For smooth spatial $\varphi$ and $g(c)=-c^2{e}^{2\varphi /c^2}d{t}^2+{\delta }_{ij}d{x}^{i}d{x}^j$, the only nonzero Christoffel symbols are ${\Gamma }^t{}_{ti}={\Gamma }^t{}_{it}=c^{-2}{\partial }_i\varphi$ and ${\Gamma }^i{}_{tt}=e^{2\varphi /c^2}{\partial }_i\varphi$. Consequently the Levi-Civita connection converges smoothly on compact sets as $c\to \infty$ although the metric itself diverges in these coordinates. The limiting torsion-free connection has only ${\Gamma }^i{}_{tt}={\partial }_i\varphi$ and gives $d^2{x}^i/d{t}^2=-{\partial }_i\varphi$ when $t$ is an affine parameter. With $R^a{}_{bcd}={\partial }_c{\Gamma }^a{}_{db}-{\partial }_d{\Gamma }^a{}_{cb}+\cdots$, its Ricci tensor is $(\Delta \varphi )dt\otimes dt$: quadratic connection terms vanish and $R^i{}_{tjt}={\partial }_j{\partial }_i\varphi$. Thus zero Ricci tensor is the Laplace equation, while full flatness would require the entire Hessian to vanish.

The product $\mu =GM$ of the gravitational constant and a central mass sets the strength of the exterior Newtonian gravitational field. Its dimensions are length cubed divided by time squared. In the exact relative two-body problem, the parameter is $G(M_1+M_2)$; using $GM$ for one source assumes the test particle approximation.

The combination is useful because orbital and falling-body observations constrain $\mu$ directly. A gravitational circular orbit of radius $a$ has oscillation period $T=2\pi \sqrt{a^3/\mu }$, so measuring $a$ and $T$ gives $\mu =4{\pi }^2{a}^3/T^2$. That measurement alone does not separately determine $G$ and $M$. Similarly, radial free fall from rest has a natural time scale $\sqrt{r_0^3/\mu }$.

Other interactions require a different model. Inverse-square radiation pressure can reduce the effective attraction for a dust Kepler orbit; calling that effective coefficient a gravitational parameter must not hide the added force.

Self-gravity is the gravitational acceleration produced by a system's own mass density. It is obtained by solving the Poisson equation for Newtonian gravity for that density, rather than treating the Newtonian gravitational potential as an externally fixed field. It competes with pressure and rotational support in gravitational instability of an astrophysical disk.

For a Newtonian gravitational potential obeying the Poisson equation for Newtonian gravity, define $T_{ij}=[{\partial }_i\Phi {\partial }_j\Phi -{\delta }_{ij}|\nabla \Phi {|}^2/2]/(4\pi G)$. Commutation of mixed partial derivatives gives ${\partial }_j{T}_{ij}=\rho {\partial }_i\Phi$. Thus the gravitational force density can be put in conservative form as $-{\partial }_j{T}_{ij}$. The sign is chosen so that $T_{ij}$ is added to the momentum flux; reversing the stress convention reverses the divergence sign.

Antiperiodic worldsheet Majorana fermions give transverse modes $b_r^I$, $r\in \mathbb{Z}+\frac{1}{2}$, with $\{b_r^I,b_s^J\}={\delta }^{IJ}{\delta }_{r+s,0}$. Positive modes annihilate the oscillator vacuum; negative modes are fermionic creation operators.

Before the GSO projection, $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k+{\sum }_{r>0}r{b}_{-r}\cdot b_r$ takes nonnegative half-integer values. The first vector at $N=1/2$ is massless for string intercept $a_{\mathrm{NS}}=1/2$.

The unprojected level-one states form a transverse vector representation and an exterior square. Fermionic anticommutation makes the two-fermion states antisymmetric. Together they restrict from an antisymmetric rank-two tensor of $SO(D-1)$, not a massive spin-two field.

Neglecting chemical potentials, the equilibrium abundance ratio is $n_n/n_p\simeq \exp [-(m_n-m_p)c^2/(k_{B}T)]$. Weak freeze-out fixes an approximate starting ratio, which is then reduced by free-neutron decay after freeze-out before most remaining neutrons bind into helium.

For nested normal linear models, the decrease in residual sum of squares obtained by adding coefficients is their Extra sum of squares. Dividing by the number of added coefficients and by the full-model residual mean square gives the partial F-test. A sequential analysis of variance attributes this decrease according to the specified order of entry; it need not equal the partial comparison when other terms are retained.

Even with a common event hazard, the exact expectation of the Nelson–Aalen estimator is ${\int }_0^t\mathbb{P}(Y(u)>0)h(u)du$, because no increments can be observed after the risk set is empty. Replacing ratios by ratios of expectations gives an approximate target and does not prove finite-sample unbiasedness. For one uncensored exponential event time, the estimator has mean $1-e^{-\lambda t}$ rather than $\lambda t$.

For untied events at $a_j$ with predictable risk set size $Y_j$, the estimated predictable variation of the Nelson–Aalen estimator is ${\sum }_{a_j\le t}{Y}_j^{-2}$. It comes from the martingale event increment variance $YdH$ and the estimator increment $dN/Y$. It applies on the range with individuals at risk; it is not an exact finite-sample unbiasedness statement.

With all subjects entering at time zero and no tied observation times, the Nelson–Aalen estimator satisfies ${\sum }_i\hat{H}(x_i)=d$. Exchanging sums makes each event increment $1/Y(t_j)$ appear in exactly $Y(t_j)$ subject hazards. The identity relies on the risk-set count equalling the number of observation times at least $t_j$, and generally needs modification for delayed entry.

A pointwise constraint on the initial derivative need not be closed in the norm controlling the value and first-derivative square integrals. For example, minimize $b{\int }_0^T{\dot{V}}^{2}dt$ with $b,L,T>0$, $V(0)=0$ and ${\int }_0^{T}Vdt=L$. The attained minimum is $3b{L}^2/T^3$, uniquely achieved by $V_{\ast }=3Lt(2T-t)/(2T^3)$, with natural condition ${\dot{V}}_{\ast }(T)=0$. Adding $\dot{V}(0)=0$ excludes this minimizer but leaves the same infimum. Multiplying $V_{\ast }$ by $1-e^{-t/\epsilon }$ and rescaling to restore the integral gives smooth admissible approximations converging in energy. The formal quadratic from the interior Euler-Lagrange equation and two initial conditions alone is therefore not a true minimizer.

The weak cosmic censorship conjecture predicts, under appropriate generic isolated initial-data and matter assumptions, that collapse singularities are hidden from distant observers by an event horizon. It motivates exterior predictability in the physical argument for the Penrose inequality. It is distinct from the strong cosmic censorship conjecture, which concerns inextendibility of the maximal Cauchy development.

A five-dimensional Myers-Perry black hole with equal rotational parameters has a metric depending on a single radial coordinate. In a fibered coordinate system, its radial functions can be written $f=1-r_0^2/r^2+{\alpha }^2{r}_0^2/r^4$, $h=1+{\alpha }^2{r}_0^2/r^4$, and $\Omega =\alpha r_0^2/(r^{4}h)$. For $0<\alpha <r_0/2$ there are outer and inner Killing horizons; at equality they form a degenerate Killing horizon. With the standard sphere identifications and $G_5=1$, its Komar mass is $3\pi r_0^2/8$.

The vector space of formal polynomials over a field $k$ in $n$ variables with total degree of a polynomial at most $d$ has the monomial basis $x_1^{a_1}\cdots x_n^{a_n}$ for nonnegative exponents with ${\sum }_i{a}_i\le d$. Introducing a slack exponent and using stars and bars counts $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ basis elements. These are formal polynomials; over a finite field, distinct formal polynomials need not define distinct polynomial functions unless suitable degree restrictions hold.

For a line $\ell =\{\mathbf{a}+t\mathbf{v}:t\in \mathbb{R}\}$ with $\mathbf{v}\ne 0$, restricting a degree-at-most-$d$ multivariate polynomial gives a univariate polynomial in $t$ of degree at most $d$. Its $d+1$ coefficients are linear functionals of the original coefficient vector. Vanishing on the whole line is therefore equivalent to at most $d+1$ homogeneous linear conditions. Alternatively, $d+1$ distinct roots force the restriction to be identically zero.

A nonzero element $a$ of a finite field is a quartic residue when $a=b^4$ for some nonzero $b$. If $q\equiv 1(\mathrm{mod}4)$, the multiplicative group of a finite field is cyclic of order divisible by four, so the quartic residues form a subgroup of index four; equivalently $a^{(q-1)/4}=1$.

For an arithmetic function, multiplicativity means $f(1)=1$ and $f(mn)=f(m)f(n)$ whenever $m,n$ are positive coprime integers. It allows values of an arithmetic function to be assembled from its values on prime powers, as explained under multiplicative function.

Specify a conditional distribution for each incomplete variable given the others, cycle through parameter draws and missing-value draws, and retain multiple completed datasets after suitable convergence checks. Preserve observed entries. Adequate models, appropriate support and the intended analysis structure matter; arbitrary conditional models need not be compatible with one joint distribution. Standard applications assume missing at random given included predictors. Use Rubin's rules to pool estimates and uncertainty.