For a Weierstrass equation of an elliptic curve, use the local parameter $t=-x/y$ and put $w=-1/y$. The formal kernel of a minimal Weierstrass equation construction gives $w=t^3+a_1{t}^4+(a_1^2+a_2)t^5+(a_1^3+2a_1{a}_2+a_3)t^6+O(t^7)$. Substitution into the elliptic curve group law yields the displayed initial coefficients and the formal inverse $-t/(1-a_{1}t-a_{3}w(t))$. In characteristic zero, the invariant differential on an elliptic curve becomes $(1+a_{1}t+(a_1^2+a_2)t^2+O(t^3))dt$, giving the formal logarithm $t+a_1{t}^2/2+(a_1^2+a_2)t^3/3+O(t^4)$. The addition coefficients are integral polynomial expressions and hence also apply in positive characteristic of a field.

Let ${\Lambda }_1,{\Lambda }_2\subset {\mathbb{R}}^d$ be full-rank Euclidean lattices, and choose $\epsilon$ smaller than every nonzero vector length in either. The shortest vectors of ${\Lambda }_i\oplus \epsilon {\mathbb{Z}}^k$ are exactly the new coordinate vectors and their negatives. Any isometry between the associated flat tori lifts to an affine Euclidean isometry, whose orthogonal part maps the lattices. It must preserve the span of the shortest vectors, hence also its orthogonal complement, and therefore maps ${\Lambda }_1$ to ${\Lambda }_2$. Consequently a nonisometric pair remains nonisometric after adding sufficiently short equal circle factors. The spectrum of a flat torus shows that adding the same factors preserves isospectral manifolds.

In a proper flat family of algebraic curves, if the first sheaf cohomology of a line bundle vanishes on every fibre of an open set, its direct image there is a vector bundle of rank given by the Riemann-Roch theorem and commutes with arbitrary base change. Locally a finite free complex computes both fibre sheaf cohomology groups. Vanishing of the last cokernel makes its last map surjective; splitting that map leaves a finite locally free kernel in degree zero. This is the mechanism behind charts of the Picard scheme of a curve and evaluation descriptions of the theta divisor.

For globally Lipschitz drift $b$, couple $d{X}^{\epsilon }=b(X^{\epsilon })dt+\epsilon dW$ and $\dot{x}=b(x)$ from the same point. The Gronwall inequality gives ${\sup }_{s\le t}|X_s^{\epsilon }-x_s|\le \epsilon e^{Lt}{\sup }_{s\le t}|W_s|$. Thus convergence is uniform on each finite horizon. The Feynman-Kac formula for generator ${\epsilon }^2\Delta /2+b\cdot \nabla +c$ has weight $\exp \int c$, with a positive sign. Bounded continuous $c$ is uniformly continuous on a compact neighbourhood of the deterministic trajectory, and bounded continuous $f$ supplies a deterministic integrable bound. The dominated convergence theorem proves the displayed limit without requiring global uniform continuity of $c$.

Let $\varphi$ be continuous, nonnegative, zero on $[0,\infty )$, and strictly positive on $(-\infty ,0)$. By continuity of Brownian paths, its occupation integral vanishes exactly on nonnegative paths. Dominated convergence gives the displayed limit. The Brownian reflection principle makes it $1-2{\mathbb{P}}_x(B_t\le 0)$ for $x\ge 0$ and zero for $x<0$. In particular the limit at $x=0$ is zero for every positive time. The choice $\varphi (x)=\min (1,\max (0,-x))$ is a bounded soft killing potential.

Time-of-flight atom detection measures particles after releasing a trapped atomic cloud and allowing it to expand for a known time. For ballistic motion, $\mathbf{r}(t)={\mathbf{r}}_0+{\mathbf{p}}_{0}t/m+\mathbf{g}{t}^2/2$, so after accounting for gravity the late position largely reflects the initial momentum when the initial cloud size is sufficiently small.

A position- and time-resolving detector can record individual arrivals and form spatial pair correlations. In fermionic atom antibunching, such correlations probe suppression of nearby detection pairs for the prepared state. Finite source size, interactions during early expansion, detector resolution and detection efficiency limit a literal initial-momentum interpretation; expansion is part of the experimental model, not merely a delay before measurement. A microchannel plate detector is one implementation for suitable metastable atoms.

A neutral chemical atom with a magnetic dipole moment can experience a force in a nonuniform magnetic field, despite having zero net electric charge. For an internal state that follows the local field adiabatically, its position-dependent Zeeman energy supplies an effective trapping potential energy.

A low-field-seeking state has increasing energy with field magnitude and can be confined near a field minimum. The atom must retain the trapped internal state: rapid changes of field direction or a vanishing splitting can permit transitions to untrapped states. The trap therefore selects both spatial motion and internal spin character. In atom-correlation experiments, releasing this trapped population provides a controlled source for time-of-flight atom detection.

The pair detection correlation divides joint detection probability by the product of single detection probabilities. The baseline $g^{(2)}=1$ means independent detection; $g^{(2)}<1$ means suppressed pairs. Its value depends on the prepared quantum state and detector resolution, so exclusion alone does not require every experimental curve to reach zero.

For a nonzero element of a Weyl algebra over a characteristic-zero field, choose a monomial $x^{\alpha }{D}^{\beta }$ of maximal total degree with coefficient $c_{\alpha \beta }\ne 0$. The displayed iterated commutators kill every lower-degree monomial. Another monomial of the same degree survives only if all its exponents dominate $(\alpha ,\beta )$, which forces equality. Hence the result is a nonzero scalar. If the element has filtered degree at most $j$, expansion of these commutators gives a sum of terms $uav$ with both $u$ and $v$ of degree at most $j$. This proves faithful finite-step action of a Weyl algebra over any characteristic-zero field.

For a nonnegative integer-valued random variable, the displayed inversion is valid when $\mathbb{E}(X{)}_s{s}^k/s!\to 0$. Indeed the gap between consecutive Jordan-Bonferroni exact-occurrence inequalities is $\mathbb{E}(X{)}_{k+m+1}/[k!(m+1)!]$, which tends to zero under this hypothesis. No exchange of an infinite signed sum with an expectation is needed.

On $E\times E$ for a complex elliptic curve, the two factor curves and the diagonal have intersection matrix $\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$, of determinant two. Their cohomology classes are therefore independent. The degree-one image of the external product of Chow classes has cohomology contained in the span of the two factor classes. The cycle class map thus shows that the diagonal is outside that image, proving failure of surjectivity.

For the blowup of a complex manifold at a point in dimension $n\ge 2$, the exceptional divisor is ${\mathbb{CP}}^{n-1}$ and $[E]{|}_E={\mathcal{O}}_E(-1)$. The local blowup is the total space of the tautological line bundle, so its zero section has that normal bundle. Restricting a power to a projective line inside $E$ gives degree $-r$; a trivial bundle has degree zero. This proves that no nonzero power of $[E]$ is trivial, without assuming global projectivity of the original manifold.

Normalize the section to the sphere bundle. Scaling it from zero to unit length gives a homotopy within the disk bundle from the zero section to this sphere-valued section. The image of the relative Thom class in absolute cohomology restricts to zero on the sphere bundle. Homotopy invariance therefore makes its zero-section pullback vanish.

// Target: geometry-and-topology.bigb

Let $\mathcal{B}(H)=B(H,\cdot )$ be the Killing form identification of a Cartan subalgebra with its dual. The root-space decomposition gives $B(H,K)={\sum }_{\beta \in \Phi }\beta (H)\beta (K)$, and therefore the displayed root expansion. On the real coroot span its coefficients are real, proving $\mathcal{B}({\mathfrak{h}}_{\mathbb{R}})\subseteq \mathbb{R}P$. For ${\mathfrak{sl}}_3$ and $H_{12}=E_{11}-E_{22}$, the formula gives $\mathcal{B}(H_{12})=6(L_1-L_2)$. The normalization is $B(X,Y)=\mathrm{tr}(\mathrm{ad}X\mathrm{ad}Y)$, not a rescaled trace form.

// Target: semisimple-lie-algebra.bigb

A split equalizer consists of $e:E\to X$ and $f,g:X\rightrightarrows Y$ with $fe=ge$, together with maps $r:X\to E$, $s:Y\to X$ satisfying $re=1_E$, $sf=er$, and $sg=1_X$. If $fh=gh$, these identities give $h=erh$, so $rh$ is its unique factorization through $e$. Thus it is an equalizer, and every functor preserves it: all the defining identities survive application of the functor. Interchanging $f,g$ gives the other orientation of the same convention.

For a planar interface, equal permeability and no singular sheet current make all magnetic components continuous. A nonzero horizontal Fourier wavenumber then gives the first three conditions in the title. Continuity of the tangential electric field supplies the last: continuous mean velocity cancels its advective contribution, and the retained alpha mean-field electromotive force is along the toroidal direction, so the other tangential electric component is $-U{\partial }_{x}A-\eta {\partial }_{z}B$. The given alpha-Omega equations retain only toroidal-to-poloidal alpha generation; adding a full isotropic alpha contribution would alter both the bulk equation and the matching problem.

Let $p=s-i\omega$ and $q^2=k^2+p/\eta$, with $\mathrm{Re}q>0$. The upper poloidal amplitude is $(A_0+Cz)e^{-qz}$ and the lower toroidal amplitude is $(B_0+cz)e^{qz}$. Their forcing equations give $2\eta qC=\alpha B_0$ and $-2\eta qc=ikV{A}_0$; interface matching gives $C=2q{A}_0$ and $c=-2q{B}_0$. Eliminating amplitudes yields the dispersion relation. Its potentially growing branch has $s=\eta k^2(\sqrt{D/32}-1)$ and $\omega =-\eta k^2\sqrt{D/32}$ with the convention $e^{ikx-i\omega t}$. The opposite branch decays.

Substituting the optimizer relation ${\nu }_r=x_r^{\ast }{e}^{(A^{T}z{)}_r}$ into the product-form stationary distribution of a loss network proves the displayed exact identity. The independent Poisson random variables give a reference law centered at the optimized occupancies, while the constraint and exponential factor account for saturated resources. An unconstrained Gaussian approximation can miss these boundary effects.

For $D_{n+1}=[D_n\to D_n]$ with recursively lifted embedding-projection pairs, form coherent sequences $x_n=p_n{x}_{n+1}$, ordered componentwise. Stage projections $P_n$ and embeddings $E_n$ satisfy $P_n{E}_n=\mathrm{id}$, $E_n{P}_n\uparrow \mathrm{id}$. A coherent family $f_n\in [D_n\to D_n]$ induces $f={\sup }_n{E}_n{f}_n{P}_n$; conversely $f_n=P_{n}f{E}_n$. These constructions are inverse. Removing the initial coordinate of the coherent sequence identifies this function-space limit with the original domain limit.

In a finite connected electrical network, run the conductance random walk from $s$ until its first visit to a distinct vertex $t$. Let $G(s,x)$ be the expected visits to $x$ before stopping, and $w_x={\sum }_y{w}_{xy}$. The expected signed number of traversals of $\{x,y\}$ is the displayed expression. Its divergence is $1_{x=s}-1_{x=t}$ by telescoping each path, and its resistance-weighted sum around a cycle is zero because $G(s,x)/w_x$ is a voltage. Uniqueness follows by summing ${\sum }_{\{x,y\}}{w}_{xy}(v_x-v_y{)}^2$: a zero-divergence voltage flow has zero energy. Thus the expected net traversals are exactly the unit source-to-sink current.