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For nonzero $\lambda$, the ordinal ${\omega }^{\lambda }$ is multiplicatively closed exactly when $\lambda$ is additively closed. One direction follows fromFor the other, the leading exponent of an ordinal product shows that products of ordinals below ${\omega }^{\lambda }$ still have leading exponent below $\lambda$.

For a cylinder of radius $a$ in a coaxial container with gap $h\ll a$, global displacement forces reverse flow through the gap and createsThe pressure drag exceeds side shear by order $a/h$. Balancing it with excess weight gives $U\simeq \Delta \rho ,g{h}^3/(6\mu a)$.

The eigenfunctions satisfying $X(0)=0$ and $X^{\prime }(L)=0$ are

For a family with a monotone likelihood ratio in $T$, a level-$\alpha$ upper-tail test in $T$ is a uniformly most powerful test for the corresponding one-sided hypothesis.

Rotational carbon-monoxide line emission traces cold molecular clouds because abundant molecular hydrogen is difficult to observe directly. A conversion factor $X_{\mathrm{CO}}$ relates integrated CO intensity to molecular-hydrogen column density.

Phase matching at a planar interface preserves frequency and tangential wavenumber. Thus waves of speeds $c_j$ and angles ${\theta }_j$ from the normal satisfy

An accessible automaton over one letter consists of a directed tail entering one directed cycle. Its minimal quotient is obtained by merging positions having the same future binary acceptance sequence.

At each perturbation order, the coefficient of a homogeneous oscillatory mode must vanish from the forcing to prevent secular growth. This solvability condition determines the slow evolution of lower-order amplitudes.

The invariant sigma-algebra of $T$ consists, modulo null sets, of measurable sets $A$ satisfying $T^{-1}A=A$.

Replacing the chosen unit normal $N$ by $-N$ replaces the shape operator $S$ by $-S$. On a surface, Gaussian curvature $K=\det S$ is unchanged, while mean curvature $H=\mathrm{tr}(S)/2$ changes sign.

A minimal surface has vanishing mean curvature vector. For a smooth surface in three-dimensional Euclidean space, this means $H=0$, equivalently $k_1+k_2=0$. Thus Gaussian curvature is $K=-k_1^2\le 0$, but neither principal curvature needs to vanish. A plane and a catenoid both satisfy the condition, with very different Gaussian curvatures.

By the first variation of area formula, a minimal surface is a stationary point of surface area for every smooth variation that fixes its boundary, or has compact support in its interior. The word “minimal” specifies this differential equation; it does not guarantee a global minimum of surface area, or even a local minimum. A stable minimal surface imposes a further condition on the second variation.

For $v,w$ in a real normed space, start with the points at distance $\Vert v-w\Vert /2$ from both endpoints and repeatedly retain those within half the previous diameter of every retained point. Reflection about $(v+w)/2$ preserves every stage, the midpoint belongs to every stage, and the diameters decrease by a factor at least two. Their intersection is therefore exactly the midpoint.

The maximal cyclotomic extension of ${\mathbb{F}}_p$ is ${\overline{\mathbb{F}}}_p$. A primitive $(p^m-1)$th root of unity generates ${\mathbb{F}}_{p^m}$ over ${\mathbb{F}}_p$, and these finite fields exhaust the algebraic closure.

The maximal cyclotomic extension of $\mathbb{Q}$ is a proper subfield of $\overline{\mathbb{Q}}$. Every finite subextension inside a cyclotomic field is Galois and abelian, whereas $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not normal.

The maximal cyclotomic extension of $\mathbb{R}$ is $\mathbb{C}$, since adjoining a primitive fourth root of unity adjoins $i$.

The Maurer-Cartan equation is $d\rho +\rho \wedge \rho =0$ for the left-invariant form $\rho =g^{-1}dg$.

Let $A=M-N$ be Hermitian positive definite. If $M^{\ast }+N$ is also Hermitian positive definite, then $M$ is invertible andThus the stationary iteration associated with the splitting converges from every initial vector.

For matrices $A$ and $B$, the Kronecker product $A\otimes B$ replaces each entry $a_{ij}$ by the block $a_{ij}B$. It satisfieswhenever the products are defined.

Given ordered bases $\mathcal{B}=(v_j)$ and $\mathcal{C}=(w_i)$, the matrix $[T{]}_{\mathcal{C}\leftarrow \mathcal{B}}$ has column $j$ equal to the $\mathcal{C}$-coordinate vector of $T(v_j)$.

A permutation matrix has exactly one entry equal to one in each row and column and zeros elsewhere. Left or right multiplication permutes coordinates, and $P^{-1}=P^T$.