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.

A bad sequence in a preorder is an infinite sequence with no indices $i<j$ satisfying $x_i\le x_j$. A preorder is a well-quasi-ordering precisely when no such sequence exists.

The Rado order has domain $R=\{(m,n)\in {\mathbb{N}}^2:m<n\}$ and $(m,n){\le }_R(k,l)$ iff either $m=k$ and $n\le l$, or $n<k$. It is a well-quasi-ordering: either one row recurs infinitely in a sequence, or the first coordinates are unbounded and yield a cross-row comparison. Its infinite rows form an infinite antichain under Hoare domination preorder, showing that arbitrary-subset lifting need not preserve well-quasi-ordering.

A decidable well-order on the natural numbers is a well-order whose comparison relation is computable. Comparing two codes is an effective finite task; the proof that the relation has no infinite descending chain is a separate mathematical assertion. Computable Cantor normal form notation gives such an order of type epsilon zero.

The coordinate conditions define a subgradient precisely when $p\in {\ell }^2$ and $J_w(u)<\infty$. They follow by varying one coordinate; summing their scalar supporting inequalities proves sufficiency. Since $w_k\ge c>0$, a subgradient can exist only at finitely supported $u$. At such a base point $v$, $D_{J_w}^p(u,v)={\sum }_k(w_k|u_k|-p_k{u}_k)$, allowing the value infinity. Merely having finite penalty does not guarantee a nonempty subdifferential in the ambient l2 sequence space.

For $L>0$, constant $\beta >0$ and $\mathrm{Re}p>0$, on $(0,L)$ put $b=\beta /2$, $\kappa =\sqrt{p+b^2}$ with the principal square root, andThese satisfy $(p-{\partial }_x^2-\beta {\partial }_x)u_p=(p-{\partial }_x^2-\beta {\partial }_x)v_p=0$ and the left and right homogeneous Neumann boundary conditions, respectively. The weighted Wronskian givesThe first derivative jumps by $-e^{-\beta y}$, so the resolvent integrates against $e^{\beta y}dy$. Its time Laplace transform interpretation is ${\mathcal{R}}_p={\int }_0^{\infty }{e}^{-pt}{K}_{\beta }(t)dt$. The factor $p$ records the zero eigenvalue and constant stationary eigenfunction.

Write $N=\Vert u{\Vert }_{\alpha }$, $r=|t|$ and $A=\alpha (1-s)>r$. Along the straight integration segment, choose $\sigma (\tau )=s+(A-\tau )/(2\alpha )$. A Cauchy estimate on a disc of radius $R(\sigma -s)$ bounds the integrand by $4\alpha N\tau /[R(A-\tau {)}^2]$. ThereforeThe integral is taken at fixed $x$, and the disc and segment lie inside the shrinking domain. Consequently $u\mapsto {\int }_0^t(i{u}_x+f)dz$ is a contraction mapping when $0<\alpha <R/4$ and the holomorphic source $f$ is bounded on the relevant closed polydisc. The source term has weighted norm at most $\alpha \sup |f|$, so the Banach fixed-point theorem gives a holomorphic solution with zero initial data.

A locally small category with all small categorical limits and a small weakly initial set has an initial object. Take the product $W$ of the weakly initial family, then the simultaneous equalizer $e:E\to W$ of all endomorphisms of $W$ and its identity. For any parallel $a,b:E\to X$, their equalizer $j:Y\to E$ receives a map $t:W\to Y$. The equation $(ejt)e=e$ forces $jte=1_E$, so $j$ is invertible and $a=b$. Weak initiality supplies existence of maps from $E$. This is the smallness mechanism in the Freyd general adjoint functor theorem.

Consistency of ZFC implies consistency of ZFC with the Generalized continuum hypothesis and no weakly inaccessible cardinals. Pass to the constructible universe. If it has an inaccessible, cut at its least one; that rank segment still models ZFC and the Generalized continuum hypothesis, but has no inaccessibles. In this model the singular cardinal enumeration is continuous at every nonzero limit ordinal index. This is a relative-consistency construction, not a deduction of a transitive model from bare consistency.

A weakly inaccessible cardinal $\kappa$ is weakly Mahlo if $\{\alpha <\kappa :\alpha =\mathrm{cf}(\alpha )\}$ is a stationary set. Intersecting this set with the club set of uncountable limit cardinals shows that the weakly inaccessible cardinals below $\kappa$ are stationary, hence unbounded.

Let $h:\mathbb{R}\to \mathbb{R}$ be continuous, odd, strictly increasing and onto. Suppose $u\in C^2({\mathbb{R}}^d)$ tends to zero at infinity and satisfies $-\Delta u+h(u)=f$, with bounded $f$. At a positive maximum $m$ of $u$, $-\Delta u\ge 0$, so $h(m)\le f\le \Vert f{\Vert }_{\infty }$. Apply the same argument to $-u$ to obtain the displayed bound, using the maximum-point argument of the weak maximum principle for elliptic operators. For $h(s)=s+\sin s$, strict monotonicity follows by integrating $h^{\prime }=1+\cos s$, whose zeros are isolated. Thus $\Vert u{\Vert }_{\infty }\le h^{-1}(\Vert f{\Vert }_{\infty })\le \Vert f{\Vert }_{\infty }+1$. The stronger bound by $\Vert f{\Vert }_{\infty }$ holds when that norm is at most $\pi$, but need not hold for larger sources.

If the physical refractive index is $n=1+\mu W$, then $n^2-1=2\mu W+{\mu }^2{W}^2$. In the parabolic wave equation, the last term produces $ik{\mu }^2\mathbb{E}[W^{2}E]/2$ in the mean equation. For unit-variance, finite-correlation fluctuations, $\mathbb{E}[W^{2}E]=m+O(\mu )$ in a weak-fluctuation expansion, giving a coherent phase drift $ik{\mu }^{2}m/2$ to second order. This term is absent if the model deliberately linearizes $n^2-1$ to $2\mu W$. It is not obtained by squaring ideal Gaussian white noise, whose pointwise square is not the finite-variance random variable $W^2$.

This is the coarsest topology making all displayed bounded-continuous test integrals continuous. On a separable metric space it is metrized by the bounded-Lipschitz metric. This probability-measure topology should be distinguished from the weak topology of a Banach space.

For massless momenta, define $T^{\alpha \beta }(a,b)=\mathrm{Tr}[/a{\gamma }^{\alpha }(1-{\gamma }^5)/b{\gamma }^{\beta }(1-{\gamma }^5)]$. The gamma matrix trace identities and the antisymmetric tensor contraction give $T^{\alpha \beta }(a,b)T_{\alpha \beta }(c,d)=256(a\cdot c)(b\cdot d)$. Reversing an antiquark's spinor order changes the momentum pairing and creates the $(1-y{)}^2$ angular factor in a weak charged-current quark scattering rate.

For a massless neutrino scattering on an unpolarized up antiquark through the Fermi interaction, $d\sigma /dy=G_F^{2}s(1-y{)}^2/\pi$. The corresponding down-quark channel has $d\sigma /dy=G_F^{2}s/\pi$. Their difference follows from the chiral weak-current trace contraction. The parton distribution functions weight these rates in a hadron.

Initial position ${\varphi }_0$ for a wave map takes values in the target, and initial velocity ${\varphi }_1$ belongs to its tangent space at ${\varphi }_0(x)$. For $S^2$ this means $|{\varphi }_0|=1$ and ${\varphi }_0\cdot {\varphi }_1=0$. Localized data agree with a constant map and zero velocity outside a compact set; the map itself need not be a compactly supported ambient function.

The conserved energy of a wave map from Minkowski spacetime to a Riemannian target is $E=\frac{1}{2}\int (|{\varphi }_t{|}^2+|\nabla \varphi {|}^2)dx$, with norms measured in the target metric. Contraction of the wave map equation with the velocity and integration by parts gives conservation.

A time-independent wave map is a harmonic map of its spatial domain into the target. For target $S^2$, its equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. In two dimensions inverse stereographic projection gives a smooth nonconstant example of finite wave map energy $4\pi$.

For smooth localized compatible wave map Cauchy data and compact Riemannian target, one-dimensional wave maps stay smooth globally. In null coordinates, the equation is $D_v{\varphi }_u=D_u{\varphi }_v=0$. Compatibility of the target metric with the covariant derivative makes $|{\varphi }_u{|}^2$ independent of $v$ and $|{\varphi }_v{|}^2$ independent of $u$. These transported derivative bounds and higher energy estimates prevent finite-time breakdown.