On the new subspace of cusp forms, a common eigenspace for the Hecke operators away from the level has dimension one. Distinct newforms, even at different primitive levels, have different systems of these eigenvalues. Consequently the commuting bad-prime operators also act by scalars on these lines. This is the multiplicity-one part of Atkin-Lehner-Li theory, not a consequence of normal operator diagonalization alone. See Stein's statement of the theorem.

For a newform of weight $k\ge 2$, level $M$ and trivial Dirichlet character, the bad-prime Hecke operator $U_p$ has the displayed possibilities. The local trace relation on the new subspace of cusp forms gives $U_p=-p^{k/2-1}{W}_p$ when $p\parallel M$, where the normalized local Atkin-Lehner operator squares to one; if $p^2\mid M$, the new-vector relation gives $U_p=0$. In particular $|a_p|<p^{k/2}$, a strict inequality useful for proving irreducibility of old blocks.

Here ${\log }^{+}x=\max (0,\log x)$. The function $m(r,f)$ measures logarithmic proximity of $f$ to infinity. For a finite target $a$, proximity is $m(r,1/(f-a))$. Logarithmic singularities on a circle are integrable.

The number $n(t,f)$ counts poles in $|z|<t$ with multiplicity, and $n(0,f)$ is the pole order at zero. Thus $N(r,f)=n(0,f)\log r+{\sum }_{0<|p|<r}\log (r/|p|)$. Applying this to $1/(f-a)$ counts occurrences of the value $a$.

The characteristic of a meromorphic function is the sum of its Nevanlinna proximity function and Nevanlinna integrated counting function. It treats large values and actual poles together. Replacing $f$ by $f-a$ changes the characteristic by at most $\log (1+|a|)$.

For a fixed finite target $a$, the displayed identity compares growth with proximity to and occurrences of $a$. If $g(z)=c{z}^{\nu }(1+O(z))$ near zero, where $\nu$ is its signed zero order, the exact reciprocal identity is $T(r,g)-T(r,1/g)=\log |c|$. It follows by factoring out zeros and poles on a disc, applying the mean value property for harmonic functions to the remaining nonvanishing factor, and using ${\log }^{+}u-{\log }^{+}(1/u)=\log u$. Translation changes $T$ by a bounded amount, proving the general assertion.

For a nonconstant meromorphic function, its deficiency at a point of the Riemann sphere is the displayed asymptotic proportion missing from the full Nevanlinna integrated counting function. The Nevanlinna first main theorem bounds $N(r;a)$ by $T_f(r)+O(1)$ and identifies deficiency with the lower limiting normalized proximity. An omitted value has deficiency one.

The Nevanlinna ramification index is the normalized lower limiting integrated excess multiplicity over a target value, with pole orders used for infinity. It is an asymptotic value-distribution quantity, distinct from the integer ramification index of a holomorphic map at one source point. It can be positive even when the target has positive Nevanlinna deficiency.

For a nonconstant meromorphic function on the plane, the displayed estimate holds outside an exceptional set of radii of finite linear measure, for large $r$. The logarithmic derivative has only simple principal parts at zeros and poles; Poisson-Jensen formula estimates and control of increasing auxiliary growth functions produce the logarithmic error. It supplies the error term in the Nevanlinna second main theorem.

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.