A weak composition of an integer $N\ge 0$ into $k\ge 1$ parts is an ordered tuple of nonnegative integers with sum $N$. Zero parts retain their positions: $(0,2,3)$ and $(2,0,3)$ differ. Fixing the length $k$ is essential when applying stars and bars, since inserting extra zero parts creates further tuples. ^[ref]

On a Euclidean real space of dimension $2n$, an orthogonal complex structure is an orthogonal endomorphism $J$ with $J^2=-I$. Orthogonal changes of basis act transitively on these structures, and the stabilizer of the standard structure is $U(n)$. The compact homogeneous space has real dimension $n(n-1)$.

// Target: geometry-and-topology.bigb

For the connected tree scattering amplitude of one particle and one antiparticle of a complex scalar field, the Dyson series inserts $-i\lambda {\varphi }^{†2}{\varphi }^2/4$. There are $2!$ assignments of the particle and antiparticle external attachments to each pair of identical field factors, giving $2!2!=4$ and hence the displayed Feynman vertex. With relativistic normalization of a one-particle state, integrating the vertex position gives $i\mathcal{T}(2\pi {)}^4{\delta }^4(p_1+q_1-p_2-q_2)$ with $\mathcal{T}=-\lambda$. Vacuum diagrams and one-particle self-energy insertions are separate disconnected contributions; normal ordering eliminates the single-vertex vacuum and tadpole contractions.

Identify ${\mathbb{CP}}^m$ with nonzero homogeneous binary forms of degree $m$, modulo nonzero scalar multiplication. Multiplication of $m$ projective linear forms gives a well-defined holomorphic map: no product is zero, and scaling one form scales the entire coefficient vector. A form with $m$ distinct roots has exactly $m!$ ordered factorizations. In affine root coordinates the derivative is a Vandermonde determinant, nonzero at distinct roots. Each such preimage contributes $+1$ with complex orientations, so the mapping degree is $m!$. This also realizes the least positive degree allowed by the factorial degree obstruction for products of two-spheres.

Every continuous self-map of ${\mathbb{CP}}^{2r}$ has a fixed point. If its pullback sends the integral degree-two generator to $d$ times itself, multiplicativity gives trace $d^j$ in degree $2j$. Thus its Lefschetz number is $1+d+\cdots +d^{2r}$, which is positive: for $d\ne 1$ it is $(1-d^{2r+1})/(1-d)>0$, and at $d=1$ it is $2r+1$. The Lefschetz fixed-point theorem applies. In particular no nontrivial finite group acts freely on this simply connected space.

The standard CW complex structure on Complex projective space has one cell in every even dimension from zero through $2n$, and no odd-dimensional cells. Its cellular chain complex therefore has zero differential. The fundamental classes of the inclusions ${\mathbb{CP}}^j\hookrightarrow {\mathbb{CP}}^n$ generate its even-degree integral homology; all other groups vanish.

In the chart $x_j=X_j/X_0$ of Complex projective space, differentiate the quotient coordinates to obtain this formula for $\pi :{\mathbb{C}}^{n+1}\setminus \{0\}\to {\mathbf{P}}^n$. The kernel is the radial line $\mathbb{C}X$. A degree-one homogeneous vector field on the punctured cone descends to a holomorphic vector field, since its scaling cancels the inverse scaling of $d\pi$. This gives the fiberwise construction of the Euler sequence on complex projective space.

Delayed adjustment of electric polarization after a change in electric field. A simple illustrative model lets the slowly adjusting part ${\mathbf{P}}_s$ obey $\tau d{\mathbf{P}}_s/dt+{\mathbf{P}}_s={ϵ}_0\Delta {ϵ}_r\mathbf{E}$, with a positive response time $\tau$. A harmonic electric field then gives a contribution $\Delta {ϵ}_r/(1+i\omega \tau )$ to complex permittivity divided by vacuum permittivity. This model is an assumption, not a claim that all dielectrics have one response time.

Conversion of electrical energy into heat in a dielectric driven by an electric field. In the stated harmonic convention, a passive scalar response has ${ϵ}^{\prime \prime }\ge 0$ for positive angular frequency. A dielectric capacitance measurement measures a dissipative component as well as capacitance; leakage from electrical conductivity can contribute to the inferred loss.

An argument of a nonzero complex number $z$ is any angle $\theta$ for which $z=|z|e^{i\theta }$. All arguments are $\theta +2\pi k$, with $k$ an integer. Zero has no argument: it has no distinguished direction. A commonly chosen representative, the principal argument $\mathrm{Arg}z$, lies in $(-\pi ,\pi ]$. Choosing representatives is part of specifying a branch of a multivalued function. These conventions are described in Jeremy Orloff, MIT 18.04, topic 1, sections 1.5 and 1.14.

For $3+4i$, the principal argument is $\arctan (4/3)\approx 0.927295$ radians, computed with the inverse tangent. But the coordinate ratio alone loses the quadrant: $-3-4i$ has the same ratio $4/3$, yet its principal argument is $\arctan (4/3)-\pi \approx -2.214297$ radians.

For nonzero $z,w$, the correct rule isOrdinary equality can fail. With $z=w=e^{3\pi i/4}$, their principal arguments sum to $3\pi /2$, but $zw=-i$ has principal argument $-\pi /2$. They differ by one full turn. As a point crosses the negative horizontal axis, the principal argument jumps by $2\pi$. A continuous choice on a suitable region therefore requires a branch cut; the jump belongs to the choice of representative, although the point moves smoothly.

An elliptic curve with complex multiplication is called a CM elliptic curve. Distinguish geometric endomorphisms from endomorphisms defined over the base field. In characteristic zero, adjoining the imaginary quadratic field selected by the tangent action makes all geometric endomorphisms defined over the base. Good split primes have ordinary reduction; good inert primes have supersingular reduction.

For a complex CM torus $\mathbb{C}/\mathfrak{a}$ and an idele $s$ of its CM field, the automorphism corresponding to $s$ by Artin reciprocity takes its lattice to $s^{-1}\mathfrak{a}$ and acts on torsion through the adelic scalar $s^{-1}$. When the curve is defined over a larger field, use the idelic norm to the CM field. Comparison with the original curve gives a scalar in the CM field, producing the Hecke character of a CM elliptic curve.

A complex CM period compares an algebraic invariant differential on a CM elliptic curve with its analytic lattice coordinate. At an ordinary prime, a $p$-adic CM period compares the same differential with the invariant differential of the multiplicative formal group: an isomorphism $\eta$ is normalized by ${\log }_{\hat{E}}\eta (T)={\Omega }_p\log (1+T)$. Matched periods make normalized critical Hecke L-function values algebraic and give precise p-adic L-function interpolation.

The Milnor-Novikov coefficient theorem identifies the coefficient ring of complex cobordism as the displayed graded polynomial ring. In cohomological grading it has no positive-degree or odd-degree coefficients. Homological grading reverses the signs: $M{U}_{\ast }$ has polynomial generators in degrees $2i$.

Represent a map $f:X\to Y$ by a smooth map, embed its graph into $Y\times {\mathbb{R}}^N$, and specify a stable complex structure on the normal bundle of that embedding, with the standard framing on the auxiliary real coordinates. Stabilization and homotopy identify equivalent choices. The auxiliary dimension is chosen to make the normal rank even; odd relative dimensions use an extra real suspension coordinate. The resulting oriented stable normal data defines the Gysin map in complex cobordism.

For a rank-$r$ real bundle with chosen stable complex normal orientation, the complex-cobordism Thom class gives the displayed isomorphism by $a\mapsto {\pi }^{\ast }a{u}_{\xi }$. Stabilizing by a trivial complex bundle suspends the Thom space and its class compatibly. The class of an actual complex rank-$n$ bundle has degree $2n$.

Pulling the external product of two normally complex submanifolds back along the diagonal gives their transverse intersection. Its normal complex structure is the direct sum of the restricted normal complex structures. Multiplication of Thom classes for a direct sum identifies this geometric representative with the cup product in complex cobordism. Empty intersection represents zero.

// Target: geometry-and-topology.bigb

The displayed product applies when $k+m\ge n$. Move one projective subspace by a unitary isotopy into transverse position. The intersection is a linear projective subspace of dimension $k+m-n$, with normal complex bundle the sum of the restricted input normal bundles. If $k+m<n$, representatives can be made disjoint and the product is zero.

A complete sufficient statistic is independent of every ancillary statistic. To prove this, let $S$ be complete and sufficient, let $A$ be ancillary, and take any bounded measurable function $h$. By sufficiency, $g(S)=E_{\theta }[h(A)\mid S]$ has a parameter-independent version; by ancillarity, $c=E_{\theta }h(A)$ is independent of $\theta$. Thus $E_{\theta }[g(S)-c]=0$ for every $\theta$. Completeness gives $g(S)=c$ almost surely. Choosing $h$ to be the indicator of each measurable event in the ancillary space proves independence. Completeness is essential: a sufficient statistic may itself contain a nonconstant ancillary, as in the minimum and maximum of a uniform location sample.

The octahedral graph is the graph obtained from $K_6$ by removing a perfect matching; equivalently it is $K_{2,2,2}$. Its chromatic number is three, and it contains two vertex-disjoint triangles in a graph. A complete bipartite graph with a single additional internal edge cannot contain it: every triangle in that host uses the additional edge.