For a spherical incompressible planetary interior, $M(r)=4\pi \rho r^3/3$ gives $dP/dr=-4\pi G{\rho }^{2}r/3$. With $P(R)=0$, $P(r)=2\pi G{\rho }^2(R^2-r^2)/3$. The surface gravity $g_s=4\pi G\rho R/3$ then yields $P_c=3g_s^2/(8\pi G)$. Add an imposed surface pressure to the entire profile if needed.

Truthful reporting is Bayesian incentive compatible when every type maximizes its expected utility over the other players' types by reporting truthfully. In an independent private values model, the displayed inequality uses the same interim allocation and payment functions for every possible true type.

Dominant-strategy incentive compatibility means truthful reporting maximizes a player's utility for every fixed profile of the other reports. It is the monetary-mechanism version of strategyproofness and implies Bayesian incentive compatibility under any independent prior.

For a perturber following $(p,0,vt)$, the impulse approximation integrates its tidal tensor to $2G{m}_p\mathrm{diag}(1,-1,0)/(v{p}^2)$. A fixed target position therefore receives $\Delta \mathbf{v}=2G{m}_p(x,-y,0)/(v{p}^2)$. The longitudinal kick vanishes after integrating the symmetric trajectory.

A real orthogonal matrix $T$ of size three with $\det T=-1$ has an eigenvector $n$ of eigenvalue $-1$. Its orthogonal complement is invariant, and the restriction there is an orientation-preserving plane rotation. Thus $T$ is a rotation about the axis $\mathbb{R}n$ composed with a reflection across the plane perpendicular to $n$. Writing the plane angle as $\varphi$, $\mathrm{tr}T=-1+2\cos \varphi$. A pure plane reflection is the case $\varphi =0$; if $\det (T-I)\ne 0$, it cannot be a pure plane reflection.

A real charge of height $z$ has a reflected image charge at $-z$. Its acceleration is $\ddot{z}=-\kappa /z^2$, with $\kappa =q^2/(16\pi {\epsilon }_{0}m)$. Released from rest at height $d$, its ideal-model arrival time isThe image follows the instantaneous reflected position. The induced-charge energy carries one-half the full real-image pair interaction factor, consistent with this force.

With initial exit rates $a$ to the advanced living state and $b$ directly to death, and death rate $c>0$ from the advanced state, the expected time to death from state 1 is $1/(a+b)+a/((a+b)c)$. The first term is initial holding time; the second weights the advanced-state holding time by the probability of reaching it.

An idempotent morphism $e:E\to E$ splits if there are $r:E\to H$ and $s:H\to E$ with $sr=e$ and $rs=1_H$. The object $H$ is a retract of $E$. In a functor category, the splitting maps must be natural transformations; objectwise splittings without compatible functorial choices are not enough.

Distinct idempotents are orthogonal when their products in both orders vanish. A finite family summing to one gives a decomposition $M={⨁}_i{e}_{i}M$ of every left module as a vector space; the summands need not be submodules unless the idempotents are central. Vertex idempotents of a path algebra supply the spaces of its quiver representation.

For an incompressible flow with constant mass density, the Elsässer variables are ${\mathbf{z}}^{\pm }=\mathbf{u}\pm {\mathbf{v}}_a$, where ${\mathbf{v}}_a$ is the Alfvén velocity. Adding and subtracting the Euler momentum equation and ideal magnetohydrodynamic induction equation yields ${\partial }_t{\mathbf{z}}^{\pm }+({\mathbf{z}}^{\mp }\cdot \nabla ){\mathbf{z}}^{\pm }=-\nabla \psi$, with $\psi =p/\rho +\Phi +|{\mathbf{v}}_a{|}^2/2$ and $\nabla \cdot {\mathbf{z}}^{\pm }=0$. Thus each field is transported by the other, a useful way to expose interactions of oppositely directed Alfvén waves.

For a static smooth ideal magnetohydrodynamics equilibrium with conservative external forces and surface terms eliminated by the boundary conditions, the linear displacement equation is $\rho \ddot{\mathbf{\xi }}=\mathcal{F}\mathbf{\xi }$, where the force operator is a self-adjoint operator in the appropriate integral pairing. Its restoring quadratic form $\delta W=-\frac{1}{2}\int {\mathbf{\xi }}^{\ast }\cdot \mathcal{F}\mathbf{\xi }dV$ is real. Multiplying the normal mode equation by the complex conjugate of the displacement and integrating gives ${\omega }^2\int \rho |\mathbf{\xi }{|}^{2}dV=2\delta W$. Hence squared frequencies are real, and a negative restoring form gives an unstable mode through the variational Rayleigh quotient, subject to the operator domain and spectral hypotheses. Background fluid flow adds terms to this argument; the static hypothesis is essential.

The smooth ideal-MHD system couples mass conservation, momentum, adiabatic pressure transport and the ideal magnetohydrodynamic induction equation, subject to the solenoidal magnetic-field constraint. Without gravity it is a quasilinear partial differential equation system. Across shocks, conservative total energy replaces smooth adiabatic pressure transport.

A useful ideal magnetohydrodynamics reduction has $\mathbf{u}=\mathbf{v}(z,t)+ax{\mathbf{e}}_y$, $\mathbf{B}=\mathbf{B}(z,t)$ and $\rho =\rho (z,t)$. The constant background shear flow supplies $a{v}_x$ in the axial momentum equation and $a{B}_x$ in axial magnetic induction. The solenoidal constraint and MHD induction equation together make $B_z$ constant.

The inviscid magnetohydrodynamic momentum equation balances inertia, pressure, gravitational acceleration and Lorentz force density. The magnetic force separates into magnetic tension and magnetic pressure. It is coupled to the continuity equation and the ideal magnetohydrodynamic induction equation.

For a fixed Newtonian gravitational potential, total energy density is $\mathcal{E}=\rho (u^2/2+\Phi )+p/(\gamma -1)+B^2/(2{\mu }_0)$. Its flux combines advected kinetic/potential energy, specific enthalpy and the Poynting vector. Dotting the ideal magnetohydrodynamic momentum equation with velocity, adding the adiabatic internal energy equation and the magnetic energy equation cancels the magnetic work. The pressure terms combine into $-\nabla \cdot (p\mathbf{u})$, proving the conservation law. A time-dependent imposed potential instead contributes $\rho {\partial }_t\Phi$.

For an ideal $I\subseteq R$, its conormal module is $I/I^2$, naturally an $R/I$-module. It records generators and relations to first order along the quotient $R/I$. If $I$ is the maximal ideal of a local ring, it is the cotangent space of a local ring over the residue field. For the augmentation ideal $(X_1-1,\dots ,X_r-1)$ of $A[X_1^{\pm 1},\dots ,X_r^{\pm 1}]$, reduction modulo $I^2$ gives $I/I^2\cong A^r$, with basis the classes of $X_i-1$.

In ${\mathbb{F}}_2[u,v]/(u,v{)}^2$, the ideals $(u)$, $(v)$ and $(u+v)$ are pairwise incomparable and nonprime. Their union is the ideal $(u,v)$: the three lines cover its two-dimensional vector space over ${\mathbb{F}}_2$. This shows why a primality hypothesis matters in prime avoidance.

The whole ring $R$, viewed as an ideal. Elements $f_1,\dots ,f_r$ generate the unit ideal exactly when coefficients $g_i\in R$ give ${\sum }_i{g}_i{f}_i=1$.

For ideals $I,J$ in a commutative ring, the ideal quotient is $(I:J)=\{r\in R:rJ\subseteq I\}$. Closure under subtraction and multiplication by arbitrary ring elements proves it is an ideal. If $I\subseteq J=(a)$, then $I=a(I:J)$: write $x=ar\in I$ and observe that $rJ=xR\subseteq I$. Consequently, if also $(I:J)=(b)$, then $I=(ab)$ is a principal ideal.

A non-null hypersurface projection annihilates the unit normal vector and fixes tangent vectors to the hypersurface. It obeys $P^2=P$. Apply it to every tensor index to remove normal components, with the metric used to lower or raise the index pair for covariant slots. For a timelike unit normal in mostly-plus signature it is the spatial projection tensor. A null normal cannot be normalized this way, and a null hypersurface requires additional choices for a corresponding projection.