A nilpotent thickening is a closed immersion defined by a nilpotent ideal sheaf of a closed subscheme. Its underlying topological map is a homeomorphism, because every prime ideal contains every nilpotent element. For example, quotienting the dual numbers by $(\epsilon )$ changes the structure sheaf without changing the single point of the spectrum.

For a continuous Loewner driving function $U$, $T_z$ is the maximal lifetime of $g_t(z)$ in the complex upper half-plane under the Chordal Loewner equation. A finite lifetime has $\mathrm{Im}{g}_t(z)\to 0$ as $t\uparrow T_z$: otherwise the continuous driver and the ordinary differential equation continue the solution. A swallowed point need not have been visited by the Loewner trace; it may lie in a disconnected pocket of the filled hull.

On the invertible-$F_X$ branch, use $y=(G\chi )/F_X-(GG)F_Y/(2F_X^2)$ and the two-component Grassmann algebra identity $(G\chi {)}^2=-\frac{1}{2}(GG)(\chi \chi )$. Then $y^2=-(GG)(\chi \chi )/(2F_X^2)$, $y^3=y^2\chi =0$ and $y^2{F}_Y-y\chi \chi =0$. These are the components of $Y^3=0$. Generally $y^2$ is nonzero, so cubic nilpotency must not be replaced by quadratic nilpotency. The allowed analytic holomorphic monomials are $1,X,Y,Y^2$.

Translation by a differential one-form $\beta$ is $(x,\xi )\mapsto (x,\xi +{\beta }_x)$. It changes the canonical one-form on a cotangent bundle by ${\pi }^{\ast }\beta$. A closed differential form gives a symplectomorphism; an exact differential form gives a Hamiltonian diffeomorphism when compact support is not required.

A section of the square of the canonical bundle, locally $q=\varphi (z)d{z}^2$; its coefficient changes by the square of the coordinate derivative. A holomorphic quadratic differential has holomorphic coefficient, and its flat metric of a quadratic differential is $|q{|}^{1/2}$.

A section of the cube of the canonical bundle, locally $c=\varphi (z)d{z}^3$ with holomorphic coefficient. The flat coordinates obtained by integrating cube roots have transitions $w\mapsto \zeta w+b$, ${\zeta }^3=1$; general real-linear deformations do not preserve these as holomorphic transitions.

For standard Brownian motion started at zero, $H_a=\inf \{t\ge 0:B_t\ge a\}$ is finite almost surely for $a\ge 0$ andThe Brownian reflection principle proves finiteness. For $u\ge 0$, the Exponential martingale for Brownian motion stopped at $H_a\wedge t$ is bounded by $e^{ua}$. The dominated convergence theorem gives $\mathbb{E}{e}^{ua-u^2{H}_a/2}=1$, and taking $u=\sqrt{2\lambda }$ gives the transform.

For standard Brownian motion, set $H_a=\inf \{t:B_t\ge a\}$ and $T_a=\inf \{t:B_t>a\}$. The Strong Markov property and immediate positive excursions give $T_a=H_a$ almost surely for each fixed $a\ge 0$. However, these level-indexed processes are not indistinguishable stochastic processes. Almost surely the random level $A={\sup }_{t\le 1}{B}_t$ is positive and exceeds $B_1$. It is first attained before time one but is not exceeded until after time one, so $H_A<1<T_A$. The simultaneous-equality event therefore has Probability zero. The strict passage process $T_a$ is the right-continuous choice needed for a subordinator.

A root string of length $\Lambda +1$ gives an irreducible sl2 Lie algebra module $R(\Lambda )$ when restricting the Adjoint representation to the sl2 subalgebra associated with a root. Include also the root's own $R(2)$ and one commuting Cartan direction $R(0)$. For $A_2$ either simple root gives $R(2)\oplus 2R(1)\oplus R(0)$. For $B_2$, the long root gives $R(2)\oplus 2R(1)\oplus 3R(0)$ and the short root gives $3R(2)\oplus R(0)$. For $G_2$, the long root gives $R(2)\oplus 4R(1)\oplus 3R(0)$ and the short root gives $R(2)\oplus 2R(3)\oplus 3R(0)$.

A first-order formula is ZF-equivalent bounded if ZF proves it equivalent, with the same free variables, to a syntactically bounded formula in set theory. It need not itself have only bounded quantifiers. Between transitive models of ZF, use the bounded equivalent and the axioms in each model to prove set-theoretic absoluteness. The syntactic bounded-formula result requires only transitivity; the provable-equivalence extension additionally requires the theory used for the equivalence.

Let a nonnegative kernel for density estimation $K$ have integral one, zero first moment and finite second moment ${\mu }_2(K)$. If a probability density function $f$ has bounded second derivative, its kernel density estimator has pointwise bias at most $h^2{\mu }_2(K)\Vert f^{\prime \prime }{\Vert }_{\infty }/2$. Apply the Taylor theorem with Lagrange remainder to $f(x-hu)$ and integrate: the first-order term vanishes and the remainder is bounded by $h^2{u}^2\Vert f^{\prime \prime }{\Vert }_{\infty }/2$. For $K=1_{[-1/2,1/2]}$, ${\mu }_2(K)=1/12$.

The law of ${\sum }_{j\ge 1}2{\omega }_j{3}^{-j}$, for independent fair binary digits ${\omega }_j$, is supported on the Cantor set. It is invariant and ergodic under $T_3$, as the image of a Bernoulli shift. Each permitted ternary cylinder of length $n$ has measure $2^{-n}$, giving entropy rate $\log 2$. The Host equidistribution theorem makes almost every point a normal number in base $2$, while its ternary digits omit $1$.

For hidden activations $h_j=\sigma (b_j+{\sum }_r{W}_{jr}{x}_r)$ and class probabilities $p_c={\mathrm{softmax}}_c(a+Vh)$, a one-hot label $t$ has log-likelihood $\ell ={\sum }_c{t}_c\log p_c$. Put ${\delta }_c=t_c-p_c$ and ${\Delta }_j=h_j(1-h_j){\sum }_c{V}_{cj}{\delta }_c$. ThenThese follow from the chain rule and give single-observation stochastic gradient descent updates for the negative log-likelihood.

For a finite morphism $f:X\to Y$, the pullback of an ample line bundle is ample. If $f$ is also surjective, ampleness is equivalent in both directions. These results hold for proper schemes over a field, including nonreduced schemes. Cohomological ampleness and the exact finite pushforward prove the forward direction; coherent-sheaf devissage gives finite-surjective descent. In particular, a finite map to projective space makes the pulled-back hyperplane bundle ample.

On a projective scheme, a line bundle is ample exactly when its restriction to every reduced irreducible component is ample. The disjoint union of those components maps finitely and surjectively to the scheme, so finite-surjective descent of the finite pullback of an ample line bundle proves this. The same argument applies to restrictions to any closed subscheme. Thus nilpotent structure does not affect ampleness.

An affine open subscheme is an open subset of a scheme with its restricted structure sheaf, which is itself an affine scheme. Every scheme has an open cover by such subschemes. On an affine scheme, the principal open subschemes give an open basis.

For an affine morphism $f:X\to Y$ and a quasi-coherent sheaf $\mathcal{F}$ on $X$, the direct image sheaf has the same sheaf cohomology on $Y$ as $\mathcal{F}$ on $X$. To prove this, take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$. Its direct images are flasque sheaves. On each affine open $V\subset Y$, the complex of sections of the direct images is $\Gamma (f^{-1}V,{\mathcal{I}}^{\bullet })$. By vanishing of quasi-coherent cohomology on an affine scheme, this augmented complex is exact. Checking on the affine basis proves that $f_{\ast }{\mathcal{I}}^{\bullet }$ resolves $f_{\ast }\mathcal{F}$. Its global sections are identical to those of the original resolution.

For the zero-truncated Poisson distribution, $21_{\{Y\text{ even}\}}$ is the unique integrable nonrandomized unbiased estimator of $p=1-e^{-\theta }$. Indeed unbiasedness gives ${\sum }_{y\ge 1}t(y){\theta }^y/y!=e^{\theta }+e^{-\theta }-2$, and power series coefficients force the parity values. Its variance is $2p-p^2$; clipping it to $1_{\{Y\text{ even}\}}$ gives mean squared error $p/2$, strictly smaller for $0<p<1$.

Normalizing the tableau lines in a compatible real phase convention gives the orthonormal basis formAdmissible swaps have $|d|\ge 2$. Nonadmissible swaps act by $+1$ within a row and $-1$ within a column. The symmetric matrix is an orthogonal involution and is useful for making the unitary representation and its phases explicit.

If $a$ is irrational and $b,c$ are real, the fractional parts of $a{n}^2+bn+c$ form an equidistributed sequence. In the Van der Corput lemma, every nonzero-lag correlation of $e^{2\pi im(a{n}^2+bn+c)}$ is a geometric series with irrational frequency $2mah$ and tends to zero. The Weyl criterion then applies.