For a scalar Landau-de Gennes free energy with $\bar{b}>0$, $\bar{c}<0$, and coupling $-\chi E^2\lambda$ with $\chi >0$, the critical endpoint of a quartic Landau free energy is reached at the displayed field strength and ${\bar{a}}_c=3{\bar{c}}^2/(8\bar{b})$, ${\lambda }_c=-\bar{c}/(4\bar{b})$. It terminates the field-biased first-order phase transition when both field and quadratic coefficient can be tuned. This is a quartic small-field mean-field approximation; higher terms may alter its location.

On a polydisc, a $\partial$-closed differential form of type (p, q) with $p>0$ is $\partial$-exact. Apply the Dolbeault-Poincaré lemma to its complex conjugate, whose antiholomorphic degree is $p$.

For a complex manifold, Bott-Chern cohomology isIts classes are represented by $d$-closed forms of pure type. A natural map sends a Bott-Chern class to its Dolbeault cohomology class.

Let $E/K$ have good reduction over a local field of residue characteristic $p$, with $p\nmid n$. All solutions of $nQ=P$, for all $P\in E(K)$ together, lie in a single finite unramified extension of degree at most $nf$, where $f=[K(E[n]):K]$. First lift a solution over the algebraic closure of the residue field to a finite unramified extension; the multiplication isomorphism of a formal group law uniquely corrects the error in $E_1$. This also puts $E[n]$ in the maximal unramified extension. If $\sigma$ is its Frobenius generator, ${\sigma }^f$ fixes $E[n]$. For $nQ=P$, set $T={\sigma }^{f}Q-Q\in E[n]$. Then ${\sigma }^{nf}Q=Q+nT=Q$, uniformly in $P$.

For an isogeny of elliptic curves $\varphi :E_1\to E_2$, write $x_2\circ \varphi =A(x_1)/B(x_1)$ with coprime polynomials $A,B$. Each $x_i:E_i\to {\mathbb{P}}^1$ is a two-to-one finite morphism, and function field degrees multiply in a tower. Consequently $2\mathrm{deg}\varphi =2\max (\mathrm{deg}A,\mathrm{deg}B)$. This computes the total degree, including an inseparable factor. The zero homomorphism is assigned degree zero separately.

On the product of an elliptic curve with itself, let $D_{+}$ and $D_{-}$ be the diagonal and the graph of negation. The principal divisor of $x(P)-x(Q)$ is $D_{+}+D_{-}-2(\{O\}\times E)-2(E\times \{O\})$: equality of the coordinates means $Q=\pm P$, while $x$ has a double pole at $O$. Pulling back along $(\varphi ,\psi )$ and taking degrees proves the parallelogram law for the degree of an isogeny when $\varphi ,\psi$ are nonzero and $\varphi \ne \pm \psi$. The exceptional cases follow from $\mathrm{deg}[2]=4$. The same argument uses the quotient by negation in characteristic two. Thus degree, with degree zero assigned to the zero map, is a positive quadratic form on $\mathrm{Hom}(E_1,E_2)$.

Let $M/K$ be a finite Galois extension, $L=M^H$, and $D$ the decomposition group of a prime $\mathfrak{P}$ above $\mathfrak{p}$. The bijectionfollows from transitivity on primes for both $M/K$ and $M/L$.

For $V=L({\omega }_1)$ with the G2 root system numbered short-first,In the chain basis of the crystal of the seven-dimensional G2 representation, normalize $v_1=f_1{v}_0$, $v_2=f_2{v}_1$, $v_3=f_1{v}_2$. Then $v_0\wedge v_1$ and $v_0\wedge v_3-2v_1\wedge v_2$ are nonzero highest-weight vectors of weights ${\omega }_2$ and ${\omega }_1$; the identities $e_1{v}_1=v_0$, $e_1{v}_3=2v_2$, $e_2{v}_2=v_1$ verify both raising conditions. The Weyl complete reducibility theorem and G2 dimension polynomial exhaust the twenty-one dimensions of the exterior square. In the symmetric square, $v_0^2$ generates the twenty-seven-dimensional summand. Self-duality supplies a Lie-invariant bilinear form, which is symmetric because a nondegenerate alternating bilinear form cannot have odd dimension; its inverse gives the remaining invariant line.

For a temporal photon wave packet $|f,0⟩=\int d^{3}pf(\mathbf{p})a_{\mathbf{p}}^{0†}|0⟩/(2\pi {)}^3$, the oscillator commutator gives $⟨f,0|f,0⟩=-\int d^{3}p|f(\mathbf{p}){|}^2/(2\pi {)}^3<0$ when $f\ne 0$. This is the negative direction of the indefinite Hermitian form, not the divergent normalization of an unsmeared momentum eigenstate. It is excluded by the physical condition in Gupta-Bleuler quantization.

For a prime $p\equiv 3(\mathrm{mod}8)$, apply two-isogeny descent to the congruent number elliptic curve $y^2=x^3-p^{2}x$ and its companion $y^2=x^3+4p^{2}x$. The first square-class image is exactly $\{1,-1,p,-p\}$, already supplied by 2-torsion. The second has only positive candidates $1,2,p,2p$. The quartic covering in a two-isogeny descent equations for $p$ and $2p$ are impossible modulo $p$, because $-1$ is not a quadratic residue. For $2$ the equation is $W^2=2U^4+2p^2{V}^4$. Coprimality forces $U,V$ odd and $W=2Z$; modulo sixteen this gives $2Z^2=1+9=10$, which is impossible. Thus the second image is trivial and the square-class index formula for two-isogeny descent gives rank zero. The same prime-reduction argument bounds its rational torsion by four, and it already has four rational 2-torsion points, so it has no rational point corresponding to a right triangle of area $p$.

Let $t$ be a Young tableau of shape $\lambda$, and let $u$ be a Young tableau of shape $\mu$, both on $n$ entries. If the Column antisymmetrizer of a Young tableau $b_t$ satisfies $b_t\{u\}\ne 0$ over any field, then $\lambda$ dominates $\mu$ in the dominance order on partitions.

If two entries from one column of $t$ lie in the same row of $u$, their transposition fixes the tabloid $\{u\}$. Pairing terms of $b_t$ by this transposition gives $b_t\{u\}=0$, including in characteristic two. Otherwise the first $r$ rows of $u$ contain at most $\min (r,{\lambda }_j^{\prime })$ entries from column $j$ of $t$. Thusfor every $r$, proving dominance.

The top-degree case of cohomology growth for nef twists is bounded independently of $m$. It fills a gap when a dimension induction hypothesis only covers intermediate sheaf cohomology degrees.

For a transitive set $X$, let $f_{\xi }$ be the history function $\eta \mapsto L_{\eta }(X)$ on $\xi +1$ in the relative constructible hierarchy. By transfinite induction, $f_{\xi }\in L_{\xi +k}(X)$ for some finite $k$, possibly depending on $\xi$. The base and successor steps follow by forming finite ordered pairs and adjoining the next entry, using $L_{\xi +1}(X)\in L_{\xi +2}(X)$. These operations require finitely many subsequent definable power sets.

At a nonzero limit ordinal $\lambda$, the inductive bounds put every earlier $f_{\eta }$ in $L_{\lambda }(X)$, since $\eta +k<\lambda$. This level satisfies finite relation closure for set-theoretic coding, independently of the availability of history functions. Consequently its coded relative constructible stage predicate computes the endpoints correctly. Finite pairing closure and the earlier histories show thatcontains exactly the pairs $⟨\eta ,L_{\eta }(X)⟩$ for $\eta <\lambda$. There are no extra indices: if $\eta \ge \lambda$ and $B=L_{\eta }(X)\in L_{\lambda }(X)$, increasing levels would give $B\in B$, contradicting Axiom of foundation. Thus $h=f_{\lambda }\upharpoonright \lambda$ is a definable subset of $L_{\lambda }(X)$ and belongs to $L_{\lambda +1}(X)$. Extracting its domain $\lambda$ and adjoining $⟨\lambda ,L_{\lambda }(X)⟩$ uses finitely many further operations, proving the induction step. Every nonzero limit $\alpha$ therefore contains all $f_{\xi }$ with $\xi <\alpha$, as required by relative constructible level recognition.

In an orthonormal basis $({\epsilon }_1,{\epsilon }_2)$, the roots are $\pm ({\epsilon }_1-{\epsilon }_2)$, $\pm ({\epsilon }_1+{\epsilon }_2)$, $\pm 2{\epsilon }_1$, and $\pm 2{\epsilon }_2$. Choose simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ and ${\alpha }_2=2{\epsilon }_2$, with ${\alpha }_1$ a short root. The fundamental weights are ${\omega }_1={\epsilon }_1$ and ${\omega }_2={\epsilon }_1+{\epsilon }_2$.

A quasi-coherent sheaf of ideals $\mathcal{I}\subseteq {\mathcal{O}}_X$ gives a closed subscheme by gluing $\mathrm{Spec}(A/J)$ on affine open subschemes where $\mathcal{I}=\tilde{J}$. Compatibility follows from exactness of localization. On the closed set $Z=\{x:{\mathcal{I}}_x\ne {\mathcal{O}}_{X,x}\}$, the structure sheaf is $i^{-1}({\mathcal{O}}_X/\mathcal{I})$, where $i:Z\hookrightarrow X$ is the inclusion. Its pushforward is ${\mathcal{O}}_X/\mathcal{I}$.

A projective complete intersection is a closed subscheme of a projective space defined by a homogeneous regular sequence of positive degrees. A length-$r$ sequence in $k[t_0,\dots ,t_N]$ gives dimension $N-r$ when $r\le N$. Sheafifying its successive quotient sequences gives exact restriction sequences of twisting sheaves on projective space, allowing computation of its sheaf cohomology. It need not be smooth or reduced.

For a holomorphic line bundle with Hermitian metric $h$, its normalized Chern curvature iswhere $e$ is a holomorphic local frame and $h_e=h(e,e)>0$. A change of frame multiplies $h_e$ by the squared absolute value of a holomorphic unit; its logarithm has zero mixed derivative. The local expressions therefore define a closed real (1, 1)-form representing the First Chern class.

Positive curvature is an additional condition on the metric, distinct from positivity of each fibre norm. On the trivial line over $\mathbb{C}$, the metric $h_e=1$ is positive definite but has zero curvature. The metric $h_e=e^{-|z{|}^2}$ instead gives ${\omega }_{(L,h)}=idz\wedge d\bar{z}/(2\pi )$, a positive real (1, 1)-form. Globally, admitting such a positively curved metric is the definition of a positive holomorphic line bundle.

Replacing $h$ by $e^{-u}h$ adds $(i/(2\pi ))\partial \bar{\partial }u$. Thus the normalized Chern curvature changes by an exact differential form, while its First Chern class stays fixed.

For $n\ge 2$, the Bn root system has simple roots ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n={\epsilon }_n$. Its Dynkin diagram is a chain with a double terminal bond pointing toward short ${\alpha }_n$. The highest root is $\theta ={\epsilon }_1+{\epsilon }_2$. In the Extended Dynkin diagram, ${\alpha }_0=-\theta$ attaches by a single bond to ${\alpha }_2$ for $n\ge 3$. For $n=2$, both long nodes $0,1$ attach by double bonds to short node $2$. Rank one is the rank-one root system; its affine diagram has two equal-length nodes joined by a double bond without a short-root arrow.

The sum of a nef real Cartier class and an ample real Cartier class on a projective scheme is ample. By Kleiman's criterion, the ample cone is the interior of the nef cone. If a ball about an ample class lies in that convex cone, translating it by a nef class still lies in the cone. The sum is consequently still an interior point.

Over the complex numbers, the signed Young permutation module isThe isomorphism is the tensor identity for induced representations. The partition index is unchanged by this twist; conjugation instead changes the labels of its irreducible constituents. For the conjugate partition ${\lambda }^{\prime }$, the character inner product of $M^{\lambda }$ and ${\tilde{M}}^{{\lambda }^{\prime }}$ is one: Mackey restriction formula reduces it to the unique zero-one matrix with row margins $\lambda$ and column margins ${\lambda }^{\prime }$.