The massive N=1 vector multiplet contains spins $1,1/2,1/2,0$. Its massless helicity content splits into a CPT-complete vector multiplet with helicities $(1,1/2)$ and $(-1/2,-1)$, and a CPT-complete chiral multiplet with helicities $(1/2,0)$ and $(0,-1/2)$. The longitudinal vector polarization becomes one of the two scalar polarizations; the other comes from the original real scalar. This is the representation-theoretic inverse of the supersymmetric Higgs mechanism.

Let independent increments ${\xi }_k$ equal $4^k$ or $-4^k$ with probabilities $2^{-k-1}$ each, and zero otherwise. Every partial sum $M_n={\sum }_{k\le n}{\xi }_k$ is integrable and is a martingale. The Borel-Cantelli lemma makes only finitely many increments nonzero almost surely, so $M_n$ has a finite almost-sure limit $Y$. Yet $\mathbb{E}|Y|=\infty$: the disjoint events that only the $k$th increment is nonzero have probabilities at least $c{2}^{-k}$ for $c={\prod }_j(1-2^{-j})>0$, and contribute at least $c{2}^k$ each. This does not contradict the martingale convergence theorem, whose integrability conclusion requires a uniform first-moment bound.

// Target: markov-process.bigb

A trace plot displays an observable along successive iterations of a Markov chain Monte Carlo algorithm. Its horizontal axis is iteration and its vertical axis is the observed parameter or function value. Trends suggest an initial transient or slow exploration, separated regimes suggest transitions between posterior regions, and exact plateaus show repeated states. The plot should be read with acceptance statistics, autocorrelation and multiple starting states; its visual appearance alone cannot prove stationarity or identify the unique cause of poor mixing.

For a chain map $f:C\to D$, use $M_i=C_{i-1}\oplus D_i$ and $d(x,y)=(d_{C}x,(-1{)}^{i}f(x)+d_{D}y)$. The chain map identity makes $d^2=0$. The short exact sequence of chain complexes $0\to D\to M\to C_{\ast -1}\to 0$ has connecting homomorphism $(-1{)}^i{f}_{\ast }$ in degree $i$. Its long exact sequence in homology shows that the homology of $M$ vanishes precisely when every $f_{\ast }$ is an isomorphism.

A spectrum homotopy between compatible maps $D\to E$ is a map from the cylinder spectrum $D\wedge [0,1{]}_{+}$ restricting to them at the endpoints. Thus the homotopies at all levels must themselves respect the structure maps.

A phantom map of spectra is a stable map $p:D\to E$ with $pu=0$ for every map $u:F\to D$ from a finite spectrum. Such a map can be nonzero even though it induces zero on all represented homology theories on based spaces.

For decreasing positive sequences in ${\ell }^1$, write $x\prec y$ if every initial partial sum of $x$ is at most the corresponding partial sum of $y$ and the finite total sums agree. The finiteness requirement matters: equality of two divergent totals interpreted as infinity is too weak to guarantee a doubly stochastic realization of summable-sequence majorization.

For Grassmann-odd four-dimensional Majorana spinors, choose $C^T=-C$ and $({\gamma }^a{)}^T=-C{\gamma }^a{C}^{-1}$. Then $C{\gamma }^a$ is symmetric whereas $C{\gamma }^{abc}$ is antisymmetric. Swapping Grassmann components therefore changes the sign of a vector bilinear but preserves a three-gamma bilinear. Integration by parts and antisymmetry of the vector indices then make the two spinor variations of the quadratic Rarita-Schwinger field action equal. Commuting auxiliary spinors have different interchange signs.

For a Majorana spinor $\lambda$ with odd Grassmann variables as components, ${\lambda }^{T}M\lambda$ sees only the antisymmetric part of $M$. With $C^T=-C$ and $({\gamma }^a{)}^T=-C{\gamma }^a{C}^{-1}$, reversing an antisymmetrized gamma product gives the displayed transpose rule. Thus the same-spinor vector and bivector bilinears vanish, whereas scalar, pseudoscalar and three-form bilinears can be nonzero. Four-dimensional Hodge duality gives the same vanishing result for a bivector multiplied by ${\gamma }_5$. Commuting spinors instead see the symmetric part, so their selection rule is different.

The locally constant function equal to one on $2{\mathbb{Z}}_2$ and minus one on its other coset has Mahler coefficients $a_n=(-2{)}^n$: at nonnegative integers its values are $(-1{)}^j$, so the forward difference operator multiplies it by $-2$. Since $|a_n{|}_2=2^{-n}$ and the binomial polynomials have absolute value at most one on the 2-adic integers, the series converges uniformly. The binomial theorem verifies its values on the nonnegative integers, and continuity extends the identity to the entire domain.

For the Mahler coefficients of a continuous function on the p-adic integers, finite binomial inversion gives $c_n={\sum }_{j=0}^n(-1{)}^{n-j}(\genfrac{}{}{0ex}{}{n}{j})f(j)$. Equivalently their exponential generating function is the displayed identity in the formal power-series ring ${\mathbb{Q}}_p[[T]]$. Each coefficient involves only finitely many terms; no claim of convergence for every $p$-adic value of $T$ is needed.

For an incompressible vertical-field mode, frozen-in magnetic tension supplies a spring of frequency ${\omega }_A$. Its displacement satisfies ${\ddot{\xi }}_x-2\Omega {\dot{\xi }}_y-4\Omega A{\xi }_x=-{\omega }_A^2{\xi }_x$ and ${\ddot{\xi }}_y+2\Omega {\dot{\xi }}_x=-{\omega }_A^2{\xi }_y$. Their determinant gives the ideal magnetorotational dispersion relation. A weak spring transfers angular momentum from the faster inner element to the slower outer element, causing inward and outward migration that increase their separation when angular velocity decreases outward. A sufficiently strong spring instead supplies a restoring force.

// Target: physics.bigb

For $0<A<2\Omega$ and equal kinematic viscosity and magnetic diffusivity, a vertical wavevector admits $\mathbf{v}=(V,V,0)$ and $\mathbf{b}=(i{\omega }_{A}V/A,-i{\omega }_{A}V/A,0)$ in Alfvén-speed units. Substitution into the magnetorotational instability equations gives the displayed frequency and growth rate. In a Keplerian disk, growth of this polarization requires $B_z^2/({\mu }_0\rho )>5\nu \Omega /4$. This is a bound for this selected channel and its wavelength, not a universal threshold for every MRI mode in an unbounded domain.

A polarization specifies the direction or trajectory of the oscillating velocity and magnetic field in a magnetohydrodynamic wave. The linear Alfvén wave has velocity perpendicular to the plane of the background magnetic field and wavevector; the magnetosonic waves have velocity in that plane. A real fixed polarization vector gives a linear oscillation. Two orthogonal components of equal amplitude in quadrature give a circular trajectory. For propagation parallel to a uniform field, combining the two degenerate transverse linear polarizations produces circular polarization; at finite amplitude its constant transverse magnetic-field magnitude removes the longitudinal magnetic pressure gradient.

An entropy mode is a perturbation of entropy and density with no leading pressure or velocity perturbation in a uniform ideal fluid. For an adiabatic equation of state define $s=\delta p/p_0-\gamma \delta \rho /{\rho }_0$. An entropy perturbation can have $\delta p=0$ and $\delta \rho /{\rho }_0=-s/\gamma$, so it has zero frequency in the local fluid frame. In anisotropic reduced magnetohydrodynamics, $Ds=0$ with $D={\partial }_t+{\mathbf{u}}_{\perp }\cdot {\nabla }_{\perp }$, and turbulence mixes its variance as a passive scalar.

Ambipolar damping is attenuation of magnetic disturbances by friction between the magnetically coupled charged fluid and a neutral fluid. In a strongly collisional mixture, the fluids nearly move together, but their small relative velocity dissipates energy. With drag coefficient ${\rho }_i{\mu }_{in}={\rho }_n{\mu }_{ni}$, the frictional energy loss is $-{\rho }_i{\mu }_{in}\int |{\mathbf{u}}_i-{\mathbf{u}}_n{|}^{2}dV$. Neglecting other dissipative processes does not remove this collisional loss.

In a planar magnetohydrodynamic shock with both velocity and magnetic field normal to the front, the normal magnetic field is continuous. Its normal Maxwell stress tensor contribution is consequently the same on both sides, and the Poynting vector vanishes because $\mathbf{u}\times \mathbf{B}=0$. The remaining jump conditions are the ordinary Rankine-Hugoniot conditions for a perfect gas. This is the purely parallel branch; an oblique or switch-on shock has additional transverse fields and is not covered by this reduction.

For a planar magnetohydrodynamic shock with normal velocity and tangential magnetic field, mass conservation gives $\rho u=m$ and the ideal magnetohydrodynamic induction equation gives $uB$ constant across the front. Hence $B/\rho$ is unchanged, expressing magnetic flux freezing. Normal momentum flux is $\rho u^2+p+B^2/(2{\mu }_0)$, and energy flux per unit mass flux is $u^2/2+\gamma p/[(\gamma -1)\rho ]+B^2/({\mu }_0\rho )$, apart from a smooth common gravitational potential. The amplified tangential field contributes both magnetic pressure and magnetic energy transport.

For a static equilibrium and a fluid Lagrangian displacement $\xi$, conservation of mass, adiabatic entropy and magnetic flux freezing give $\delta p=-\xi \cdot \nabla p-\gamma p\nabla \cdot \xi$ together with the displayed relations. Linearizing the pressure/tension form of the momentum equation yields $\rho \ddot{\xi }=-\nabla \delta \Pi -g\delta \rho e_z+{\mu }_0^{-1}(\delta B\cdot \nabla B+B\cdot \nabla \delta B)$, where $\delta \Pi =\delta p+B\cdot \delta B/{\mu }_0$. Static force balance removes the zeroth-order terms; the density multiplying acceleration is the equilibrium density.

For a static ideal magnetohydrodynamics equilibrium in fixed gravity, let $\nabla \cdot \xi =0$ and $\Pi =p+B^2/(2{\mu }_0)$. Frozen flux gives $b=(B\cdot \nabla )\xi -(\xi \cdot \nabla )B$ and continuity gives ${\rho }^{\prime }=-\xi \cdot \nabla \rho$. Multiplying the linear momentum equation by the conjugate displacement and integrating by parts gives the displayed identity when surface terms vanish. The two magnetic cross terms cancel; the remaining magnetic equilibrium derivative combines with the mass density-gradient force to give the two Hessian matrices. Real symmetric Hessian matrices and the nonnegative magnetic tension term make ${\sigma }^2$ real.