A finite nonempty vertex set in an amenable graph is a Følner set with tolerance $\epsilon$ if its external boundary has size at most $\epsilon$ times its volume. A graph Følner sequence consists of such sets with tolerances tending to zero. For a bounded-degree graph, every fixed-thickness boundary layer also has negligible relative volume along this sequence.

An alternating trilinear form is a scalar-valued map, linear in three arguments, which vanishes whenever two arguments agree. In characteristic zero, interchanging two arguments negates its value, while cyclic permutation preserves it. For a symmetric bilinear form $B$ on a Lie algebra, the tensor $B([x,y],z)$ is alternating exactly when $B$ is an invariant bilinear form on a Lie algebra.

A connected finite hypergraph of three-element supports, with a three-cycle on each support, generates the full alternating group. To merge a new triple meeting an old alternating support in two letters, conjugate over unordered pairs of old letters. For one-letter overlap compare $(abc)$ and $(a^{\prime }bc)$: their quotient is $(ab{a}^{\prime })$, reducing to two two-letter-overlap steps.

Except in degree six, the $3$-cycles are characterized among order-three elements by the largest centralizer. Their cyclic subgroups correspond to three-element supports. Two different such subgroups commute exactly for disjoint supports; with one common point their generator products have order five, and with two common points those products have order two or three. Thus an automorphism induces an automorphism of $J(n,3)$. The maximal cliques of a Johnson graph reconstruct a permutation of the points. After undoing that permutation, an automorphism fixes every cyclic $3$-subgroup. Product orders force its choices of generator inversion to be consistent on the connected graph, and inversion of every $3$-cycle would reverse a noncommuting product. Hence it fixes all $3$-cycles, which generate $A_n$.

Every maximal proper subgroup normalizes one of its minimal normal subgroups. As $A_5$ has no proper nonabelian simple subgroup, that subgroup is elementary abelian. Its possible types are $C_2,C_2^2,C_3,C_5$. Their normalizers have orders $4,12,6,10$ respectively. The order-four normalizer lies in an order-twelve point stabilizer; the other three types are maximal and each forms a single conjugacy class. There are five, six and ten maximal subgroups of orders twelve, ten and six, respectively.

For an optically thick alpha disk dominated by gas pressure, vertical hydrostatic equilibrium gives $H\propto T^{1/2}/\Omega$ and $\rho \propto \Sigma \Omega T^{-1/2}$. The specified opacity therefore scales as $\kappa \propto {\Sigma }^{-3/2}{\Omega }^{-3/2}{T}^2$. Viscous heating scales as $Q^{+}\propto \alpha \Sigma T\Omega$, while radiative diffusion gives $Q^{-}\propto T^4/(\kappa \Sigma )\propto {\Sigma }^{1/2}{\Omega }^{3/2}{T}^2$. Their balance gives $T\propto \alpha {\Sigma }^{1/2}{\Omega }^{-1/2}$, hence $\nu \propto {\alpha }^2{\Sigma }^{1/2}{\Omega }^{-3/2}$. Fixed $\alpha$ and Keplerian rotation yield the displayed law.

The local thermal stability of an accretion disk follows because, at fixed surface density and radius, cooling has logarithmic temperature slope two while heating has slope one. The viscous stability of an accretion disk follows from ${\partial }_{\Sigma }(\nu \Sigma )>0$ on the thermal-equilibrium branch. Both conclusions apply within this pressure, opacity and transport closure.

Let $\mathcal{R}=k/({\mu }_m{m}_p)$ and choose $h^6=3\alpha \kappa {\Sigma }^2{\mathcal{R}}^4/(16\sigma {\Omega }^5)$. Set $z=h\zeta$, $\rho =(\Sigma /h)D$, $p=\Sigma {\Omega }^{2}hP$, $T=({\Omega }^2{h}^2/\mathcal{R})\Theta$ and $F=\alpha \Sigma {\Omega }^3{h}^{2}Q$. Vertical hydrostatic equilibrium, viscous dissipation, radiative diffusion and the ideal gas law reduce to $P^{\prime }=-D\zeta$, $Q^{\prime }=9P/4$, ${\Theta }^{\prime }=-DQ/{\Theta }^3$, $P=D\Theta$. Midplane symmetry gives $Q(0)=0$; a regular radiative-zero surface has $P=\Theta =0$ at a universal ${\zeta }_s$, with $2{\int }_0^{{\zeta }_s}Dd\zeta =1$. All physical parameters have disappeared. Uniqueness of this normalized boundary-value problem makes every physical profile a rescaling of one dimensionless solution.

// Target: astrophysics.bigb

Let $\mathcal{R}=k/({\mu }_m{m}_H)$, take constant opacity $\kappa$, and prescribe the dynamic viscosity $\mu =\alpha p_g/\Omega =\alpha \mathcal{R}\rho T/\Omega$ in a Keplerian disk. The pressure supporting the disk may contain both gas pressure and radiation pressure. Combining viscous dissipation and radiative diffusion gives ${\partial }_m^2{T}^4+A\Omega T=0$ in the vertical mass coordinate of a disk, where $A=27\alpha \kappa \mathcal{R}/(16\sigma )$. Set $\zeta =2m/\Sigma$, $T=T_{\ast }t(\zeta )$ and $T_{\ast }^3=A\Omega {\Sigma }^2/4$. With a radiative-zero disk surface, $t$ satisfies $(t^4{)}^{\prime \prime }+t=0$, $t^{\prime }(0)=t(1)=0$. Writing $I_t={\int }_0^{1}td\zeta$, the density-weighted mean kinematic viscosity is $\bar{\nu }=\alpha \mathcal{R}{T}_{\ast }{I}_t/\Omega$, giving the displayed formula. This calculation does not assume gas-pressure domination: the closure depends on gas pressure while hydrostatic support can be mixed.

The three principal aqueous glycine forms differ by loss of one proton at each step:Call these glycine forms $G^{+}$, $G^0$ and $G^{-}$, respectively: a cation, a zwitterion and an anion, with net electric charges $+e$, $0$ and $-e$, where $e$ is the elementary charge. $G^0$ specifically denotes the zwitterion, not an uncharged amino/carboxyl drawing. The first dissociation removes the carboxyl proton; the second removes a proton from the positively charged nitrogen. The chemical equilibrium changes are reversible, not three permanently separate kinds of glycine. ^[ref]

For all calculations and both diagrams here, explicitly stipulate the pKa values $\mathrm{p}{K}_{a1}=2.35$ and $\mathrm{p}{K}_{a2}=9.78$. These are illustrative model inputs, not measurements from the glycine titration experiment or a claim of universal constants. Actual acid dissociation constants depend on temperature and on the convention used for thermodynamic activity; apparent concentration-based values also depend on ionic strength. For comparison, the inspected textbook tabulates $2.34$ and $9.60$ for glycine, which would give a different isoelectric point. ^[ref]

A proteinogenic amino acid is incorporated into proteins during translation. The conventional twenty include glycine, alanine, methionine and proline; they are a subset of amino acids, not a definition of the whole family. ^[ref]

A sequence converges completely to $X$ when ${\sum }_n\mathrm{Pr}(|X_n-X|>ϵ)<\infty$ for every $ϵ>0$. The first Borel-Cantelli lemma implies Almost sure convergence. For independent $X_n$, any limit in Probability is almost surely constant; the second Borel-Cantelli lemma then gives the converse from almost sure to complete convergence. independence is essential: $X_n=1_{\{U\le 1/n\}}$ for one uniform random variable $U$ converges almost surely to zero but its exceedance probabilities have a divergent sum.

A locally decomposable complex $n$-form $\Omega$ on a real $2n$-manifold, with $\Omega \wedge \overline{\Omega }$ nowhere zero, determines an almost complex structure. The intrinsic rank-$n$ subbundle $W=\{\alpha \in T^{\ast }M\otimes \mathbb{C}:\alpha \wedge \Omega =0\}$ is locally spanned by the factors of $\Omega$. The nonvanishing condition gives $T^{\ast }M\otimes \mathbb{C}=W\oplus \overline{W}$. Set $J^{\ast }=i$ on $W$ and $J^{\ast }=-i$ on $\overline{W}$. This commutes with conjugation, so comes from a unique real bundle endomorphism with square $-I$.

A maximal algebraic torus is an algebraic torus not properly contained in another algebraic torus of the ambient algebraic group. Over an algebraically closed field, maximal algebraic tori in a connected reductive algebraic group are conjugate. A Borel subgroup contains one, and it splits as $B=T⋉R_u(B)$ with its unipotent radical.

For $G{L}_n$, the invertible diagonal matrices form a maximal algebraic torus isomorphic to ${\mathbb{G}}_m^n$. Choosing a different basis conjugates this example. The dimension of a maximal algebraic torus is the absolute rank of the connected reductive group.

Over a general field, geometric conjugacy does not assert conjugacy by rational points. In $S{L}_2(\mathbb{R})$, the split diagonal algebraic torus has noncompact real points, while the rotation norm-one algebraic torus has compact real points, so they cannot be conjugate by a real matrix. Both become maximal split algebraic tori after passing to the complex numbers.^[ref]

The multiplicative group scheme represents units: its $R$-points are $R^{\times }$. Multiplication is specified by $t\mapsto t\otimes t$ on the coordinate ring. It is the principal open subset $D(t)$ of the affine line and the one-dimensional split algebraic torus.

// Target: ringed-space.bigb

An isogeny is a surjective morphism of algebraic groups with finite kernel. For abelian varieties it is a finite surjective group morphism. An isogeny of elliptic curves is its one-dimensional abelian-variety specialization.

The scheme $C^{(d)}$ parametrizes effective degree-$d$ Cartier divisors on a smooth algebraic curve. Near a Cartier divisor $\sum m_i{P}_i$, its completed local ring is a power-series ring in the elementary symmetric coordinates of $m_i$ local parameters at each $P_i$. It is therefore smooth of dimension $d$, even when the characteristic of a field divides $d!$. For projective $C$ it is projective.

An affine algebraic curve is an affine variety of Krull dimension one. An irreducible affine curve has a finitely generated one-dimensional domain as its coordinate ring. Quotienting this ring by a nonzero element gives a zero-dimensional Noetherian ring, hence an Artinian ring. Consequently a nonzero regular function has finitely many zeros. No nonsingularity assumption is part of the definition.

For an affine plane algebraic curve $F(x,y)=0$, a point over an algebraic closure is nonsingular when the displayed condition holds. Its Zariski tangent space has dimension one, and a local coordinate can be chosen in the direction of a nonzero partial derivative. A smooth algebraic curve has all points nonsingular. Over a complete local field, the Hensel lemma lifts a nonsingular residue point by solving for a coordinate with unit partial derivative.

An algebraic curve which is projective, reduced and irreducible. Over an algebraically closed field, its only global regular functions are constants and its arithmetic genus is $h^1({\mathcal{O}}_C)\ge 0$.

The smooth dual consists of the smooth vectors of the algebraic contragredient representation. Their union is a vector subspace because finite intersections of open stabilizers remain open; it is invariant because translation conjugates stabilizers.