs from specifying a skew shape by two nested proper partitions.

The full bracket of a zero-framed Hopf link colored i and j is the displayed pairing. Its normalized matrix is the finite sine transform with parity signs. Sine orthogonality gives $H^2={\mathcal{D}}^{2}I$, where ${\mathcal{D}}^2={\sum }_j{d}_j^2=r/(2{\sin }^2(\pi /r))$. Thus the pairing is nondegenerate in the reduced theory.