A product of scalar superfields is a scalar superfield: the pullback of a product is the product of the pullbacks at the same transformed superspace point. Infinitesimally, the even supersymmetry variation obeys $\delta (ST)=(\delta S)T+S(\delta T)$. This follows from the graded Leibniz rule for the odd supercharges, after including their odd parameters.

A function $S(x,\theta ,\bar{\theta })=\varphi (x)$ is an allowed special configuration of an unconstrained superfield. The subspace where all other components remain zero is not invariant: $\delta S=(i\theta {\sigma }^{\mu }\overline{ϵ}-iϵ{\sigma }^{\mu }\bar{\theta }){\partial }_{\mu }\varphi$ produces components depending on Grassmann variables whenever $\varphi$ is nonconstant. Thus an isolated spacetime scalar field is not closed under supersymmetry. A constant gauge-singlet value is a trivial invariant exception.

The complement of a multiplicative subset is a union of prime ideals exactly when the subset is saturated. If $a\notin S$, saturation ensures $(a)\cap S=\varnothing$; extend this to a prime ideal avoiding a multiplicative subset. Conversely, a product in $S$ cannot have a factor in a prime ideal contained in the complement.

In $\mathbb{Z}$, the saturated multiplicative subset $\{-1,1\}$ has complement equal to the union of $(p)$ over all positive primes. No finite subcollection of prime ideals covers this complement: a prime outside the finite list is a nonunit not in any listed ideal.

For independent bond percolation on the square lattice at $p=1/2$, square crossing probabilities bounded below by self-duality extend to a uniform positive bound for rectangles of aspect ratio $3/2$. Expose an extremal vertical square crossing, reflect its geometry, and attach it to an exterior side through an unexposed independent region. Reflection symmetry bounds the attachment probability below. Two attachments and a horizontal square crossing join by the Harris-FKG inequality. Overlapping rectangles and transverse overlap-square crossings then give all fixed aspect ratios. Reflecting a geometrical graph path does not require its reflected edges to be open.

For independent bond percolation on the square lattice, a positive uniform lower bound for square crossings implies a positive uniform lower bound for rectangle crossings of every fixed aspect ratio $\rho >1$. The RSW reflection extension lemma first extends a square to aspect ratio $3/2$. Overlapping rectangles and transverse overlap-square crossings then glue using the Harris-FKG inequality. The required constant depends on $\delta$ and $\rho$, not on scale. At a self-dual parameter, dual crossings also give a uniform upper bound below one.

For stress-free impermeable fixed-temperature plates, thermal-time growth rate $s$, horizontal wavenumber $k$ and vertical index $n$, put $a=k^2+n^2{\pi }^2$ and $P=\mathrm{Pr}$. Vertical velocity, temperature and vertical vorticity amplitudes satisfy $a(s/P+a)W+\sqrt{\mathrm{Ta}}n\pi Z=R{k}^2\Theta$, $(s/P+a)Z=\sqrt{\mathrm{Ta}}n\pi W$, $(s+a)\Theta =W$. Taking their determinant givesThis is a cubic and remains valid without dividing by a potentially zero factor. Setting $s=0$ or $s=i\omega$ determines the stationary or admissible oscillatory neutral curves.

For the free-slip plane layer, onset selects the smallest Rayleigh number across all admissible stationary and oscillatory normal modes. The first vertical mode wins because increasing vertical index raises viscous cost and reduces oscillatory admissibility. Oscillatory minimization replaces the right-hand side of the stationary wave-number equation by $P^2\mathrm{Ta}{\pi }^2/(1+P{)}^2$. For the oscillatory minimum also require positive angular frequency squared. Where both selected curves are admissible, at rapid rotation the ratio of the minimized thresholds is $R_{o,c}/R_{s,c}\sim 2P^{4/3}/(1+P{)}^{1/3}$. Equality gives $8P^4-P-1=0$, with positive root about $0.67660$; below it sufficiently rapid rotation can select oscillatory primary onset.

A saturated convection roll is not stable merely because its single-amplitude cubic coefficient is positive. Perturbations include phase sidebands, transverse bending, differently oriented convection rolls, oscillatory modes and large-scale Eulerian mean flow. Their growth rates follow from the corresponding enlarged amplitude equations or the full linearization about the convection roll. Küppers–Lortz instability and small-angle instability of rotating convection rolls are distinct orientation-dependent mechanisms.

Near an oscillatory onset of rotating Rayleigh-Bénard convection, opposite travelling roll waves have coupled Hopf bifurcation amplitudes. Symmetry gives the displayed cubic amplitude equations, with generally complex self-coupling and cross-coupling coefficients. Write $a_s=\mathrm{Re}{g}_s$ and $a_c=\mathrm{Re}{g}_c$. For $r>0,a_s>0$, a single travelling wave has intensity $r/a_s$ and is stable against its counterpropagating competitor when $a_c>a_s$. Equal standing-wave intensities are $r/(a_s+a_c)$; with $r>0$ and positive denominator they are stable to intensity differences when $a_s>a_c$. Phase symmetries give orbital, rather than strict phase-decay, stability. Spatial modulation, mean flows and oblique convection rolls require additional stability tests.

The polynomial $\rho (w)=(w-1)(w^2+w+a)$ satisfies the root condition for a multistep method exactly for $0\le a\le 1$. For $a<0$ a root is below $-1$; for $a>1$ the quadratic root product forces a root outside the unit disc. The endpoints have simple unit-modulus roots, and the repeated root at $a=1/4$ is strictly inside. A consistent method with this polynomial is convergent throughout the closed interval.

For a circular orbit, fixed total mass, transfer from $M_2$ to $M_1$, negligible spin and $R_L\propto a(M_2/M{)}^{1/3}$, the orbital angular momentum $J=M_1{M}_2\sqrt{Ga/M}$ givesIf the donor follows a radius sequence with stellar radius response exponent $\zeta$, has no additional radius change at fixed mass, and maintains contact, equating its radius derivative to this expression proves the displayed contact equation. Negative external $\dot{J}$ drives donor mass loss when $\zeta +5/3-2q>0$. The appropriate equilibrium or adiabatic response and its timescale must be specified separately; the equation alone is not a universal dynamical-stability criterion.

Case A mass transfer starts while the donor star is on the main sequence, with core hydrogen burning. Its onset precedes giant-envelope formation; donor structure and the binary mass ratio still determine stability.

Case B mass transfer starts after core hydrogen exhaustion and before core helium ignition. It is subdivided by donor structure; early case B mass transfer occurs before a deep convective giant envelope develops.

In nonconservative binary evolution, matter can escape the system and/or external processes change its orbital angular momentum. Escaping overflow, a stellar wind and a common envelope can invalidate the fixed-total-mass and period-product invariants of conservative mass transfer.

For $R_L=Ca(M_d/M{)}^{1/3}$, the period when a donor reaches radius $R$ is $P=2\pi \sqrt{R^3/(C^{3}G{M}_d)}$. Requiring overflow before a specified evolutionary radius therefore bounds the initial period independently of companion mass in this lobe approximation. If that radius scales as $M_d^{5/3}$, the bound scales as $M_d^2$.

For a decaying harmonic function on a semi-infinite strip with equal outward Robin boundary conditions and reflection-symmetric end data, the top and bottom value traces agree. The global relation and its reflected spectral companion express their transforms through $D(k)$. With $H(k)=\frac{1}{2}{\int }_0^{\ell }{e}^{ky}g(y)dy$, the relation is $\psi (-ik)+\psi (ik)=2H(k)/D(k)$. Real zeros of $D$ require compatible data; unrestricted negative Robin parameters do not guarantee a decaying solution.

A bijective ring endomorphism. Its inverse is again a ring homomorphism, so it preserves the ring structure in both directions.

A base-change invariant polynomial identity holding on a rigid representation holds on its dense orbit and hence throughout the representation space. A path through nonzero vertex spaces cannot vanish identically there: assign every involved arrow a rank-one map carrying a chosen source vector to the chosen target vector. This works for repeated arrows. Consequently a nonzero path image in a rigid representation cannot be killed by an outgoing arrow into a nonzero vertex space.

A shear-horizontal wave mode has ${\omega }_n(k)=c_s\sqrt{k^2+(n\pi /h{)}^2}$. At $x=Vt$ with $0<|V|<c_s$, a nondegenerate stationary Fourier phase gives generic $t^{-1/2}$ decay. Faster rays have no stationary point; compactly supported data give eventual zero by finite propagation speed.