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For zero initial velocity and Fourier amplitude $a(k)$, observation along $x=Vt$ with $0\le V<c$ selects the two stationary wavenumbers $\pm k_0$, whereWriting $s=\sqrt{c^2-V^2}$, the leading oscillation has angular frequency $As$ and amplitude proportional to $t^{-1/2}$.

In a flat fundamental square, straight horizontal, vertical and rational-slope lines project to closed geodesics. The central horizontal glide axis is one-sided, a vertical line is two-sided, and the line $(2t,t)$ closes with one transverse self-intersection.

Forthe transformations with even $c$ form an index-two translation subgroup. Its quotient is a torus, giving the orientation double cover $T^2\to K$.

If spatial reflection leaves a Stokes configuration and forcing equivalent to flow reversal while preserving one candidate velocity component, uniqueness forces that component to vanish.

The Penrose process extracts rotational energy by allowing a fragment with negative stationary Killing energy to enter the black hole while another fragment escapes with more energy than the original body carried.

Kernel principal component analysis diagonalizes a centered kernel matrix to perform principal component analysis in an implicit feature space.

For three transverse families of $R^{1/2}\times R^{1/2}\times R$ tubes in ${\mathbb{R}}^3$,

A generalized eigenvector lies in the kernel of a positive power of A minus lambda I.

Over an algebraically closed field, the characteristic polynomial gives the total size of the Jordan blocks for each eigenvalue, while the minimal polynomial gives the largest block size. For total dimension at most three, these two numbers determine the partition into block sizes: a multiplicity-three eigenvalue has partition $3$, $2+1$, or $1+1+1$ according as its largest block has size $3$, $2$, or $1$. Hence two matrices of size at most three with the same characteristic and minimal polynomials are similar.

If $\beta$ has eigenvalues $\lambda ,\mu \ne 0$ and is diagonalizable on a two-dimensional complex vector space, thenIf $\beta$ has one size-two Jordan block, then

A Jordan block has one eigenvalue on its diagonal and ones on its superdiagonal.

The geometric multiplicity of an eigenvalue is the dimension of its eigenspace. In Jordan form it equals the number of blocks carrying that eigenvalue.

The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. In Jordan form it is the sum of the sizes of all blocks carrying that eigenvalue.

If $F$ and $G$ have no explicit time dependence and satisfy $\{F,H\}=\{G,H\}=0$, the Jacobi identity gives $\{\{F,G\},H\}=0$. Their Poisson bracket is therefore also conserved.

For real $0<k<1$, the period lattice of $\mathrm{sn}(u,k)$ is generated byThe periods arise by composing the monodromy reflections of its inverse elliptic integral of the first kind.

If ${\phi }_n(x,t)\sim c_n(t)e^{-{\kappa }_{n}x}$ as $x\to +\infty$, the time equation $Q=0$ givesand hence $c_n(t)=c_n(0)e^{4{\kappa }_n^{3}t}$.

The translation length of an isometry $g$ of a metric space is $\ell (g)={\inf }_{x}d(x,gx)$. For a hyperbolic tree isometry, it is attained exactly on the axis of a tree isometry.

A hyperbolic tree isometry has positive translation length and preserves a unique bi-infinite geodesic.

An elliptic tree isometry fixes a vertex. Its fixed-point set is a nonempty convex subtree.

Every simple module over a commutative ring $R$ is isomorphic to $R/\mathfrak{m}$ for a maximal ideal $\mathfrak{m}$; its annihilator is therefore maximal.