A number field is totally real when every embedding into the complex numbers has image in the real numbers. The maximal real subfield of a cyclotomic field is a basic example. Ordinary Hilbert class fields split at real places and therefore retain this property.

In a planar warped spacetime, contraction of the Ricci tensor with a null vector in an orthonormal frame gives $R_{ab}{k}^a{k}^b=-2(\log A{)}^{\prime \prime }(k^3{)}^2$. The Einstein field equations identify this with $8\pi T_{ab}{k}^a{k}^b$; metric terms, including a cosmological constant, vanish in a null contraction. Thus the null energy condition is equivalent to concavity of $\log A$ in the proper transverse coordinate $z$. Null directions confined to a planar slice saturate the condition.

A minimal normal subgroup is a nontrivial normal subgroup containing no smaller nontrivial normal subgroup of the ambient group. It need not be a minimal nontrivial subgroup. In a finite soluble group, it is elementary abelian: its derived subgroup must be trivial, a characteristic Sylow subgroup selects one prime, and the characteristic subgroup of elements killed by that prime gives exponent $p$.

Here $\Delta \psi =\psi (v^{\prime })+\psi (v_{\ast }^{\prime })-\psi (v)-\psi (v_{\ast })$ is the pairwise collision change, not a Laplace operator. Average over the two particle labels and use collision reversal to obtain the formula. Every collision invariant has $\Delta \psi =0$, giving mass, momentum and kinetic energy conservation. A second symmetrization replaces $f{f}_{\ast }\Delta \psi /2$ by $(f^{\prime }{f}_{\ast }^{\prime }-f{f}_{\ast })(-\Delta \psi )/4$.

The angular collision rate is integrable at each nonzero relative velocity, with any needed speed-growth bounds stated separately. This makes the separate gain and loss integrals available under suitable velocity integrability. Non-cutoff kernels can have infinite angular collision rate and require cancellation rather than the same naive gain-loss split.

Write $f=M(1+h)$ around a fixed Maxwellian and expand the nonlinear Boltzmann collision operator to first order. The weak Boltzmann collision identity gives the nonnegative dissipation form in $L^2(Mdv)$, and polarization proves symmetry on an appropriate common domain. Its zero modes are the collision invariants when the scattering kernel is nondegenerate. A spectral gap, and self-adjointness of an unbounded realization, require additional domain and kernel hypotheses.

Suppose an action is invariant under a local transformation $\delta {\Phi }_a=R_a\alpha +R_a^{\mu }{\partial }_{\mu }\alpha$ for every compactly supported function $\alpha$. Variation and integration by parts give the off-shell identityThus arbitrary local symmetry parameters imply dependencies among the field equations. In gauge theory, these identities accompany constraints rather than independent physical charges for each function. This complements the conserved-current statement of Noether theorem.

The open NS action has integer bosonic modes and half-integer fermionic matter modes, together with even Virasoro and odd supercurrent multipliers. Opposite fermion signs at the two endpoints make the doubled fermion antiperiodic and remove a matter zero mode. The $b_r$ matter variables are distinct from the $bc$ reparameterization FP ghosts. Canonical quantization yields $[{\alpha }_k^m,{\alpha }_l^n]=k{\eta }^{mn}{\delta }_{k+l,0}$ and $\{b_r^m,b_s^n\}={\eta }^{mn}{\delta }_{r+s,0}$.

The lowest fermionic excitation has level $1/2$. Its old-covariant physical conditions are ${\alpha }^{\prime }{p}^2+1/2-a=0$ and $p\cdot A=0$. For nonzero real momentum, timelike $p$ gives a positive polarization space, null $p$ gives a nonnegative space with a longitudinal null direction, and spacelike $p$ permits a negative-norm time polarization. The limiting $a=1/2$ state is a massless gauge vector after quotienting null polarizations. This level-specific test does not prove positivity of every higher NS level.

A fixed finite monoid operation has a constant-size Boolean truth-table Boolean circuit on its constant-bit encodings. Reduce $n$ elements in a balanced binary tree to get linear size and logarithmic depth. Associativity ensures that tree grouping does not change the product. MOD3 uses addition in the three-element residue group, with input-position signs handled at the leaves.

For a binary numeral, its remainder is the sum of its bits weighted alternately by $+1$ and $-1$ from the least significant position. A balanced finite-monoid reduction circuit adds these residues modulo three in two-bit encodings. A final zero test gives a uniform linear-size logarithmic-depth Boolean circuit, proving membership in NC1.

The disagreement point specifies each player's payoff if bargaining fails. It supplies the baseline from which the Nash product measures gains. In a game-based example it may be set to the vector of security level payoffs; this choice must be recomputed when the underlying payoff matrices change.

For an essential Nash bargaining problem, maximizing the Nash product gives a unique vector: on positive gains its logarithm is a strictly concave function. The rule satisfies the four bargaining axioms, and the supporting triangle for Nash bargaining proves their converse characterization. The nonessential case requires additional conventions, since a zero product may have many maximizers.

When the feasible payoff set and disagreement point are invariant under interchange of players, the solution must give both players the same payoff.

Transforming each utility by an independent positive affine map must transform the chosen payoff by the same map. The Nash product changes only by a positive factor, so the Nash bargaining solution has this invariance.

If an admissible smaller feasible set with the same disagreement point contains the original chosen vector, deleting the other outcomes must leave the choice unchanged. The Nash bargaining solution satisfies this axiom because its unique product maximizer remains feasible.

A payoff is individually rational for a Nash bargaining problem when neither player receives less than its disagreement point payoff. Essentiality permits strict improvement for both. Maximizing a positive Nash product therefore selects a strictly individually rational payoff.

Over a field of characteristic zero, an alternating polynomial changes sign when two variables are interchanged. It vanishes when two coordinates agree, so each factor $x_i-x_j$ divides it. These distinct linear factors are relatively prime in the polynomial ring, hence their product, the Vandermonde determinant, divides it. The quotient is a symmetric polynomial.

An algebra representation is a unital algebra homomorphism into the endomorphisms of a vector space. It is equivalent to making that vector space a left module over the algebra, with $av=\rho (a)v$.

For a prime $p$ and $k\ge 1$, write $a=b+p^{k}h$. The binomial theorem shows $a^p-b^p$ is divisible by $p^{k+1}$: the linear term contains the extra factor $p$ and every higher term contains at least $p^{2k}$. Iteration gives the displayed modular congruence for every $r\ge 0$, including $p=2$. Applied to Fermat's little theorem, it yields $a^{p^n}\equiv a^{p^{n-1}}(\mathrm{mod}{p}^n)$.