A thin axisymmetric coating of physical thickness $d=ϵah$ on a rigid cylinder of radius $a$ has leading curvature $a^{-1}-d/a^2-d_{\bar{z}\bar{z}}$. With constant surface tension $\gamma$, negligible Newtonian gravity, no-slip boundary condition at the wall and negligible exterior stress, lubrication theory gives $q=-d^3{p}_{\bar{z}}/(3\mu )$. Set $z=\bar{z}/a$ and $t=\gamma {ϵ}^3\bar{t}/(3\mu a)$ to obtain the displayed thin-film equation. A perturbation of a uniform film $h_0$ proportional to $e^{st+ikz}$ hasThus the fastest Rayleigh–Plateau instability has $|k|=1/\sqrt{2}$, or physical wavelength $2\pi \sqrt{2}a$.

A wide, shallow drop with cross-sectional area $A$ has the gravity-driven volume flux $q=-g{h}^3{h}_x/(3\nu )$ when substrate no-slip and zero surface shear are imposed. Mass conservation yields a porous medium equation, $h_t=[g/(3\nu )](h^3{h}_x{)}_x$. Its symmetric source-type similarity solution has half-width $\ell$ in the display and height $h=[9\nu ({\ell }^2-x^2)/(10gt){]}_{+}^{1/3}$, where $I={\int }_0^{\pi /2}{\cos }^{5/3}\theta d\theta$. Integrating the profile recovers $A$. The contact-edge slope diverges, so the bulk lubrication approximation requires an inner edge regularization if contact-line microphysics or surface tension is retained.

For gap $h\ll \ell$ and speed $U$, shear stress scales as $\mu U/h$ and viscoelastic normal stresses as ${\psi }_i{U}^2/h^2$. Their force contributions scale respectively as $\mu U/h^2$ and ${\psi }_i{U}^2/(h^2\ell )$. Thus the displayed criterion permits neglecting the normal-stress corrections in the leading lubrication approximation. It is weaker than requiring the normal stresses themselves to be smaller than the shear stress. Rapid time dependence can impose an additional material-time criterion.

The pressure force on a plug-and-film control volume is $\pi a^2\Delta p$. Its opposing wall traction is $2\pi a\mu \int w_y(0)dx$, where $w_y(0)=4U/h-6Q/h^2$. Equating them for a freely moving body proves the displayed formula. Writing $s=2Q/U$, $h=sH$ and $x=\sqrt{s/\kappa }X$ converts it into $H(-\tilde{L})-H(\tilde{L})=C\lambda \int [4/(3H)-1/H^2]dX$, with $C=s/a$ and $\lambda =6\mu U/(\alpha \sqrt{\kappa }{s}^{5/2})$.

An asymptotic scalar field is a free field describing stable incoming or outgoing particles of an interacting scalar field theory. With a field that overlaps the stable one-particle state, the scattering asymptotic condition is $\varphi \to \sqrt{Z}{\varphi }_{\mathrm{in}}$ in the remote past and $\varphi \to \sqrt{Z}{\varphi }_{\mathrm{out}}$ in the remote future. These are scattering limits of suitably smeared matrix elements, not unrestricted strong pointwise limits of local fields. With covariant state normalization $⟨p^{\prime }|p⟩=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })$, the free-field vacuum-to-one-particle matrix element is one. Translation invariance and the asymptotic condition then give $⟨p|\varphi (0)|0⟩=\sqrt{Z}$ after choosing its phase positive. The wave-function renormalization residue $Z$ measures that overlap.

On a measure space of positive total measure, the L-infinity norm of an essentially bounded measurable function is $\Vert f{\Vert }_{\infty }={\mathrm{esssup}}_{\mu }|f|$. The absolute value matters: a constant function $f=-3$ has L-infinity norm three, even though its essential supremum is $-3$. The norm belongs to the equivalence class in Lp space; different representatives equal almost everywhere give the same answer. ^[ref].

For the function $g$ in essential infimum and essential supremum, $\Vert g{\Vert }_{\infty }=1$ under Lebesgue measure, whereas its supremum norm is four. Replacing the exceptional values by the ordinary values $x$ leaves every finite Lp norm unchanged as well: $\Vert g{\Vert }_p=({\int }_0^1{x}^{p}dx{)}^{1/p}=(p+1{)}^{-1/p}$. This tends to one as $p\to \infty$. That agreement explains why the endpoint with exponent $p=\infty$ uses an essential bound rather than a pointwise bound.

A finite probability space makes the distinction between zero and small measure explicit. Let $f(a)=1$, $f(b)=3$, and $f(c)=10$. Assign weights $\mu (a)=1/2-\epsilon$, $\mu (b)=1/2$, $\mu (c)=\epsilon$, where $0\le \epsilon <1/2$. For $1\le p<\infty$,When $\epsilon =0$, the singleton $\{c\}$ is a null set and $\Vert f{\Vert }_{\infty }=3$. For every $\epsilon >0$, however small, $\Vert f{\Vert }_{\infty }=10$. For example, $\epsilon =0.001$ gives $\Vert f{\Vert }_1=2.009$ and $\Vert f{\Vert }_2=\sqrt{5.099}$, yet the large-$p$ Lp norms approach ten. The ordinary pointwise supremum norm is ten in both cases.

Why can a rare value dominate? For a bounded function on a probability space, an almost everywhere bound $|f|\le M$ gives $\Vert f{\Vert }_p\le M$. If $0<t<\Vert f{\Vert }_{\infty }$, the set $A_t=\{|f|>t\}$ has positive probability, so $\Vert f{\Vert }_p\ge t\mu (A_t{)}^{1/p}$. As $p\to \infty$, the factor $\mu (A_t{)}^{1/p}$ tends to one. These two bounds squeeze the limit to $\Vert f{\Vert }_{\infty }$. The probability measure assumption keeps total measure finite; this explanation does not assert the same finite-$p$ integrability on arbitrary infinite-measure spaces.

The Sorgenfrey plane is the product topology on two copies of the Sorgenfrey line. It is separable because ${\mathbb{Q}}^2$ meets every nonempty basic rectangle $[a,b)\times [c,d)$. Its antidiagonal $\{(x,-x):x\in \mathbb{R}\}$ is an uncountable discrete subspace: $[x,x+ϵ)\times [-x,-x+ϵ)$ meets it only at $(x,-x)$. In a discrete space every dense subset is the whole space, so this subspace is not separable. Thus separability need not pass to subspaces of nonmetrizable spaces.

// Target: analysis.bigb

For a finite matrix with singular values in nonincreasing order, truncating its singular value decomposition after $k$ terms minimizes the Frobenius norm error, whose square is the sum of the discarded squared singular values. It also minimizes the operator norm error, which equals the next singular value, or zero if none remain. See Mazeika, Section 4.3: Low rank approximation.

The problem of replacing a matrix by one of lower matrix rank is explicit in Eckart and Young's 1936 paper, ^[ref]. The publication predates the modern image compression example, showing how a general matrix question can support later applications.

A spacetime metric is the Lorentzian metric on a spacetime. It determines proper time along timelike paths and the null directions that describe local light propagation.

For $J=\mathrm{diag}(1,-1)$, the matrices satisfying $A^{T}JA=J$ and $A_{11}>0$ form the time-orientation-preserving Lorentz group in one spatial dimension. They are exactly $A=H(t)D_{\epsilon }$, where $H(t)=\left(\begin{array}{cc}\cosh t& \sinh t\\ \sinh t& \cosh t\end{array}\right)$, $D_{\epsilon }=\mathrm{diag}(1,\epsilon )$ and $\epsilon =\pm 1$. Preservation of the Minkowski metric makes the first column a future-pointing unit timelike vector; Minkowski inner product orthogonality determines the second up to sign. The identity $D_{\epsilon }H(s)=H(\epsilon s)D_{\epsilon }$ gives the multiplication law $(t,\epsilon )(s,\eta )=(t+\epsilon s,\epsilon \eta )$ and inverse $(-\epsilon t,\epsilon )$.

For a processed prefix $(x_1,\dots ,x_i)$, the minimum-tail table for increasing subsequences storesThere is no slot for an unattainable length. The table records minimum endings, rather than the terms of one longest increasing subsequence. Replacing the value comparisons by $<$ gives the corresponding table for strictly increasing subsequences, whose entries are strictly increasing.

For uniform vortex-free superconductivity, the London equation and stationary Maxwell equations give ${\nabla }^2\mathbf{B}=\mathbf{B}/{\lambda }_L^2$. A field tangential to a planar superconducting half-space decays as $\mathbf{B}(z)=\mathbf{B}(0)e^{-z/{\lambda }_L}$. A nonzero normal boundary field is incompatible with this translation-invariant decaying Meissner solution, because divergence-free magnetic flux would require that component to remain constant.

This map equals an explicit unit-step update of a logistic predator-prey differential equation. Its coexistence fixed point is $(b,(1-a-b)/a)$, requiring $a+b<1$. Its trace and determinant are $T=2-b/a$ and $D=(1-2b)/a$. The Schur stability criterion gives stability precisely when $b>(1-a)/2$ and $b<a+1/3$. The coexistence flip at $b=a+1/3$ is subcritical on $1/9<a<1/3$, with effective cubic coefficient $-9/[(3a+1{)}^2(9a-1)]$ when the $-1$ eigenvector has first component one. The unit-circle crossing $a+2b=1$ is a supercritical Neimark–Sacker bifurcation away from the strong resonances of a planar map $a=1/7,1/5$; in the eigenvector normalization $q_1=1$ its cubic radial coefficient is $-1/[a^2(1-a)]$. The whole nonnegative quadrant is not forward invariant, because $x^{\prime }$ can become negative. Local fixed-point stability is therefore not a global population-dynamics classification.

// Destination: dynamical-systems.bigb

For a fixed catch rate $h$, natural logistic growth can replace the catch at two abundances if $0<h<rK/4$. The upper equilibrium $N_{+}=K[1+\sqrt{1-4h/(rK)}]/2$ attracts, and the lower $N_{-}$ is an unstable survival threshold. They meet in a saddle-node bifurcation at $h=rK/4$, $N=K/2$. Above this catch every positive population declines to zero. Below it a population initially under $N_{-}$ also becomes extinct. A constant catch is stopped at extinction to avoid negative abundances; it is distinct from per-capita harvesting proportional to $N$.

For normalized residuals $A_i=(Y_i-\bar{Y})/s$, Fisherian conditional inference uses the conditional distribution of $(\bar{Y},s)$ given the observed $A=a$. For almost every attainable ancillary value, the joint conditional density of $\bar{Y}=m$ and $s=t>0$ is proportional toThe factor $t^{n-2}$ is the radial Jacobian on the $(n-1)$-dimensional centered residual space. After setting $u=(m-\mu )/\sigma$ and $v=t/\sigma$, the conditional density is proportional to $v^{n-2}{\prod }_i{f}_0(u+v{a}_i)$, which is parameter-free. Conditioning on a continuous ancillary uses a regular conditional distribution, rather than division by the probability of a singleton ancillary value.

If $W$ has probability density function $f(w;q)$ and cumulative distribution function $F_W(w;q)$, then $T=\exp (\alpha +\sigma W)$, $\sigma >0$, has density $f((\log t-\alpha )/\sigma ;q)/(\sigma t)$ and distribution function $F_W((\log t-\alpha )/\sigma ;q)$ for $t>0$. The factor $1/(\sigma t)$ follows from the change-of-variables formula for a probability density. Upper truncation at $M$ divides the density by $F_W((\log M-\alpha )/\sigma ;q)$.

In a locally path-connected space, every path component is open: a path-connected open neighbourhood of a point lies within that point's path component. The complement of a path component is a union of the other open path components, hence is also open. In a nonempty connected space, a nonempty subset that is both open and closed must be the whole space. Thus a connected space that is locally path-connected is path-connected.

A multiplicative subset $S$ of a ring contains $1$ and is closed under multiplication. It may contain zero, in which case its localization is the zero ring.

Here $S$ consists of the non-zero-divisors of a commutative ring $A$. The localization map $A\to Q(A)$ is injective: a fraction $a/1$ is zero only if some $s\in S$ annihilates $a$. For a reduced ring with finitely many minimal prime ideals, its zero divisors form the union of these primes. Indeed, if $ab=0$ and $a$ avoids every minimal prime, reduction to each domain quotient forces $b$ into their intersection, which is zero. Conversely, for $a$ in a minimal prime ${\mathfrak{p}}_i$, choose $b$ in every other minimal prime but outside ${\mathfrak{p}}_i$, using their incomparability and multiplying suitable elements. Then $b\ne 0$ and $ab=0$.

An operation that applies a unitary operator to one party's subsystem and the identity to the other is a local unitary operation. Several separated parties may each apply a known unitary, resulting in a product of their local unitaries. Such operations preserve the squared Schmidt coefficients across the relevant bipartition.