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A propagating chemical front can persist when only one species diffuses. For $u_t=u(1-u-\beta v)+u_{xx}$, $v_t=-\delta uv$, set $z=x-ct$, $k^2=1-\delta /2$, $c=-\delta /(2k)$ and $\beta v_0=2-\delta$, with $0<\delta <2$. The displayed profiles satisfy both equations because $U^{\prime }=kU(1-U)$ and $V^{\prime }/V=-2kU$. They connect $(u,v)=(0,v_0)$ to $(1,0)$: the second concentration changes by local reaction, even without its own diffusion.

The dimensionless bacterial growth and substrate depletion system $b_t=b_{xx}+gb$, $g_t=-gb$ has travelling-wave equations $b\prime \prime +cb\prime +gb=0$, $cg\prime =gb$ in coordinate $y=x-ct$. For a front with $(b,g)\to (b_0,0)$ behind and $(b,g)\to (0,1)$ ahead, its first integral is $b\prime +c(b+g)=c$, forcing $b_0=1$. Positivity of the leading edge requires $c\ge 2$.

A travelling front is a travelling wave joining distinct limiting states ahead of and behind a moving interface. Its spatial profile depends on a coordinate such as $x-ct$.

For $c>2\eta$, the triangle $0\le a\le \eta$, $-\eta a\le b\le 0$ traps the unstable branch leaving the saddle $(\eta ,0)$ toward lower population. On its top edge $b^{\prime }<0$. On the sloping edge, $(b+\eta a{)}^{\prime }=\eta (c-\eta )a-a({\eta }^2-a^2)/(1+a^2)>0$, so that edge is also inward. The right edge has $a^{\prime }=b\le 0$. Monotonicity of $a$ inside the compact region leaves only $(0,0)$ as the forward limit. This proves a nonnegative monotone travelling front. For $0<c<2\eta$, the target equilibrium is a focus, so any approaching nonconstant trajectory crosses into negative population and cannot be a physical front.

For a designated root vertex in an undirected complete graph, a 1-tree consists of a spanning tree on all other vertices together with two distinct root-incident edges. It is connected and has exactly one cycle, which passes through the root. Its minimum weight is the cost of an independent minimum spanning tree plus the two cheapest root edges, computable in quadratic time on a dense graph by Prim's algorithm.

A transport port is a numbered identifier used by a transport protocol to distinguish communication endpoints at a host. TCP and UDP have separate port spaces.

Data retransmission sends data again to recover from loss or detected corruption. Transport sequence numbers let a receiver recognize a duplicate if the original transmission arrived but its acknowledgement was lost.

A transport sequence number identifies a data position within an exchange, supporting ordering, duplicate rejection and precise acknowledgements. TCP numbers bytes rather than simply numbering packets.

An acknowledgement reports receipt according to the protocol's rules. A TCP cumulative acknowledgement names the next expected byte; it does not establish that the application has durably acted on all preceding data.

A sliding window protocol permits a range of numbered data to be outstanding or accepted, advancing that range as transmission makes progress.

Flow control limits a sender according to a receiver's buffering or processing capacity, rather than the capacity of the intervening computer network.

Scala orders a class and its inherited traits to resolve member lookup and super calls. For two independent traits sharing a base, the rightmost mixed-in trait comes first, followed by the earlier trait and then the common base. Ordinary virtual calls start at the receiver's most specific override; an unqualified super call starts after the implementation's owner in that receiver's linearization. It does not necessarily call the textual parent of the trait in isolation.^[ref]

A Hausdorff compactification is a compact Hausdorff space $K$ with a topological embedding $j:X\to K$ whose image is dense. It retains the original topology on $X$ while adjoining boundary points. For a locally compact noncompact Hausdorff space, its one-point compactification is an example. For every Tychonoff space, the Stone-Čech compactification extends all continuous maps into compact Hausdorff targets.

A space is zero-dimensional when its topology has a base of clopen sets. In the Hausdorff setting these sets separate distinct points, so the space is totally disconnected. Conversely a compact Hausdorff totally disconnected space has a clopen base, by the Šura-Bura component lemma and finite subcovers of the complement of a neighborhood.

Fix a point $a$ of an uncountable set $X$ and declare $F$ closed when $a\in F$ or $F$ is countable. Finite unions and arbitrary intersections satisfy the closed-set axioms. Every convergent sequence is eventually constant: the countable set of its values different from the limit is closed, so its complement is a neighborhood of the limit. A real-valued map constant at $f(a)$ outside a countable set is continuous, because each closed inverse image either contains $a$ or lies in that countable exceptional set.

For a family of functions $f_i:X\to Y_i$ into topological spaces, the initial topology on $X$ is the coarsest topology making them all continuous. Its subbasis consists of $f_i^{-1}(U)$ for open subsets $U\subseteq Y_i$. A map into $X$ is continuous exactly when all its composites with the $f_i$ are continuous.

The +2 oxidation state of tin. The actual aqueous species can depend on pH, hydrolysis and ligands; a bare ionic formula is an analytical shorthand.

The +4 oxidation state of tin. The actual aqueous species can depend on pH, hydrolysis and ligands; a bare ionic formula is an analytical shorthand.

A tin oxide whose hydrated forms may precipitate through hydrolysis of suitable tin(IV) solutions.