A Banach limit is a positive linear functional on the real l-infinity sequence space that extends the ordinary limit on the convergent sequence space, has operator norm one, and is invariant under the left shift on bounded sequences. The Hahn-Banach theorem constructs one by separating the constant sequence $1$ from the subspace $(I-S){\ell }^{\infty }$, whose distance from $1$ is one. The functional is generally not unique.

For a finite-dimensional Hermitian operator $A$ with decreasing eigenvalues ${\lambda }_j$ and integer $0\le r\le d$, maximizing $\mathrm{Tr}(AB)$ over positive contractions $B$ with trace $r$ gives the sum of its leading $r$ eigenvalues. In an orthonormal eigenbasis, the diagonal entries of $B$ lie in $[0,1]$ and sum to $r$. Moving their weight to the largest eigenvalues can only increase the objective. An orthogonal projection onto their eigenvectors attains the maximum.

Split a uniform set family into the section avoiding a coordinate and the section containing it, with that coordinate removed. Replace each section by a colexicographic initial segment of the same size. Induction on the ground-set size, and nesting of initial segments, show that this operation cannot enlarge the lower shadow.

For a possibly noncommutative ring and central $t_i$, the formal exterior basis gives free bimodules with differential $d(e_{i_1}\wedge \cdots \wedge e_{i_k})={\sum }_a(-1{)}^{a-1}{t}_{i_a}{e}_{i_1}\wedge \cdots \widehat{e_{i_a}}\cdots \wedge e_{i_k}$. Centrality makes this a bimodule differential. Pairwise cancellation proves $d^2=0$.

An $L^2$ observable of an invertible probability measure-preserving system is almost periodic when its Koopman operator orbit is totally bounded in the global $L^2$ norm. A noninvertible system uses its forward orbit. This is the Hilbert-space dynamical definition, distinguished from uniform almost periodicity for continuous functions of a real variable.

If a cumulative distribution function is constant between consecutive lattice points spaced by $h$ near zero, then its Kolmogorov distance from the standard normal distribution function is at least half the increase of that function across one such interval. This gives a lower bound of order $h$.

For the initial field $\varphi (x)=\tanh (x+c)-\tanh (x-c)-1$ and zero initial time derivative, evaluate the scalar-field momentum flux at the midpoint: ${\varphi }_x(0)=0$ and $\varphi (0)=2\tanh c-1$. The force on the left kink is the displayed positive value, attractive toward the right antikink. This exact initial expression has the large-separation asymptotic $32e^{-2d}$ for $d=2c$. If the initial velocity at the cut is $\psi (0)$, subtract $\psi (0{)}^2/2$.

For downward settling speed $u_s(1-\varphi )$ in upward coordinate $z$, particle flux is $j=-u_s\varphi (1-\varphi )$. The concentration-characteristic speed is $j^{\prime }=u_s(2\varphi -1)$ and differs from particle speed. Since $j^{\prime \prime }=2u_s>0$, decreasing concentration with height creates a compressive shock, while increasing concentration creates a rarefaction wave.

Use the quadratic hindered-settling flux $j(c)=-u_{s}c(1-c)$ in upward coordinate $z$, where $c$ is particle volume fraction normalized by the packed value. Initially $c=2a$ for $0<z<z_m$ and $c=a$ for $z_m<z<H$, with $0<a<1/2$. Clear fluid lies above, and a stationary packed deposit with $c=1$ forms below.

The Rankine-Hugoniot condition gives the clearing, internal and deposit-front trajectories before interactions:The internal jump is compressive because the lower concentration is larger. If the internal and deposit fronts meet first, the remaining deposit/upper-suspension jump joins $c=1$ to $c=a$ and advances at $u_{s}a$. If the clearing and internal fronts meet first, their replacement joins $c=2a$ to clear fluid. Either ordering conserves the same particle volume:Thus final packed thickness does not depend on which pair of fronts interacts first, although the intermediate wave pattern does.

Particle mass conservation gives $h_c=[{\varphi }_1(H-z_m)+{\varphi }_2{z}_m]/{\varphi }_{\max }$. Averaging concentration over the entire original container gives $\bar{\varphi }={\varphi }_{\max }{h}_c/H$. This average differs from the concentration inside the deposit itself.

A moving discontinuity of particle volume fraction in kinematic sedimentation. Conservation of particle volume gives the Rankine-Hugoniot condition $U[\varphi ]=[j]$ in a consistently oriented coordinate. A clearing front and a deposit front can both be sedimentation shocks. A stationary deposit uses zero particle flux; its compacted branch need not obey the suspension flux law.

A particle suspension contains two particle species with separate particle volume fractions. In an independent-settling model, each species obeys its own flux conservation equation, while a shared stationary deposit can couple the jump conditions through its maximum total packing fraction. Such independence is an imposed model; general hindered settling need not decouple species.

A disrupted binary star with positive post-impulse specific orbital energy ${\epsilon }^{\prime }$ has relative speed $V=\sqrt{2{\epsilon }^{\prime }}$ at infinite separation, because the Newtonian gravitational potential energy tends to zero. With the notation of the kick-direction binary survival criterion, $V=v\sqrt{1+2\alpha \cos \psi +{\alpha }^2-2\beta }$. This is not a systemic recoil velocity. A marginal parabolic Kepler orbit has $V=0$ at infinity.

Vertical shear instability destabilizes differential rotation that varies with height, when buoyancy restoration is absent or sufficiently weakened. In the homogeneous incompressible flow model with background $U_y=-3\Omega x/2+q\Omega z$, an axisymmetric perturbation with wavevector $(k_x,0,k_z)\ne 0$ hasThe pressure-free radial/vertical momentum equation and the azimuthal equation give this result after using incompressibility. Growth requires $2q{k}_x{k}_z>k_z^2$, so a small vertical shear favors nearly radial wavevectors. Here $q$ is the vertical shear amplitude, distinct from the orbital shear parameter $3/2$. Stratification, cooling and boundaries can change this simplified criterion.

The three-form field strength of the Kalb–Ramond field is $H=dB$. Its components are $H_{mnp}={\partial }_m{B}_{np}+{\partial }_n{B}_{pm}+{\partial }_p{B}_{mn}$. In flat spacetime the free equation is ${\partial }^m{H}_{mnp}=0$.

When ${\partial }_{-}$ is invertible, choose $B_{-m}=0$. Residual gauge parameters are exact one-forms and act trivially on $B$. The remaining independent modes are antisymmetric transverse components $B_{IJ}$, satisfying $\square B_{IJ}=0$; longitudinal components obey $B_{+I}={\partial }_{-}^{-1}{\partial }^J{B}_{IJ}$. There are $(D-2)(D-3)/2$ physical polarizations.

For the harmonic oscillator equation $y^{\prime \prime }+{\omega }^{2}y=J\delta (t-t_{\ast })$, with $\omega >0$, the displacement is continuous and the velocity increases by $J$ at the impulse. If the incoming displacement is zero and $J=-y^{\prime }(t_{\ast }-)$, the outgoing displacement and velocity both vanish. The subsequent homogeneous solution is identically zero. Equivalently, the causal forced response $(J/\omega )H(t-t_{\ast })\sin (\omega (t-t_{\ast }))$ cancels the pre-existing oscillation, where $H$ is the Heaviside step function.

The commutator action of $X_s$ is diagonalizable on endomorphisms: on $\mathrm{Hom}(V_{\lambda },V_{\mu })$ it acts by $\mu -\lambda$. The action of $X_n$ is nilpotent by nilpotence of commutation by a nilpotent endomorphism. The two actions commute, so uniqueness of the Additive Jordan decomposition identifies them as the parts of $\mathrm{ad}X$.

The model matrix $B=\left(\begin{array}{cc}0& 1\\ ϵ& d\end{array}\right)$ has a defective double eigenvalue when $d=ϵ=0$. Its characteristic polynomial is ${\lambda }^2-d\lambda -ϵ$, giving the displayed roots. For fixed nonzero $d$ the small root is $-ϵ/d+O({ϵ}^2)$ and the other is $d+ϵ/d+O({ϵ}^2)$. These expansions cease to be ordered at $d=O(\sqrt{ϵ})$, the distinguished limit in which both roots split on a square-root scale. This is the generic leading behavior when a perturbation closes a two-dimensional Jordan block through a nonzero lower-left entry; it need not occur for a perturbation that leaves that entry zero.

For a connected based CW complex with free integral homology, the Pontryagin ring of its looped reduced suspension is the indicated tensor algebra. The generator inclusion comes from $X\to \Omega \Sigma X$, and multiplication corresponds to concatenation of words in the James reduced product.