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For a profinite group $G$ and a complete discrete valuation ring $I$ with a continuous ${\mathbb{Z}}_p$-algebra structure, an $I$-valued measure is a bounded $I$-linear functional on continuous $I$-valued functions, with the uniform norm. Equivalently it assigns finitely additive values in $I$ to clopen sets. Locally constant functions are uniformly dense, so integration extends uniquely from finite sums. Countable additivity in the usual real-measure sense is not imposed. The coefficient ring may have infinite residue field and need not be compact.

For a measure on a topological space in which compact sets are measurable, inner regularity means that the measure of each measurable set is the supremum of the measures of its contained compact sets. Every Borel probability measure on a compact metric space is inner regular and has outer regularity. Combining the two properties allows approximation of an indicator function by a continuous function in the $L^1$ norm.

A standard Borel space is a measurable space isomorphic to a Borel set of a Polish space, with its Borel sigma-algebra. This measurable regularity permits the usual constructions of random samples and random counting measures, without requiring a particular compatible topology.

For a uniform traction-free sphere, first-order self-gravity is $b=-gx/a$. Static averaging gives $\overline{\sigma }=-\rho gaI/5$. Homogeneous linear elasticity then gives $\overline{e}=-\rho gaI/(15\kappa )$. The boundary strain identity identifies $\overline{e}=u(a)I/a$, proving the displayed positive radius decrease. The local strain is not uniform, and the cancellation of the shear modulus concerns only the mean contraction.

The mean molecular weight per ion counts nuclei rather than nuclei plus free electrons: $\rho /({\mu }_i{m}_u)$ is the ion number density. For species with mass fractions $X_j$ and mass numbers $A_j$, ${\mu }_i^{-1}={\sum }_j{X}_j/A_j$. It is useful when ions provide a thermal ideal gas pressure but electron degeneracy pressure supplies a separate contribution.

In stellar ideal gas equations, the stellar gas constant $\mathcal{R}=k_B/m_u$ converts a dimensionless mean molecular weight $\mu$ into the specific thermal relation $P=\mathcal{R}\rho T/\mu$. Here $m_u$ is the reference atomic mass unit, commonly approximated by the proton mass in simplified hydrogen-helium models. It has units of energy per mass per temperature. It is the molar gas constant divided by the molar mass corresponding to $m_u$, rather than the molar gas constant used with a dimensional molar molecular weight.

The logarithm of the stretch after $N$ independent uniform renovations is a sum of independent identically distributed bounded increments. For singular stretching $e^{\gamma }$, $\gamma =\mathrm{arsinh}(|a|/2)$, the increment has Fourier expansion $r+{\sum }_{n\ge 1}(-1{)}^{n+1}{q}^n\cos (2n\theta )/n$, with $q=\tanh \gamma$ and $r=\frac{1}{2}\log (1+a^2/4)$. Orthogonality proves the displayed variance. The central limit theorem gives an approximate central log-normal distribution of ${\lambda }_N$, not an exact description of extreme tails. For $|a|\ll 1$, both the mean and variance of $\log {\lambda }_N$ are $N{a}^2/8$ to leading order. Direct expansion of fixed-power moments gives $⟨{\lambda }_N^p⟩=\exp [Np(p+2)a^2/16+O(N{a}^4)]$; in particular mean stretch grows faster than typical stretch.

For a ferromagnetic Ising model with coordination number of a lattice $z$, its mean-field self-consistency equation is $M=\tanh (zJM/(k_{B}T))$. Put $A=T_c/T$. For $A\le 1$, the only solution is zero. For $A>1$, strict concavity of $\tanh (AM)-M$ on $M>0$ gives one positive solution and its spin-reversed partner. Expanding $\tanh (AM)=AM-(AM{)}^3/3+O(M^5)$ gives $M^2=3(A-1)/A^3+O((A-1{)}^2)$, proving a continuous onset with order-parameter critical exponent $\beta =1/2$. The Bragg-Williams free energy of the Ising model has a positive quartic coefficient at $T_c$, so these are the stable ordered minima rather than a first-order jump.

For an ideal soap film with equal constant surface tension $\gamma$ on both faces, the effective tension is $\tau =2\gamma$. Choose a unit normal $N$ from the minus side to the plus side, and $\Delta p=p_{-}-p_{+}$. Translating the Young–Laplace equation to $S=-dN$ givesA single interface uses $\tau =\gamma$. An outward-oriented spherical soap bubble has $H=-1/R$, giving $\Delta p=4\gamma /R$. The positive-convex curvature sum in the Young–Laplace equation article is $-2H$. ^[ref]

An open film with the same resting gas on both sides has zero pressure jump. Neglecting film weight, soap film pressure balance gives $H=0$: a minimal surface. A pressurized bubble instead has nonzero mean curvature. These statements assume equilibrium, constant surface tension, and negligible normal viscous stress. Surfactant gradients or motion require the fuller normal stress balance. ^[ref]

The variational outline is short. With fixed boundary, a normal displacement $fN$ gives $\delta A=-2{\int }_{M}HfdA$ by the first variation of area formula. For a closed interface oriented outward, $\delta V={\int }_{M}fdA$, so stationarity of $\tau A-\Delta pV$ gives $-2\tau H-\Delta p=0$. This is a first variation calculation; testing for a stable minimal surface needs the second variation, and deciding a global minimum needs a comparison of competitors.

Scale the catenoid by its waist radius $a>0$ and choose a unit normal with outward radial component:The hyperbolic cosine controls the profile $r(z)=a\cosh (z/a)$. Direct differentiation yields the first fundamental form and second fundamental formThe coordinate tangent vectors are orthogonal, so the meridional and circumferential principal curvatures areThe opposite principal curvatures cancel at every point, not just at the waist. This is precisely why the catenoid is a minimal surface, while its nonzero second fundamental form and negative Gaussian curvature show that it is not a plane.

For a sphere of radius $R>0$, its outward unit normal is $N(p)=p/R$. Applying the shape operator gives $S(v)=-v/R$, so the two principal curvatures coincide:For the lateral surface of a circular cylinder with outward unit normal, moving around a circle turns $N$ at rate $1/R$, while moving along the axis leaves $N$ constant. Thus the circumferential and axial principal curvatures giveHere $K$ is Gaussian curvature. The circular cylinder bends in only one principal direction, so its mean curvature is half that of the equally sized sphere. These formulas concern the smooth lateral surface, excluding any edges where end caps meet it.

The common length scale is $\ell$, equal to the radius $R$ for the sphere and circular cylinder, and to the waist radius $a$ for the catenoid. The mean curvature marker is halfway between the two principal curvature markers. Gaussian curvature has a different unit and is listed separately beneath each panel.

For an $n$-dimensional embedded submanifold, let $A$ denote its vector-valued second fundamental form and $e_1,\dots ,e_n$ an orthonormal basis of its tangent space. This branch uses the trace mean curvature vectorAn alternative averaged mean curvature vector is $\mathbf{H}/n$; on a smooth surface in three-dimensional Euclidean space, these are $2HN$ and $HN$, respectively. Reversing $N$ reverses $H$, so their product is independent of the chosen orientation of a surface. With the trace convention, the first variation of area formula has no additional factor $n$. For an outward-oriented sphere of radius $R$, $\mathbf{H}=-2N/R$: it points toward the centre even though the scalar's sign depends on conventions.

For a metric space with the same metric on both factors, the diagonal is a closed subset of its product. At an off-diagonal point $(x,y)$ let $\delta =d(x,y)>0$. A maximum-metric ball of radius $\delta /3$ misses the diagonal, since a common coordinate $z$ would give $d(x,y)\le d(x,z)+d(z,y)<2\delta /3$. Equality of underlying sets without equality or compatibility of the metrics is a different assertion.

On a compact parameter space, suppose the expected log-likelihood is continuous, uniquely maximized at the true parameter, and the normalized empirical log-likelihood converges uniformly in probability to its expectation. On the compact complement of every neighbourhood of the truth, the expected criterion has a strictly positive gap below its maximum. Uniform approximation by less than one-third of this gap excludes every empirical maximizer from that complement. Identifiability and nonnegativity of Kullback-Leibler divergence supply the unique expected maximizer. The proof works for any measurable choice of maximizer.

For independent $p$-variate multivariate normal observations with unknown mean and unrestricted positive-definite covariance matrix, the centered scatter has rank at most $n-1$. If it is singular, choose the mean to be the sample mean and shrink the covariance on a null direction of the scatter. The quadratic likelihood term stays fixed while the determinant tends to zero, so the density likelihood tends to infinity. Thus a nonsingular covariance maximum likelihood estimate exists only when the centered scatter is positive definite; for a nonsingular Gaussian population this occurs almost surely exactly when $n>p$.

For independent normal distribution observations with mean $\mu >0$ and variance ${\mu }^2$, put $S_1={\sum }_i{X}_i$, $S_2={\sum }_i{X}_i^2$. The Fisher-Neyman factorization theorem makes $(S_1,S_2)$ a sufficient statistic, since the likelihood is proportional to ${\mu }^{-n}\exp [-S_2/(2{\mu }^2)+S_1/\mu ]$. Its logarithmic derivative has numerator $S_2-S_1\mu -n{\mu }^2$. When $S_2>0$, this changes sign once on the positive half-line and gives the displayed global maximum-likelihood estimator. The zero sample is a probability-zero exception with unbounded likelihood as $\mu \downarrow 0$.

In a flow network for treatment facilities, the amount treated at a facility is represented by a directed arc from it to the sink, with that arc's capacity equal to its treatment limit. Untreated fluid may pass through the facility independently of how much is treated there. Thus the treatment limit is not a bound on total incoming flow. Flow conservation at a facility equates incoming flow with outgoing untreated flow plus its sink-arc flow.

For a league in which every remaining game gives exactly one win, first give the target team all its remaining wins, obtaining $W$. Make a node for each unordered pair of other teams, with source capacity equal to their remaining games. Route each game-node flow to one of its two teams, and cap the team's total additional wins at $W-w_i-1$ for a strict target victory. A negative cap immediately rules out success. Otherwise an integral maximum flow saturating every game-source arc specifies the winners of all games and proves feasibility. Each actual game is counted once; duplicating ordered pairs changes the problem.

For a distribution function with tail $1-F(x)=1/(x\log x)$, let $a_n>1$ solve $a_n\log a_n=n$. Then $n(1-F(a_{n}x))\to 1/x$ for $x>0$, so the normalized sample maximum tends to the shape-one Fréchet distribution. The inequalities $n/\log n<a_n<n/(\log n-\log \log n)$ for large $n$ show $a_n\sim n/\log n$, allowing that simpler scale too.

Above any proper normal subgroup of a finitely generated group, a chain of proper normal subgroups has proper union. Otherwise that union would contain the finite generating set, all of which would already lie in one member of the chain. The Zorn lemma therefore provides a maximal proper normal subgroup containing the given one.

// Target: algebra.bigb