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The matrix logarithm near the identity maps $Sp(n)=\{A\in U(2n):AJ{A}^t=J\}$ onto an open set of its real compact symplectic Lie algebra $\{X:X^{\ast }=-X,XJ+J{X}^t=0\}$. Its inverse is the matrix exponential. These statements follow by applying the logarithm to $A^{\ast }=A^{-1}$ and $A^t=J^{-1}{A}^{-1}J$, and differentiating $e^{tX}J{e}^{t{X}^t}$. Left translations provide manifold charts at every group element. In block form $X=\left(\begin{array}{cc}P& Q\\ -\bar{Q}& \bar{P}\end{array}\right)$ with $P^{\ast }=-P$ and $Q^t=Q$, giving real dimension $n^2+n(n+1)=n(2n+1)$.

The barycenter of a probability measure on a compact convex set is the point satisfying the displayed affine integral identities. It exists by approximating the measure by finitely supported measures and using compactness of their finite convex combinations. It is unique because continuous affine functions separate points. In a Banach-space setting it agrees with the appropriate vector integral when that integral is defined.

For a bounded real function on a compact convex set, this envelope is the infimum of its continuous affine majorants. It is finite, concave and upper semicontinuous, and dominates the function. For continuous functions it describes the maximal integral among probability measures with a specified barycenter. Its homogeneity, subadditivity and affine-translation identities give the supporting measure lemma for affine upper envelopes.

Every point of a compact metrizable convex subset of a Hausdorff locally convex space is the barycenter of a Borel probability measure concentrated on its extreme points. The theorem is an existence assertion; uniqueness requires additional hypotheses such as a simplex structure.

In two dimensions, the outgoing harmonic Helmholtz equation kernel has magnitude proportional to $(k_{0}r{)}^{-1/2}$. Combining this spreading with the compact acoustic quadrupole source and $k_0\ell =O(M)$ changes the density-amplitude Mach number power to $7/2$ when geometric range is separated.

With $\mathcal{H}=aH=\dot{a}$, the Friedmann acceleration equation givesThe coefficient is positive when $a>0$, $\rho >0$ and $H\ne 0$. Thus growth of the comoving Hubble radius and linear growth of departures from flatness have the same sign criterion in an expanding universe. This local criterion is distinct from a global particle horizon, which depends on the whole past history.

The structure map of a comodule; a right coaction satisfies $(\rho \otimes 1)\rho =(1\otimes \Delta )\rho$ with coherent parentheses, and $(1\otimes \epsilon )\rho$ is the inverse right unitor.

A comonoid morphism $f:H\to K$ induces a functor $f_{\ast }$ on right comodules, replacing $\rho$ by $(1\otimes f)\rho$ and preserving underlying objects and arrows. For bimonoids, it is strict monoidal exactly when $f$ is also a monoid morphism.

A natural underlying transformation of tensor bifunctors on right comodules over a field has components ${\phi }_{M,N}(u\otimes v)=\sum u_{(0)}\otimes v_{(0)}\gamma (u_{(1)},v_{(1)})$. Recover $\gamma =(\epsilon \otimes \epsilon ){\phi }_{H,H}$; colinearity and monoidal axioms impose further equations.

The vector fields ${\partial }_t$ and ${\partial }_i$ generate time and space translations and commute with the flat d'Alembert operator. They form the unweighted part of the commutation vector fields for the wave equation.

The vector field ${\Omega }_{ij}=x_i{\partial }_j-x_j{\partial }_i$ is an infinitesimal spatial rotation. It commutes with the flat d'Alembert operator and supplies angular derivatives in the Klainerman-Sobolev inequality.

For Minkowski spacetime with unit light speed, $L_i=t{\partial }_i+x_i{\partial }_t$ generates a Lorentz boost. It commutes with the d'Alembert operator and controls derivatives transverse to time slices in the vector field method for wave equations.

The vector field $S=t{\partial }_t+x\cdot \nabla$ generates simultaneous spacetime dilation. For the flat d'Alembert operator, $[\square ,S]=2\square$. Consequently it commutes with the homogeneous wave equation at the level of its solution set, although its operator commutator is nonzero.

For commuting actions of finite groups $G,H$ with coprime orders, the canonical map $A^G/H\to (A/H{)}^G$ is a bijection. It is always injective. On any $G$-stable $H$-orbit, all $G$-orbits have the same size, dividing both $|G|$ and the $H$-orbit size, hence dividing $|H|$. Coprimality forces that size to be one, proving surjectivity. Thus categorical limits indexed by $G$ commute with colimits indexed by $H$ in the Category of sets.

If groups $G,H$ have a common nontrivial quotient $K$, act on $K$ by left multiplication through $G$ and right inverse multiplication through $H$. The actions commute. There are no fixed points of a group action for $G$, but the $H$ orbit set is a singleton. The comparison $A^G/H\to (A/H{)}^G$ is therefore the map from the empty set to a singleton, which is not an isomorphism.

For small $J,K$ and a functor $D:J\times K\to \mathcal{C}$ into a complete category, the two iterated categorical limits are canonically isomorphic. They have the same projections to every $D(j,k)$ and the same universal property. The result follows also from preservation of categorical limits by the limit functor.

A communicating class of a Markov chain is open when a state in the class has positive probability of transitioning to a state outside it. Once a path leaves, it cannot return to the class: a return path would make the outside state communicate with the class. Every open class consists of transient states. Conversely, a finite transient communicating class is open; an infinite closed communicating class can be transient.

At fixed stellar mass $M_{\star }$, planetary mass density ${\rho }_p$, age $t_{\star }$ and initial ratio $\kappa =a_0/a_p$, the order-unity Safronov number comparison and comet energy diffusion give boundariesTheir logarithmic slopes in planetary mass versus radius are $-3/2$ and $3/4$. Above both, strong kicks and a short estimated diffusion time favour ejection; below the escape boundary, repeated weak encounters favour collision statistically. Below the age boundary, the stated diffusion mechanism has not completed on its characteristic clock. Collision lifetimes require additional cross-section and encounter-rate information, so this map alone cannot establish retention or accretion within the age.

Linearize the Collisionless Boltzmann equation and the perturbation Poisson equation about a homogeneous velocity distribution $f_0$. A plane wave gives $f_1={\varphi }_1\mathbf{k}\cdot {\nabla }_v{f}_0/(\mathbf{k}\cdot \mathbf{v}-\omega )$. Integrating this density and using $-k^2{\varphi }_1=4\pi G\int f_1{d}^{3}v$ yields $1+(4\pi G/k^2)\int [\mathbf{k}\cdot {\nabla }_v{f}_0/(\mathbf{k}\cdot \mathbf{v}-\omega )]d^{3}v=0$. Growing modes have $\mathrm{Im}\omega >0$ and no real-velocity pole; real and damped frequencies require the causal continuation prescription. The Newtonian homogeneous background requires the Jeans swindle.

In physical polar velocity components, a planar rotating system has accelerations ${\dot{v}}_R=2{\Omega }_b{v}_{\varphi }+v_{\varphi }^2/R+{\psi }_{\mathrm{eff},R}$ and ${\dot{v}}_{\varphi }=-2{\Omega }_b{v}_R-v_R{v}_{\varphi }/R+{\psi }_{\mathrm{eff},\varphi }/R$. The Collisionless Boltzmann equation is constant distribution along these characteristics. Its physical phase-space measure includes the spatial Jacobian $R$.