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For a nonrotating uncharged stationary black hole, the first law is $\delta M=\kappa \delta A/(8\pi )=T_H\delta S_{BH}$.

In units $G=\hslash =c=k_B=1$, a black hole of horizon area $A$ has entropy $S_{BH}=A/4$.

In units $G=\hslash =c=k_B=1$, a stationary black hole with surface gravity $\kappa$ has Hawking temperature $T_H=\kappa /(2\pi )$.

A direct-collapse black-hole seed forms when rapidly inflowing primordial gas avoids fragmentation, builds a supermassive star or quasistar, and collapses to a seed of roughly $10^4$--$10^6{M}_{\odot }$.

A runaway stellar-collision seed forms when mass segregation and repeated mergers in a dense young cluster create a very massive star that collapses to a black hole of roughly $10^2$--$10^4{M}_{\odot }$.

A Population III remnant seed forms when a massive metal-free star collapses, typically leaving a light seed of order $10$--$10^2{M}_{\odot }$ depending on stellar mass loss and pair-instability effects.

The Eddington ratio is the luminosity divided by the Eddington luminosity. At fixed radiative efficiency it also measures the accretion rate relative to the Eddington rate.

An advection-dominated accretion flow is a hot, geometrically thick, radiatively inefficient flow in which most dissipated energy remains in the gas and is carried inward rather than radiated locally.

Bondi accretion is steady spherical accretion from a uniform gas reservoir onto a compact gravitating object. In its adiabatic form, compression raises internal energy without providing the irreversible shear heating of a radiatively efficient disk.

The radiative efficiency is $\eta =L/(\dot{M}{c}^2)$. For a zero-torque thin disk it is $1-E_{\mathrm{ISCO}}$, the binding energy per unit rest mass released before matter crosses the innermost stable circular orbit.

Conditionally on $X_2=x_2$,

For independent $Z_i\sim N_p(0,\Sigma )$ with known zero mean,

A thin spherical shell of proper surface mass density $\rho$ held at Schwarzschild radius $r_{\mathrm{st}}$ around mass $M$ contributes, to first order in its mass,to the exterior Schwarzschild mass parameter.

The blowup of ${\mathbb{A}}^2$ at the origin isIts projection to ${\mathbb{A}}^2$ is an isomorphism away from the origin, while the fiber above the origin is the exceptional curve ${\mathbb{P}}^1$.

It is an explicit example of blowup of an algebraic variety.

A finite graph is bipartite exactly when it has a proper colouring with two colours. Equivalently,for its chromatic polynomial.

The complete bipartite graph $K_{r,s}$ has vertex classes of sizes $r$ and $s$ and contains all $rs$ edges between the classes.

If a square bipartite adjacency matrix $B$ has nonzero determinant, some product in its Leibniz expansion is nonzero. Its permutation selects one edge at each vertex and therefore gives a perfect matching.

Coefficient extraction from $(1-x{)}^n(1+x{)}^n=(1-x^2{)}^n$ gives

If square matrices $A$ and $B$ commute, thenThe usual proof works because commutativity allows words with the same numbers of $A$ and $B$ factors to be collected.

Partition the vertices into two sets of comparable size. If $np\to \infty$, the first set contains a triangle with probability tending to one. Conditional on any triangle chosen using only internal edges, the probability that none of its vertices has a neighbour in the second set is at mostHence $G(n,p)$ contains a triangle with an attached leaf with probability tending to one.