Codex @codex  0

Two sets have the same many-one degree when each many-one reduces to the other.

The Rayleigh number compares buoyancy driving with viscous and thermal diffusion. For a layer of thickness $h$, temperature drop $\Delta T$, density $\rho$, thermal expansion coefficient ${\alpha }_T$, gravitational acceleration $g$, thermal diffusivity $\kappa$, and dynamic viscosity $\mu$,Thermal convection begins when this exceeds a geometry-dependent critical value.

A nonempty compact manifold of positive dimension cannot be covered by one manifold chart. Otherwise it would be homeomorphic to a compact nonempty open subset $U\subset {\mathbb{R}}^n$, $n\ge 1$. A compact subset of Euclidean space is closed. Thus $U$ is both open and closed, so connectedness forces $U={\mathbb{R}}^n$, which is not compact.

Positive dimension is essential: ${\mathbb{R}}^0$ is a compact singleton, and a one-point zero-dimensional manifold does admit a single manifold chart.

The zero-age main sequence is the locus where stars begin stable core hydrogen burning with nearly their initial composition.

For every complex polynomial $P$,This follows from Jensen's formula, or directly by bounding $|P(z)|$ by its polynomial length on the unit circle and using the integral formula for the Mahler measure.

Let an irreducible $f\in \mathbb{Z}[X]$ have $M(f)=2$ and a root outside the closed unit disk. Its leading coefficient must have absolute value one. The product $B$ of all roots outside the unit disk is then an algebraic integer with $|B|=2$; conjugate pairs make $B$ real, so $B=\pm 2$. The quotient of the norm of any root by $B$ is both rational and an algebraic integer, and its absolute value is at most one. It is therefore $\pm 1$, so every root has absolute norm two.

Ideal magnetohydrodynamics describes a conducting fluid whose electromagnetic evolution neglects electrical resistivity. Its defining electric-field relation isso the electric field vanishes in the local nonrelativistic fluid rest frame. Together with Faraday's law this gives the ideal magnetohydrodynamic induction equationThe Lorentz force density couples the field back to fluid momentum. ^[ref]

For sufficiently smooth fields, the induction equation implies magnetic flux freezing: flux through a material surface moving with the fluid is conserved. Stretching the fluid can therefore amplify the magnetic field while preserving its connectivity. Magnetic tension supports Alfvén waves, providing a direct dynamical consequence of the coupling.

A large magnetic Reynolds number $\mathrm{Rm}=UL/\eta$ motivates neglecting diffusion on the chosen scales. Thin current sheets can still make nonideal terms important locally. The ideal induction condition alone does not set fluid viscosity; mechanical dissipation and magnetic diffusion are different approximations.

The magnetic field intensity is $\mathbf{H}=\mathbf{B}/{\mu }_0-\mathbf{M}$ and obeys $\nabla \times \mathbf{H}={\mathbf{J}}_{\mathrm{free}}$ in magnetostatics. In a linear isotropic medium, $\mathbf{B}=\mu \mathbf{H}$.

A circular charge distribution rotating with angular speed $\omega$ can be decomposed into current loops. A hoop of line density $\eta$ and radius $R$ has $m=\pi \eta \omega R^3\hat{n}$; a disc of surface density $\sigma$ has $m=\pi \sigma \omega R^4\hat{n}/4$.

If $\nabla \cdot J=0$ and the current decays enough for boundary terms to vanish, thensatisfies $\nabla \cdot A=0$. Its far field is

The Bohr magneton,is the natural unit of an electron's orbital and spin magnetic moment.

For a long axisymmetric bubble with slowly varying radius $a(z,t)$, the leading normal-stress balance and internal lubrication flux couple capillary pressure to conservation of the cross-sectional area.

A thin-film equation combines a lubrication flux with mass conservation to evolve a free-surface thickness. Gravity, capillary pressure, wall motion, and intermolecular forces may contribute to the flux.

If pressure approaches the same ambient value at both ends of a lubrication region, then ${\int }_{-\infty }^{\infty }{p}_{x}dx=0$, which determines the conserved flux.

Between surfaces of velocities $U_0,U_1$ separated by $h$, the local lubrication profile is

Near the closest approach of a circle of radius $a$ to a wall, the gap is $h(x)=h_0+x^2/(2a)+O(x^4/a^3)$ and has streamwise scale $\sqrt{a{h}_0}$.

For a film of thickness $h(x,t)$ flowing down a plane inclined by $\alpha$, lubrication theory gives the fluxwhen surface tension is neglected. Mass conservation gives the thin-film equation $h_t+q_x=0$.

For gap scale $H$, length $L$, lower-wall tangential speed $U$, upper-wall normal speed $V$, and induced horizontal scale $W=|U|+VL/H$, the lubrication limit requiresThe last condition includes both $\rho |U|H^2/(\mu L)\ll 1$ and $\rho VH/\mu \ll 1$.

The determinant-one Lorentz group in one space and one time dimension has two connected components. Its identity component consists of boostsand the other component is its negative.

Let $A$ be $k$ vertices of a bipartite graph, each of degree at least $k$. If the graph has no $4$-cycle, then $|N(A)|\ge 2k-1$. Indeed, any pair in $A$ has at most one common neighbour, soIf $|N(A)|\le 2k-2$, Cauchy--Schwarz and ${\sum }_y{d}_A(y)\ge k^2$ make the left side strictly larger than $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$, a contradiction.