If $v\in H_0^1(\Omega )$ and $g:\mathbb{R}\to \mathbb{R}$ is Lipschitz continuous with $g(0)=0$, then $g(v)\in H_0^1(\Omega )$. The Lipschitz version of the Sobolev chain rule controls its weak derivatives, and its boundary trace is $g(0)=0$. Equivalently, approximate $v$ by compactly supported smooth functions, apply the derivative bound to their compositions, and pass weakly in $H^1$ and strongly in $L^2$. The closed linear space $H_0^1$ is weakly closed. This allows positive-part and bounded truncations in weak formulations without losing the Dirichlet boundary condition.

On a nonempty bounded domain where the Poincaré inequality holds, no $a\in H_0^1(\Omega )$ can satisfy $a\ge a_{-}>0$ almost everywhere. For $g(s)=\min (\max (s,0),a_{-})$, Lipschitz truncation preserves zero-boundary Sobolev spaces, so $g(a)\in H_0^1(\Omega )$. But $g(a)$ would be the nonzero constant $a_{-}$, whose zero gradient contradicts the Poincaré inequality. In a uniformly elliptic divergence-form elliptic operator, the homogeneous boundary condition belongs to the unknown and test functions, not to a coefficient bounded below by a positive constant.

A near Young tableau fills a Young diagram with distinct entries from a totally ordered alphabet, increasing along rows and down columns. Its entries need not be the initial interval $1,\dots ,n$. It is the natural intermediate object for row insertion of a new entry.

Insert $x$ into the first row of a Young tableau by replacing its leftmost entry strictly greater than $x$, and carry the displaced entry into the next row. If no entry is greater, append $x$ and stop. The procedure preserves a semistandard Young tableau; when all entries are distinct it preserves a near Young tableau. It underlies the Robinson–Schensted correspondence.

Insert the successive letters of a permutation by row insertion to obtain an insertion tableau $P$, recording the insertion times in the new boxes of $Q$. This is a bijection between permutations and pairs of standard Young tableaux of a common shape. Reverse insertion removes the box marked by the latest recording time. The first row of $P$ has length equal to the longest increasing subsequence.

The Young branching graph is the graded cover graph of the Young lattice. Its level $n$ consists of partitions of $n$, and its edges add one box. It is also the branching graph for complex Specht modules under the restriction branching rule for a symmetric group.

The law $t_y=BD+C/D$ for positive $B,C$ has a minimum at $D_0=\sqrt{C/B}$ with value $2\sqrt{BC}$. This size differs from the minimum of the ratio of removal and collision times.

For $t_c=(A/M)D^{1/2}$, the ratio $t_y/t_c$ has its minimum at $D_{\ast }=\sqrt{3C/B}$. Removal beats collisions somewhere exactly when $M<M_y=(3^{3/4}/4)A{B}^{-3/4}{C}^{-1/4}$, provided the relevant diameters are present. This criterion uses the collision-only cascade as a reference.

Let $n_b(D)$ be the differential number of belt objects per size interval, and $t_y(D)$ their removal timescale through Yarkovsky removal. Suppose removed objects feed an external population with size-independent mean residence time $T$. A simple population balance isFor a maintained belt inventory, the steady solution is $n_{\mathrm{out}}=T{n}_b/t_y$. Starting from an empty external population adds the transient factor $1-e^{-t/T}$. The steady formula therefore requires a sufficiently long-lived supply; it is not an instantaneous identity for arbitrary initial data.

If $n_b\propto D^{-p}$ and $t_y\propto D^a$, the steady differential number spectrum is $n_{\mathrm{out}}\propto D^{-(p+a)}$. If the belt is itself in a removal-dominated steady state with fragment injection rate $S(D)$, its balance instead gives $n_b/t_y\simeq S$. The external spectrum then follows $TS(D)$. These two balances explain why a removal law can either change a supplied population's size slope or disappear from its final steady spectrum.

A real potential $V$ for a scalar field is invariant when $V(R(U)\Phi )=V(\Phi )$ for every gauge transformation. In a unitary representation, $m^{2}h(\Phi ,\Phi )+\lambda h(\Phi ,\Phi {)}^2$ is always an example. Particular representations may admit additional invariant interactions.

With $D_{\mu }={\partial }_{\mu }+\rho ({\mathcal{A}}_{\mu })$ and $\Phi \mapsto R(U)\Phi$, a local $U=1+ϵ$ gives $\delta {\mathcal{A}}_{\mu }=-{\partial }_{\mu }ϵ+[ϵ,{\mathcal{A}}_{\mu }]$, $\delta \Phi =\rho (ϵ)\Phi$ and $\delta {\mathcal{F}}_{\mu \nu }=[ϵ,{\mathcal{F}}_{\mu \nu }]$. Derivative terms cancel in $\delta (D_{\mu }\Phi )=\rho (ϵ)D_{\mu }\Phi$. Here the gauge coupling is absorbed into the connection.

The trace-free metric variation is $(P_{1}c{)}_{ab}={\nabla }_a{c}_b+{\nabla }_b{c}_a-{\gamma }_{ab}{\nabla }_d{c}^d$. A symmetric trace-free antighost exponentiates the Faddeev-Popov determinant. In light-cone worldsheet coordinates its two chiral kinetic terms are $b_{++}{\partial }_{-}{c}^{+}$ and $b_{--}{\partial }_{+}{c}^{-}$.

A scalar Wiener-Hopf equation couples two Half-range Fourier transforms analytic in complementary half-planes. After dividing by a kernel factor, additive splitting separates the known forcing. Source poles, boundary terms and contour prescriptions are part of the problem data.

A scalar Wiener-Hopf factorization writes a transform kernel on its contour as $K=K^{+}{K}^{-}$. The plus factor is holomorphic and nonzero in the designated upper domain, and the minus factor has the corresponding property in the lower domain. Normalizations and any index or polynomial factors must be specified along with these domains.

For real $a,b>0$, a concrete real-line example isThe possible zero and pole of $K^{+}$ lie below the real axis, so $K^{+}$ is holomorphic and nonzero in the complex upper half-plane; those of $K^{-}$ lie above, giving the lower-domain property. Both factors tend to one at infinity, and multiplying them verifies the kernel exactly.

The allocation of zeros, poles and branch cuts is essential in a Wiener-Hopf equation. An outgoing continuation fixes how physical modal poles and square-root branches are assigned; a factor with a zero in its designated analytic domain is not the required nonvanishing factor. Products and quotient conventions should be declared before using a table of factors.

An integral homology equivalence between simply connected spaces is a weak homotopy equivalence. A first nonzero relative homotopy group would contradict the Relative Hurewicz theorem. Between CW complexes the ordinary Whitehead theorem then gives a homotopy equivalence.

For maps between simply connected spaces, a rational homology equivalence is a rational homotopy equivalence. It follows from the relative Hurewicz theorem modulo torsion, or from rationalization and the homological Whitehead theorem.

A white noise sequence has zero mean, a common finite variance and zero autocovariance at every nonzero lag. This requires uncorrelated random variables, rather than independent random variables.

A sequence of independent and identically distributed random variables with zero mean and finite variance is strong white noise. It is weak white noise, but need not have a normal distribution.

Finite rooted trees are well-quasi-ordered by rooted-tree homeomorphic embedding. Consequently every infinite sequence contains an earlier tree embedding into a later one. Size-controlled finite forms follow by applying König infinity lemma to the finitely branching tree of bad prefixes.

Given a size bound with finitely many possible objects at each position, put finite bad sequences in a tree ordered by extension. The tree is finitely branching. Arbitrarily long bad sequences would give an infinite bad sequence by König infinity lemma. This compactness argument converts an infinite well-quasi-ordering theorem into a uniform finite length bound.