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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 Fourier transform supported in a bounded frequency interval. An integrable bandlimited function has a bounded continuous representative when its Fourier transform is integrable, and that representative extends analytically using the compact-frequency integral. Scaling arguments scales its frequency support inversely, while multiplying two such functions adds their support intervals through convolution.

A complex-coordinate parametrization of the two-dimensional Fourier transform. With $z=x+iy$, $k=k_1+i{k}_2$, and $({\xi }_1,{\xi }_2)=(2k_2,-2k_1)$, its exponent is $-i\xi \cdot (x,y)$. Since $d\xi =4dA(k)$, its inverse is $q(z)={\pi }^{-2}\int e^{k\bar{z}-\bar{k}z}Q(k)dA(k)$. The spectral equation $({\partial }_{\bar{z}}-k)F=q$ gives ${\partial }_{\bar{k}}F=-e^{k\bar{z}-\bar{k}z}Q(k)/\pi$, linking Fourier inversion to a spectral dbar equation.

A sinusoidal component at an integer multiple of a fundamental angular frequency. For a $2\pi$-periodic angular coordinate $\varphi$, real Fourier series use $\cos (k\varphi )$ and $\sin (k\varphi )$. Summing over $q$ equally spaced phases removes every component whose integer $k$ is not a multiple of $q$, since ${\sum }_{j=0}^{q-1}{e}^{2\pi ikj/q}=0$ in that case. Indeed, the geometric series has ratio $w=e^{2\pi ik/q}\ne 1$ and sum $(1-w^q)/(1-w)=0$. When $q$ divides $k$, all $q$ terms are one. This cancellation isolates higher-order feedback in resonant conjunction geometry.

The Wiener algebra consists of periodic functions with absolutely summable Fourier coefficients, with norm ${\sum }_k|a_k|$. Multiplication corresponds to discrete convolution of coefficients, so the norm is submultiplicative. Consequently $e^f={\sum }_{n\ge 0}{f}^n/n!$ also belongs to the algebra when $f$ does. Every such Fourier series is uniformly convergent and defines a continuous function.

The absolutely convergent sine series $S(\xi )={\sum }_{l\ge 1}\sin (l\xi )/l^2$ has the integral representation $S(\xi )=\sin \xi {\int }_0^{\infty }t{e}^{-t}/(1-2e^{-t}\cos \xi +e^{-2t})dt$. The integrand is positive for $0<\xi <\pi$, proving $S>0$ there without incorrectly maximizing individual Fourier terms.