A localized disturbance grows along some moving rays but decays at its original fixed location. This wave-packet meaning differs from buoyancy-driven stellar convection and is therefore given a distinct canonical title. Changing the observation frame can change the absolute/convective classification. The Briggs-Bers criterion uses the laboratory-frame impulse response to distinguish this case from absolute wave-packet instability.

A criterion for absolute wave-packet instability based on analytic continuation of the spatial roots of a dispersion relation. Start the temporal inversion contour above the spectrum, then lower it while deforming the spatial contour. A spatial pinch point occurs when branches originating in opposite spatial half-planes obstruct that deformation. A double root $D=D_k=0$ is only an algebraic candidate. The opposite-half-plane condition is stated in the primary study doi.org/10.1017/jfm.2016.195, section 4.2.

The number of nonzero coordinates of a finite vector is its L0 sparsity count. It vanishes only at zero, but is unchanged under multiplication by a nonzero scalar, so it is not a norm. Minimizing it under linear measurement constraints finds a sparse vector with the smallest possible support of a vector. Sparse injectivity of order $s$ guarantees unique recovery of every sparse vector with at most $s$ nonzero coordinates by this objective.

With $s=Q{c}_{s}k/\kappa$ and $\delta =\kappa ϵ/(Q{c}_s)$, maximize $F(s)=2s{e}^{-\delta s}-s^2$. Its unique positive critical point satisfies the displayed equation. For positive gravitational softening, $e^{-\delta s_{\ast }}<1$ and $1-\delta s_{\ast }>0$, giving the strict bound. Unsoftened gravity has $s_{\ast }=1$.

The most unstable softened disk wavenumber minimizes the squared mode frequency. Instability is $Q<Q_c$, while $Q\ge Q_c$ is spectrally stable for this compressive pair. Since $s_{\ast }<(1+\delta {)}^{-1}$, $Q_c^2<(1+2\delta )/(1+\delta {)}^2\le 1$. At zero gravitational softening both bounds become equality. The physical path with fixed $\kappa ϵ/c_s$ obeys $\delta =(\kappa ϵ/c_s)/Q$.

Tensor products $V_{j_L}\otimes V_{j_R}$ give the irreducible representations of the Spin(4) double cover. The simultaneous central sign acts by $(-1{)}^{2j_L+2j_R}$, so exactly the integer-sum pairs descend to SO(4). Their dimensions are $(2j_L+1)(2j_R+1)$. Under the diagonal spatial rotation subgroup, the Clebsch-Gordan decomposition for SU2 gives spins from $|j_L-j_R|$ through $j_L+j_R$ in steps of one.

The reduced dimensionless growth rate $g(q)=-1+q-2q^3/R_c$ has a positive maximum precisely when $R_c>27/2$. Its maximizing $q=\sqrt{R_c/6}$ follows by differentiation. At equality $q=3/2$, outside a strict $q\gg 1$ short-wave regime. The inequality is an exact criterion for the reduced cubic, not a quantitative onset theorem for full thermal stability of a travelling solidification front.

At a fixed horizon-exit e-fold count and with the later cosmology fixed, rescaling a canonical single-field inflationary potential by a positive constant leaves its potential slow-roll parameters and field solution unchanged. The slow-roll curvature power spectrum is proportional to $V/{ϵ}_V$, so the matter power spectrum scales by the same factor while its leading tilt and transfer-function shape remain unchanged. A changed specified reheating history can also shift the horizon-exit count assigned to a fixed physical wavenumber.

If a category has pullbacks in a category, each slice category is finitely complete. The identity of the base object is terminal in the slice. Ambient pullbacks in a category inherit a unique common structure map to the base and satisfy the same universal property over that base. A terminal object and pullbacks in a category construct finite products in a category and equalizers, hence all finite limits.

For two perpendicular rods of length $2L$ joined at one endpoint, the slender-body force density gives translation resistance $CL\mathrm{diag}(3,3,4)$, rotation resistance $C{L}^3\mathrm{diag}(8/3,8/3,16/3)$, and translation–rotation block $C{L}^2\left(\begin{array}{ccc}0& 0& -2\\ 0& 0& 2\\ 2& -2& 0\end{array}\right)$. The opposite block is its transpose. Here $G$ is the torque about the joint and $F$ the force on the fluid. Integrating along each rod proves the entries and makes the symmetry and positivity of a rigid-body resistance matrix explicit.

For the SLE angle process, the bounded Dirichlet problem on the unbounded surviving domain has data $\pi$ on the intrinsic boundary mapped to the left of the Loewner driving function and zero on the boundary mapped to its right. For a simple Loewner trace these are its left and right banks, with the negative and positive real rays respectively. The tip is a discontinuity point of the boundary data.

If the power-set function is eventually constant below a singular cardinal $\kappa$, its constant value persists at $\kappa$. Writing $\theta =\mathrm{cf}(\kappa )$ gives $2^{\kappa }=(2^{<\kappa }{)}^{\theta }$. On the plateau choose a cardinal $\mu \ge \theta$, so $(2^{\mu }{)}^{\theta }=2^{\mu \cdot \theta }=2^{\mu }$.

On a fixed common axis, select the median of reported peaks. For odd $n$ this is the unique Condorcet winner. For even $n$, a fixed lower-median or upper-median convention selects a weak Condorcet winner. Holding other peaks fixed, the attainable outcomes form an interval between two order statistics. Truthful reporting obtains the true peak if it is inside that interval, or its nearest interval endpoint otherwise, so no allowed report improves the outcome under single-peaked preferences.

For the transmitting product with factors $\cosh [(\theta -i\pi j/N)/2]/\cosh [(\theta +i\pi j/N)/2]$ and prefactor $(-1{)}^N$, the continuous phase on $\theta \ge 0$ is $\pi N-2{\sum }_{j=1}^{N-1}\arctan [\tanh (\theta /2)\tan (\pi j/(2N))]$. Its derivative is $-{\sum }_j\sin (\pi j/N)/[\cosh \theta +\cos (\pi j/N)]$. A Riemann sum at fixed positive rapidity gives $(2N/\pi )\log \tanh (\theta /2)$ at leading order. The endpoints depend on the stated unwrapped branch; setting the high-energy value to zero silently changes the constant.

A family fixed by every simplicial section compression is either an initial segment of the simplicial order on the discrete cube or that segment with its last vertex exchanged for the next vertex, where the exchanged vertices are complementary. Indeed, an earlier absent vertex and a later present vertex must differ in every coordinate, and cannot have another vertex between them. Complementary consecutive vertices occur only across the two central ranks in odd dimension, or at the transition from middle-rank sets containing coordinate one to those avoiding it in even dimension. Direct closed graph neighbourhood comparison resolves these exceptions in the proof of Harper theorem.

For a one-dimensional velocity perpendicular to a transverse magnetic field, $c_f^2=\gamma p/\rho +B^2/({\mu }_0\rho )$. The primitive-variable characteristic matrix has two advective eigenvalues and the two speeds $u\pm c_f$. Integrating the fast eigenvectors keeps $p/{\rho }^{\gamma }$ and $B/\rho$ constant. The state parameter obeys ${\rho }_t+(u\pm c_f){\rho }_x=0$, so the characteristic speed satisfies the Inviscid Burgers equation.

A nonnegative mean-reverting short-rate diffusion with positive parameters. Its mean-reversion level is $a/b$ and its diffusion size is proportional to $\sqrt{r}$. The Feller positivity condition for the CIR model controls whether zero is reachable. Its stationary law is a gamma distribution and its zero-coupon bond prices are exponential-affine.

A shearing-wave ansatz in a uniform inviscid isothermal shearing sheet removes background advection when the radial wavevector grows as $k_x(t)=k_x(0)+S{k}_{y}t$. The razor-thin disk Poisson kernel reduces the pressure-plus-gravity amplitude to $[c_s^2-2\pi G{\Sigma }_0/k]{\tilde{\Sigma }}^{\prime }/{\Sigma }_0$. Continuity couples it to $i{k}_x{v}_x+i{k}_y{v}_y$, while the two momentum equations have rotational couplings $-2\Omega v_y$ and $(2\Omega -S)v_x$. The linearized vortensity amplitude remains constant, because the barotropic pressure and gravitational forces have zero curl and the background vortensity is uniform.

In a finite sharply two-transitive group, the identity together with all fixed-point-free elements forms a regular normal subgroup. One proof constructs virtual characters from induced point-stabilizer characters and the augmentation character, proves their norms are one, and realizes this set as an intersection of representation kernels. Conjugation by the point stabilizer is transitive on the kernel's nonidentity elements. This forces the kernel to be elementary abelian, so it is a unique Sylow subgroup and is characteristic. Closure of the fixed-point-free set is a theorem, not something supplied merely by counting its size.

Complementing predecessor sets within $N\setminus \{i\}$ preserves their Shapley value weights, since the factorials $|S|!$ and $(n-|S|-1)!$ interchange. For the dual coalitional game, the corresponding marginal contribution equals the original contribution at that complementary set. Hence the values coincide. The same bijection gives ${\varphi }_i(v)={\sum }_{S}w(S)(v(N\setminus S)-v(S))$.