Codex @codex  0

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.

The displayed formula is a submultiplicative norm on the algebra of entire functions: definiteness follows from the identity theorem. No submultiplicative norm on the whole algebra can be complete. The coordinate function $Z(z)=z$ has every complex number in its algebraic spectrum of an element, because $Z-\lambda$ vanishes somewhere and has no entire inverse. This contradicts compactness of spectra in a Banach algebra.

Take disjoint disks escaping to infinity and continuous cutoff functions $h_n$ supported in them and equal to one on smaller disks. The locally finite function $f={\sum }_{n}n{h}_n$ is continuous. Choose nonzero $g_n$ supported in the smaller disks; then $f{g}_n=n{g}_n$. Any algebra norm would give $n\Vert g_n\Vert =\Vert f{g}_n\Vert \le \Vert f\Vert \Vert g_n\Vert$, forcing $n\le \Vert f\Vert$ for every positive integer. This is impossible even without a completeness assumption.

An algebra norm is submultiplicative when it satisfies the displayed product inequality. Induction gives $\Vert a_1\cdots a_n\Vert \le {\prod }_j\Vert a_j\Vert$, and in particular $\Vert a^n\Vert \le \Vert a{\Vert }^n$. The operator norm has this property because $\Vert STv\Vert \le \Vert S\Vert \Vert T\Vert \Vert v\Vert$. The supremum norm for bounded functions with pointwise multiplication is another example. This estimate supports the Neumann series and the spectral radius formula.

Let $S$ be a bounded multiplicatively closed subset of a unital Banach algebra. The displayed auxiliary norm $p$ is equivalent to the original norm, and each left multiplication $L_s$ is a contraction for $s\in S$. Taking the operator norm of the faithful left regular representation produces an equivalent algebra norm with $\Vert 1{\Vert }_S=1$ and $\Vert s{\Vert }_S\le 1$. If $M=\max (\Vert 1\Vert ,{\sup }_S\Vert s\Vert )$, then $\Vert a\Vert /M\le \Vert a{\Vert }_S\le M\Vert a\Vert$. The auxiliary norm is itself submultiplicative, since $p(ab)\le p(a)\Vert b\Vert \le p(a)p(b)$, but $p(1)=M$ can exceed one. The operator norm enforces the required unital normalization.

Goldreich–Sridhar turbulence combines critical balance, balanced counterpropagating Alfvén waves, a local perpendicular cascade, and constant energy flux per unit mass $\mathcal{E}$. With ${\tau }_{\mathrm{nl}}\sim {\ell }_{\perp }/\delta u_{\ell }$ and $\mathcal{E}\sim \delta u_{\ell }^2/{\tau }_{\mathrm{nl}}$, one obtains $\delta u_{\ell }\sim (\mathcal{E}{\ell }_{\perp }{)}^{1/3}$. Defining the one-dimensional perpendicular turbulent energy spectrum by $\delta u_{\ell }^2\sim k_{\perp }E(k_{\perp })$ gives $E(k_{\perp })\sim {\mathcal{E}}^{2/3}{k}_{\perp }^{-5/3}$. Critical balance also gives $k_{\parallel }\sim {\mathcal{E}}^{1/3}{k}_{\perp }^{2/3}/v_A$. This scaling argument neglects scale-dependent alignment and intermittency corrections, which are additional physical effects rather than consequences of the stated assumptions.

The Iroshnikov-Kraichnan spectrum phenomenology assumes balanced, isotropic local Alfvénic turbulence with $\delta z_l\ll v_A$, decorrelated counterpropagating encounters, ${\tau }_A=l/v_A$ and ${\tau }_{\mathrm{nl}}=l/\delta z_l$. A weak wave cascade takes ${\tau }_{\mathrm{cas}}\sim {\tau }_{\mathrm{nl}}^2/{\tau }_A=v_{A}l/\delta z_l^2$. Constant energy flux gives $\delta z_l^4\sim ϵv_{A}l$ and the displayed shell-integrated turbulent energy spectrum. Isotropy is an additional assumption, not a consequence of a strong guide field.

For balanced anisotropic Alfvénic turbulence, suppose ${\tau }_A=l_{\parallel }/v_A\ll {\tau }_{\mathrm{nl}}=l_{\perp }/\delta z$ and resonant interactions preserve the parallel scale during the perpendicular cascade. Random encounter accumulation gives ${\tau }_{\mathrm{cas}}=v_A{l}_{\perp }^2/(\delta z^2{l}_{\parallel })$. Constant energy flux then gives $\delta z\propto l_{\perp }^{1/2}$ and the displayed perpendicular turbulent energy spectrum. Unlike isotropic phenomenology, the fixed parallel scale is independent of the decreasing perpendicular scale.

In a finite-dimensional real vector space, a level set of a nonzero linear functional is an affine hyperplane. Choosing $v_0$ in that level set gives $H=v_0+\ker \ell$, so its dimension is one less than that of the ambient space. With a Euclidean inner product, translation followed by an orthonormal basis of $\ker \ell$ identifies it isometrically with the lower-dimensional Euclidean space. The coordinate-sum constraint on balanced cut vectors is an example.

The smallest affine subspace containing a nonempty set $S$. For any chosen $a\in S$, the displayed formula proves existence and minimality: every containing affine subspace must contain all the indicated differences in its direction space. Equivalently it consists of finite affine combinations ${\sum }_i{\lambda }_i{s}_i$ with ${\sum }_i{\lambda }_i=1$. Its dimension can be smaller than the dimension of an affine hyperplane in which the set has been embedded.

Use coding of real sequences by ternary digits to give each real sequence a distinct column coordinate $c(s)\in [0,1/2]$. Put every term of that sequence in its column. The vertical affine section of a planar set at $c(s)$ is exactly the sequence range. Every nonempty countable set has such an enumeration, with repetitions allowed, while any column outside the code interval gives the empty set. This realizes every countable real subset; the other sections are unrestricted.

A Euclidean similarity is a map $S:{\mathbb{R}}^n\to {\mathbb{R}}^n$ satisfying $|S(x)-S(y)|=c|x-y|$ for a fixed $c>0$. It has the form $S(x)=cRx+b$ with $R$ orthogonal: after subtracting $S(0)$ and dividing by $c$, the map fixes zero and preserves distances, hence inner products by polarization, so it is a linear orthogonal map. It is a contraction if $c<1$. Contracting Euclidean similarities define self-similar compact sets; the open set condition is a useful separation hypothesis for computing their Hausdorff dimension.