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The cellular cochain complex of finite Real projective space consists of alternating maps zero and two. Its groups can be tensored using the Künneth theorem; the integral Künneth torsion polynomial efficiently counts all tensor and Tor functor summands. A product of factors of dimensions two, three and four has free-rank polynomial $1+t^3$ and torsion polynomial $3t^2+3t^3+5t^4+6t^5+5t^6+4t^7+2t^8+t^9$.

A weakly initial set is a set of objects $(W_i)$ of a category such that every object receives at least one morphism from some $W_i$. No uniqueness is required. A solution-set condition asserts this property in suitable comma categories.

A maximal Cauchy development is the largest globally hyperbolic solution, unique up to an appropriate isometry, determined by admissible initial data, with the initial hypersurface a Cauchy hypersurface. It can sometimes be extended as a spacetime across a Cauchy horizon even though that extension is no longer globally determined by the initial hypersurface.

A Finkelstein diagram plots areal radius $r$ against $T=v-r$ in ingoing Eddington-Finkelstein coordinates. Its radial null directions show explicitly that both future light-ray families decrease $r$ inside a Schwarzschild black hole. It is not a compactified Penrose diagram.

For a homogeneous canonical scalar field, the cosmological perfect-fluid continuity equation reduces to $\dot{\varphi }(\ddot{\varphi }+3H\dot{\varphi }+V^{\prime })=0$. It implies the inflaton equation of motion wherever $\dot{\varphi }\ne 0$, and by continuity at isolated turning points. It cannot by itself exclude a frozen field on a time interval. Indeed, choose $\varphi ={\varphi }_0$ with $V({\varphi }_0)>0$ but $V^{\prime }({\varphi }_0)\ne 0$; a de Sitter scale factor in flat slicing with $H^2=V({\varphi }_0)/(3M_{\mathrm{Pl}}^2)$ satisfies the homogeneous Friedmann equation and Friedmann acceleration equation, but not the scalar equation. Varying the inflaton action in an expanding universe supplies the missing equation also for a frozen field, requiring $V^{\prime }({\varphi }_0)=0$.

For the one-dimensional Itô equation $dX=b(X)dt+\sigma (X)dW$, the infinitesimal generator on smooth test functions is $L=b{\partial }_x+{\sigma }^2{\partial }_{xx}/2$. The Kolmogorov backward equation is $u_t=Lu$, while its formal adjoint produces the Fokker-Planck equation.

The Yosida averaging of a semigroup gives $x_h=h^{-1}{\int }_0^{h}S(t)xdt\in D(A)$ with $x_h\to x$, proving that the generator domain is dense. For $x\in D(A)$, the Bochner integral identity $S(t)x-x={\int }_0^{t}S(s)Axds$ holds. If $x_n\to x$ and $A{x}_n\to y$, pass to the limit in this identity and divide by $t$. The strongly continuous semigroup property makes the resulting limit $y$, so $x\in D(A)$ and $Ax=y$. This proves that the infinitesimal generator of a semigroup is a closed operator.

For proper convex functions whose infimal convolution is well defined, $(f\square g{)}^{\ast }=f^{\ast }+g^{\ast }$. The proof separates a supremum over independent variables after writing $x=u+y$. No attainment of the infimum is needed.

For a real symmetric bilinear form, let $n_{+}$ and $n_{-}$ be the dimensions of its positive and negative diagonal subspaces. This pair gives the first two components of the inertia of a bilinear form and is independent of basis by the Sylvester's law of inertia. For a nondegenerate form their sum is the full dimension; otherwise the nullity is an additional invariant. Another convention calls the difference $n_{+}-n_{-}$ the signature of a quadratic form. The pair convention describes the positive and negative harmonic spaces of the real de Rham intersection form in dimension four.

For a plane inertia-gravity wave on a resting layer, $\eta ={\eta }_0\cos (kx-\omega t)$, the horizontal velocities are $u=U\cos \theta$ and $v=V\sin \theta$, where $U=\omega {\eta }_0/(k{h}_0)$ and $V=f{\eta }_0/(k{h}_0)$. First-order integration at a fixed equilibrium coordinate gives $\xi =-U\sin \theta /\omega$, $\upsilon =V\cos \theta /\omega$. The particle trajectory is an ellipse with axis ratio $|f|/\omega$. For positive $f$, the orbit is clockwise in the $(x,y)$ plane.

Interim individual rationality requires a nonnegative expected participation utility for each private type, averaging over the other types with the player's conditional beliefs. It differs from requiring nonnegative utility at each realized profile, which is ex post individual rationality.

Ex post individual rationality requires nonnegative participation utility at every realized profile of types. A truthful threshold allocation with its critical-value payment satisfies this condition for nonnegative valuations.

For a finite simple graph, $\alpha (G)\ge {\sum }_{v\in V(G)}1/(1+\mathrm{deg}v)\ge |V(G){|}^2/(|V(G)|+2|E(G)|)$. Randomly order its vertices and retain each vertex preceding all its graph neighbours. This gives an independent set with the first expression as its expected value. The second inequality is the Cauchy-Schwarz inequality.

For a uniform-density sphere with incompressible flow of radius $R$ with one surface spherical harmonic displacement ${\xi }_r(R)=lA{R}^{l-1}{Y}_l^m$, the bulk density perturbation vanishes but the surface contributes $\delta \rho =\rho {\xi }_r(R)\delta (r-R)$. Continuity of the Newtonian gravitational potential and the Poisson equation give $[{\partial }_r\delta \Phi {]}_{\mathrm{out}-\mathrm{in}}=4\pi G\rho {\xi }_r(R)$. The regular interior and decaying exterior solutions areThe coefficient follows because the derivative jump is $-(2l+1)C{R}^{l-1}{Y}_l^m$. This surface source is why neglecting the bulk density perturbation does not justify neglecting the perturbed gravity.

The Kelvin stellar modes are the potential incompressible stellar surface modes of a nonrotating self-gravitating uniform-density sphere with incompressible flow. For a regular displacement potential $U=A{r}^l{Y}_l^m$, the free-surface Lagrangian pressure perturbation condition and the surface gravity perturbation of a uniform-density sphere giveIn detail $\delta p/\rho ={\omega }^{2}U-\delta \Phi$ and $\delta p/\rho =(GM/R^2){\xi }_r$ at $r=R$, so ${\omega }_l^2=lGM/R^3-4\pi G\rho l/(2l+1)$. The $l=1$ branch is rigid translation with zero restoring force. This potential family does not exhaust vortical zero-frequency displacements: $\mathbf{\xi }=\mathbf{\Omega }\times \mathbf{r}$ is tangent to the sphere and has zero restoring force but nonzero curl.

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.