At the Batchelor scalar microscale, molecular diffusion rate $\alpha /r^2$ balances the smooth Kolmogorov strain rate $(ϵ/\nu {)}^{1/2}$. For $\alpha \ll \nu$ this scale is much smaller than the Kolmogorov length scale.

For $\tau =t_{\ast }-t$, balance $\mu a_t/a\sim \gamma /a$ to obtain radius proportional to $\tau$. Axial lubrication conservation then makes the axial length proportional to $\tau$ as well. The displayed variables reduce the lubrication equation for a bubble with viscous exterior to $\zeta A{A}^{\prime }-A^2=\frac{1}{16}\{A^4[(2\zeta A^{\prime }+1)/A{]}^{\prime }{\}}^{\prime }$. The physical interface slope is $\sqrt{\lambda }{A}^{\prime }$, compatible with a slender neck when $\lambda \ll 1$ and the dimensionless profile varies on order-one scales.

Let a continuous function satisfy both viscosity inequalities for the Laplace operator. On any relatively compact ball, take its harmonic replacement $h$ with matching boundary values. If $u-h$ has a positive maximum, the function $h+\epsilon (R^2-|x-a{|}^2)$ still lies below $u$ somewhere for sufficiently small positive $\epsilon$, while matching it on the boundary. At an interior maximum of their difference it is a valid upper test, but its Laplacian is $-2n\epsilon <0$, a contradiction. The opposite perturbation excludes a negative minimum. Thus $u=h$ throughout the ball and is a smooth harmonic function. A locally defined test is extended near its touching point using a smooth cutoff function when global tests are required.

Write the virtual character as $\psi ={\sum }_{\chi }{a}_{\chi }\chi$ with integer coefficients and irreducible characters $\chi$. Character orthogonality gives $⟨\psi ,\psi ⟩={\sum }_{\chi }{a}_{\chi }^2$. Norm one forces exactly one coefficient to be $1$ or $-1$, and positive degree selects the positive sign.

A complex class function $f$ on a finite group is a generalized character if and only if $f_E$ is a generalized character of every elementary subgroup. Equivalently, $⟨f_E,\alpha {⟩}_E\in \mathbb{Z}$ for every such $E$ and $\alpha \in \mathrm{Irr}(E)$. Ordinary characters additionally have nonnegative global irreducible multiplicities; integrality of the virtual-character lattice is the conclusion of this criterion.

Critical closed bosonic strings have independent left- and right-moving Virasoro constraints. Positive modes in both sectors annihilate physical kets, while both zero-mode conditions impose the mass shell and closed-string level matching. Quotient by physical null string states, equivalently use the appropriate BRST cohomology. With $L_0={\alpha }^{\prime }{k}^2/4+N$, the result is $N=\tilde{N}$ and $M^2=4(N-1)/{\alpha }^{\prime }$. Negative matter modes are adjoints rather than additional ket-annihilation constraints.

For cylindrical rotation independent of height, write $W_{r\varphi }=\int T_{r\varphi }dz$. Define the effective kinematic viscosity by $W_{r\varphi }=\bar{\nu }\Sigma r{\Omega }^{\prime }$, where $\Sigma$ is the surface density of a disk. For Keplerian rotation, $\bar{\nu }=-2W_{r\varphi }/(3\Sigma \Omega )$, and the outward viscous torque in an accretion disk is $-2\pi r^2{W}_{r\varphi }=3\pi \bar{\nu }\Sigma \sqrt{GMr}$. A Newtonian fluid stress tensor gives the density-weighted viscosity of a disk; the stress definition can also parameterize an effective turbulent or magnetic transport stress.

// Target: astrophysics.bigb

A nonzero imaginary root of the vertical-field magnetoconvection dispersion relation requires $0<\zeta <1$ and $Q>\zeta (1+\sigma )s^2/[\sigma (1-\zeta ){\pi }^2]$. Its marginal curve is $R^{(o)}=C{s}^3/k^2+DQ{\pi }^{2}s/k^2$, with $C=(\sigma +\zeta )(1+\zeta )/\sigma$ and $D=\zeta (\sigma +\zeta )/(1+\sigma )$. A formal minimum of that algebraic curve is a physical oscillatory threshold only if the frequency squared there is positive.

Both steady and oscillatory neutral curves have the form $R=C(k^2+p{)}^3/k^2+DQp(k^2+p)/k^2$, with $p={\pi }^2$. Their exact minimizing condition is $x^2(2x+3p)=p^3+(D/C)Q{p}^2$, where $x=k^2$. For fixed positive coefficients and large Chandrasekhar number, $x\sim [D{p}^{2}Q/(2C){]}^{1/3}$ and $R_{\min }=DpQ+3C^{1/3}(D{p}^{2}Q/2{)}^{2/3}+O(Q^{1/3})$. The optimum balances transverse diffusion against the magnetic penalty, leading to increasingly narrow rolls.

At the minimum of the steady vertical-field magnetoconvection dispersion relation, stationarity gives $R_c=2(k_c^2+{\pi }^2{)}^3/{\pi }^2$. With nonmagnetic value $R_0=27{\pi }^4/4$, eliminating the critical wavenumber gives the stated relation and $k_c^2={\pi }^2[\frac{3}{2}(R_c/R_0{)}^{1/3}-1]$. These hold for every nonnegative Chandrasekhar number, including its zero-field limit.

For one stress-free planar roll mode, thermal-time rescaling reduces the linear dynamics to $a^{\prime }=\sigma (-a+rb-\zeta qc)$, $b^{\prime }=a-b$, $c^{\prime }=a-\zeta c$. Its characteristic polynomial is $(s+\sigma )(s+1)(s+\zeta )-\sigma r(s+\zeta )+\sigma \zeta q(s+1)$. For $\sigma ,\zeta >0$ and $q\ge 0$, steady marginality is $r_s=1+q$. Oscillatory marginality is $r_H=(\sigma +\zeta )(1+\zeta )/\sigma +\zeta (\sigma +\zeta )q/(\sigma +1)$, with ${\omega }_H^2=\zeta [\sigma (1-\zeta )q/(\sigma +1)-\zeta ]$. A genuine Hopf bifurcation onset requires ${\omega }_H^2>0$; a formal imaginary-root formula with negative frequency squared is not an oscillation. The Routh-Hurwitz stability criterion selects the first crossing, while nonlinear coefficients determine criticality.

For an indecomposable module $M$ with vertex of an indecomposable module $D$, a source is an indecomposable $RD$-module $S$ which is a direct summand of ${\mathrm{Res}}_D^{G}M$ and for which $M$ is a direct summand of ${\mathrm{Ind}}_D^{G}S$. It has vertex $D$ as a $D$-module. Sources at a fixed vertex are conjugate by its normalizer.

Let $N_G(D)\le H\le G$. Set $\mathcal{X}=\{D\cap {}^{g}D:g\notin H\}$ and $\mathcal{Y}=\{H\cap {}^{g}D:g\notin H\}$. Green correspondence pairs indecomposable modules with vertex $D$ for $G$ and $H$: restriction has a unique vertex-$D$ summand, with the other summands relatively projective for $\mathcal{Y}$; induction has a unique vertex-$D$ summand, with the other summands relatively projective for $\mathcal{X}$. The two assignments are inverse and preserve sources. Its more general form applies to vertices contained in $D$ but not conjugate into any member of $\mathcal{X}$.

Every noncyclic finite finite p-group $P$ has a quotient $C_p\times C_p$. Over any field of characteristic $p$, define a square-zero representation with basis $v_0,\dots ,v_r,w_1,\dots ,w_r$ by $x{v}_i=w_i$ for $1\le i\le r$, $x{v}_0=0$, $y{v}_i=w_{i+1}$ for $0\le i<r$, $y{v}_r=0$, and $xW=yW=0$. Here $x=g-1$, $y=h-1$, and $W$ is the span of the $w_i$. An endomorphism induces the same scalar on the top and on $W$; the remaining maps from the top to $W$ form a square-zero ideal. Thus the endomorphism ring is local and the module is indecomposable. Inflating to $P$ and choosing $p\nmid 2r+1$ gives infinitely many distinct dimensions with vertex $P$: the trace of a relative-trace identity from any proper subgroup would force dimension zero in the field. This construction works over finite fields too.

At high energy with $x=-t/s\in (0,1)$ fixed, reflection and Stirling formulas for the gamma function give the displayed exponential envelope, away from real-axis resonance poles or after suitable analytic continuation. It is softer than the power-law fixed-angle behavior of typical point-particle interactions and exposes the finite string length. This limit differs from the fixed-$t$ Regge limit and does not by itself prove finiteness of every loop amplitude.

For a separation parallel to a unit vector $\mathbf{n}$, the longitudinal velocity correlation is $Q_{LL}=⟨u_L(\mathbf{x})u_L(\mathbf{x}+r\mathbf{n})⟩=u^{2}f(r)$. In isotropic incompressible turbulence the transverse normalized correlation is $g=f+rf\prime /2$ and the trace is $C=u^2(3f+rf\prime )$.

The Kármán-Howarth equation evolves two-point correlations in homogeneous isotropic turbulence. In trace form it is ${\partial }_{t}C=r^{-2}{\partial }_r[r^{-1}{\partial }_r(r^4{u}^{3}K)]+2\nu r^{-2}{\partial }_r(r^{2}C\prime )$. Integrating with radial volume weight reduces conservation of the Saffman integral to vanishing nonlinear and viscous boundary fluxes.

A gauge connection on superspace has both vector and spinor components. Subtracting the flat-frame torsion from the graded commutator of its covariant derivatives defines its supercurvature. The conventional four-dimensional $\mathcal{N}=1$ spinor-curvature constraints leave one real vector superfield as a prepotential. Its chiral field-strength superfield contains the ordinary gauge curvature, gaugino and real auxiliary field.

For a smooth vector subbundle $F\subset E$, the quotient has fibers $(E/F{)}_p=E_p/F_p$. Extend a local frame of a vector bundle of $F$ to one of $E$; the classes of the complementary frame vectors trivialize the quotient. Adapted transition functions of a vector bundle are block upper triangular, and their lower-right blocks give the quotient transitions. These are smooth and satisfy the cocycle identity, defining a smooth vector bundle. The projection $E\to E/F$ is a smooth vector bundle morphism.

A fiber metric on a smooth vector bundle $E$ gives a smooth orthogonal projection $P:E\to F$ onto any vector subbundle. In a local frame of $E$, let $G$ be the metric matrix and $A$ the columns of a frame of $F$. Then $P=A(A^{t}GA{)}^{-1}{A}^{t}G$, proving smoothness. The complementary subbundle $F^{\perp }=\ker P$ maps isomorphically onto the quotient vector bundle. Explicitly $v\mapsto (Pv,[v])$ is a vector bundle isomorphism $E\to F\oplus (E/F)$, with inverse $(f,[v])\mapsto f+(I-P)v$. This inverse is independent of the representative $v$, but the splitting depends on the chosen metric.