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The compactness theorem says that a set $T$ of first-order sentences has a model if and only if every finite subset of $T$ has a model. One implication follows by taking the same model. Conversely, if $T$ had no model, then by the Godel completeness theorem it would prove a contradiction. A formal proof uses only finitely many assumptions, so some finite subset of $T$ would already have no model. This contradiction proves compactness.

The Upward Lowenheim-Skolem theorem says that if an $L$-theory $T$ has an infinite model, then it has models of arbitrarily large cardinality; more precisely, it has a model of cardinality at least $\kappa$ for every cardinal $\kappa$. Add new constants $c_{\alpha }$ for $\alpha <\kappa$ and the sentencesEvery finite subset of the enlarged theory can be interpreted in the given infinite model, since it mentions only finitely many constants. Compactness supplies a model of the whole enlarged theory, in which the $c_{\alpha }$ are pairwise distinct. Its reduct to $L$ is a model of $T$ having at least $\kappa$ elements. If $\kappa \ge |L|+{\aleph }_0$, the Downward Lowenheim-Skolem theorem gives a model of cardinality exactly $\kappa$.

Fick's first law gives the diffusive fluxLocal conservation says $C_t+\nabla \cdot \mathbf{J}=0$, and therefore

For $D(C)=kC$, conservation of the deposited amount requiresThe dimensions satisfy $[M]=[C]L^2$ and $[k]=L^2/([C]T)$. Thus $Mkt$ has dimension $L^4$, so the similarity length is $(Mkt{)}^{1/4}$. Requiring the concentration scale to have dimension $[C]$ givesHence

In polar coordinates, substitution into
$C_t=k{r}^{-1}(rC{C}_r{)}_r$ givesMultiplying by $\xi$ and integrating once, with regularity and zero radial flux at the origin, yieldsWhere $F>0$, this reduces to $F^{\prime }=-\xi /4$, soNonnegativity and zero flux at the moving front give the compactly supported profileThe normalization givesand hence $A=1/\sqrt{2}$ and ${\xi }_0^4=32$. This is the Two-dimensional Barenblatt solution with diffusivity proportional to concentration, and its support radius isso $N=32$.

Now add the linear reaction term and writeSubstitution cancels the terms $aC$ and leavesChoosingproduces $G_{\tau }=k\nabla \cdot (G\nabla G)$, as in the linear-reaction time change for quadratic nonlinear diffusion. Therefore

For $a=0$, the front grows like $t^{1/4}$ while the central concentration decays like $t^{-1/2}$. For $a>0$, $\tau \sim e^{at}/a$, so the front grows like $e^{at/4}$ and the central concentration like $e^{at/2}$. For $a<0$, $\tau \to 1/|a|$: the front approaches a finite limiting radius while the concentration decays exponentially to zero.

A state $q$ is accessible whenfor some word $w\in {\Sigma }^{\ast }$. States $p$ and $q$ are equivalent, or indistinguishable, whenfor every continuation $w$.

This equivalence is preserved by transitions: if $p\sim q$, then $\delta (p,a)\sim \delta (q,a)$. Hence define $D/\sim$ with states $[q]$, initial state $[q_0]$, transitionand accepting classes $[q]$ with $q\in F$. Induction on word length givesso $D$ and its quotient deterministic finite automaton by indistinguishable states accept the same language. If two quotient states were equivalent, their representatives would be equivalent in $D$, so the classes would be equal. Thus no two distinct quotient states are equivalent.

For the unary alphabet, accessibility means that all states occur on the orbitFiniteness makes this orbit a directed tail entering a directed cycle. The quotient retains exactly one accepting state and merges precisely those positions having the same future acceptance pattern.

More explicitly, if the unique accepting state lies before the cycle, every state after it can never reach acceptance and these states collapse to one rejecting sink. The minimal diagram is then a directed chain from the initial state through the unique accepting state and onward to that sink, which has a self-loop.

If the accepting state lies on the cycle, all cycle states are distinct because their next visits to the accepting state occur in different residue classes modulo the cycle length. Any remaining tail is a chain feeding the cycle. In a minimal diagram with a nonempty tail, the accepting state is the cycle vertex immediately preceding the entry vertex; otherwise the final tail state has the same future acceptance sequence as a cycle state and would be merged. A pure cycle with one accepting vertex is also possible. These are exactly the minimal accessible unary DFAs with one accepting state.