For a map $f:{\mathbb{RP}}^{2k+1}\to {\mathbb{CP}}^k$ nontrivial on degree-two mod-two cohomology, an immersion in the homotopy class of $(f,f)$ has a rank-$(2k-1)$ normal bundle with total Stiefel–Whitney class $(1+x{)}^{2k+2}$. Its forbidden degree-$2k$ Stiefel–Whitney class is $(k+1)x^{2k}$, so $k$ must be odd.

For a complex semisimple Lie algebra and an integer $k\ge 0$, the Weyl character formula and Weyl denominator formula giveApply the dilation $e^{\mu }\mapsto e^{(k+1)\mu }$ to the denominator identity to obtain the numerator. Cancellation leaves a finite geometric series for each positive root of a root system. Evaluating the resulting polynomial at all formal exponentials equal to one gives the dimension, without taking an undefined quotient at the identity.

If $t$ is the $k$th largest rank weight $u_i$, exactly $h$ levels have weight greater than $t$, and exactly $s$ have weight $t$, then there are $\left(\genfrac{}{}{0ex}{}{s}{k-h}\right)$ maximizing families. For fixed $n,k$, the possible counts are exactly $\left(\genfrac{}{}{0ex}{}{s}{j}\right)$ with $1\le j\le k$ and $j\le s\le n+1-k+j$.

Let $B$ be a continuously differentiable solenoidal vector field on an open star-shaped set containing the origin. Thenis a vector potential for $B$. For $D={\int }_0^{1}tB(tx)dt$, differentiation gives $\nabla \cdot D=0$ and $\nabla \times (D\times x)=2D+(x\cdot \nabla )D={\int }_0^1{\partial }_t(t^{2}B(tx))dt=B(x)$. Boundedness near the origin removes the lower endpoint. A field defined only on a punctured domain need not satisfy these hypotheses.

Let $H$ be a very ample effective Cartier divisor on a projective scheme $X$, and suppose $L{|}_H$ is ample. Then $H^i(X,L^m)=0$ for $i\ge 2$ and sufficiently large $m$. By uniform Serre vanishing for two ample twists, cohomology of $L^m(tH){|}_H$ vanishes uniformly for $t\ge 0$. The restriction sequence identifies the higher groups of $L^m(tH)$ and $L^m((t+1)H)$ for $i\ge 2$. For fixed $m$, large $t$ kills them by Serre vanishing for $H$; descend to $t=0$. No $H^1$ vanishing is claimed.

For a hyperplane $H\subset {\mathbf{P}}^n$ with inclusion $i$, multiplication by its defining linear homogeneous polynomial gives the short exact sequence of sheavesThe long exact sequence in sheaf cohomology relates twists in consecutive degrees and consecutive dimensions. Surjectivity of restriction on global homogeneous polynomials, together with eventual positive-twist vanishing, proves vanishing of intermediate cohomology by descending induction. The top-degree dimensions satisfy $h_n(m-1)=h_n(m)+h_{n-1}(m)$ for $n\ge 2$, where $h_r(m)=\dim H^r({\mathbf{P}}^r,\mathcal{O}(m))$.

Over any field, the canonical bundle of projective space is $\mathcal{O}(-n-1)$. Indeed, dualizing the Euler sequence gives $0\to {\Omega }_{{\mathbf{P}}^n}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to \mathcal{O}\to 0$; taking determinants gives the result. Hence Serre duality identifies $H^i({\mathbf{P}}^n,\mathcal{O}(m))$ with the dual of $H^{n-i}({\mathbf{P}}^n,\mathcal{O}(-m-n-1))$.

For positive integral $k$, the rational torsion subgroup of $y^2=x^3+kx$ has order dividing four. Indeed, torsion-freeness of the formal group over Qp for odd p, reduction of torsion points on an elliptic curve, and the point-count criterion for y squared equals x cubed plus k x bound its order by $p+1$ at all good primes $p\equiv 3(\mathrm{mod}4)$. The Dirichlet theorem on primes in arithmetic progressions excludes every odd torsion prime and bounds the two-primary part by four, by choosing $p\equiv 3(\mathrm{mod}8)$. There is exactly one nonzero rational 2-torsion point, namely $(0,0)$. A rational half of it must have $x^2=k$ by the elliptic-curve addition formula. Writing $k=s^2$ with $s>0$, its equation is $y^2=2s^3$, so $s=2t^2$ and $k=4t^4$. Conversely $(2t^2,4t^3)$ has order four. Thus the cyclic group of order four occurs exactly for $k=4t^4$, $t\ge 1$. The order bound itself holds for every nonzero integral $k$ of either sign: positivity was used only to classify the rational points of orders two and four.

For integral $m\ge 2$, suppose $E:y^2=x(x+1)(x+m^2)$ has good reduction at five or seven. At five, $m^2\equiv 4$; at seven, $m^2\equiv 2$ or $4$. Each resulting reduced elliptic curve has eight points. These restrictions follow from $\Delta =16m^4(m^2-1{)}^2$ and $c_4=16(m^4-m^2+1)$: if an odd prime divides $m(m^2-1)$, the j-invariant of an elliptic curve has negative valuation. The torsion-freeness of the formal group over Qp for odd p bounds the whole rational torsion by eight. The point $(m,m(m+1))$ doubles to $(0,0)$, and $(-1,0)$ is an independent 2-torsion point. These generate the stated group of order eight.

A quantum time-reversal operator is an antiunitary operator reversing spatial momentum and angular momentum. For a time-reversal-invariant Hamiltonian operator, its conjugation of $i$ makes it intertwine forward and backward unitary time evolution without reversing the energy spectrum. A scalar momentum eigenstate may transform as $\hat{T}|\mathbf{p}⟩=e^{i\chi (\mathbf{p})}|-\mathbf{p}⟩$. With unit-modulus phases and a complete normalized basis, expansion of arbitrary states proves antiunitarity.

For a closed smooth submanifold $M\subseteq N$ of codimension $c>0$, a tubular neighborhood and the homological Thom isomorphism theorem identifySubstitution in the long exact sequence in relative homology gives the displayed embedding Gysin sequence. It does not require orientability over ${\mathbb{F}}_2$.

A smooth endomorphism $A$ of the tangent bundle defines the tensor derivation $D_A$ by $D_{A}f=0$ and $D_{A}Y=A(Y)$. On a differential one-form, $D_A\omega =-\omega \circ A$. On a general tensor field, it acts by $A$ in each vector factor and by the negative dual action in each covector factor. The two actions cancel in each contracted pairing, proving compatibility with tensor contraction.

For a smooth vector field $X$, the Lie derivative is the tensor derivation determined by ${\mathcal{L}}_{X}f=X(f)$ and ${\mathcal{L}}_{X}Y=[X,Y]$. The identity $[X,fY]=f[X,Y]+X(f)Y$ permits its unique extension. On a differential one-form,On differential forms it agrees with the usual Lie derivative of a differential form and Cartan's magic formula.

For the usual recursive syntactic forcing relation, $p{⊩}^{\ast }\exists x\phi (x)$ means that conditions $q\le p$ forcing $\phi (\sigma )$ for some forcing name $\sigma$ are dense below a forcing condition $p$. This density clause proves the existential step of the forcing theorem from the truth lemma for each named instance. A generic filter containing $p$ meets that dense set after it is augmented by conditions incompatible with $p$.

For forcing names $\sigma ,\tau$ and a generic filter $G$, ${\sigma }^G\in {\tau }^G$ exactly when some $p\in G$ syntactically forces $\sigma \in \tau$. Assuming the equality truth lemma, interpreted membership supplies an active pair $(\rho ,r)\in \tau$ and a condition in $G$ forcing $\sigma =\rho$; a common strengthening forces membership. Conversely the defining equality witnesses occur dense below a forcing condition forcing membership. The dense-below generic meeting lemma supplies one in $G$, and the equality truth lemma recovers interpreted membership.

For the defining four-dimensional representation $V$ of the symplectic Lie algebra ${\mathfrak{sp}}_4$, with the short simple root numbered first, its highest weight is ${\omega }_1$. Its tensor square decomposes aswith dimensions $10,5,1$. The first summand is the symmetric square; the other two form the exterior square, split by symplectic contraction of an exterior square.

Over $\mathbb{C}$, the symplectic Lie algebra on a four-dimensional symplectic vector space acts faithfully on its five-dimensional primitive exterior square, preserving its symmetric bilinear form. Faithfulness follows because an operator killing all wedges $e_i\wedge e_j$ has all off-diagonal coefficients zero and satisfies $A_{ii}+A_{jj}=0$ for every pair, hence is zero. The source and target Lie algebras both have dimension ten, so this faithful map is an isomorphism to the Special orthogonal Lie algebra ${\mathfrak{so}}_5$.

The primitive exterior square of a symplectic vector space is the kernel of its symplectic contraction of an exterior square. In dimension four, the invariant inverse-form bivector $\Omega$ has nonzero wedge square. Choose $\mathrm{vol}=\Omega \wedge \Omega /2$; the symmetric bilinear form $\xi \wedge \eta =B(\xi ,\eta )\mathrm{vol}$ is nondegenerate on ${\Lambda }^{2}V$. The primitive subspace is ${\Omega }^{\perp }$, and therefore inherits a nondegenerate form of dimension five. The symplectic Lie algebra acts on it by infinitesimal orthogonal transformations.

This sheaf consists of holomorphic functions vanishing at $p$ on opens containing $p$, and all holomorphic functions on opens not containing $p$. Evaluation gives $0\to {\mathcal{I}}_p\to {\mathcal{O}}_M\to {\mathbb{C}}_p\to 0$, with a skyscraper sheaf as quotient. On an elliptic curve, $H^0({\mathcal{I}}_p)=0$, $H^1({\mathcal{I}}_p)\cong \mathbb{C}$, and all higher cohomology vanishes.

A probability measure-preserving transformation is strong mixing exactly when $\mu (T^{-n}A\cap A)\to \mu (A{)}^2$ for every measurable $A$. The converse follows by applying decay of autocorrelation implies weak convergence of an observable to $f=1_A$ and then testing against $1_B$. Diagonal polarization alone controls a sum of the two directed cross-correlations and does not suffice as a proof.