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A morphism of schemes is étale when it is flat, unramified and locally of finite presentation. For a morphism between smooth varieties of the same dimension, an invertible differential gives this property locally. A finite étale morphism over an algebraically closed field has reduced finite fibres, and their cardinalities equal its degree when the source and target are connected.

A morphism of schemes is smooth when it is locally of finite presentation, flat and has geometrically regular fibres. Smooth fibres alone do not imply that a morphism is smooth: flatness and the dimension behaviour matter as well. The Abel map of an algebraic curve has smooth projective space fibres whose dimensions can jump.

A formal scheme is a locally ringed space with a sheaf of topological rings that is locally a formal spectrum of a complete adic ring. An ordinary scheme is the special case of the discrete topology. A formal completion of a scheme retains all infinitesimal neighbourhoods of a closed subscheme, rather than just that subscheme.

// Target: ringed-space.bigb

Given schemes $X_i$ and isomorphisms between designated open subschemes satisfying inverse and cocycle conditions, identify the underlying open spaces and glue their structure sheaves. The resulting locally ringed space is locally one of the $X_i$, hence a scheme. The same construction works for formal schemes and topological structure sheaves. Compatible morphisms out of the charts glue uniquely.

// Target: ringed-space.bigb

The functor of points describes a scheme through its morphisms from test schemes. On commutative rings, it is covariant because a ring homomorphism reverses the corresponding affine-scheme map. A functor on rings is represented by $X$ in the algebraic-geometric sense when it is naturally isomorphic to $h_X$. The representing scheme need not be affine; this is different from demanding a representable functor in the category of rings itself.

// Target: module-theory.bigb

Let a real measurable Fourier transform equal one on $|t|<2\pi /3$, vanish on $|t|\ge 4\pi /3$, and satisfy the displayed complementary identity on the positive transition band. On a fundamental interval, either only the central translate is nonzero, or exactly the two complementary translates contribute. Thus orthonormal translates and Fourier periodization holds. The periodic mask defined on $[-\pi ,\pi ]$ by $m(t)=f(2t)$ for $|t|<2\pi /3$ and $m(t)=0$ otherwise satisfies the scaling refinement equation in Fourier form. A concrete even transition is $f(t)=\cos (\pi (3|t|/(2\pi )-1)/2)$ on the transition bands. For complex-valued $f$, ordinary squares cannot replace absolute squares: transition values $i$ and $\sqrt{2}$ have squares summing to one but squared moduli summing to three.

For a matter power spectrum $P(k)=A{k}^n$ at a fixed epoch, the Poisson equation gives potential Fourier amplitudes proportional to ${\delta }_k/k^2$, and the growing-mode peculiar velocity is proportional to ${\delta }_k\mathbf{k}/k^2$. With a spherical top-hat window function, their variances are proportional to $\int k^{n-2}W(kR{)}^{2}dk$ and $\int k^{n}W(kR{)}^{2}dk$, respectively. Rescaling $kR$ gives the displayed powers when the integrals converge: $1<n<5$ for the averaged potential and $-1<n<3$ for the bulk velocity. Absolute averaged potential has infrared sensitivity; band-limited fluctuations or potential differences can have different convergence requirements. The density variance converges only for $-3<n<1$, so these three absolute averages cannot all converge for one unbroken power law at every scale.

A supersymmetry transformation mixes the bosonic and odd coordinates of superspace. A bare derivative with respect to an odd coordinate therefore acquires a spacetime-derivative term under the coordinate Jacobian and is not a covariant spinor superfield. Supersymmetric covariant derivatives add terms linear in the other odd coordinate; their graded anticommutation with the supercharges makes their application to a scalar superfield transform covariantly.

Let $c=\sqrt{a^2+b^2}$ with independent $a,b\sim N(0,{\Gamma }^2)$, and let $D\sim N(0,{\Lambda }^2)$ be independent of them. Then $c$ has Rayleigh distribution and $\alpha =|D|/c$ has the displayed cumulative distribution function for $A\ge 0$ and density ${\Lambda }^2\Gamma ({\Gamma }^2{A}^2+{\Lambda }^2{)}^{-3/2}$. Indeed, conditioning on $D$ gives $\mathbb{P}(c\ge |D|/A)=\exp [-D^2/(2{\Gamma }^2{A}^2)]$, and a Gaussian integral completes the calculation. Independence is essential: choosing $D=(\Lambda /\Gamma )a$ has the same Gaussian marginal but makes the ratio bounded by $\Lambda /\Gamma$.

In the long-time growing horizontal-gradient branch, independent vertical shear noises combine as $\cos \varphi d{W}_1+\sin \varphi d{W}_2=dW$. The signed inclination $\beta =k_3{e}^{-ct}/k_0$ satisfies the displayed Ornstein-Uhlenbeck stochastic differential equation. For $c>0$ its stationary density is $\sqrt{c/(\pi g^2)}{e}^{-c{B}^2/g^2}$, with variance $g^2/(2c)$. The Fokker-Planck probability current vanishes at infinity and gives this Gaussian by one integration. At $c=0$ and nonzero noise there is no normalizable stationary density.

For one real phi-fourth theory field, each of the three labeled one-loop four-point channels has symmetry factor $1/2$. In $d=4-2ϵ$, the displayed expression is its amplitude through the finite part. The three poles require $\delta \lambda =3{\lambda }^2/(32{\pi }^2ϵ)$, and differentiating ${\lambda }_B={\mu }^{2ϵ}(\lambda +\delta \lambda )$ at fixed bare coupling gives $\beta (\lambda )=3{\lambda }^2/(16{\pi }^2)+O({\lambda }^3)$ in four dimensions. The minimal subtraction scheme retains $\log (4\pi )-{\gamma }_E$; modified minimal subtraction absorbs that constant. The Feynman-parameter logarithm is left unevaluated.

If no $k$-set were shattered, the Sauer-Shelah lemma would bound the number of traces by ${\sum }_{i<k}\left(\genfrac{}{}{0ex}{}{2k}{i}\right)=(2^{2k}-(\genfrac{}{}{0ex}{}{2k}{k}))/2<2^{2k-1}$. This strict middle-layer gap contradicts the displayed lower bound. Therefore a family with such traces at every even size has unbounded finite VC dimension.

For $r>b$ and $P=|\mathbf{r}-{\mathbf{r}}_0|$, the displayed kernel satisfies $\varphi ={\int }_r^{\infty }ds/[s|s\hat{\mathbf{r}}-{\mathbf{r}}_0|]$. The limit at $b=0$ is $1/r$. Differentiating along $\mathbf{Q}\times {\mathbf{r}}_0$, which preserves $b$, gives $(\mathbf{Q}\times {\mathbf{r}}_0)\cdot {\nabla }_{{\mathbf{r}}_0}\varphi =(\mathbf{Q}\times {\mathbf{r}}_0)\cdot \mathbf{r}/F$. Thus it generates the normalized magnetic scalar potential in the Sarvas formula.

The sampling fraction is the sample size divided by the finite population size. In simple random sampling, it determines how much the finite population correction reduces the variance relative to $S^2/n$.

A finite nonzero Saffman integral gives $E(k)=L{k}^2/(4{\pi }^2)+o(k^2)$ in the standard isotropic spectrum normalization. This follows by expanding the radial spectral transform at small wavenumber. A further analytic term requires additional correlation moments.

With nonzero invariant $L$, a self-preserving integral scale of turbulence gives $L\propto \mathcal{E}{\ell }^3$. Combining this with freely decaying high-Reynolds-number viscous dissipation $ϵ\propto {\mathcal{E}}^{3/2}/\ell$ gives $\mathcal{E}\propto (t-t_{\ast }{)}^{-6/5}$ and $\ell \propto (t-t_{\ast }{)}^{2/5}$. The closure assumptions, not the invariant alone, determine this exponent.

For a nonempty integer set with doubling constant at most $K$, Plünnecke-Ruzsa inequality bounds $|2A-2A|$ by $K^4|A|$. Embed the set modulo a sufficiently large auxiliary prime $q$, randomly dilate it, and select the more populated of two half-intervals of representatives. Differences of two pair sums on that half have absolute value less than $q$, so the lift preserves original relations. Every nonzero original difference is uniformly dilated among nonzero residues; avoiding all residues represented by nonzero multiples of $p$ of absolute value less than $q$ has positive probability when $p>4|2A-2A|$. Reduction modulo $p$ then yields a Freiman 2-isomorphism on the selected half.

For $|A+A|\le K|A|$, one can take $q=8|8A-8A|\le 8K^{16}|A|$. Embed in a much larger prime cyclic group and choose a dilation avoiding the residues of all nonzero multiples of $q$ of absolute value below that prime for every nonzero eightfold difference. A union bound ensures such a dilation. Retain a most-populated ninth-interval of representatives. Eight-term sums then have range shorter than the prime, and reduction modulo $q$ both preserves and reflects eight-term additive relations.

A nonempty finite integer set has a subset of the displayed size Freiman s-isomorphic to a subset of ${\mathbb{Z}}_N$. Embed in a sufficiently large auxiliary prime modulus and randomly dilate. Avoid the residues of every nonzero multiple of $N$ of absolute value less than that prime for all nonzero members of $kA-kA$; a union bound gives a successful dilation. Retain a most-populated one of $k+1$ short intervals of representatives. Its $k$-term sum differences have absolute value less than the auxiliary prime, so reduction modulo $N$ both preserves and reflects relations. Under $|kA-kA|\le C|A|$, this gives $N\le 4C|A|+1$.