The small-$\lambda$ limit of superheated two-phase Stefan similarity gives $a\simeq A\sqrt{\kappa t}$, where $A=2/[\sqrt{{\vartheta }^2/\pi +2S}+\vartheta /\sqrt{\pi }]$ and $\vartheta =\Theta /\Delta T$. For fixed superheat at large latent-to-sensible heat ratio, $A\sim \sqrt{2/S}$. For much larger superheat, $A\sim \sqrt{\pi }/\vartheta$. The approximation retains the leading effect of a hot interior and its coefficient need not be exact when $S$ is only order one.

The two quadratic relations form a regular sequence in $\mathbb{Q}[a,b]$, so their Koszul complex has only the quotient-ring cohomology. Represent the two degree-two classes by rational polynomial forms and choose primitives for the exact relations to obtain a quasi-isomorphism. There are no further generators.

A substructure $M\subseteq N$ is simply closed in $N$ when every formulawith quantifier-free formula $\theta$ and parameters $\mathbf{a}$ from $M$, that is true in $N$ is already true in $M$. This is existential closure in the specified extension. It is weaker than being an elementary substructure, which preserves arbitrary first-order formulas.

One quantified variable with a quantifier-free formula as matrix does not suffice. Let $M$ be an infinite edgeless simple graph, and let $N$ add two new vertices joined to each other and to no old vertex. Every one-variable quantifier-free formula over a finite parameter tuple from $M$ realized by a new vertex is realized by an old vertex outside that tuple. Nevertheless $N$ satisfies $\exists x\exists yE(x,y)$ and $M$ does not. Iteration of one-variable reflection fails because the remaining existential matrix need not be a quantifier-free formula.

Every real sequence has a monotone subsequence, without a boundedness assumption. Call an index a peak when its term is at least every later term. Infinitely many peaks give a nonincreasing subsequence; beyond finitely many peaks each term has a larger later term, which recursively gives an increasing subsequence.

In deep radiation domination, an oscillatory radiation density source in the matter equation has order ${\Delta }_r/{\tau }^2$ and frequency of order $k$. Its rapidly varying particular response is therefore of order ${\Delta }_r/(k\tau {)}^2$. For $k\tau \gg 1$ it does not control the secular matter growth, though forcing near horizon entry sets the constant and logarithmic homogeneous amplitudes.

A thin layer of subglacial till idealized as a Newtonian fluid of dynamic viscosity $\lambda \mu$ and thickness $l$ undergoes Couette flow between stationary bedrock and ice moving at speed $u$. Its resisting basal shear stress has magnitude ${\tau }_b=\lambda \mu u/l$ and acts opposite to the sliding motion. Combining this drag with the hydrostatic driving force gives the bulk approximation used in the friction-dominated grounding-line evolution law.

A gravitational driving force associated with a dense descending slab. The slab's buoyancy and ability to remain coherent matter, so increased planetary gravity alone does not determine whether subduction occurs.

A nonzero complex fundamental doublet has trivial stabilizer in the gauged $SU(2)$. Its three angular directions are gauge directions, absorbed by the Higgs mechanism. For vacuum ${\varphi }_0=(0,v/\sqrt{2}{)}^T$, the three gauge bosons have equal mass $gv/2$; one radial scalar remains. There is no unbroken photon-like gauge generator in this theory.

For field strength $F_{\mu \nu }^a=f_{\mu \nu }^a-g{ϵ}^{abc}{A}_{\mu }^b{A}_{\nu }^c$, the square produces ${\mathcal{L}}_3=g{ϵ}^{abc}({\partial }_{\mu }{A}_{\nu }^a)A^{b\mu }{A}^{c\nu }$ and ${\mathcal{L}}_4=-g^2{ϵ}^{abc}{ϵ}^{ade}{A}_{\mu }^b{A}_{\nu }^c{A}^{d\mu }{A}^{e\nu }/4$. The cubic Feynman vertex has one derivative; the quartic one has none. No higher pure-gauge vertex occurs.

Every compactly supported distribution on an open set $X$ is a finite sum $u={\sum }_{\alpha }{\partial }^{\alpha }{f}_{\alpha }$ with continuous $f_{\alpha }$ compactly supported inside $X$. One can first write $u={\partial }^{\gamma }F$ for a continuous global convolution primitive, then multiply by a cutoff function equal to one near the support. The identity $\chi {\partial }^{\gamma }F={\sum }_{\beta \le \gamma }(-1{)}^{|\beta |}(\genfrac{}{}{0ex}{}{\gamma }{\beta }){\partial }^{\gamma -\beta }[({\partial }^{\beta }\chi )F]$ localizes every coefficient. Finite global representation can fail for a distribution of unbounded order.

An inaccessible cardinal $\iota$ is an inaccessible limit of inaccessibles if the inaccessible cardinals below it are unbounded in $\iota$. A rank cutoff at the first inaccessible limit of inaccessibles shows that existence of unboundedly many inaccessibles does not prove existence of such a limit, provided the former theory is consistent.

Every tripartite density operator satisfies $S(AC)+S(BC)\ge S(A)+S(B)$. Purify to $ABCD$ and apply Strong subadditivity of Von Neumann entropy to $ACD$, conditioned on $A$. The Schmidt decomposition gives $S(AD)=S(BC)$ and $S(ACD)=S(B)$, proving the inequality. The original tripartite state need not be pure.

Apply data-processing inequality for quantum relative entropy to ${\rho }_{ABC}$ and ${\rho }_A\otimes {\rho }_{BC}$, tracing out $C$. The resulting comparison is $D({\rho }_{ABC}\Vert {\rho }_A\otimes {\rho }_{BC})\ge D({\rho }_{AB}\Vert {\rho }_A\otimes {\rho }_B)$. Expand the logarithms of the product marginals to obtain $S(AB)+S(BC)-S(B)-S(ABC)\ge 0$, exactly Strong subadditivity of Von Neumann entropy.

A strong monoidal functor whose comparison maps are identities. It preserves the tensor, unit and structural constraints exactly; its source need not be a strict monoidal category.

If ${\sup }_n\mathbb{E}|X_n{|}^2<\infty$ and the $X_n$ are pairwise uncorrelated random variables, then $n^{-1}{\sum }_{k=1}^n(X_k-\mathbb{E}{X}_k)\to 0$ almost surely. Independence is unnecessary for this version of the strong law of large numbers. A uniformly bounded martingale difference sequence meets its hypotheses, since the differences have mean zero and are orthogonal in $L^2$.

A map $e:X\to Y$ has this property if every commutative square with $e$ as its left side and a monomorphism $m:A\to B$ as its right side has a diagonal filler $t:Y\to A$. The filler is unique because $m$ is monic. This definition does not itself stipulate that $e$ is an epimorphism; binary products imply that conclusion by testing against the categorical diagonal.

A jointly proper convex function $f(x,z)$ represents an optimization problem at perturbation $z=0$ and a family of nearby problems at other $z$. Its primal value function is $p(z)={\inf }_{x}f(x,z)$. Relaxing an inequality by $z$ gives the example $f(x,z)=J(x)+{\delta }_{(-\infty ,0]}(r(x)-\sigma -z)$, using an indicator functional. The resulting convex perturbation duality expresses multipliers as supporting subgradients of the value function.

In nonrotating Rayleigh-Bénard convection with fixed-temperature, stress-free boundary conditions, the stress-free convection growth-rate polynomial has real roots on the destabilizing side. Consequently the first instability is stationary, rather than oscillatory. Minimizing the free-slip convection neutral curve selects ${\lambda }_c=\pi /\sqrt{2}$ and ${\mathrm{Ra}}_c=27{\pi }^4/4$.

The orientation of a force-free, torque-free infinitely slender straight rod follows background rotation plus the transverse component of strain. The last term preserves $|\mathbf{p}|=1$. Finite aspect ratio changes the strain coefficient and permits periodic tumbling in a simple shear flow.

If $S\le T$ are stopping times and $A\in {\mathcal{F}}_S$, then $U=S{1}_A+T{1}_{A^c}$ is a stopping time. The key identity isThis uses the ordering $S\le T$: without it, $A$ need not be known when $T$ occurs. Such pastings test the martingale identity through expectations at bounded stopping times.