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For distinct fixed target values and a nonconstant meromorphic function, the truncated second main theorem has the displayed form, with $S_f=O({\log }^{+}{T}_f+\log r)$ outside a set of finite linear measure. The truncated Nevanlinna counting function retains only one unit at each preimage. For finite-order functions the error is $O(\log r)$; for transcendental functions it is $o(T_f)$ along the nonexceptional radii.

This theory compares the area, boundary length and Euler characteristic of a surface mapped holomorphically to another surface, using the pulled-back metric. It extends covering-degree and Riemann-Hurwitz formula estimates when some boundary maps into the interior of the target. The Ahlfors second fundamental theorem and Ahlfors five islands theorem are important consequences.

A compact Lagrangian submanifold $L$ with vanishing first de Rham cohomology cannot be a fibre of a smooth positive-dimensional fiber bundle whose nearby fibres are also Lagrangian submanifolds. In a Weinstein neighborhood, nearby fibres are one-form graphs of closed forms. They are exact by the cohomology assumption, so meet the zero section at a critical point of their potential, contradicting disjointness of distinct fibres.

Put $P=A_1+i{A}_2$, $Q=A_1-i{A}_2$ and $R=A_3$, and defineDirect expansion gives $[L,M]=-i[P,R]+i[P,Q]\lambda +i[R,Q]{\lambda }^2$. The Nahm equations imply $\dot{P}=-i[P,R]$, $2\dot{R}=i[P,Q]$ and $-\dot{Q}=i[R,Q]$, so $\dot{L}=[L,M]$. In any finite-dimensional matrix representation, cyclicity of the matrix trace givesThus every coefficient of the degree-at-most-$2p$ matrix trace polynomial is conserved. At projective infinity the polynomial is interpreted as a section of $\mathcal{O}(2p)$, not a globally holomorphic scalar function on the complex projective line.

For an indecomposable $kK$-module $V$ with vertex $D$ and ${\mathrm{Br}}_D(e)V=V$, every indecomposable summand of $(1-e){\mathrm{Ind}}_K^{G}V$ has vertex conjugate into $D\cap {}^{g}D$ for some $g\notin N_G(D)$. The refinement keeps the intersection subgroups from Green correspondence, rather than asserting only that the remaining vertices are smaller than $D$.

The epoch lengths in Kingman's coalescent are independent random variables. Conditional on these lengths, mutation counts on disjoint tree pieces are independent Poisson random variables. Their conditional laws depend only on the corresponding epoch length, so integrating the product of conditional laws against the independent epoch laws leaves independent unconditional epoch mutation counts. This justifies adding their variances, rather than assuming independence from conditional independence alone.

Integrating the Poisson distribution probability generating function over the exponential distribution of the epoch length givesThus $Y_j$ has a geometric distribution on $\{0,1,\dots \}$, with expectation $\theta /(j-1)$ and variance $\theta /(j-1)+{\theta }^2/(j-1{)}^2$. The support convention matters: the count includes zero mutations.

The independent mutation counts in coalescent epochs have the geometric mutation count in a coalescent epoch laws. Summing their expectations and variances gives the displayed formulas, where $H_m={\sum }_{i=1}^{m}1/i$ and $H_m^{(2)}={\sum }_{i=1}^{m}1/i^2$. The extra quadratic term records randomness of the ancestral tree, beyond the conditional Poisson distribution variance.

If the displayed covariance is finite and the mean obeys $m^{\prime }(x)+A(x)=0$, initialize the Musiela forward-curve equation with the corresponding Gaussian random field, independently of future noise. The resulting curve is stationary in time. Indeed shifting the initial covariance contributes the integral over $[t,\infty )$ and fresh noise contributes the integral over $[0,t]$, so their sum is $K$. A deterministic initial curve generally is not stationary, and nondecaying volatility may make $K(x,x)$ infinite. Exponentially decaying finite-factor volatilities give stationary Ornstein-Uhlenbeck processes.

For nonsingular within-group scatter $E$ and between-group scatter $H$, the statistic is ${\prod }_j(1+{\theta }_j{)}^{-1}$, where ${\theta }_j$ solve the generalized eigenvalue problem $Hv=\theta Ev$. It is small when the group means differ strongly relative to within-group variation. Maximizing a common-covariance normal likelihood under equal and unrestricted means gives the likelihood ratio ${\Lambda }^{N/2}$. Its determinant factors cancel under any nonsingular change of response coordinates. For one response it is equivalent to the analysis of variance F-test.

For groups with sizes $n_r$, means ${\bar{x}}_r$, and overall mean $\bar{x}$, define $W={\sum }_{r,i}(x_{ri}-{\bar{x}}_r)(x_{ri}-{\bar{x}}_r{)}^T$ and $B={\sum }_r{n}_r({\bar{x}}_r-\bar{x})({\bar{x}}_r-\bar{x}{)}^T$. Expanding $x_{ri}-\bar{x}=(x_{ri}-{\bar{x}}_r)+({\bar{x}}_r-\bar{x})$ gives total scatter $T=W+B$ because each within-group residual sum vanishes. Both matrices are positive semidefinite, and $\mathrm{rank}B\le \min (p,g-1)$. They separate within-group noise from between-group mean separation.

The subspace filtration and quotient filtration in a short exact sequence are good when the associated graded algebra is Noetherian. Their cumulative Hilbert functions add in every degree. Positive leading coefficients make the degree of the sum the larger degree; when both degrees agree, factorial times the leading coefficients add. Thus equal-dimensional nonzero factors consume positive multiplicity of a filtered module. With a standard graded algebra normalization these multiplicities are integral.

Let $G_N(B)=E(20N[1+2B(1-B)],21N)-B$. The proportional scaling limit of the Erlang loss formula makes its limits at $B=1/100$ and $B=1/8$ respectively $-1/100$ and $7/520$. Since $G_N(0)>0$ and $G_N(1)<0$, sufficiently large $N$ give alternating signs at these four points. The intermediate value theorem supplies three distinct zeros in the intervening intervals. These are multiple solutions of the Erlang fixed point approximation, not multiple stationary laws of an exact finite irreducible Markov chain.

For an elliptic curve over a number field $K$, the Mordell-Weil theorem writes $E(K)$ as a finite torsion subgroup plus a free abelian group of finite rank of an abelian group $r$. This integer is the rank of the elliptic curve over $K$. It can increase under field extension. A two-isogeny descent supplies rank bounds from rational square-class images; the canonical height pairing can certify independence of rational points.

The saturation of a subgroup $H$ of the Mordell-Weil group consists of all points with a positive multiple in $H$. Saturation includes the full torsion subgroup. For a full-rank $H$ containing that torsion, it equals $E(K)$, but the index of $H$ may exceed one. Checking that $\ell P\in H$ forces $P\in H$ for every prime divisor of a certified index bound proves that $H$ is already the whole group. A canonical height pairing proves independence of found points; saturation answers the separate question of whether a full-rank subgroup is the full group.

For the monodisperse circular ash runout, integrate the local particle deposition flux ${\rho }_p{w}_{s}c(t)$ from first arrival to infinite time. Here $s=\max (s_0,r^2/R_{\infty }^2)$ for $0\le r<R_{\infty }$ and $s_0=R_0^2/R_{\infty }^2$. The displayed $\Sigma$ is deposited particle mass per horizontal area, and vanishes outside the limiting current. It is constant under the initial disk, then decreases to zero; integrating $2\pi r\Sigma$ gives the initial ash mass ${\rho }_p{c}_0\mathcal{V}$. A deposit thickness additionally requires its packing fraction or bulk deposit density.

For a compact string coordinate of radius $R$, choose $p_L=n/R+wR/{\alpha }^{\prime }$ and $p_R=n/R-wR/{\alpha }^{\prime }$, where the left-moving oscillators are denoted by tildes. The two classical zero-mode Virasoro constraints give $M^2=p_L^2+4\tilde{N}/{\alpha }^{\prime }=p_R^2+4N/{\alpha }^{\prime }$. Their difference yields the displayed shifted closed-string level matching condition. Their average is $M^2=(n/R{)}^2+(wR/{\alpha }^{\prime }{)}^2+2(N+\tilde{N})/{\alpha }^{\prime }$. Classically the center-of-mass momentum can be continuous; the integer momentum number belongs to compact-coordinate quantization. Quantum intercepts shift the mass formula but cancel out of level matching.

In the critical closed bosonic string, compactification of one coordinate gives $M^2=n^2/R^2+w^2{R}^2/{\alpha }^{\prime 2}+2(N+\tilde{N}-2)/{\alpha }^{\prime }$ and $\tilde{N}-N+nw=0$. At oscillator level zero, level matching forces $nw=0$. Thus a pure momentum ground state becomes a massless scalar field at $R=|n|\sqrt{{\alpha }^{\prime }}/2$, while a pure winding ground state does so at $R=2\sqrt{{\alpha }^{\prime }}/|w|$. These are threshold states descended from the bosonic tachyon; they are additional to the neutral graviton, two-form, vectors, dilaton and radius modulus. At the self-dual circle, the four charges $(\pm 2,0),(0,\pm 2)$ contribute to the enhanced massless scalar field spectrum.

Symmetrize the $p$ upper and $q$ lower indices separately, and remove every upper-lower trace. The resulting representation has highest weight $p{\omega }_1+q{\omega }_2$. Its dimension isThe subtraction is valid because contraction is onto: in the polynomial realization its adjoint, for the monomial-factorial inner product, is multiplication by the nonzero polynomial ${\sum }_i{x}_i{y}_i$, an injective map. The resulting kernel contains the highest-weight constituent and has precisely its Weyl dimension formula dimension, hence is irreducible. This highest-weight constituent is not always the largest-dimensional constituent of the full mixed tensor.

For $T^{abc}$, split its first pair into symmetric and antisymmetric parts. The fully symmetric part $S^{abc}$ is a symmetric cubic representation of SU(3). Write $M^{abc}=T^{(ab)c}-S^{abc}$ and $O^a{}_d={ϵ}_{dbc}{M}^{abc}$. Then $O$ is traceless and $M^{abc}=({ϵ}^{bcd}{O}^a{}_d+{ϵ}^{acd}{O}^b{}_d)/3$. The antisymmetric part is $T^{[ab]c}={ϵ}^{abd}{N}^c{}_d$, with $N^c{}_d={ϵ}_{dab}{T}^{abc}/2$. Splitting $N$ into its traceless part and trace gives the second adjoint representation of SU(3) and the singlet. These maps use invariant tensors, so they commute with the group action and explicitly realize the four irreducible summands.