Parts per million, written ppm, expresses a dimensionless ratio multiplied by $10^6$. A mass fraction of $3\times 10^{-6}$ is 3 ppm by mass; an amount fraction of the same numerical value is 3 ppm by amount of substance. The two compare different quantities and need not describe the same mixture composition.

A concentration written in milligrams per litre is a mass concentration, not automatically a parts-per-million ratio. Its approximate numerical equality to ppm by mass for a dilute aqueous sample uses an additional mass density approximation of $1kg/L$. Stating the basis prevents a unit conversion from silently becoming an assumption about the sample.

A computer file whose logically zero-filled regions need not occupy physical file data blocks. The file offset and byte length describe the logical content independently of the blocks actually allocated.

The left and right identity laws and the associativity law for a computational monad. They justify regrouping sequencing while preserving the order and multiplicity of effects.

A construction taking a computational monad $M$ to another computational monad $TM$, normally with a lift from $MA$ to $TMA$. The order of monad transformers can affect which information survives an exception or a control transfer.

A computation represented by a function from an initial program state to a result and a final program state. Its sequencing passes each output program state to the next computation.

The computational monad whose result is the identity function and whose sequencing is ordinary application. It adds no effects and provides a starting point for a stack of monad transformers.

A computational monad for continuation-passing style. Result invokes the supplied continuation immediately; sequencing replaces it with a continuation that runs the next computation.

Give a nonhalting Turing machine computation halting time infinity. The formal language of pairs satisfying the displayed inequality is a co-recursively enumerable language but is not a recursively enumerable language. A violation has a finite witness: $N_1$ has halted on some input after $t$ moves while $N_2$ has not halted by that time. For the other direction, reduce non-halting by making $N_1$ simulate a fixed computation on every input and making $N_2$ run forever. The pair satisfies domination exactly when the fixed computation never halts. The convention $\infty \ge \infty$ is essential to this many-one reduction.

It is undecidable whether a Turing machine recognizes a formal language containing exactly one word. Reduce the halting problem by a Turing machine that rejects every nonempty input and, on the empty word, simulates a fixed computation and accepts if that computation halts. Its recognized formal language is a singleton exactly when the fixed computation halts.

A continuous choice of complex logarithm on a negative real interval has a fixed integer $k$ in the displayed expression. The choice with argument $\pi$ is the upper-bank boundary value of the principal logarithm; the usual analytic principal-branch domain itself excludes the negative axis. Consequently, integrating a real weight times that logarithm adds $(2k+1)i\pi$ times the integral of the weight to the real logarithmic integral. The branch choice must be stated.

For $\dot{n}=n(1-n-m/2)$, $\dot{m}=qm(1-m-n/2)$, choose $k=2/3$ and weights $1,1/q$. Then $\dot{V}=-(n-k{)}^2-(m-k{)}^2-(n-k)(m-k)$, strictly negative away from coexistence. The function diverges at the positive-quadrant boundary and at infinity, keeping trajectories in compact interior level sets. This proves global convergence for positive initial states, while boundary-axis data retain their single-species dynamics.

This generation-based compartmental model replenishes the susceptible compartment by $B>0$ per step and transfers $\alpha S_t{I}_t$ individuals into the next infectious generation. Members of the current infectious compartment leave it after one step. In scaled variables $x=\alpha S$, $y=\alpha I$, $r=\alpha B$, the update is $x^{\prime }=x+r-xy$, $y^{\prime }=xy$.

Its only fixed point is $(S_{\ast },I_{\ast })=(1/\alpha ,B)$. There is no finite disease-free fixed point, since susceptible recruitment continues when $I=0$. The Jacobian matrix at the fixed point is $M=\left(\begin{array}{cc}1-r& -1\\ r& 1\end{array}\right)$, with determinant one and trace $2-r$. For $0<r<4$, its two distinct eigenvalues have modulus one, so the linearized iterates are bounded and oscillatory. At $r=4$, the repeated eigenvalue $-1$ has a nontrivial Jordan block, making generic linearized iterates grow linearly. For $r>4$, one eigenvalue has modulus greater than one.

Thus $0<r<4$ gives neutral linear stability, not asymptotic stability. Unit-modulus multipliers do not determine nonlinear stability. Recruitment can restore susceptibility above $1/\alpha$ after a wave, allowing repeated infection growth; a closed SIR model without replenishment has a different susceptible balance.

A compartment is one of the state categories in a compartmental model. Its state variable measures the number or amount in that category, and transfer terms record exchanges between compartments.

After a balanced comparison tournament, the second-largest element is the maximum among the winner's $m$ direct losers. The third-largest must have lost directly to the winner or to that second-largest element: no smaller element could defeat it. Excluding the known top two leaves at most $(m-1)+(m-1)$ candidates, because the second-largest won at most $m-1$ earlier comparisons. The three stages use at most $(n-1)+(m-1)+(2m-3)$ key comparisons, proving the displayed bound. Defeat records must retain their earlier subtrees; keeping only the overall winner's direct-loser values discards information used by the last stage.

A single comparison returning one of three outcomes: less than, equal to, or greater than. A comparison model may count this as one operation even when a programming language expresses the outcomes using multiple Boolean tests. Binary search on ordered keys uses one such comparison per inspected position.

A component specified by possible CSP events and successor behaviours. $\mathrm{STOP}$ has no transitions. CSP prefixing, CSP choice and CSP parallel composition construct larger components.

A representation of a CSP process by specified kinds of observable behaviour. The CSP traces model, CSP stable-failures model and CSP failures-divergences model record progressively more information about refusals and internal activity.

For a suitably guarded recursion, two solutions of the same defining equation denote the same CSP process. There is also a mutually recursive version, comparing tuples of solutions.

Rules deriving the labelled transition system of a composite CSP process from transitions of its components.

A chemical substance capable of sustaining combustion with an appropriate oxidizing agent under suitable conditions. Combustibility depends on the conditions and reaction partner.