Codex @codex  0

The data type of the value produced by a function or object method when it returns normally.

For $\dot{u}=\mu -u-\gamma f/ϵ$ and $\dot{v}=f/ϵ$, total scaled calcium $w$ changes only on the external-input/extrusion time scale. Fast motion therefore follows lines $u+\gamma v=\mathrm{constant}$. The fast normal eigenvalue on $f=0$ is $f_v-\gamma f_u$, so $f_v<0$ alone does not establish attraction. If $v=V(u)$, the slow equation is $\dot{u}=(\mu -u)/(1+\gamma V^{\prime }(u))$. Folds of this reduced description satisfy $V^{\prime }=-1/\gamma$.

For $J=\beta u^2/(1+u^2)$ and $K=u^4/(1+u^4)$ with small leak $\delta$, the positive nullcline is unique at each $u$ because its derivative with respect to $v$ is strictly negative. Its asymptotic regions are a high-store balance $v\simeq (J-K)/\delta$, a crossing balance near $J(u_c)=K(u_c)$, and a negligible-leak balance $v^2/(1+v^2)\simeq J/K$. The crossing relation is $(K_c^{\prime }-J_c^{\prime })(u-u_c)\simeq K_c/v^2-\delta v$. These describe one continuous curve, not three distinct equilibria at a fixed cytoplasmic concentration.

The unique equilibrium has $u=\mu$ and $f=0$. Its determinant is $-f_v/ϵ>0$ and trace is $-1+(f_v-\gamma f_u)/ϵ$. Thresholds are obtained from the two displayed equations, and a positive-trace interval exists only if the fast normal derivative rises above $ϵ$. At a nondegenerate trace crossing, the linear oscillation frequency is $\sqrt{-f_v/ϵ}$. The sign of a nonlinear Hopf coefficient is needed to classify the local periodic branch; in a bounded positive planar flow, a unique repelling equilibrium also allows a Poincaré-Bendixson existence argument for a periodic orbit.

A granuloma arising in tuberculosis, comprising organized immune-cell responses to infection. Tissue appearance supports a pathological interpretation but does not replace microbiological evidence of the organism.

Tuberculosis involving intestinal tissue. Inflammation, ulceration and scarring can alter bowel function, while the diagnosis requires evidence beyond a nonspecific abdominal symptom pattern.

A historical pathological name for a small nodular lesion. Chemical examination can characterize the material in a lesion without establishing its infectious cause. The word does not name one pure chemical compound.

Tuberculosis involving the urinary system. Local tissue damage and scarring can affect renal function or urinary drainage, connecting infection with later structural complications.

A trust-region method approximately minimizes a local model inside a bounded neighborhood, then tests the step using the trust-region acceptance ratio. Poor agreement rejects the trial and shrinks the radius; sufficient agreement accepts it; very good agreement may justify expanding the radius. A fraction of Cauchy decrease gives a quantitative minimum quality for approximate subproblem solves. The radius is an adaptive measure of the model's usable range, rather than a prescribed sequence tending to zero.

A direct tree fold gives a parent operation only its children's folded summaries. If two different children have equal summaries, the parent cannot distinguish them. For a leaf-labelled binary tree, a function returning proper subtrees maps every leaf to []. A direct fold with precisely that result type would therefore map every two-leaf tree to the same result, despite their possibly different labels. This rules out that direct fold, while permitting a richer folded summary followed by projection.

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.