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Under the continuous common-distribution null of the Wilcoxon rank sum test, pooled labels are exchangeable. The product records every choice of ranks: choosing $z{t}^k$ assigns rank $k$ to the first sample. Its indicated coefficient counts subsets of size $m$ and rank sum $w$. Divide by $\left(\genfrac{}{}{0ex}{}{m+n}{m}\right)$ and sum the lower tail for the one-sided P-value. Equivalently the counts satisfy $a_{k,h,w}=a_{k-1,h,w}+a_{k-1,h-1,w-k}$ with $a_{0,0,0}=1$, allowing an exact integer computation.

With $N=m+n$ and no ties, sampling $m$ ranks without replacement gives the displayed mean and variance. For observed rank sum below its mean, a continuity-corrected lower-tail statistic is $Z=[W_X-m(N+1)/2+1/2]/\sqrt{mn(N+1)/12}$. Ties of sizes $t_g$ replace the variance by $mn[(N+1)-{\sum }_g(t_g^3-t_g)/(N(N-1))]/12$. These formulas concern the raw rank sum; the Mann–Whitney U test statistic subtracts $m(m+1)/2$ from both the observation and its mean.

A collar neighborhood identifies a neighborhood of the boundary with $\partial W\times [0,\epsilon )$, with the boundary at parameter zero. It allows a manifold's boundary to be pushed slightly inward by a homotopy.

At a good split prime, the one-prime CM division tower has a pro-$p$ Galois group isomorphic to ${\mathbb{Z}}_p$. Its prime-ramified Iwasawa module is torsion on each finite-character component. The main conjecture identifies its characteristic ideal with the corresponding elliptic-unit p-adic L-function. The class-field unit sequence and equality of characteristic ideals of global-units modulo elliptic-units and of class groups prove the identification. The cyclotomic Selmer formulation requires the appropriate CM twist and two-variable specialization; the one-prime tower is not the cyclotomic tower.

For the real p-power cyclotomic tower, take norm limits of local principal units and closed cyclotomic-unit subgroups. The corrected Coleman logarithm identifies $U_{\infty }^1$ with the Iwasawa algebra of $G={\mathbb{Z}}_p^{\times }/\{\pm 1\}$. On a symmetric cyclotomic unit it gives $([b]-[a]){\zeta }_p$. A topological generator generates the cyclotomic-unit limit and the augmentation ideal, yielding the quotient description. This local calculation does not alone identify the characteristic ideal of the global Galois module.

In the real cyclotomic tower, the class-field unit sequence gives $0\to E/C\to U/C\to X\to Y\to 0$, where $X$ is the p-ramified Iwasawa module and $Y$ the unramified one. The modules are torsion over each one-variable character component. Splitting the four-term sequence into two short exact sequences proves the displayed identity by multiplicativity of characteristic ideals. The Iwasawa theorem on local cyclotomic units computes $\mathrm{char}(U/C)$ analytically. Thus the main conjecture for $X$ is equivalent to $\mathrm{char}(E/C)=\mathrm{char}(Y)$; local interpolation alone does not prove this remaining global equality.

This lacunary series is analytic for $|z|<1$ and satisfies $f_l(z^l)=f_l(z)-z$. At roots of unity of $l$-power order, its radial tail diverges, giving a dense set of singularities on the unit circle and excluding algebraicity as a function. The Mahler method proves transcendence of its values at every algebraic $z$ with $0<|z|<1$. At $z=1/2$, reduced truncations have denominators $2^{l^m}$ and error less than twice their $-l$th power; for $l\ge 3$, the Roth theorem gives a shorter proof.

In inviscid magnetostrophic balance within an impenetrable axisymmetric container, with no azimuthal buoyancy or external mechanical torque, the cylindrically integrated Lorentz force density must vanish on every cylinder coaxial with rotation. The azimuthal pressure derivative integrates to zero, and the cylindrical Coriolis integral is proportional to net radial flux, zero by incompressibility and impermeability. Multiplying by cylinder area and lever arm gives the equivalent zero-torque form. This rotating-fluid condition is distinct from magnetic-helicity relaxation to a force-free plasma state. Its original primary source is Taylor's 1963 rotating-fluid paper.

For an imposed field $B_0$ in a layer of depth $d$, the Chandrasekhar number compares magnetic tension with viscous and magnetic-diffusive effects. Here $\eta$ is magnetic diffusivity and $\nu$ is kinematic viscosity. In thermal-diffusion units the linear magnetic-force coefficient is $Q\zeta$, where $\zeta =\eta /\kappa$ and $\kappa$ is thermal diffusivity.

For a stress-free, fixed-temperature layer with vertical magnetic perturbation at its boundaries, take vertical velocity and temperature proportional to $\sin \pi z$, vertical magnetic perturbation proportional to $\cos \pi z$, and horizontal magnetic perturbations proportional to $\sin \pi z$. With $s=k^2+{\pi }^2$, the vertical velocity, temperature and magnetic amplitudes form a three-variable linear system whose determinant is the stated polynomial. The Prandtl number is $\sigma$, magnetic-to-thermal diffusivity ratio is $\zeta$, and field strength is the Chandrasekhar number $Q$. The steady marginal curve is $R^{(e)}=[s^3+Q{\pi }^{2}s]/k^2$.

In thermal-diffusion units, let an imposed field point along $\mathbf{c}=(\sin \alpha ,0,\cos \alpha )$ and consider a Fourier mode of wavevector $(k,0,m)$. Put $q=k^2+m^2$ and $d=k\sin \alpha +m\cos \alpha$. The growth exponent $s$ of its buoyant branch satisfieswhere $\sigma$ is the Prandtl number, $\zeta$ the magnetic-to-thermal diffusivity ratio, $Q$ the Chandrasekhar number, and $R$ the Rayleigh number. The resistive induction equation gives $\hat{\mathbf{b}}=id\hat{\mathbf{u}}/(s+\zeta q)$, and thermal transport gives $\hat{\theta }=\hat{w}/(s+q)$. Projecting momentum perpendicular to the wavevector removes pressure; the vertical component of the projected buoyancy is $R(k^2/q)\hat{\theta }$. Substitution gives the displayed polynomial. The stationary neutral curve is $R=q^3/k^2+Qq{d}^2/k^2$. No vertical boundary conditions are imposed in this plane-wave calculation.

In nonlinear magnetoconvection, magnetic flux separation is coexistence of weak-field regions with vigorous broad convection and strong-field regions with inhibited or small-scale convection. Flux expulsion provides a mechanism for segregation. Conserved mean vertical magnetic flux implies $f{B}_c+(1-f)B_q=B_0$ for an idealized two-region area fraction $f$. Weak field in convecting regions therefore requires enhanced field elsewhere. The existence and stability of the segregated state depend on parameters and boundaries.

Subcritical magnetoconvection means finite-amplitude convection on the stable side of the linear conductive-state threshold. At large magnetic Reynolds number, flux expulsion can reduce interior magnetic inhibition enough to sustain a nonlinear convective cell even when infinitesimal disturbances decay in the unredistributed imposed field. Stable subcritical states can produce hysteresis and magnetic flux separation. A schematic saturated amplitude equation $\dot{A}=\mu A+c|A{|}^{2}A-d|A{|}^{4}A$, $c,d>0$, has an unstable lower branch and stable upper radial branch for $-c^2/(4d)<\mu <0$, coexisting with stable $A=0$. This illustrates bistability; it is not a derivation of coefficients for all magnetoconvection geometries.

The ratio of magnetic diffusivity $\eta$ to thermal diffusivity $\kappa$ controls their relative relaxation times. In nonrotating vertical-field magnetoconvection, oscillatory linear onset in the simple stress-free layer requires $0<\zeta <1$, so thermal perturbations diffuse faster than magnetic perturbations. This is distinct from the magnetic Prandtl number, $\nu /\eta$.

For a unit-depth Boussinesq layer with stress-free boundary conditions, fixed boundary temperatures, imposed field along $y$, and horizontal magnetic perturbations at the boundaries, put $a_h^2=k^2+m^2$ and $\Lambda =a_h^2+{\pi }^2$. Poloidal vertical velocity, temperature and vertical field proportional to $e^{st+ikx+imy}\sin \pi z$ have amplitudes satisfyingThe determinant gives the stated dispersion relation. Here $\sigma$ is the Prandtl number, $\zeta$ the magnetic-to-thermal diffusivity ratio, $Q$ the Chandrasekhar number and $R$ the Rayleigh number. The steady neutral curve is $R={\Lambda }^3/a_h^2+Q{m}^2\Lambda /a_h^2$.

// Target: fluid-mechanics.bigb

A Galerkin method approximation can retain one roll streamfunction mode, two temperature modes and two magnetic-flux perturbation modes in vertical-field magnetoconvection. Projection onto these five modes, rather than pointwise equality of the truncated fields, gives a closed system of five ordinary differential equations. Products generate higher spatial harmonics that are discarded by orthogonal projection. The system has symmetry $(a,b,c,d,e)\mapsto (-a,-b,c,-d,e)$ and hence admits paired steady roll branches.

When the magnetic field supplies instantaneous damping rather than an evolving magnetic perturbation, the roll amplitudes obey a two-variable linear system with trace $-[s+\sigma (s+Q{\pi }^2/s)]<0$. Its determinant vanishes at the displayed stationary threshold. Therefore its initial instability cannot be a Hopf bifurcation. This reduced model must not be confused with a three-variable magnetoconvection model carrying finite magnetic diffusivity and an additional oscillatory mode.

Use depth $d$, time $d^2/\kappa$, stream-function scale $\kappa$, temperature perturbation scale $\Delta T$, and magnetic flux-function scale $B_{0}d$. With $\mathbf{u}=(-{\psi }_z,0,{\psi }_x)$, $\mathbf{B}=(-A_z,0,A_x)$ and $A=x+\chi$, advection is $J(\psi ,f)={\psi }_x{f}_z-{\psi }_z{f}_x$. The temperature background contributes ${\psi }_x$ to the heat equation and the imposed vertical field contributes ${\psi }_z$ to the perturbation induction equation. Taking the curl of the magnetohydrodynamic momentum equation gives a magnetic term $\sigma \zeta QJ(A,{\nabla }^{2}A)$, where $\sigma =\nu /\kappa$, $\zeta =\eta /\kappa$, $R=g\alpha \Delta T{d}^3/(\nu \kappa )$ and $Q=B_0^2{d}^2/({\mu }_0\rho \nu \eta )$.

A Galerkin method using one convective streamfunction mode and two Cartesian magnetic flux function modes yields the displayed system. With $\alpha =\pi /\sqrt{3}$, use $\psi =4\sqrt{2}a\sin (\alpha x)\sin (\pi z)$ and $A=(3\sqrt{2}/\pi )b\sin (\alpha x)\cos (\pi z)+(\sqrt{3}/\pi )c\sin (2\alpha x)$, time $\tau =(4{\pi }^2/3)t$, $q=9Q/(16{\pi }^2)$ and $r=9R/(64{\pi }^4)$. The origin's planar trace is $\sigma (r-1)-\zeta$ and determinant $\sigma \zeta (1+q-r)$; the third eigenvalue is $-\zeta$. The stationary threshold is $r=1+q$, and the oscillatory threshold is $r=1+\zeta /\sigma$ when $q>\zeta /\sigma$. They meet in a symmetric double-zero point at $(q,r)=(\zeta /\sigma ,1+\zeta /\sigma )$. Higher spatial harmonics generated by the quadratic terms are discarded, so this is a projected approximation rather than an exact invariant subspace of the original equations.

For a smooth Anosov magnetic flow on a closed oriented surface, a smooth function plus a smooth one-form can be a smooth flow coboundary only when the function is zero and the one-form is exact. Equivalently their sum has zero integral along every closed orbit precisely in that case, using the smooth Livšic theorem. This is the nontrivial magnetic transform kernel theorem. Exact one-forms give the converse directly, since $F(w\circ \pi )=dw(v)$. The theorem does not require pointwise negative Gaussian curvature.