For a nonconstant meromorphic function on the unit disc and $q\ge 3$ distinct spherical values, this version of the Nevanlinna second main theorem holds outside a set $E$ with ${\int }_{E}dr/(1-r)<\infty$, as $r\uparrow 1$. The truncated Nevanlinna counting function counts distinct preimages rather than multiplicity. The boundary-distance error is essential and need not be small relative to the characteristic. It follows from the disk Nevanlinna logarithmic derivative lemma and the growth lemma in the variable $\log (1/(1-r))$. The disk error estimate and weighted exceptional-set argument appear in mathweb.tifr.res.in/Documents/Publications/Lectures/tifr17.pdf .

For a nonnegative increasing continuously differentiable function not identically zero, choose $x_0$ with $\varphi (x_0)>0$. On $I=\{x\ge x_0:{\varphi }^{\prime }>{\varphi }^2\}$, $1<{\varphi }^{\prime }/{\varphi }^2$. Thus $|I|\le {\int }_{x_0}^{\infty }{\varphi }^{\prime }/{\varphi }^2\le 1/\varphi (x_0)$. Adding the finite initial interval controls all exceptions. For the zero function there are none. Taking logarithms turns derivative growth into logarithmic auxiliary-function growth in value-distribution estimates.

The truncated counting function counts each distinct $a$-point of a meromorphic function once, rather than with its local multiplicity. It uses the same logarithmic radial weights as the Nevanlinna integrated counting function, including an origin contribution $\log r$ when appropriate. The difference $N-\overline{N}$ measures integrated excess multiplicity.

The rational composition law for the Nevanlinna characteristic gives $T_f(r)=3r/\pi +O(1)$. Its zeros are precisely $2\pi i\mathbb{Z}$, all of multiplicity two, so $N(r;0)=2r/\pi +O(\log r)$ and $\overline{N}(r;0)=r/\pi +O(\log r)$. Hence both its Nevanlinna deficiency and Nevanlinna ramification index at zero equal one third.

For a degree-$d$ rational function $U$ of the Riemann sphere, choose relatively prime homogeneous polynomials $(P,Q)$ of degree $d$. Their joint norm on the unit three-sphere has positive lower and finite upper bounds. Thus $\log \Vert (P,Q)(v)\Vert -d\log \Vert v\Vert$ is bounded. Averaging over a circle using a nonvanishing entire homogeneous lift of a meromorphic function proves the displayed law for the spherical characteristic. Jensen's formula and $0\le \frac{1}{2}\log (1+x^2)-{\log }^{+}x\le \frac{1}{2}\log 2$ show that it also holds for the ordinary Nevanlinna characteristic.

The growth order of a meromorphic function is the upper power exponent of its Nevanlinna characteristic, with constants assigned order zero. It is different from the local signed order at a zero or pole. A Nevanlinna integrated counting function has an analogous growth order. The Nevanlinna first main theorem bounds every counting order by the function order.

With the standard string oscillator algebra and $G_0={\sum }_m{\alpha }_m\cdot {\psi }_{-m}$, pairing positive and negative modes gives $G_0^2={\alpha }_0^2/2+{\sum }_{m>0}({\alpha }_{-m}\cdot {\alpha }_m+m{\psi }_{-m}\cdot {\psi }_m)$. This is the vacuum-normal-ordered Ramond sector generator $L_0^R$, so $\{G_0,G_0\}=2L_0^R$. Rescaling the superconformal current rescales its square quadratically; its normalization cannot be ignored when comparing it with a fixed Virasoro generator.

The graph $z=x^2+y^2$ in Euclidean three-space has the displayed induced Riemannian metric and positive Gaussian curvature. It is smooth at its vertex; the polar-coordinate singularity is only a coordinate artifact. The metric dominates the Euclidean plane metric, so upward stability of Riemannian completeness proves completeness. It has infinite diameter. In dimension two Ricci curvature equals Gaussian curvature times the metric, so this is a counterexample to compactness under merely pointwise positive Ricci curvature. Its curvature tends to zero at infinity.

Here $\psi$ is a nontrivial additive character of the finite field. For nontrivial $\chi$, substituting $x=ty$ in the squared absolute value givesThe inner sum is $q-1$ at $t=1$ and $-1$ otherwise; multiplicative character orthogonality completes the identity.

When $\chi ,\eta ,\chi \eta$ are nontrivial, changing variables in a product of Gauss sums of finite-field characters gives $G(\chi )G(\eta )=J(\chi ,\eta )G(\chi \eta )$, hence $|J|=\sqrt{q}$. If at least one character is nontrivial and their product is trivial, or the other character is trivial, the absolute value is one. Thus all non-main terms in a two-character cyclotomic intersection count are $O(\sqrt{q})$.

The first coefficient of the alpha tensor is ${\alpha }_{ij}^{(1)}={ϵ}_{ikl}⟨u_k{\partial }_j{u}_l⟩$. Its contraction with the Levi-Civita symbol isby incompressibility and vanishing averages of total derivatives. Every antisymmetric second-rank tensor in three dimensions is detected by this contraction, so the coefficient is a symmetric matrix. The proof needs no Fourier analysis.

// Target: fluid-mechanics.bigb

For $\mathbf{C}=\mathbf{u}\times (\mathbf{B}\cdot \nabla )\mathbf{u}$, the cross product identity $(\mathbf{u}\times \nabla \times \mathbf{C}{)}_i=u_j{\partial }_i{C}_j-u_j{\partial }_j{C}_i$ and integration by parts give ${\mathcal{E}}_i^{(2)}=-⟨{\partial }_i{u}_j{C}_j⟩$. Its alpha tensor isInterchanging $i,j$ reverses the cross product, showing that the coefficient is an antisymmetric matrix. Such a coefficient represents a turbulent magnetic pumping contribution rather than the symmetric part of the alpha effect.

// Target: fluid-mechanics.bigb

For independent exponential distributions with means $\lambda e^{\psi x_i}$ and a nonconstant fixed design, put $A(\psi )={\sum }_i{Y}_i{e}^{-\psi x_i}$. The constrained maximum-likelihood estimate is ${\tilde{\lambda }}_{\psi }=A(\psi )/n$ and the profile log-likelihood is ${\ell }_p(\psi )=-n\log [A(\psi )/n]-\psi {\sum }_i{x}_i-n$. Use the ancillary statistic $a_i=\log Y_i-\log \hat{\lambda }-\hat{\psi }{x}_i$ to specify sample coordinates. At fixed $\hat{\psi },a$, the Jacobian determinant $\partial {\tilde{\lambda }}_{\psi }/\partial \hat{\lambda }={\tilde{\lambda }}_{\psi }/\hat{\lambda }$, while $j_{\lambda \lambda }(\psi ,{\tilde{\lambda }}_{\psi })=n/{\tilde{\lambda }}_{\psi }^2$. Therefore the modified profile likelihood satisfiesThe adjustment is independent of $\psi$. Keeping only the observed information adjustment would miss this cancellation.

For independent Inverse Gaussian distributions with shape $\psi$ and mean $\lambda$, the constrained and full nuisance maximum-likelihood estimates both equal $\bar{Y}$, giving a unit sample-coordinate Jacobian determinant. The nuisance observed information there is $n\psi /{\bar{Y}}^3$. Hence subtracting half its logarithm from the profile log-likelihood replaces the coefficient $n/2$ of $\log \psi$ by $(n-1)/2$. For a nonconstant sample the modified fit is $(n-1)/({\sum }_i{Y}_i^{-1}-n/\bar{Y})$.

Conjugate the collocation matrix by the diagonal matrix with entries $(-1{)}^i$. The resulting negative-off-diagonal matrix has a nonnegative inverse, as a convergent Neumann series shows. The absolute inverse row sums are therefore the entries of its inverse applied to the all-ones vector. They solve $4u_i-u_{i-1}-u_{i+1}=6$ with $u_0=u_{n+1}=0$, whose solution is displayed. The maximum occurs at the central row or central two rows and equals the inverse operator norm on ${\ell }^{\infty }$. Consequently the associated spline interpolation operator has norm at most three independently of dimension.

Cover the sphere by complements of its two poles. Each open set is contractible, and their intersection deformation retracts to the equatorial sphere. The Poincaré lemma and homotopy invariance of de Rham cohomology make the two neighboring groups in the Mayer--Vietoris sequence for de Rham cohomology zero, so its connecting map is an isomorphism. Its inverse takes a top-degree form to the difference of its local primitives on the equator, up to the connecting-map sign convention.

Lorentz generators produce infinitesimal Lorentz transformations. In four dimensions they split into three rotation and three boost generators. Their action on supercharges uses the appropriate spinor representation, while their action on translations is the mixed Poincare algebra commutator. Signs in component formulas must agree with whether the generators act on states, operator columns or dual rows.

Suppose the mapped increments $K_{s,t}$ have real-centred enclosing radii bounded by a nondecreasing modulus $\omega (t-s)\to 0$. Nested closures define the unique growing real point ${\xi }_s$. For $v\in K_{t,t+h}$, its preimage $z=g_{K_{s,t}}^{-1}(v)$ belongs to $K_{s,t+h}$. The uniform displacement estimate for a mapping-out function gives $|v-z|\le C\omega (t-s)$, while $|z-{\xi }_s|\le 2\omega (t-s+h)$. Letting $h\downarrow 0$ with $h\le t-s$ proves the displayed estimate. It proves uniform continuity of the Loewner transform on each bounded time interval without evaluating a conformal map on a rough boundary.

For Euclidean vectors $x_i$ of length at least one, their labeled subset sums can be partitioned into pairwise distance-at-least-one blocks, with $\left(\genfrac{}{}{0ex}{}{n}{r}\right)-\left(\genfrac{}{}{0ex}{}{n}{r-1}\right)$ blocks of size $n-2r+1$. To extend a separated block by a vector $x$, choose its member $y$ minimizing inner product with $x$. The translated block together with $y$ remains separated, since $(z+x-y)\cdot x\ge \Vert x{\Vert }^2$. The untransformed block with $y$ removed remains separated too. Block sizes change from $\ell$ to $\ell +1,\ell -1$, the same recurrence as a symmetric chain decomposition of a Boolean lattice. There are exactly $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$ nonempty blocks.

For real coefficients of absolute value at least one, an open interval of length four contains at most the sum of the two largest binomial coefficients of signed sums. The corresponding subset family has no three-element inclusion chain. Its Lubell mass is therefore at most two, giving the bound by allocating mass to the two largest levels. Unit coefficients attain it at a center midway between the two chosen sum levels.