Let $h_i$ represent a nonzero two-torsion point of a complex torus, and write $e_i=\wp (h_i)$. The product $(\wp (z+h_i)-e_i)(\wp (z)-e_i)$ is an elliptic function whose apparent poles cancel against double zeros. It is therefore constant by Liouville theorem. Its value at $z=0$ is ${\wp }^{\prime \prime }(h_i)/2$. The Weierstrass elliptic differential equation gives ${\wp }^{\prime \prime }(h_i)/2=3e_i^2-g_2/4=(e_i-e_j)(e_i-e_l)$, proving the formula.

Let $A_j$ be the weights of the cubic Stokes dispersion symmetry and set $F(k)={\sum }_j{A}_j(k){\hat{q}}_0({\nu }_j(k))$. Orient $\partial D_{+}$ with $D_{+}$ on its left. Evaluating the global relation at its two lower-half-plane roots and interpolating the linear expression $ik{G}_1+G_2$ removes both unknown boundary traces. The resulting Fokas method formula isThe discarded transform of the solution is holomorphic in $D_{+}$ and has zero integral after closing its contour for $x>0$. Initial compatibility and sufficient smoothness and spatial decay are required. At the boundary, use a fixed time horizon later than $t$ before Fourier inversion, to avoid an endpoint half-value.

For $q_t=q_{xx}+\alpha q_x$ with $\alpha >0$, the dispersion relation is $\omega =k^2-i\alpha k$ and its symmetry is $\nu =i\alpha -k$. On the upper domain $\mathrm{Re}\omega <0$, $\mathrm{Im}k>\alpha$ and therefore $\mathrm{Im}\nu <0$. After removing the unknown trace through the global relation, deform the boundary contour to the displayed two rays, with $0<\theta <\pi /4$, oriented from left infinity to $i\alpha$ to right infinity. The factor $e^{ikx}$ decays like $e^{-rx\sin \theta }$, while $e^{-\omega t}$ has Gaussian decay for large $r$. Apparent poles introduced by splitting a finite-time forcing transform must be treated as removable singularities of the unsplit expression.

The baryon octet has rows $Y=1/2,0,-1/2$ with isospins $1/2$, $1\oplus 0$, $1/2$. The central weight has two states, ${\Sigma }^0$ and ${\Lambda }^0$. The baryon decuplet has rows $Y=1/2,0,-1/2,-1$ with isospins $3/2,1,1/2,0$. The three-quark flavour singlet has one state at the origin. Both abstract octet copies in $3^{\otimes 3}$ share the same weights.

The Haar projection replaces a function by its mean on each dyadic interval of length $2^{-j}$. For a function satisfying Lipschitz continuity with constant $L$, every value in a cell differs from its cell mean by at most $L{2}^{-j}$. This bound holds for all points with a consistent endpoint convention, and requires no global L2 norm assumption when the projection is defined by local averages.

For a pure scalar growing density contrast with time-independent boundary conditions, ${\nabla }^2\psi ={\delta }_{\ast }$ fixes the scaled peculiar gravitational potential up to an irrelevant constant. The linearized continuity equation fixes the longitudinal scaled velocity to $-\nabla \psi$. The remaining homogeneous velocity is ${\mathbf{u}}_h=\mathbf{C}(\mathbf{x})/(a^2\dot{b})$; a divergence-free such field does not change the growing density at linear order. Excluding this decaying velocity component is necessary for time-independent $\mathbf{u}$. In an Einstein-de Sitter universe, $b=a$ and ${\mathbf{u}}_h\propto a^{-3/2}$, furnishing a counterexample if the pure growing-velocity hypothesis is omitted.

For each treatment year let $C$ be cycles, $S$ singleton deliveries, $T$ twin deliveries and $H$ deliveries of three or more babies. A singleton-per-cycle model uses $S\sim \mathrm{Bin}(C,p)$. A multiple-delivery model conditional on a live delivery instead uses $T+H\sim \mathrm{Bin}(S+T+H,q)$. The latter denominator counts deliveries, not babies, and it excludes failed cycles. Confusing these denominators changes the statistical question.

For independent $Y_{ij}\sim \mathrm{Bin}(m,p_i)$ with $k$ replicates per group and $\mathrm{logit}(p_i)=\mu +{\alpha }_i$, ${\alpha }_1=0$, let $S_i={\sum }_j{Y}_{ij}$ and $N=km$. The likelihood equations give ${\hat{p}}_i=S_i/N$, $\hat{\mu }=\log (S_1/(N-S_1))$, and ${\hat{\alpha }}_i=\mathrm{logit}({\hat{p}}_i)-\hat{\mu }$ when $0<S_i<N$. Boundary totals give infinite logits. With $0<p_1<1$ fixed and $N\to \infty$, the central limit theorem and delta method giveThe other groups do not increase information about the baseline logit when they have free offsets. For two groups, the information matrix for $(\mu ,{\alpha }_2)$ is $\left(\begin{array}{cc}{w}_1+w_2& w_2\\ w_2& w_2\end{array}\right)$ with $w_i=N{p}_i(1-p_i)$, whose inverse has first diagonal entry $1/w_1$.

This subgroup uses Hamiltonians whose support in $[0,1]\times M$ is compact. Their flows are the identity outside a fixed compact set. On closed manifolds this support condition imposes no extra restriction. On an open symplectic manifold, it excludes nontrivial whole-fiber cotangent fiber translations, even when their generating one-form is exact.

If the closed span of the Rademacher functions were a complemented subspace of $L^1(0,1)$, compose a projection with the inverse of its l2 sequence space parametrization. The Grothendieck theorem for L1 to Hilbert operators would make this map an absolutely summing operator. Its composition with the parametrization would make the identity on the infinite-dimensional l2 sequence space absolutely summing, contradicting its coordinate-vector test.

A local two-body and three-body mean-field energy density $V_0{n}^2/2+W_0{n}^3/3$ generates cubic and quintic terms in the Gross–Pitaevskii equation. Variation with respect to ${\psi }^{\ast }$ gives $(V_{0}n+W_0{n}^2)\psi$, with $n=|\psi {|}^2$. A uniform stationary state's chemical potential is $\mu =V_0{n}_0+W_0{n}_0^2$. Its positive-compressibility condition is $V_0+2W_0{n}_0>0$. If $\mu >0$, scaling amplitude by $\sqrt{n_0}$ and distance by $\hslash /\sqrt{2m\mu }$ gives coefficients $\alpha =V_0{n}_0/\mu$, $\beta =W_0{n}_0^2/\mu$ and $\alpha +\beta =1$. Negative chemical potential requires a different sign in the dimensionless linear term.

Factorizing a normalized condensate field as ${\Psi }_{3D}(x,y,z)=\psi (x,y)\chi (z)$ with $\int |\chi {|}^{2}dz=1$ reduces its contact interaction energy to $(g_{2D}/2)\int |\psi {|}^{4}dxdy$, with the displayed coupling. Thus $g_{2D}$ has units of energy times area, whereas $g_{3D}=4\pi {\hslash }^{2}a/m$ has units of energy times volume for a physical scattering length $a$. For a normalized axial Gaussian $\chi (z)={\pi }^{-1/4}{a}_z^{-1/2}{e}^{-z^2/(2a_z^2)}$, direct Gaussian integration gives $g_{2D}=g_{3D}/(\sqrt{2\pi }{a}_z)$. A model with density normalized per unit axial length can instead retain $g_{3D}$ with a line-density normalization.

Let $D_n={\partial }_t+{\mathbf{v}}_n\cdot \nabla$ be the material derivative of the normal fluid, and $\zeta >0$. The displayed phenomenological density damping preserves the condensate continuity equation because its coefficient is real. Under a Galilean boost, both velocities shift by the boost velocity and the field acquires the Schrödinger phase; $D_n|\psi {|}^2$ is invariant. Thus the normal-fluid derivative, rather than a derivative along the condensate flow, is essential to the model.

Apply Green theorem with $P=-y/2$ and $Q=x/2$ to a positively oriented regular planar region. Since $Q_x-P_y=1$, its area is the displayed line integral. Clockwise orientation changes the sign; holes have the opposite boundary orientation to the outer curve.

If $N_G(D)\le H\le G$, Brauer correspondence pairs the blocks of a group algebra of $RG$ and $RH$ having defect group of a block $D$. This is Green correspondence for their bimodules, with vertex of an indecomposable module $\Delta D$ in $G\times G$. The normalizer of $\Delta D$ lies in $H\times H$, and the corresponding restriction and induction have unique summands with that vertex.

The Green correspondent is the unique indecomposable summand of restriction or induction having the selected vertex of an indecomposable module, in the setting of Green correspondence. The two correspondence maps are inverse and preserve sources of an indecomposable module up to the appropriate normalizer conjugacy.

Let $N_G(D)\le H\le G$ and let $U$ be indecomposable with vertex $D$ over $H$. The Mackey restriction formula writes ${\mathrm{Res}}_H^G{\mathrm{Ind}}_H^{G}U=U\oplus T$, where the vertices in $T$ lie in $H\cap {}^{g}D$ for $g\notin H$. None of these subgroups contains an $H$-conjugate of $D$: equality would put $g$ in $N_G(D)H=H$. Every summand of the induction with a $G$-conjugate of vertex $D$ contributes a vertex-$D$ summand on restriction. Krull-Schmidt decomposition therefore forces exactly one such summand, with multiplicity one. Choosing sources and using transitivity of induction proves the inverse correspondence and preservation of sources.

Map a site $x\in {\mathbb{Z}}^d$ to its height $h(x)={\sum }_i{x}_i$. Maintain one infected representative in height $n$ for every infected site of an auxiliary one-dimensional contact process. The representative's death makes that auxiliary site recover. Its $d$ outgoing arrows to each neighbouring height, each of rate $\lambda$, make an auxiliary infection at total rate $d\lambda$, using the arrow's target as the new representative. Arrows to an already infected auxiliary height do nothing there. Distinct active representatives use distinct death and outgoing-arrow clocks, so their conditional rates are exactly those of the one-dimensional process. Every auxiliary infection has a live representative upstairs, proving the survival comparison.

For the one-dimensional contact process with recovery rate one and per-neighbour infection rate $\lambda$, use parity sites $(x,n)$ with $x+n$ even. Call a site good if $x$ has no recovery in $[(n-1)\delta ,(n+1)\delta ]$ and sends an arrow to each neighbour during $[n\delta ,(n+1)\delta ]$. The goodness indicators are independent: recovery intervals for successive parity sites at the same spatial point only touch at endpoints, and all outgoing-arrow intervals and clocks are disjoint. Their common probability is the displayed $q$. A path of good sites carries infection from each time layer to the next, because both its source and target avoid recovery during the transmitting interval. Taking $\delta ={\lambda }^{-1/2}$ makes $q\to 1$. The incoming-edge coupling of site and bond percolation and $p_c^{\mathrm{bond}}<1$ imply $p_c^{\mathrm{site}}<1$, so this proves a finite critical infection rate.

An indexed lambda term assigns a natural number to every subterm occurrence; one also admits a constant $\perp$ with empty interpretation. In the graph model of lambda calculus, every syntax constructor is interpreted as usual and then intersected with $\{0,\dots ,n\}$ at its occurrence index $n$. All indexed interpretations are finite subsets of the unindexed interpretation, and their directed union recovers that interpretation.