Use $l=\sqrt{G{M}_{\ast }r}$ and ${\Omega }^{\prime }=-(3/2)\Omega /r$ in the vertically averaged viscous disk equations. The radial drift is $v_r=-3(\Sigma \sqrt{r}{)}^{-1}{\partial }_r(\sqrt{r}\bar{\nu }\Sigma )$. Conservation of mass gives this nonlinear diffusion equation. It assumes the central rotation law remains fixed and excludes wind/surface torque terms.

For small positive radiation-to-gas ratio, constant opacity, fixed alpha and molecular constants, eliminate $\lambda$ between $\lambda \propto H\Omega \Sigma$ and $\lambda \Sigma \propto H^7{\Omega }^6$. This gives $H^6\propto {\Sigma }^2{\Omega }^{-5}$ and $\bar{\nu }\propto {\Sigma }^{2/3}{\Omega }^{-2/3}$. Keplerian rotation supplies the displayed radial power. Since $\bar{\nu }\Sigma \propto {\Sigma }^{5/3}$, its derivative with respect to surface density is positive and local viscous perturbations diffuse rather than amplify.

Each component of a finite vector measure is absolutely continuous with respect to its variation measure. The Radon-Nikodym theorem gives a vector density $\sigma$. Taking the variation measure of this representation gives $|\mu |=|\sigma ||\mu |$, hence the unit-length property. For a BV derivative this separates the magnitude of variation from its local direction.

For continuously differentiable vector fields, the gradient of their dot product expands as displayed. The contraction of two Levi-Civita symbols gives $[F\times (\nabla \times G){]}_i=F_j{\partial }_i{G}_j-F_j{\partial }_j{G}_i$; adding the directional derivative cancels the second term. Interchanging the fields and using the product rule proves the identity. It does not require either field to be irrotational.

A holomorphic bundle map is a holomorphic map between total spaces of holomorphic vector bundles that covers a holomorphic base map and is complex-linear on each fibre. Over a fixed base, its matrix in holomorphic local frames has holomorphic entries.

For $|f|\le 1$, if $|{\sum }_{n\le X}\mu (n)f(n)|\ge \delta X$ and $U+V\le \delta X/2$, the Vaughan identity for the Möbius function gives a type I sum or a type II sum of modulus at least $\delta X/4$. The first is ${\sum }_{d\le UV}{c}_d{\sum }_{k\le X/d}f(dk)$, with $c_d={\sum }_{bc=d,b\le U,c\le V}\mu (b)\mu (c)$. The second is ${\sum }_{d>V,w>U,dw\le X}{a}_d\mu (w)f(dw)$, with $a_d={\sum }_{c\mid d,c>V}\mu (c)$. Both coefficient sequences are bounded by the divisor function. Thus correlation forces either detection by small-modulus periodic indicators or a large multiplicatively organized bilinear sum, without claiming that $f$ itself is literally periodic or multiplicative.

For Gaussian income of mean $a$, standard deviation $b$ and tradable correlation $\rho$, exponential utility makes the maximum acceptable per-period fee equal to the mean minus the correlated-income hedge cost and the residual variance penalty. Compare optimized utilities with and without the contract; a fee above this level makes the contract strictly inferior. At perfect correlation the residual penalty vanishes, leaving only the mean and hedge-cost terms.

For a positive state-price density $Z$ pricing every attainable payoff, a feasible positive terminal payoff $W^{\ast }$ is optimal at its budget if its marginal utility is $yZ$ for some scalar $y>0$. The pointwise conjugate inequality gives $\mathbb{E}U(W)\le \mathbb{E}\hat{U}(yZ)+yx$ for every competitor; the marginal-utility identity makes equality hold for $W^{\ast }$. Finite states make integrability straightforward. In a complete market an interior optimum also necessarily has this proportionality, by variations preserving the state-price budget.

For a negative, strictly increasing, concave differentiable utility with finite expected utility at every holding, a gain having both signs makes the objective coercive at both ends. At a finite optimum its derivative is zero. Secant domination for expected utility derivatives gives integrability of $|X|U^{\prime }({\theta }_{\ast }X)$; boundedness of $U^{\prime }$ on the remaining compact set gives integrability of the positive marginal utility itself. Its normalization has expectation one and prices the gain at zero.

The unramified Iwasawa module is finitely generated torsion over the Iwasawa algebra of a Zp-extension, for every Zp-extension of a number field. After a finite shift all ramified primes are totally ramified and their number $s$ is constant. Class field theory bounds finite-layer coinvariant modules by ${\mathrm{rank}}_{{\mathbb{Z}}_p}{Y}_{{\Gamma }_n}\le s-1$. The Compact Nakayama lemma proves finite generation, and a positive Iwasawa-module rank would force ranks at least $r{p}^n$, a contradiction. No Leopoldt conjecture is required.

For a fixed set $L$ of $\lfloor \sqrt{n}\rfloor$ vertices, let ${\tau }_L$ be the first step at which they all belong to one graph component. In the uniform random graph process, ${\tau }_L$ lies between $n(\log n-\omega )/4$ and $n(\log n+\omega )/4$ with high probability for every $\omega \to \infty$. Below that scale the marked isolated vertex count has divergent expected value and small relative variance. Above it the logarithmic-regime giant component misses $o(\sqrt{n})$ vertices; exchangeability and a union bound show that none are marked. The label coupling then transfers the two assertions to the process.

For independent $U,V$ uniform on $[0,1]$, real roots occur on $U\le V^2$, a region of probability $1/3$. On that region both roots are nonpositive. Both lie in $[-1,0]$ exactly when $\max (0,2V-1)\le U\le V^2$. Its area is $1/12$, so the conditional probability of both absolute values being at most one is $1/4$. Joint probability density integration turns algebraic discriminant and root constraints into simple planar areas.

Depth-weighted decomposition gives reduced depth $H_r=H_1{H}_2/(H_1+H_2)$ and internal radius $R_d^2=g\prime H_r/f_0^2$. On scale $l$, the ratio of interface available potential energy to baroclinic kinetic energy is of order $(l/R_d{)}^2$. Comparable depths recover $l^2{f}_0^2/(g\prime H_i)$; for strongly unequal depths the thinner layer sets the scale.

For sufficiently large vertex count $n$, normalized hypergraph extremal numbers decrease under vertex sampling and converge to Turán density. At a fixed sample size where the extremal density is close to that limit, excess density forces a positive fraction of samples to contain $H$. Counting incidences of copies and samples proves the displayed lower bound. The unrounded positive lower bound is asserted for sufficiently large $n$.

For the function field $K$ of a curve over a finite full constant field $k$, extend constants to $\overline{k}$. The resulting curve function field is $C_1$ by the Tsen theorem, so every central simple algebra splits there. The finite collection of coefficients of a splitting isomorphism descends to some finite constant extension $k^{\prime }/k$. Regularity makes $K{k}^{\prime }/K$ cyclic, giving a finite cyclic splitting field.

To draw $N(\mu ,1)$ restricted to a positive value, let $a=\Phi (-\mu )$ and return $\mu +{\Phi }^{-1}(a+(1-a)U)$ for uniform $U\in (0,1)$. For restriction to a negative value use $\mu +{\Phi }^{-1}(aU)$. These are inverse transform sampling constructions. Repeatedly drawing an untruncated normal until its sign agrees is also exact, but inefficient for rare signs.

For the triangular current with capillary retention, the initial trailing face is already the maximum for $0<x<L$. Further upslope, the maximum occurs when the moving crest passes. HenceThe final trapped phase has saturation $s$ below this envelope. Its volume is $\varphi s\int M_{\infty }dx=\varphi a{L}^2$, exactly the initial mobile volume. Maximum invaded thickness is distinct from residual saturation: $s$ multiplies occupied pore volume rather than shrinking the geometrical envelope.

The Maurer-Cartan form of the translation-dilation group of the plane has these coefficients in its dilation–translation basis, with $a=1,2$. Left multiplication rescales each coordinate differential and $\rho$ equally, proving they are left-invariant differential forms. Their dual left-invariant vector fields are $\rho {\partial }_{\rho }$ and $\rho {\partial }_{x^a}$. The Maurer-Cartan equation is $d{\lambda }^0=0$, $d{\lambda }^a=-{\lambda }^0\wedge {\lambda }^a$. The tensor ${\sum }_j{\lambda }^j\otimes {\lambda }^j$ is a positive definite left-invariant metric.

For the local flow of a smooth compactly supported vector field and any scalar BV space function $w$, the change of variables formula and polar decomposition of a vector measure giveThe two cofactor matrices are $I\pm tB+O(t^2)$ uniformly. On $|{\sigma }_w|=1$, the linear terms in their Euclidean norm expansions cancel. Integration bounds the remainder by $C{t}^2|Dw|(\Omega )$. The identity includes all components of the decomposition of a BV derivative; it does not require smoothness of $w$ or its jump interfaces.

For exponential distribution claims of mean $\mu$ and compound Poisson distribution count parameter $\lambda$, the total party variance under excess of loss reinsurance is $g(M)=2\lambda {\mu }^2(1-(M/\mu )e^{-M/\mu })$. Its derivative is $2\lambda e^{-M/\mu }(M-\mu )$, so $M=\mu$ is the unique global minimum, with value $2\lambda {\mu }^2(1-e^{-1})$.