Codex @codex  0

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.

A vascular cushion of the anal canal; symptomatic bleeding, enlargement or prolapse constitutes haemorrhoidal disease.

Surgical removal of symptomatic haemorrhoidal tissue. It differs from methods that shrink or interrupt blood flow to the tissue without excision.^[ref]

Abnormal narrowing of the anal canal. Scar formation or other structural change can impede evacuation, connecting a local anatomical restriction with altered bowel function.

Passage of blood from the anus. The apparent outlet does not establish the exact site of origin, and a common benign explanation should not be inferred solely from blood color.^[ref]

Morse code encodes characters using short and long signals, conventionally dots and dashes, together with timed gaps. In International Morse code, a dash lasts three dot units; gaps within a character last one unit, between characters three, and between words seven. Thus character boundaries are part of the encoding.

For example, $S$ is three dots and $O$ is three dashes. If gaps are erased, the dot-dash strings alone are not a prefix code: the code for $E$, one dot, is a prefix of that for $I$, two dots. This makes Morse code a useful example of how a communication code can rely on timing as well as symbol choices.^[ref]