A kernel specifying the jump rate from state $x$ for each displacement $z$. With the displacement convention the infinitesimal generator is $L\phi (x)=\int (\phi (x+z)-\phi (x))K(x,dz)$. The destination convention instead records a rate measure on the new state, translating each displacement by $x$. For finite total rate, that total is the exponential holding rate and the normalized measure is the next-jump distribution.

Let $T$ be a rate-one exponential random variable independent of a uniform random variable $U$ on $(0,1)$. In its natural filtration, this Markov jump process jumps once from zero to its mark and then stays there. Its destination rate kernel is $q(x,dy)=1_{\{x=0\}}{1}_{(0,1)}(y)dy$. For a Borel mark set $B$, the martingale $1_{\{T\le t,U\in B\}}-|B|(t\wedge T)$ follows directly from the memoryless residual lifetime. Information about the future mark or lifetime must not be inserted into the initial filtration.

This finite-state Markov jump process models two opposing populations, each suffering losses at a rate proportional to the surviving opposing population. Both coordinate axes are absorbing. Its embedded jump chain removes one individual at each step, with probabilities $b/(a+b)$ and $a/(a+b)$. The corresponding deterministic equations $\dot{a}=-\kappa b$, $\dot{b}=-\kappa a$ conserve $a^2-b^2$; finite stochastic populations satisfy a near-invariant instead.

A diffusion limit rescales a sequence of stochastic processes so that its limit is a diffusion process. For suitably rescaled independent small jumps, the first two jump moments determine the limiting drift and diffusion coefficient. Brownian motion arises when the limiting drift is zero and the variance grows linearly in time.

A driftless square-root diffusion process is the nonnegative solution of $d{X}_t=\sqrt{X_t}d{W}_t$, with zero an absorbing boundary. For elapsed time $h>0$, starting from $x\ge 0$, its Laplace transform isTo prove this, fix the final time $h$ and set $F(t,x)=\exp [-xu/(1+u(h-t)/2)]$. Differentiation gives $F_t+\frac{1}{2}x{F}_{xx}=0$, while $F(h,x)=e^{-ux}$. By Itô formula, $F(t,X_t)$ is a local martingale; it takes values in $[0,1]$, so localization and bounded convergence make it a true martingale. Taking expectations at the two endpoints proves the formula. The Markov property gives the same formula conditionally at any starting time.

In a one-Brownian-factor arbitrage-free market, every traded derivative with nonzero diffusion exposure $s$ has the same excess-drift/exposure ratio. Otherwise holdings $s_2$ in the first and $-s_1$ in the second, financed in the bank account, cancel all diffusion risk while leaving a nonzero excess drift. At zero exposure, the correct statement is $m-rV=\theta s=0$, not division by zero. The short rate itself need not be a traded asset, so its physical drift does not alone fix this risk price.

A positive state-price density prices a productive stock through a conditional dividend sum plus a future deflated-price term. The displayed condition removes that terminal term, yielding transversality and fundamental dividend prices. An individual consumption budget instead has terminal term $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$, so its vanishing must be imposed or proved for the particular holdings as well.

For nested compact H-hulls whose closures avoid $x\in \mathbb{R}$, the boundary derivative is an excursion avoidance probability. Avoiding the larger hull is a subset of avoiding the smaller one, which proves the derivative inequality. For a hull contained in $\{|z|\le R\}$ and $|x|>R$, comparison with the filled half-disc gives the sharp bound $1-R^2/x^2\le g_A^{\prime }(x)\le 1$, since that half-disc has mapping-out function $z+R^2/z$. The empty hull and filled half-disc attain the two bounds.

Decreasing strictly positive summable sequences with equal totals and $x\prec y$ admit an infinite doubly stochastic matrix realizing the displayed relation. Choose $k$ with $y_k\ge x_1>y_{k+1}$ and average the pair to obtain $x_1$ and $y_k+y_{k+1}-x_1$. The remaining decreasing sequence majorizes the remaining target sequence: short initial sums contain values at least $x_1$, and longer sums are the old sums minus $x_1$. Repeat on the tail, freezing one normalized finite-support row at each stage. Limit column sums are at most one; Tonelli's theorem, equal finite totals and strict positivity of every $y_j$ force every column sum to equal one.

For anticommuting components and an antisymmetric numerical matrix $A$, $\delta ({\psi }^{T}A\psi )=2\delta {\psi }^{T}A\psi$: move the second varied field past the first and use $A^T=-A$. Applying this to $C^{-1}$ and to $C$ cancels the factor $1/2$ in $\mathcal{L}=\bar{\psi }i/\partial \psi +\frac{1}{2}(m{\psi }^T{C}^{-1}\psi -m^{\ast }\bar{\psi }C{\bar{\psi }}^T)$. Varying the Dirac adjoint gives the first displayed equation. Integration by parts in the spinor variation gives $i{\gamma }^{\mu T}{\partial }_{\mu }{\bar{\psi }}^T+m{C}^{-1}\psi =0$; multiplication by $C$ gives the second. Squaring the coupled equations yields $(\square +|m{|}^2)\psi =0$, so the physical Majorana fermion mass is $|m|$.

For one-dimensional flow obeying an isothermal equation of state with $u_z=B_z=0$, the primitive variables $(\rho ,u_x,u_y,B_y)$ form a four-variable hyperbolic system and $B_x$ is constant. The speeds relative to $u_x$ satisfy $c^4-(c_s^2+v_a^2)c^2+c_s^2{v}_{ax}^2=0$. These are the fast and slow magnetosonic waves. The independent out-of-plane Alfvén wave polarization and a freely advected entropy mode are absent from this reduced isothermal system.

For sound speed $v_s$ much larger than Alfvén speed $v_a$, and angle $\theta$ between the wavevector and magnetic field, the fast magnetosonic wave has $v_f^2=v_s^2+v_a^2{\sin }^2\theta +O(v_a^4/v_s^2)$. The slow magnetosonic wave has $v_{\mathrm{slow}}^2=v_a^2{\cos }^2\theta [1-(v_a^2/v_s^2){\sin }^2\theta +O(v_a^4/v_s^4)]$. The fast mode is predominantly longitudinal sound. The slow mode becomes almost incompressible, polarized in the plane of the wavevector and magnetic field; the Alfvén wave is polarized perpendicular to that plane. The latter two are predominantly magnetic-tension waves with equal leading phase speeds.

For two incompressible streams of equal density with a common tangential magnetic field parallel to their velocities, decaying normal modes and interface displacement/total-pressure continuity give $\rho (\sigma +k_x{U}_1{)}^2+\rho (\sigma +k_x{U}_2{)}^2=2k_x^2{B}^2$ in units ${\mu }_0=1$. The displayed dispersion relation follows, with $v_A^2=B^2/\rho$. Thus $|\Delta U|<2v_A$ excludes exponential Kelvin-Helmholtz instability; equality is marginal. Above threshold the growth rate is $|k_x|\sqrt{(\Delta U{)}^2/4-v_A^2}$. The transverse horizontal wavenumber affects the decay length through $k=(k_x^2+k_y^2{)}^{1/2}$ but not this growth threshold. Magnetic tension supplies the stabilizing restoring force.

In a steady axisymmetric magnetohydrodynamic wind, axisymmetric pressure and Newtonian gravity exert no azimuthal torque. The azimuthal momentum equation is $\rho {\mathbf{u}}_p\cdot \nabla (s{u}_{\varphi })={\mu }_0^{-1}{\mathbf{B}}_p\cdot \nabla (s{B}_{\varphi })$. Mass conservation and the solenoidal vector field condition imply $\nabla \cdot {\mathbf{F}}_J=0$. With magnetohydrodynamic mass loading $\rho {\mathbf{u}}_p=\kappa {\mathbf{B}}_p$, the transported angular momentum per unit poloidal magnetic flux is $\ell =\kappa s{u}_{\varphi }-s{B}_{\varphi }/{\mu }_0$. It is constant on connected regular poloidal flux surfaces. The usual specific magnetohydrodynamic angular-momentum invariant is $L=\ell /\kappa$, provided $\kappa \ne 0$; confusing these two normalizations loses a factor of mass loading in the wind torque.

The shell-integrated magnetic energy spectrum distributes magnetic energy over wavenumbers. With a solenoidal isotropic spectral tensor normalized to ${\hat{C}}_{ii}=2H(k)$ and the inverse Fourier transform measure $d^{3}k/(2\pi {)}^3$, one has $\mathbb{E}|\mathbf{B}{|}^2={\pi }^{-2}\int k^{2}H(k)dk$. In Gaussian units, $E_B(k)=k^{2}H(k)/(8{\pi }^3)$ integrates to the energy density $\mathbb{E}|\mathbf{B}{|}^2/(8\pi )$. A spectrum per unit mass or normalized to $B^2/2$ has different constant factors. Scale-local fluctuation estimates give $\delta B_l^2$ proportional to $k{E}_B(k)$ at $k\sim l^{-1}$ when the spectrum is normalized consistently.

For magnetic concentration by an incompressible stagnation flow, one possible normalization gives $(g^2{)}^{\prime }=2\eta -2\omega g^2$, $f^{\prime }/f=2\omega -\lambda \eta /g^2$, and $b^{\prime \prime }+(q^{-1}+q)b^{\prime }+(\lambda -m^2/q^2)b=0$. The mode parameter $\lambda$ is not fixed by decay at infinity alone. The regular axisymmetric Gaussian has $\lambda =2$, $b=e^{-q^2/2}$ and conserved axial magnetic flux, with energy proportional to $g^{-2}$. The radial equation also has regular algebraic-tail solutions.

For a horizontal field, set $C=B^2/{\mu }_0$. Static force balance gives $\delta \Pi =\rho g{\xi }_z-(\gamma p+C)\nabla \cdot \xi +Ci{k}_y{\xi }_y$. Given smooth compactly supported vertical and along-field displacements, the displayed choice makes total-pressure perturbation zero while ${\xi }_x=O(k_x^{-1})$. The nonnegative pressure and transverse-tension terms vanish as $|k_x|\to \infty$. This constructs the infimum needed for the magnetohydrodynamic energy principle; a strictly negative limit is already negative at some finite wavenumber.

Every antichain in $\mathcal{P}([n])$ has at most $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$ members. Equality is attained only by a full middle level, with either middle level possible when $n$ is odd.

A nonzero steady Lorenz model of thermosolutal convection branch obeys $r(x)=(1+x)+r_s\tau (1+x)/({\tau }^2+x)$, with $x=a^2$. For $0<\tau <1$, it has an interior saddle-node bifurcation precisely when $r_s>{\tau }^3/(1-{\tau }^2)$. Differentiating gives $x_{\ast }=-{\tau }^2+\sqrt{r_s\tau (1-{\tau }^2)}$ and the stated minimum. The weaker-versus-stronger threshold distinction matters: a sufficient bound need not be the sharp subcriticality condition. Steady-branch existence alone does not decide oscillatory stability.

Linearization of the Lorenz model of thermosolutal convection at conduction yields $(\lambda +\sigma )(\lambda +1)(\lambda +\tau )-\sigma r(\lambda +\tau )+\sigma r_s(\lambda +1)=0$. A Hopf bifurcation requires the cubic coefficient identity $a_3=a_1{a}_2$ and positive frequency squared. Its threshold is the stated $r_H$, with ${\omega }^2=\sigma r_s(1-\tau )/(\sigma +1)-{\tau }^2>0$. If $r_s={\beta }^2\tau$ and $\delta =\sigma /(1+\sigma )$, then $r_H=1+\tau /\delta +{\beta }^2\tau \delta +O({\tau }^2)$. Comparing with the steady fold of a thermosolutal Lorenz model gives a leading nonnegative difference $\tau (\beta \delta -1{)}^2/\delta$ for $\beta \ge 0$.