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For $G$ with $n$ independent standard Gaussian rows in dimension $k$, combine a constant-radius sphere metric net, the quadratic form net bound, the chi-squared concentration inequality, and a union bound. This gives $P(\Vert G^{T}G/n-I_k{\Vert }_{\mathrm{op}}>1/2)\le 2\exp (k\log B-n/1024)$ for a numerical $B$. Taking $n$ larger than a constant times $k\log p$ gives exponential decay in $k\log p$. The subspace is fixed, so no union over supports is needed.

With independent standard-normal priors on the two coefficients, the conditional AR(2) likelihood gives precision $\Lambda =I+{\sum }_t{v}_t{v}_t^T$, where $v_t=(x_{t+1},x_t{)}^T$, and mean ${\Lambda }^{-1}{\sum }_t{v}_t{x}_{t+2}$. Positive definiteness holds even for a deficient design. If $\Lambda =\left(\begin{array}{cc}A& C\\ C& D\end{array}\right)$ and the linear term is $(r,s)$, the conditional means are $(r-Cb)/A$ and $(s-Ca)/D$, with variances $1/A$ and $1/D$.

For independent and identically distributed random variables $X,Y$ with a centered multivariate normal distribution, a continuously differentiable function $f$ with bounded gradient, and a convex function $\Psi$ for which the expressions are integrable, the displayed inequality holds. First apply Jensen inequality conditionally to $f(X)-f(Y)$. For the Gaussian rotation of independent copies, the chain rule gives $f(X)-f(Y)={\int }_0^{\pi /2}⟨\nabla f(U_{\theta }),V_{\theta }⟩d\theta$. A second application of Jensen inequality to the uniform measure on $[0,\pi /2]$, followed by the Gaussian rotation of independent copies, proves the inequality. The useful choice $\Psi (v)=e^{\lambda |v|}$ yields a Gaussian concentration inequality through an exponential moment of an absolute standard normal variable.

For an embedded submanifold of a Riemannian manifold, use $R(v,w)y={\nabla }_v{\nabla }_{w}y-{\nabla }_w{\nabla }_{v}y-{\nabla }_{[v,w]}y$. The Gauss formula and tangential derivative of a normal field giveSubtract the expression with $v,w$ interchanged and the bracket derivative, whose normal part pairs to zero. This proves the displayed curvature identity. For flat Euclidean ambient space, its ambient curvature term vanishes and one obtains the ordinary Gauss equation. Declaring the four-slot convention $R(x,y,v,w)=⟨R(v,w)y,x⟩$ avoids sign ambiguities when permuting arguments.

For a Ricci-flat spacetime with the stated Gauss–Codazzi equations for a spatial hypersurface and induced orientation ${\widetilde{ϵ}}_{abc}=n^d{ϵ}_{dabc}$, the electric part of the Weyl tensor and magnetic part of the Weyl tensor satisfyThe electric relation comes from the vanishing spatial projection of the Ricci tensor. The magnetic relation combines the normal curvature projection with the sign ${ϵ}_{acde}{n}^c=-{\widetilde{ϵ}}_{ade}$.

For an optically thick diffusion-controlled wind, put $\beta =P_{\mathrm{gas}}/(P_{\mathrm{gas}}+P_{\mathrm{rad}})$. Diffusion gives $(\log T{)}^{\prime }=-g\Gamma \beta /[4a_g^2(1-\beta )]$. At $\Gamma =1$, the numerator of the gas and radiation pressure stellar wind equation is $2a_g^2/r+g\beta /[4(1-\beta )]>0$ for finite-temperature gas and $0<\beta <1$. A subsonic branch then decelerates and cannot pass a regular sonic point. This statement is specific to subsonically launched winds in this transport model, not to every supersonic solution or every form of stellar-wind driving.

A closed nowhere dense family can meet every cone $[M{]}^{\omega }$ in continuum many sets, as the gap-doubling closed family of infinite subsets does. Well-order the cones and choose two fresh candidates in each, permanently reserving one of each colour. The red family meets every cone and its complement contains every reserved blue candidate, so it is not Ramsey. As a subset of the closed nowhere dense family, it is still nowhere dense and has the Baire property in the ordinary infinite-subset topology. This recursion requires no regularity assumption on the continuum cardinal.

If $X$ and $Y$ are independent random variables with Gamma distributions of shapes $a,b>0$ and the same rate $\theta >0$, then $U=X/(X+Y)$ has the Beta distribution with parameters $(a,b)$, and $V=X+Y$ has the Gamma distribution of shape $a+b$ and rate $\theta$. Moreover $U$ and $V$ are independent. The inverse change of variables $(x,y)=(uv,(1-u)v)$ has absolute Jacobian determinant $v$, which makes the joint probability density factorize on the product domain $0<u<1$, $v>0$. The common rate is essential for this factorization.

Choose an oriented lattice basis with $t={\omega }_2/N$ modulo the lattice. Changing such a basis is exactly the Gamma 1 congruence subgroup action, so the weight law makes the displayed function independent of the choice. It satisfies $F(uL,ut)=u^{-k}F(L,t)$. Holomorphy on the half-plane and at all modular cusp degenerations identifies the functions coming from modular forms.

Sum over index-$p$ overlattices in which the marked point retains exact order $N$. There are $p+1$ summands at good primes and $p$ at bad primes. With homogeneity $F(uL,ut)=u^{-k}F(L,t)$, the coefficient $1/p$ is the normalization giving the standard Hecke operator Fourier action. Cusp holomorphy under rational slash operators proves preservation of modular forms.

Multiplication of a marked point by a unit modulo $N$ gives the diamond action. In modular coordinates this is the slash action of a lift in ${\Gamma }_0(N)$ with lower-right entry $d$. Its Dirichlet character eigenspaces give nebentypus modular forms. For a nonzero form, the action of $-1$ imposes $\chi (-1)=(-1{)}^k$.

For a nearest-neighbour fortune on $\{0,\dots ,N\}$, with win probability $p$, loss probability $q=1-p$ and $p\ne q$, put $\rho =q/p$. First-step analysis gives $m_i=1+p{m}_{i+1}+q{m}_{i-1}$, with zero endpoint values. Solving the recurrence gives the displayed mean duration. Its symmetric limit is $i(N-i)$.

For a spherical isotropic bound distribution $f(E)$ with $E=\psi -v^2/2$, no unbound population and $\rho (0)=0$, velocity integration gives $\rho (\psi )=4\pi \sqrt{2}{\int }_0^{\psi }f(E)\sqrt{\psi -E}dE$. Differentiation followed by Abel transform inversion yields $f(E)=(\sqrt{8}{\pi }^2{)}^{-1}(d/dE){\int }_0^E{\rho }^{\prime }(\psi )/\sqrt{E-\psi }d\psi$. The factor is $1/(\sqrt{8}{\pi }^2)$, with ${\pi }^2$ outside the square root. Regularity sufficient for these operations and nonnegativity of the resulting distribution are separate requirements; an arbitrary density-potential pair need not give a physical distribution.

A regular phase-mixed distribution $F(H,p_{\varphi },I)$ in an axisymmetric Stäckel potential is even separately in $v_{\lambda }$ and $v_{\mu }$, because these integrals depend on those components only through squares. Odd integration then gives all three mixed second moments zero, even if the $p_{\varphi }$ dependence produces azimuthal streaming. Bare stationarity alone is insufficient in degenerate resonant potentials with additional integrals. For example the isotropic harmonic Hamiltonian has the extra conserved $K=v_x{v}_y+{\omega }^{2}xy$; a positive Gaussian $\exp (-aH+ϵK)$ with $a>|ϵ|>0$ is a steady tracer and has nonzero $⟨v_x{v}_y⟩$. Its harmonic gravitational potential is a Stäckel potential in these spheroidal coordinates. This tracer need not itself source that harmonic gravitational potential. The alignment claim thus needs the phase-mixed three-integral hypothesis, not merely the weak form of Jeans theorem.

An augmented density $\rho (r,\psi )$ treats radius and relative potential as independent variables and reduces to the physical density on $\psi =\psi (r)$. Its extension away from that curve contains additional orbital-anisotropy information. Different extensions can give the same physical density; physical realizability additionally requires a nonnegative galactic distribution function.

The density and self-consistent potential of a spherical galaxy need not determine its galactic distribution function. Different angular-momentum dependence can reproduce the same density with different velocity-anisotropy parameters. An isotropic Plummer model and its positive anisotropic Plummer distribution with unit anisotropy radius provide an explicit example. A chosen full augmented density is additional information.

An Osipkov-Merritt spherical distribution has $F=f(Q)$ for $Q=E-L^2/(2r_a^2)>0$ and vanishes otherwise. Rescaling transverse velocities by $\sqrt{1+r^2/r_a^2}$ makes its velocity integral isotropic in the rescaled variables. Its velocity-anisotropy parameter is $\beta =r^2/(r^2+r_a^2)$.

For a stationary collisionless stellar system confined to a plane, velocity isotropy makes the galactic distribution function depend on speed through $u=v^2/2$. The Collisionless Boltzmann equation implies ${\nabla }_{x}f=f_u{\nabla }_x\Phi$, hence $f=F(E)$ with specific orbital energy $E=u+\Phi$. This conclusion requires stationarity as well as isotropy.

The streaming velocity is the local mean stellar velocity obtained by averaging with the galactic distribution function. In an axisymmetric disk it usually denotes $⟨v_{\varphi }⟩$. It differs from the circular speed, which is set by the force field, and from the velocity dispersion, which measures fluctuations about the mean.

For stable positive detector pixel means $N_i$ and uniform detector conversion gain $g$, high-count independent Poisson distribution noise gives $\mathrm{Var}(A_i/B_i)\simeq 2/(g{N}_i)$. Thus substituting a global mean in the equal-signal formula yields $g_{\mathrm{naive}}=g/[⟨N_i⟩⟨1/N_i⟩]\le g$, by the Jensen inequality. Equality holds for uniform means. The flat-field correction pattern cancels from the ratio mean but not from the signal-dependent noise. Local signal bins, or a variance fit after normalization of differences by the square root of their local mean, avoid this leading bias.