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A nonzero Boolean integral domain has only the elements zero and one, so it is ${\mathbb{F}}_2$. The quotient criterion then makes every prime quotient of a Boolean ring a field, and every prime ideal maximal.

A Boolean function is a function on a Boolean hypercube, often with codomain $\{-1,1\}$ or $\{0,1\}$.

The edge boundary of a vertex set $A$ consists of graph edges with exactly one endpoint in $A$. In a $d$-regular graph,where $e(A)$ counts edges with both endpoints in $A$.

If a nonzero function on the Boolean hypercube has degree at most $k$, then its support has measure at least $2^{-k}$. This sharp bound follows by induction on the dimension.

One form of the hypercontractive inequality is $\Vert T_{1/\sqrt{3}}f{\Vert }_4\le \Vert f{\Vert }_2$, where $T_{\rho }$ is the noise operator on the Boolean hypercube.

Let $G=\mathbb{Z}/N\mathbb{Z}$ have prime order, and let $B(\Gamma ,\rho )$ use chord width $0<\rho \le 2$ and positive rank $d=|\Gamma |\ge 1$. Simultaneous character approximation gives a centered arithmetic progression inside it of length at leastThe prime-order condition makes every nonzero step have order $N$, so the required short arithmetic progression can use distinct residues. If $\Gamma$ is empty, the Bohr set is the whole group and contains an arithmetic progression of $N$ distinct residues; the expression $N^{1/d}$ should not be used with $d=0$. For larger chord widths the Bohr set is again the whole group. The nonwrapping progression in a cyclic Bohr set supplies a weaker exponent that also works at composite orders.

For a Bohr set in chord-distance convention, with rank $d$ and width $0<\rho \le 2$,Partition each character circle into $M=\lceil 2\pi /\rho \rceil$ arcs. One of the $M^d$ resulting boxes contains at least $|G|/M^d$ images. Subtract a fixed group element in that box from all the others. Their character differences have chord distance at most $\rho$, giving that many distinct points in the Bohr set. Since $M\le 8/\rho$ in this width range, the displayed estimate follows.

For $\rho >2$ every character constraint is automatic and the Bohr set is all of $G$. One may therefore replace $\rho$ by $\min (\rho ,2)$ in a bound valid at every width. Without a width restriction the original right side could exceed $|G|$.

Three Hermitian expectation values determine an arbitrary qubit state exactly when the traceless parts of the three observables span the three-dimensional space generated by the Pauli matrices. Mere linear independence as Hermitian matrices is insufficient: $I,{\sigma }_x,{\sigma }_z$ are independent but cannot detect the sign or magnitude of the Bloch $y$ component.

For a standard European call option, the Black-Scholes implied volatility is the value $\hat{\sigma }$ solving $C_{\mathrm{BS}}(\hat{\sigma })=C_{\mathrm{observed}}$, with spot, strike, remaining maturity, interest rate and dividend convention held fixed. In the non-dividend model with $S,K,\tau >0$,Positive option vega makes the call price continuous and strictly increasing in $\sigma >0$. Thus there is exactly one finite positive Black-Scholes implied volatility when the observed call price lies strictly between these bounds. The lower boundary corresponds to zero volatility; the upper boundary is approached only as volatility tends to infinity. Outside the bounds there is no solution in this model.

Monotonicity makes bracketed root finding a direct numerical inversion. This is a model-implied parameter; calls with different strikes can imply different values even though the basic Black-Scholes model assumes one constant volatility. For more general claim payoffs, uniqueness requires a separate monotonicity argument.

A delta hedge holds ${\Delta }_t={\partial }_{s}V(t,S_t)$ units of the risky asset. In the Black-Scholes model, this choice cancels the claim's Brownian exposure.

The option delta is one of the Greeks (finance); hedging is the trading procedure that uses this sensitivity.

The price $V(t,s)$ of a sufficiently regular European claim satisfies

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.