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Normalize the target in the function field of the generically finite Gauss map of a theta divisor; its finite map has an intrinsic branch locus. For a nonhyperelliptic smooth projective curve, this locus is the projective dual variety of the canonical curve. For a hyperelliptic curve of genus at least three, it is the union of the dual hypersurface of its canonical rational normal curve and the $2g+2$ hyperplanes corresponding to branch points. Locally, ramification exchanges two roots of a hyperplane section, or exchanges the two lifts of a root passing through a branch point. Biduality then recovers the curve or its double-cover data.

In a gas syringe, the graduated barrel reading $V_{\mathrm{read}}$ can omit a fixed connecting or sensor volume $V_d$ occupied by the same sealed gas. For a common settled temperature, Boyle's law then requiresThis second relation makes a graph of $1/p$ against $V_{\mathrm{read}}$ a straight line; its intercept divided by its slope estimates $V_d$ under these assumptions. Unequal temperatures between the barrel and the connecting space require a different correction, analogous to the gas thermometer dead-space correction.

For an original hypothetical example, let $V_d=1\mathrm{c}{\mathrm{m}}^3$, $V_{\mathrm{read},1}=20\mathrm{c}{\mathrm{m}}^3$, $p_1=100\mathrm{kPa}$, and $V_{\mathrm{read},2}=10\mathrm{c}{\mathrm{m}}^3$. The total volume falls from $21$ to $11\mathrm{c}{\mathrm{m}}^3$, so $p_2=100(21/11)\mathrm{kPa}\approx 190.91\mathrm{kPa}$, rather than $200\mathrm{kPa}$. The omitted volume makes the total compression ratio smaller than the barrel-reading ratio. These stipulated values are independent of the manufacturer's apparatus correction and are not measurements.

The tensor product of Lie algebra representations $3\otimes 3$ splits into the symmetric $6$ and antisymmetric $\overline{3}$. Then $6\otimes 3=10\oplus 8$ and $\overline{3}\otimes 3=8\oplus 1$, as highest-weight character subtraction verifies. The decuplet is fully symmetric under quark permutations, the singlet fully antisymmetric, and the two octets constitute the two-dimensional mixed permutation multiplicity space. Applying the Pauli constraint on three-quark flavour multiplets gives one ground-state spin-one-half octet and one spin-three-halves decuplet.

// Target: physics.bigb

Split the two dual factors into antisymmetric and symmetric tensors. The invariant Levi-Civita symbol identifies the antisymmetric pair with $3$, giving $6\oplus \overline{3}$ after multiplying by the remaining triplet. On the symmetric part, contraction $T^a{}_{ba}$ maps onto $\overline{3}$; the equivariant splitting is $T^a{}_{bc}=({\delta }_b^a{v}_c+{\delta }_c^a{v}_b)/4$. Its trace-free kernel has dimension $18-3=15$ and highest weight with Dynkin labels $(1,2)$. The Weyl dimension formula gives dimension $15$, so this is exactly the irreducible kernel. It is conjugate to the other fifteen-dimensional module with labels $(2,1)$; dimension alone does not distinguish the two.

In an adapted real orthonormal coframe, the fundamental form of a Hermitian manifold is $\omega ={\sum }_j{e}^j\wedge f^j$ and the Riemannian volume form is ${\omega }^n/n!$. The complex-linear Hodge star sends each summand to the product of all the other oriented two-dimensional factors. Summation gives the displayed formula, valid for every Hermitian manifold, without a Kähler assumption.

For a directed set of Scott continuous maps $f_i:D\to E$, its least upper bound is $f(x)={\sup }_i{f}_i(x)$. The two directed suprema over $i$ and over inputs commute, proving that $f$ remains Scott continuous. Thus the function space of complete partial orders is a complete partial order.

In the unique decomposition $f={\sum }_{i=0}^{p-1}(1+T{)}^i\phi (f_i)$, define $\psi (f)=f_0$. Averaging translates over the p-th roots of unity annihilates every nonzero basis index and retains the zeroth, proving the displayed identity and its integrality. Thus $\psi \phi =1$ and $\psi (\phi (g)f)=g\psi (f)$. Under the Amice transform, $\psi =0$ characterizes measures supported on ${\mathbb{Z}}_p^{\times }$.

For $R={\mathbb{Z}}_p[[T]]$ and $\varphi (f)=f((1+T{)}^p-1)$, reduction modulo $p$ is the injective substitution $\overline{f}(T)\mapsto \overline{f}(T^p)$. Thus $\varphi (f)\in pR$ implies $f\in pR$. Dividing by $p$ and repeating proves the displayed equality for every $k\ge 1$. In particular, a ring congruence of Frobenius substitutions on cyclotomic power series reflects back to the original formal power series.

For an elliptic curve over ${\mathbb{F}}_q$,

If an isogeny of elliptic curves $\pi :E_1\to E_2$ satisfies $\pi {\varphi }_1=-{\varphi }_2\pi$ for their Frobenius isogenies, then $\pi (1-{\varphi }_1)=(1+{\varphi }_2)\pi$. Multiplicativity of the degree of an isogeny gives $\mathrm{deg}(1-{\varphi }_1)=\mathrm{deg}(1+{\varphi }_2)$. The degree parallelogram law then gives the displayed point-count identity, since $\mathrm{deg}(1-{\varphi }_i)=\#E_i({\mathbb{F}}_q)$ and $\mathrm{deg}{\varphi }_i=q$.

For an integral one-dimensional formal group law over a finite extension of the p-adic field, the inverse of ${\partial }_{Y}F(T,0)$ has integral coefficients because its constant term is one. Integrating this invariant differential gives ${\log }_F(T)=T+{\sum }_{n\ge 2}{c}_n{T}^n$ with $n{c}_n$ integral. For $v_K(\pi )=1$ and $e=v_K(p)$ the displayed bound follows. The estimate $v_p(n)\le (n-1)/(p-1)$ makes the nonlinear part strictly contracting on ${\pi }^s{\mathcal{O}}_K$ for $s>e/(p-1)$, yielding the deep logarithm subgroup of a formal group.

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.