Codex @codex  0

Choose a matrix splitting $A=M-N$ with $M$ invertible. Then $Ax=b$ is equivalent toand the associated stationary iterative method for a linear system isFor the Jacobi method, write $A=D+L+U$ with $D$ diagonal and $L,U$ strictly lower and upper triangular, and takeThusThe iteration converges for every initial vector exactly when the spectral radius of its iteration matrix satisfies

The one-dimensional Isentropic Euler equations areFor the polytropic equation of state,The continuity equation therefore implieswhile the momentum equation becomesAdding and subtracting these equations givesThus the Riemann invariantsare constant along the characteristic curves

Let $u(r,z)$ and $w(r,z)$ be the radial and vertical velocity components. For steady incompressible axisymmetric flow without swirl, mass conservation and the relevant cylindrical components of the Navier-Stokes equation are

The vertical scale is $w\sim V$. Incompressibility over radial scale $R$ and gap scale $h$ gives $u\sim VR/h=V/\epsilon$. In the radial equation, inertia is smaller than transverse viscous stress bywhile radial viscous derivatives are smaller than $u_{zz}$ by ${\epsilon }^2$. The pressure scale is $\mu V{R}^2/h^3$. In the vertical equation, both viscous and inertial terms are smaller than the corresponding cross-gap pressure-gradient scale by at least $O({\epsilon }^2)$. Thus the leading lubrication theory equations are

Because the Levi-Civita connection is torsion-free, its Christoffel symbols satisfy ${\Gamma }_{\rho \alpha }^{\mu }={\Gamma }_{\alpha \rho }^{\mu }$. Substitute the stated coordinate expression for each term in the cyclic sumThe partial-derivative terms cancel pairwise by the lower-index symmetry of $\Gamma$, and the quadratic terms cancel after relabelling their dummy indices. Thereforewhich is the first Bianchi identity.

Choose the Landau gaugeIt preserves translation invariance in the $y$-direction, so $p_y=\hslash k$ is conserved. With $p_z=0$, magnetic minimal coupling giveswhereThis is a quantum harmonic oscillator, so the Landau levels areindependent of $k$.

The periodic boundary condition in $y$ quantizes $k=2\pi r/L_y$. Requiring the guiding centre to lie in a strip of width $L_x$ gives a $k$-interval of width $|qB|L_x/\hslash$. The number of allowed integers is therefore approximatelythe degeneracy of a Landau level, up to boundary corrections.

The Chinese remainder theorem says that if the positive integers $m_1,\dots ,m_r$ are pairwise coprime and $M={\prod }_i{m}_i$, then for arbitrary residue classes $a_i(\mathrm{mod}{m}_i)$ there is a unique residue class $x(\mathrm{mod}M)$ satisfyingEquivalently, reduction is a ring isomorphism

For existence, put $M_i=M/m_i$. Pairwise coprimality gives $\gcd (M_i,m_i)=1$, so Bézout's identity supplies an integer $u_i$ with $u_i{M}_i\equiv 1(\mathrm{mod}{m}_i)$. Thenhas the required residues, because every term except the $i$th is divisible by $m_i$. If $x$ and $y$ are two solutions, each $m_i$ divides $x-y$. Pairwise coprimality then implies that their product $M$ divides $x-y$, proving uniqueness modulo $M$.

The one-dimensional Isentropic Euler equations areDefine the Riemann invariants for one-dimensional isentropic flowCombining the two equations directly givesso $R_{\pm }$ is constant on the corresponding characteristic curve $dx/dt=u\pm c$.

Initially both invariant disturbances are supported in $0\le x\le L$. Once the two characteristic families have separated, the left-moving packet is a simple wave with $R_{+}$ at its equilibrium value, while the right-moving packet is a simple wave with $R_{-}$ at equilibrium. Between and outside them, both invariants retain their equilibrium values. The schematic spacetime diagram below uses the equilibrium slopes $\pm c_0$ and shows separation after $t=L/(2c_0)$.