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A probability measure is invariant for a Markov process with transition semigroup $P_t$ when starting with that law gives the same law at every later time. Equivalently, $\int P_{t}fd\nu =\int fd\nu$ for every bounded measurable function. Invariant laws may be constructed by weak convergence of probability measures of evolved laws when the Feller semigroup property permits passing the semigroup through the limit, as for the extremal invariant measures of the contact process.

A positive linear operator on bounded measurable functions that preserves the constant function one. A Markov semigroup is a time-indexed family of such operators with the semigroup law.

Each emitted symbol identifies the state of a stationary finite Markov chain. Here the alphabet is the state space itself. Its transition matrix $P$ and stationary distribution $\pi$ determine the entropy rate of a fully observed Markov source.

For a clean spherical gas bubble with negligible internal dynamic viscosity and thermal conductivity, a weak temperature gradient produces the displayed migration velocity, where ${\gamma }_T=d\gamma /dT$. The exterior Papkovich–Neuber representation can use only the scalar potential $\chi =-a^3\mathbf{U}\cdot \mathbf{r}/(2r^3)$. Its surface normal velocity equals $\mathbf{U}\cdot \mathbf{n}$, and its tangential traction balances minus the surface surface tension gradient from the temperature dipole around an insulating sphere. For the usual ${\gamma }_T<0$ the bubble moves toward warmer liquid. Small Reynolds number, Péclet number, capillary number, and fractional surface-tension variation justify the approximation; surfactants can oppose the temperature-driven Marangoni stress.

The mapping torus of a torus map with this monodromy has fundamental group generated by $x,y,t$ with $x$ central and $ty{t}^{-1}=x^{b}y$. Normal forms $x^m{y}^n{t}^k$ multiply byThis proves torsion-freeness and nilpotence of class at most two. For $b\ne 0$ its center is $⟨x⟩$ and commutator subgroup $⟨x^b⟩$; for $b=0$ it is ${\mathbb{Z}}^3$. These realize all noncyclic torsion-free nilpotent closed orientable three-manifold groups.

The main cardioid consists of quadratic parameters with an attracting fixed point. If that fixed point has multiplier $\lambda$, then it is $\lambda /2$ and $c=\lambda /2-{\lambda }^2/4$, $|\lambda |<1$. The map is injective on the open unit disc: equality of two parameter values forces either equal multipliers or their sum to be two, impossible for distinct interior multipliers. Its boundary is the cardioid obtained from $\lambda =e^{it}$. At $\lambda =1$ the parameter derivative vanishes, giving the cusp $c=1/4$. Parameters just to the real right of that cusp have monotonically escaping critical orbits, while the cusp's critical orbit tends to $1/2$, so it is on the Mandelbrot set boundary.

The period-two bulb is the open disc $|c+1|<1/4$ of quadratic parameters with an attracting exact two-cycle. The points in that cycle are roots of $z^2+z+c+1$, whose product is $c+1$. Its multiplier is therefore $4(c+1)$, giving precisely the disc inequality. At $c=-3/4$ the two roots merge into the fixed point $-1/2$, whose original multiplier is minus one; the second iterate has multiplier one and a triple fixed-point root. This boundary point is the attachment to the main cardioid.

Near a distinct centre of mass $M$, write $H=M/r+h+rF(r,\theta ,\varphi )$, where $h$ is the regular part evaluated at that centre and $F$ is analytic in local angular charts. Set $A=M+hr$, $Q=rH=A+r^{2}F$ and $dv=dt+(h+M/r{)}^{2}dr$. The metric becomesAll coefficients extend analytically across $r=0$, with $Q(0)=M>0$ and two-dimensional determinant $-1$. The surface is null, its generator is ${\partial }_v$, and the double zero of its squared norm gives zero surface gravity. The construction uses a constant $h$ so that the coordinate differential is integrable despite the nonspherical regular part of $H$.

The main-sequence turnoff is the location in a coeval population's Hertzsprung-Russell diagram where stars are leaving the main sequence after exhausting their core hydrogen. More massive stars generally have larger luminosity per unit mass and shorter nuclear lifetimes, so the most massive stars still on the main sequence constrain the population's age. In a simple model, equating the stellar nuclear timescale $t\sim qM/L(M)$ to the common age determines a turnoff mass; detailed calibration requires composition and stellar models.

Angular magnification is the ratio of the signed viewing angle with an optical instrument to the signed viewing angle in a stated reference arrangement. Measure both as apparent directions from the eye, positive toward the same side of the optical axis; then an inverted view has negative $M_{\theta }$. If only positive angular sizes are compared, the result is $|M_{\theta }|$ and carries no orientation information. ^[ref].

For the upright virtual image of a converging thin lens, let the eye be at $x=e\ge 0$ and take the unaided reference to be the same small object at a distance $D$ from the eye. Within the small-angle approximation,Specify $D$: keeping the object in its original position means $D=e+u$, whereas the customary magnifier rating uses a near-point reference, often $D=25\mathrm{cm}$. A reference distance is a convention, not a claim that every eye has that near point. ^[ref].

For the virtual image diagram, $m=4$, $v=-30\mathrm{cm}$ and $e=5\mathrm{cm}$. With $D=25\mathrm{cm}$, $M_{\theta }\simeq 20/7$, whereas keeping the object at $D=e+u=12.5\mathrm{cm}$ gives $M_{\theta }\simeq 10/7$. Thus neither viewing comparison gives the transverse magnification four. A person may also be unable to focus on the unaided object at the shorter reference distance.

At $u=f$, each outgoing bundle is parallel with slope $-h_o/f$; its apparent viewing direction has angle ${\theta }_i\simeq h_o/f$. Consequently $M_{\theta }\simeq D/f$ remains finite even though $m$ is undefined at a finite image plane. Equivalently, in the limit $u\to f^{-}$, the factors $m$ and $e-v$ both grow without bound but their ratio tends to $1/f$. For $f=10\mathrm{cm}$ and $D=25\mathrm{cm}$, the relaxed-view prediction is $M_{\theta }\simeq 2.5$. ^[ref].

Transverse magnification is the ratio of corresponding signed lengths perpendicular to the optical axis. For an ideal image of a small arrow of nonzero height,Use the same upward-positive height convention in both planes. Thus an upright $0.50\mathrm{cm}$ arrow imaged as $-0.25\mathrm{cm}$ has $m=-1/2$: inverted and half as tall. The sign describes orientation; real image versus virtual image describes ray intersections. They are separate properties, even when a particular thin lens setup correlates them.^[ref]

For a thin lens with object coordinate $-u$ and image coordinate $v$, the undeviated centre light ray is the straight line through $(-u,h_o)$ and $(0,0)$, so its height at $x=v$ is $h_i=-h_{o}v/u$. This is the signed similar triangles argument. Combining it with the thin lens equation givesFor $f>0$, $u>2f$ gives $-1<m<0$; $u=2f$ gives $m=-1$; $f<u<2f$ gives $m<-1$; and $0<u<f$ gives $m>1$. At $u=f$, the image is conventionally “at infinity”: the divergent expressions for $v$ and $m$ describe the limiting behaviour, not a finite image of infinite height. Different object points produce parallel bundles in different directions, which is why angular magnification remains meaningful.

The real image experiment tests $m$ in two independent ways: signed ratios of measured corresponding intervals, and $-v/u$ from measured longitudinal distances. As a further prediction, moving from $u=30\mathrm{cm}$ to $u=15\mathrm{cm}$ for $f=10\mathrm{cm}$ swaps $u,v$, changing $m=-1/2$ to $m=-2$. The two image sizes differ by a factor of four, while both images remain inverted. This reversal follows from the symmetric thin lens equation and connects to the Bessel lens displacement method.

Under proper rotations a third-rank isotropic tensor is proportional to the Levi-Civita symbol. Its contraction with a field vector is antisymmetric in the two stress indices. A physical symmetric stress therefore vanishes in this linear constitutive model. Under the full orthogonal group an ordinary isotropic third-rank tensor is already zero; the orientation-sensitive Levi-Civita tensor requires the usual pseudotensor distinction.

For an axisymmetric magnetic field $\mathbf{B}=\psi {\mathbf{e}}_{\varphi }/(r\sin \theta )+\nabla \times [\chi {\mathbf{e}}_{\varphi }/(r\sin \theta )]$, the poloidal magnetic flux function gives $B_r={\chi }_{\theta }/(r^2\sin \theta )$ and $B_{\theta }=-{\chi }_r/(r\sin \theta )$. Force balance implies locally $p=p(\chi )$ and $\psi =\psi (\chi )$ on regular connected flux surfaces. Direct calculation gives $(\nabla \times \mathbf{B})\times \mathbf{B}=-({\Delta }_{\ast }\chi +\psi {\psi }^{\prime })\nabla \chi /(r^2{\sin }^2\theta )$, with ${\Delta }_{\ast }\chi ={\chi }_{rr}+\sin \theta {\partial }_{\theta }({\chi }_{\theta }/\sin \theta )/r^2$. Equating this to ${\mu }_0\nabla p$ proves the displayed magnetostatic equilibrium equation, including its sign and dimensions.

An induced magnetosphere is maintained by the interaction of an external magnetized plasma flow with a conducting ionosphere or atmosphere. Its currents and draped magnetic field do not establish an internal planetary dynamo. Venus is a central example; Mars combines this interaction with localized crustal fields.

The Lehnert number compares an Alfvén speed to a rotation speed on length scale $L$. With $\mathrm{Le}\ll 1$, magnetic-wave frequencies are small compared with the rotation frequency, allowing nearly geostrophic flow. Ideal weakly damped magnetic waves also require $\eta /L^2$ and $\nu /L^2$ to be small compared with their wave frequency; a small Lehnert number alone does not guarantee weak diffusion.

An incompressible Boussinesq fluid with uniform magnetic field, uniform vertical rotation and uniform temperature gradient supports waves combining magnetic tension, Coriolis force and buoyancy. Put $K=|\mathbf{k}|$, $h^2=K^2-k_z^2$, $a^2=(\mathbf{B}\cdot \mathbf{k}{)}^2/({\mu }_0\rho )$ and $H=g\alpha \beta$, with ${\partial }_z{T}_0=-\beta$. Eliminating pressure and the temperature and magnetic perturbations gives the displayed dispersion relation for nonzero frequencies. Stable thermal stratification has $H<0$. Clearing frequency denominators requires checking any resulting zero roots in the original equations; a stationary magnetostatic-buoyancy balance can exist independently of the propagating branches.

Electron magnetohydrodynamics neglects ion motion and electron inertia while retaining transport of the magnetic field by the electron current. At constant electron number density, ${\mathbf{u}}_e=-\mathbf{J}/(en)=-c\nabla \times \mathbf{B}/(4\pi en)$. Flux freezing in this velocity givesFor a uniform guide field, the solenoidal linear equation implies ${\omega }^2=(c{B}_0/4\pi en{)}^2{k}_{\parallel }^2{k}^2$. With $v_A=B_0/\sqrt{4\pi n{m}_i}$ and $d_i=c\sqrt{m_i/(4\pi e^{2}n)}$, the coefficient is $c{B}_0/(4\pi en)=v_A{d}_i$.

Hall magnetohydrodynamics retains bulk ion inertia but freezes magnetic flux into the electron velocity ${\mathbf{u}}_e=\mathbf{u}-\mathbf{j}/(en)$. With constant density, negligible electrical resistivity and electron inertia, Gaussian units give $\mathbf{j}=c\nabla \times \mathbf{B}/(4\pi )$ and ${\partial }_t\mathbf{B}=\nabla \times [(\mathbf{u}-c\nabla \times \mathbf{B}/(4\pi en))\times \mathbf{B}]$. The ion velocity still satisfies the magnetohydrodynamic momentum equation. Neglecting that velocity gives electron magnetohydrodynamics, a further approximation rather than a synonym for the complete Hall system.

A theta pinch is a cylindrical plasma configuration with axial magnetic field and azimuthal electric current. For a static equilibrium without gravity, the radial Lorentz force gives $d[p+B^2/(8\pi )]/dr=0$ in Gaussian units. This straight-field configuration has no field-line curvature drive of the usual interchange instability.

A kinematic magnetic dynamo studies magnetic field amplification by a prescribed fluid flow, neglecting the field's feedback on the velocity through the Lorentz force. With incompressibility and zero magnetic diffusivity, a Lagrangian field vector evolves by its local velocity-gradient matrix. A growth law derived in this approximation cannot hold indefinitely once magnetic stresses significantly change the flow.