The Hermitian Lefschetz operator has injective powers in the indicated range. The commutator formula for powers of the Lefschetz operator proves this by induction on $r+k$: a form in the kernel can be written as $L\alpha$ using injectivity of the smaller power, and the lower-degree form $\alpha$ is then in the kernel of another power covered by the induction hypothesis.

On a compact Kähler manifold, the Lefschetz operator preserves harmonic forms, so the indicated power acts on harmonic representatives of Dolbeault cohomology. It is injective by injectivity of powers of the Lefschetz operator; Hodge symmetry and Hodge duality make its finite-dimensional source and target have equal dimension, so it is an isomorphism.

For a reduced connected alternating knot diagram, its knot signature is $s_A(D)-c_{+}(D)-1$, where $s_A$ counts circles in the all-$A$ bracket smoothing state and $c_{+}$ counts positive crossings. The convention is that the $A$-smoothing at a positive braid crossing is the oriented smoothing, and a positive trefoil knot has knot signature $-2$. Reversing the global knot signature convention negates the formula.

For $[e_1,e_2]=a{e}_3$, $[e_2,e_3]=b{e}_1$, and $[e_3,e_1]=c{e}_2$, the Adjoint representation matrices give the displayed Killing form. Cross traces vanish. Nonzero $a,b,c$ make it nondegenerate and hence the algebra semisimple. If the three constants have the same sign it is negative-definite; mixed signs give signature $(2,1)$. In particular $(a,b,c)=(2,2,2)$ is the compact $\mathfrak{su}(2)$ case, whereas $(2,2,-2)$ is the split real traceless two-by-two algebra. This gives a direct compactness criterion from the Killing form in a small explicit example.

A Hodge metric is a Kähler metric whose Kähler form represents an integral cohomology class in real de Rham cohomology. Equivalently, its periods over integral two-cycles are integers. Some conventions require the class of $\omega /(2\pi )$ to be integral instead; this changes the normalization but not existence.

On a nonsingular equation $y^2=x^3+a_2{x}^2+a_{4}x+a_6$ in characteristic different from two, $\omega =dx/(2y)$ is regular and nowhere zero. For a chord of slope $h$ giving $P+Q=(X,Y)$, differentiating the line-intersection identity yields $dX/dx=Y/y$, so $dX/(2Y)=dx/(2y)$. Extension across the exceptional cases proves invariance under every translation on an elliptic curve. The addition map consequently satisfies ${\mu }^{\ast }\omega ={\mathrm{pr}}_1^{\ast }\omega +{\mathrm{pr}}_2^{\ast }\omega$, and $[n{]}^{\ast }\omega =n\omega$.

For a density with finitely many jumps ${\Delta }_r$ at separated points and the unit-width box kernel, take $h$ smaller than all consecutive jump spacings. The smoothing bias is supported in disjoint length-$h$ neighbourhoods of the jumps. At each jump its two sides form triangles of height $|{\Delta }_r|/2$, and exact integration givesCombined with the integrated variance of a kernel density estimator, this gives expected $L^2$ error $O(n^{-1/4})$ at $h\asymp n^{-1/2}$.

The product of all hook lengths in a Young diagram is the denominator in the hook-length formula, and the scalar in the quasi-idempotence identity for its Young symmetrizer.

Every closed topological manifold with the homotopy type of $S^d$ is homeomorphic to $S^d$. This is a classification theorem about homeomorphisms, and does not assert a diffeomorphism between prescribed smooth structures. It is the precise final step needed after proving that a cone link is a simply connected integral homology sphere in the manifold criterion for a suspension.

For a line bundle $L$ on an abelian variety $A$, this closed subgroup consists of $a$ such that $T_a^{\ast }L\cong L$. It is a stabilizer of a Picard group class, which need not be the same as the stabilizer of an individual Weil divisor representing that class.

On a connected regular radial interval, trivial holonomy around circular fibres forces the continuous function $f^{\prime }$ to be integer-valued, hence constant. The resulting Riemannian metrics have $f(r)=nr+b$, with $n\in \mathbb{Z}$ and $f>0$ on the interval. Their Gaussian curvature is $-f^{\prime \prime }/f=0$. For $n\ne 0$, $\rho =f/|n|$ and $\theta =|n|\varphi$ give a local isometry to the Euclidean plane; for $n=0$, $r,b\varphi$ give a flat cylindrical chart. The polar developing map is injective on the regular domain when $|n|=1$, and is a multiple covering when $|n|>1$. Trivial holonomy therefore does not imply a global Euclidean chart. The slope is a discrete parameter, not an unrestricted real one.

For quadratic and cubic interactions, assigning the couplings chiral spurion transformations makes each monomial formally covariant. The ordinary $U(1)$ leaves $\theta$ neutral, while the R-symmetry gives it charge one and the superpotential charge two. The neutral ratio $g\Phi /m$ and the covariant factor $m{\Phi }^2$ give the symmetry-allowed holomorphic form $m{\Phi }^{2}f(g\Phi /m)$. Perturbative regularity and tree matching are additionally needed for superpotential non-renormalization; symmetry alone leaves an arbitrary function. Fixed numerical couplings need not enjoy these formal symmetries as actual field symmetries.

The derivative of a section defining a regular zero locus patches intrinsically along that locus because changes of trivialization introduce derivative terms multiplied by the vanishing section. Its tangent kernel and surjectivity give $N_V\cong E{|}_V$. Combining this with adjunction for a smooth submanifold gives $K_V=(K_M\otimes \det E){|}_V$.

In a holomorphic local frame, a meromorphic section has a meromorphic scalar coefficient; the coefficients transform by the bundle's transition functions of a vector bundle. A nonzero meromorphic section has a divisor on a complex manifold recording its zero and pole orders.

Equilibrium fails where the helium-3 relaxation time becomes comparable to stellar age. In a fixed-density pp-I scaling with reaction exponents four and sixteen, the equilibrium abundance scales as $T^{-6}$ and the relaxation time as $T^{-10}$. Central normalization then places the transition near six million kelvin; using the full supplied exponential factors gives a somewhat higher estimate. A precise radius or temperature requires a stellar mass density/composition profile and the second branch contribution.

For a real rank-$r$ vector bundle $E\to B$, the Thom isomorphism theorem with ${\mathbb{F}}_2$ coefficients identifies the relative cohomology sequence of its disk bundle and sphere bundle withThe multiplier is the top Stiefel–Whitney class, also called the mod-two Euler class. No orientation of $E$ is required. For the real tautological line bundle, the sphere bundle is the antipodal cover of Real projective space; this sequence proves that all powers of the degree-one generator through the dimension of the base are nonzero.

For $W={\mathbb{A}}_k^2\setminus \{0\}$, restriction to $D(x)$ and $D(y)$ givesThe intersection is taken in $k(x,y)$ and the equality follows from unique factorization: a reduced denominator dividing powers of both $x$ and $y$ is a unit. The inclusion $W\hookrightarrow {\mathbb{A}}^2$ induces an isomorphism on global functions but is not an isomorphism, so $W$ is not an affine variety.

For two smooth plane conics meeting transversely, the operation of crossing a chord of the second conic tangent to the first and choosing the other tangent through the new endpoint acts as a translation on an elliptic curve on their incidence curve. Each switch is an involution of a degree-two map from a genus one curve. One periodic orbit means that the translating point is torsion, so every orbit is periodic with the same least period. The construction extends through coincident choices at ramification points using the regular involutions.

For the special linear Lie algebra ${\mathfrak{sl}}_3(\mathbb{C})$ with upper triangular positive roots, the simple coroots are $\mathrm{diag}(1,-1,0)$ and $\mathrm{diag}(0,1,-1)$. Their dual fundamental weights are ${\omega }_1(\mathrm{diag}(t_1,t_2,t_3))=t_1$ and ${\omega }_2(\mathrm{diag}(t_1,t_2,t_3))=t_1+t_2=-t_3$. In simple-root coordinates, ${\omega }_1=(2{\alpha }_1+{\alpha }_2)/3$ and ${\omega }_2=({\alpha }_1+2{\alpha }_2)/3$. The primitive vectors $e_1\in {\mathbb{C}}^3$ and $e_1\wedge e_2\in {\Lambda }^2{\mathbb{C}}^3$ realize these Fundamental representations of sl3.

The form ${\omega }_{\mathrm{int}}=\frac{i}{2\pi }\partial \bar{\partial }\log (1+\sum |z_j{|}^2)$ patches on Complex projective space. The dual tautological metric on $\mathcal{O}(1)$ identifies it with normalized Chern curvature, so $[{\omega }_{\mathrm{int}}]=c_1(\mathcal{O}(1))$. With the unnormalized convention $i\partial \bar{\partial }\log (1+\sum |z_j{|}^2)$, divide by $2\pi$ to obtain this integral class.