A pure cubic number field is a degree-three number field generated by a root of $X^3-m$ for a rational number $m$ that is not a rational cube. For a positive square-free integer $m>1$, this polynomial is an Eisenstein polynomial at any prime dividing $m$. The ring of integers of a number field depends on congruences at $3$, as expressed by the integral basis of a nonexceptional pure cubic field.

If $k_a={\nabla }_{a}S$ is a null gradient, symmetry of the Hessian gives $k^a{\nabla }_a{k}_b=k^a{\nabla }_b{k}_a=\frac{1}{2}{\nabla }_b(k^2)=0$. Its integral curves are therefore affinely parametrized null geodesics, rather than merely null curves with a nonaffinity term.

For an affine null geodesic congruence with future tangent $U$, choose a future unit timelike vector $T$ on a transverse section, put $E=-U\cdot T$, and set $N=T/E-U/(2E^2)$ there. It has $N^2=0$ and $U\cdot N=-1$. Parallel transport along each generator preserves these contractions by metric compatibility and ${\nabla }_{U}U=0$. The construction is local up to caustics and supplies the auxiliary normal for a screen-space projector.

If a null vector tangent satisfies ${\nabla }_{K}K=\kappa K$, then $U=hK$ is affine when $K(\log h)=-\kappa$. Choose $h=1$ on the initial transverse surface. The screen-space projector kills derivative terms proportional to $K$, so the initial optical tensor and its null expansion agree with those of $K$. In ingoing spherical coordinates, $K={\partial }_v+(f/2){\partial }_r$ has $\kappa =f^{\prime }/2$.

The null convergence condition requires $R_{ab}{k}^a{k}^b\ge 0$ for every null vector. Through the Einstein field equations, it follows from the null energy condition, since all terms proportional to the metric tensor vanish on a null vector.

In a normed vector space, this set consists of all points at a fixed positive norm distance $r$ from $x_0$. It also makes sense in an infinite-dimensional Hilbert space, unlike a specifically Euclidean sphere. In a real Hilbert space, the sphere in a normed vector space $\Vert u{\Vert }^2=2E_0>0$ has tangent variations $v$ satisfying $(u,v)=0$; this follows by differentiating the fixed squared norm. This is the geometric constraint used in fixed-energy initial-condition optimality.

On continuous functions on a compact interval, evaluation at an interior point is not continuous in the integral norm. Narrow triangular peaks have fixed point value and integral tending to zero. Subtracting such a peak from the constant function one gives functions vanishing at the point and converging in the integral norm to one. Thus the evaluation kernel is not closed, although it is closed in the uniform norm.

Here is a fully computed endpoint version, now using the L2 norm on the actual continuous functions $C([0,1])$. For each integer $n\ge 1$, set $f_n(x)=\max (1-nx,0)$. Its point evaluation at zero is $E(f_n)=f_n(0)=1$, whileConsequently $f_n\to 0$ in the L2 norm but $E(f_n)\nrightarrow 0$: point evaluation is discontinuous on this normed space. Quantitatively $|E(f_n)|/\Vert f_n{\Vert }_2=\sqrt{3n}$ is unbounded. At $n=100$, the L2 norm is approximately $0.057735$, the integral is $0.005$, and the point value is still $1$. In the supremum norm, $\Vert f_n{\Vert }_{\infty }=1$ and $|E(f)|\le \Vert f{\Vert }_{\infty }$, so $E$ has operator norm one instead.

Set $g_n=1-f_n$. Then $E(g_n)=0$ and $\Vert g_n-1{\Vert }_2=1/\sqrt{3n}\to 0$, although $E(1)=1$. Thus the kernel of $E$ fails to be a closed set. On the full L2 space, a further distinction is essential: its elements are equivalence classes under equality almost everywhere, and changing a representative at zero does not change its class. Arbitrary point evaluation is therefore not even well-defined there. Restricting first to continuous functions makes the discontinuity statement precise.^[ref]

For a scalar function on a finite abelian group, its coefficient at an additive character $\chi$ is the displayed uniform expectation. On ${\mathbb{F}}_3^d$, write ${\chi }_{\xi }(x)=e^{2\pi i\xi \cdot x/3}$. The coefficient at the constant character is the mean of $f$, and the Parseval identity on a finite group becomes ${\sum }_{\chi }|\hat{f}(\chi ){|}^2=\mathbb{E}|f{|}^2$ after relabelling characters to match the transform convention.

For a subset $A$ of a finite abelian group of density of a finite subset $\alpha$, take normalized Fourier coefficients on a finite abelian group and an unnormalized sum over frequencies. The bound follows from $|\widehat{1_A}(\chi )|\le \alpha$ and the Parseval identity on a finite group ${\sum }_{\chi }|\widehat{1_A}(\chi ){|}^2=\alpha$. It bounds the energy of the normalized convolution on a finite group $1_A\ast 1_A$.

If $N=\{1,z\}$ is a normal subgroup of a group $G$, conjugation by any element of $G$ preserves $N$ and fixes $1$. It must therefore fix $z$ as well. Thus $N$ lies in the center of a group. This fact can turn a small normal subgroup into a commuting factor without assuming the whole group is abelian.

For a moving level surface, $n=\nabla S/|\nabla S|$, $v_n=-S_t/|\nabla S|$, and the surface delta distribution is ${\delta }_s=|\nabla S|\delta (S)$. Piecewise classical conservation laws acquire a distributional flux defect $[b]\cdot n-v_n[a]$. Vanishing of this defect is the Rankine-Hugoniot condition.

A vector $w$ is a regular normal at $z\in C$ when ${\mathrm{limsup}}_{z^{\prime }\to z,z^{\prime }\in C\setminus \{z\}}⟨w,z^{\prime }-z⟩/\Vert z^{\prime }-z\Vert \le 0$. At an isolated point all vectors satisfy this condition. It is the local first-order supporting cone.

The limiting normal cone consists of limits $w_k\to w$ of vectors from the Fréchet normal cone $w_k\in {\hat{N}}_C(z_k)$ with $z_k\to z$ in $C$. It need not be convex. Including normals from nearby graph pieces is essential to the Mordukhovich criterion.

independent additive noise $Y=X+ϵ$ multiplies characteristic functions: ${\phi }_Y={\phi }_X{\phi }_{ϵ}$. When the noise characteristic function is nonzero, division recovers ${\phi }_X$. A smoothing kernel limits the frequencies used, controlling the amplification of sampling error where $|{\phi }_{ϵ}|$ is small.

The complex Ginzburg–Landau equation is a nonlinear complex-amplitude evolution equation with linear and cubic terms and generally complex coefficients. The driven-condensate form ${\Psi }_t=(\alpha -\beta |\Psi {|}^2)\Psi +i({\nabla }^2-g|\Psi {|}^2+s)\Psi$ has gain/loss coefficients $\alpha ,\beta$ and conservative interaction/frequency coefficients $g,s$. A uniform nonzero state has number density $\alpha /\beta$ when $\alpha ,\beta >0$.

A complex Ginzburg–Landau equation describes a complex number disturbance amplitude through linear amplification, advection, complex diffusion equation terms, and often cubic amplitude saturation. A constant-coefficient example is ${\psi }_t+U{\psi }_x=\mu \psi +(1+i{c}_d){\psi }_{xx}-|\psi {|}^2\psi$. Here real $U,\mu ,c_d$ describe transport, linear growth rate and dispersive diffusion equation terms. Omitting the cubic term gives the linear complex Ginzburg-Landau equation.

For a finite group, choose one unitary irreducible representation $\rho$ of degree $d_{\rho }$ from each equivalence class. One consistent normalized Fourier transform on a finite group convention is $\hat{f}(\rho )={\mathbb{E}}_{x}f(x)\rho (x)$. The Schur orthogonality relations giveThe normalized convolution on a finite group satisfies $\widehat{f\ast g}(\rho )=\hat{f}(\rho )\hat{g}(\rho )$ in this convention. Using $\rho (x{)}^{\ast }$ in the transform instead reverses that product order for scalar functions with the same normalized convolution on a finite group convention. The Fourier transform on a finite group has matrix-valued components even when the original function is scalar-valued.

A finite group is $m$-quasirandom if every nontrivial irreducible representation over the complex numbers has degree at least $m$. The trivial representation is excluded. Large $m$ suppresses correlations of products of arbitrary subsets, through product mixing in a quasirandom group.

For $f:G\to M_n(\mathbb{C})$ on a finite group, use uniform expectation over the solutions of $x{y}^{-1}z{w}^{-1}=e$ and the ordinary, unnormalized trace to define the displayed fourth-order quantity. If $T_{\rho }$ is its matrix Fourier block, this quantity equals ${\sum }_{\rho }{d}_{\rho }\mathrm{Tr}((T_{\rho }{T}_{\rho }^{\ast }{)}^2)$, so it is real and nonnegative. For a scalar function on an abelian group, it reduces to the usual fourth power of the Gowers uniformity norm of order two. The unnormalized trace means a unitary matrix-valued function can have fourth-order quantity as large as $n$.

A prime filtration of a module over a commutative ring is a finite increasing chain from zero to the module with each nonzero successive quotient module isomorphic to $R/P$ for a prime ideal $P$. Every finitely generated module over a Noetherian ring has one. In each nonzero quotient, a maximal annihilator of a module element is prime supplies a cyclic submodule $R/P$; lift it and repeat. The ascending chain condition for a Noetherian module makes the process terminate. This is not necessarily a composition series: the prime ideals need not be maximal.

For a smooth incompressible flow at a fixed flat impermeable no-slip boundary condition, the velocity is zero at the wall for all time. Consequently both its time derivative and its convective acceleration vanish there. Write the Navier-Stokes equation per unit mass aswhere $p$ is kinematic pressure, $\nu$ is kinematic viscosity, and $\mathbf{b}$ is the specified body force per unit mass. The wall-normal projection therefore fixesThe sign of a buoyancy contribution follows its sign in the momentum equation. In particular, if the chosen perturbation convention has $\mathbf{b}=-{\mathrm{Ri}}_B\theta \hat{\mathbf{y}}$, this becomes ${\partial }_{n}p=\nu \mathbf{n}\cdot \Delta \mathbf{u}-{\mathrm{Ri}}_B\theta \mathbf{n}\cdot \hat{\mathbf{y}}$. A convention with the opposite body force reverses that term.

A homogeneous Neumann boundary condition for pressure is compatible only when this right-hand side vanishes. It does not follow from the no-slip boundary condition: zero wall velocity does not require the wall value of its second spatial derivatives to vanish. This compatibility condition supplies the wall data for the pressure Poisson equation for incompressible flow.