A subgame perfect equilibrium is a strategy profile inducing a Nash equilibrium in every subgame, including subgames not reached during equilibrium play. A proper subgame starts at a node that forms a singleton information set, includes all its descendants, and does not cut any other information set; one cannot treat indistinguishable nodes as separate games.

The requirement excludes noncredible threats. Suppose an entrant chooses Out, giving both players $1$, or Enter. After Enter, an incumbent chooses Fight with payoffs $(-1,-1)$ or Accommodate with payoffs $(2,0)$, with entrant payoff first. Out together with a prescribed Fight is a Nash equilibrium: the threat deters entry and the incumbent's off-path action does not change its realized payoff. It is not a subgame perfect equilibrium, because after entry the incumbent prefers $0$ to $-1$. The subgame-perfect outcome is Enter followed by Accommodate.

In finite perfect-information games, backward induction in games constructs such profiles. More generally, continuation subgames may require solving simultaneous-move games in mixed strategies, as in an all-pay contest. Node-by-node optimization cannot ignore information sets in games of imperfect information.

A normal-form game specifies each player's strategy set and their payoff for every strategy profile. In a finite two-player game these data form a bimatrix game; if the two payoffs always sum to zero, one payoff matrix describes the matrix game. The representation specifies choices and incentives, without a temporal order of observed moves.

Two strategies are payoff-equivalent for a player when they give that player the same payoff against every strategy of the opponents. If every player's payoff is also unchanged, replacing one by the other preserves the game outcomes entirely. This is distinct from weakly dominated strategies, whose definition requires a strict comparison somewhere.

Cooperative game theory studies what groups of players can achieve together and how their joint value can be allocated. A transferable utility game assigns a real value to each coalition. The core of a cooperative game asks which allocations no coalition can improve upon; the Shapley value averages marginal contributions across player orderings.

The absolute Galois group of a field $K$ is $G_K=\mathrm{Gal}(K^{\mathrm{sep}}/K)$. It is a profinite group, the inverse limit of the Galois groups of finite Galois extensions. Its continuous actions on discrete modules are the coefficients for Galois cohomology.

In the strict tower convention, a radical extension is a field obtained through finitely many adjunctions $a_i^{n_i}\in K_{i-1}$, $K_i=K_{i-1}(a_i)$. Some texts instead use the term for any subfield of such a tower. This distinction matters: solvability by radicals implies containment in a tower, and need not give a radical tower entirely inside the original field.

A finite separable extension is solvable when its normal closure has a solvable group as Galois group. In characteristic zero it is contained in a radical tower after adjoining suitable roots of unity; containment need not mean it is itself a tower of radical adjunctions.

A galactic disk is a flattened stellar or gaseous component of a galaxy, commonly supported mainly by ordered rotation. Its radial mass profile determines its galaxy rotation curve; thickness, truncation and external forces also matter. A galactic disk need not be an exponential galactic disk, and its neutral-gas absorption radius need not equal a stellar scale length.

An operator algebra is an algebra of linear operators, with multiplication given by composition. In quantum examples it can be generated by creation and annihilation operators subject to the canonical commutation relations or canonical anticommutation relations. Its analytic realization requires specifying a common domain for unbounded operators.

A negatively curved universe with pressureless matter and no dark energy expands forever. With $a_0=1$ and $0<{\Omega }_{m,0}<1$, its scale factor is $a={\Omega }_{m,0}(\cosh \eta -1)/[2(1-{\Omega }_{m,0})]$ and its cosmic time is $t={\Omega }_{m,0}(\sinh \eta -\eta )/[2H_0(1-{\Omega }_{m,0}{)}^{3/2}]$. The parameter is conformal time scaled by $\sqrt{-K}$ and measured from the Big Bang.

A flat universe with separately conserved pressureless matter and a coasting fluid satisfies $a^{\prime 2}=H_0^2[{\Omega }_{m}a+(1-{\Omega }_m)a^2]$. For both densities positive, choosing the Big Bang at zero conformal time gives $a={\Omega }_m[\cosh (\alpha \tau )-1]/[2(1-{\Omega }_m)]$, with $\alpha =H_0\sqrt{1-{\Omega }_m}$. Physical time follows by integrating $dt=ad\tau$. The hyperbolic parametrization is formally like the open dust solution, but the spatial geometry here is flat.

In a flat universe containing separately conserved radiation in cosmology and pressureless matter, the Friedmann equation becomes $(a^{\prime }{)}^2=H_0^2({\Omega }_{r,0}+{\Omega }_{m,0}a)$ in conformal time. Setting $a=0$ at $\tau =0$ gives the stated quadratic-plus-linear scale factor. The linear term controls radiation domination and the quadratic term controls matter domination. The formula presumes $a_0=1$, ${\Omega }_{r,0}+{\Omega }_{m,0}=1$ and no cosmological constant.

If $f\in L^1(\mathbb{R})$ and $x^{k}f\in L^1(\mathbb{R})$ for $0\le k\le r$, then its Fourier transform is $r$ times continuously differentiable and the displayed formula holds. The dominated convergence theorem permits each derivative under the integral. Thus $r+1$ vanishing moments give a zero Taylor polynomial through degree $r$ at the origin. Unweighted $L^1$ membership only guarantees a continuous function as the Fourier transform.

Fourier space labels a field by the frequency or wavevector variables of its Fourier transform, rather than by its original coordinates. Spatial gradients act by multiplication by $i\mathbf{k}$, and the negative Laplacian acts by $k^2$.

The unnormalized sinc function is $\mathrm{sinc}(x)=\sin x/x$, extended by value one at zero. The normalized sinc function is $\sin (\pi x)/(\pi x)$, with the same removable value. These arise, with convention-dependent scaling, as inverse Fourier transforms of frequency-interval indicator functions, so they describe ideal low-pass filters and the Shannon scaling function.

The Fourier sine transform is a half-line transform adapted to a homogeneous Dirichlet boundary condition. With the displayed normalization, inversion is $f(x)=(2/\pi ){\int }_0^{\infty }\sin (kx)S(k)dk$. It is equivalent to the Fourier transform of an odd extension. If $f(0)=0$ and boundary terms vanish, two integrations by parts give the transform of $f^{\prime \prime }$ as $-k^{2}S(k)$.

The Fourier transform with negative exponential and inverse factor $(2\pi {)}^{-n}$ is a continuous isomorphism of the Schwartz space. Weighted frequency derivatives are transforms of derivatives of weighted input functions, so their suprema are controlled by finitely many Schwartz seminorms. Fourier inversion gives ${\mathcal{F}}^2=(2\pi {)}^n\mathcal{R}$, where $\mathcal{R}f(x)=f(-x)$. Transposition gives a continuous isomorphism on tempered distributions as well.

With the $e^{ikx}$ convention, a right-supported decaying function has a transform analytic in the upper half-plane and a left-supported one in the lower half-plane. Incident modes continued with small imaginary frequency can have poles outside the initial convergence strip; these poles must be tracked separately from outgoing scattered waves.

A bandlimited function has its Fourier transform supported in a bounded frequency region. For $f\in L^1(\mathbb{R})$ with support of $\hat{f}$ contained in $[-B,B]$, the transform is bounded and hence integrable. Fourier inversion supplies the bounded continuous representativeThe same formula for complex $x=z$ defines an entire function: derivatives can pass under the finite-interval integral, uniformly on compact subsets. It obeys $|f(z)|\le (2\pi {)}^{-1}\Vert \hat{f}{\Vert }_1{e}^{B|\mathrm{Im}z|}$.

For $a\ne 0$, the exact scaling law isThus $f(ax)$ has bandwidth $|a|B$, while the stretched function $f(x/a)$ has bandwidth $B/|a|$. This fixes the otherwise ambiguous phrase that time and bandwidth scale inversely. Products add frequency-support intervals through the convolution theorem.

Band limitation differs from compact spatial support. A nonzero entire representative cannot vanish on an open real interval by the identity theorem. Consequently a nonzero integrable bandlimited function cannot also have compact support. This matters when interpreting a sharply started signal or a wave-packet front.

A complex spectral Fourier transform parametrizes the two-dimensional Fourier transform by complex coordinates. Put $z=x+iy$, $k=k_1+i{k}_2$ andSince $\bar{k}z-k\bar{z}=-i(2k_{2}x-2k_{1}y)$, the real frequency vector is $({\xi }_1,{\xi }_2)=(2k_2,-2k_1)$. Its Jacobian is four, givingBoth constants follow from the declared angular-frequency convention.

For smooth compactly supported $q$, a specified Cauchy-kernel solution of $({\partial }_{\bar{z}}-k)F=q$ isThe identity ${\partial }_{\bar{z}}(1/(\pi z))={\delta }_0$ verifies the equation in the distribution sense. Its local singularity is integrable. Differentiation in $\bar{k}$ cancels $z-w$ and yieldsThis is the link with a spectral dbar equation. It uses the displayed normalized Cauchy transform solution; the spatial differential equation alone leaves homogeneous solutions and does not force this spectral derivative for an arbitrary choice of $F$.