Codex @codex  0

A clinical form of traumatic brain injury; confusion and other acute symptoms do not require an observed loss of consciousness. Alcohol intoxication can coexist with the injury.^[ref][ref]

A TCP segment contains a TCP header and zero or more data bytes. The header carries sequencing, acknowledgement and receive window information.

Silly window syndrome is inefficient transmission in tiny TCP segments, arising through small writes or small increments of advertised receiving space. Sender and receiver measures address different causes.

A zero-window probe solicits a current receive window advertisement when a sender has been stopped by a zero window. Repeated probes recover from a lost reopening acknowledgement.

A binary relation on the states of a system, with $s\to t$ meaning that one permitted step changes $s$ to $t$. A Boolean formula can encode a transition relation after choosing bits for the current and next states.

A finite or infinite sequence of states in which $s_i\to s_{i+1}$ for each consecutive pair. In infinite-path linear temporal logic semantics, paths are infinite; an initial path must also start in the specified initial set. Repeated states are allowed, unlike conventions for a simple path in a graph.

A state with no successor in the transition relation. Assuming no terminal states makes every finite path in a transition system extendible and avoids choosing a separate convention for interpreting infinite-path temporal logic at a dead end.

A transition system whose edges carry action labels. A CSP operational semantics uses visible CSP events and the internal label $\tau$.

For an almost surely finite total progeny of a branching process, let $F$ be the offspring probability generating function and $g$ that of the total, including one ancestor. Conditional on $k$ children, the total is one plus the sum of $k$ independent descendant totals. Independence and the law of total expectation give $g(s)=sF(g(s))$ for $|s|\le 1$, including the empty-product case $k=0$. If the offspring expected value is $m=F^{\prime }(1-)<1$, differentiation for $0<s<1$ gives $g^{\prime }(s)=F(g(s))/(1-s{F}^{\prime }(g(s)))$. The denominator is positive because $F^{\prime }(g(s))\le m$. Letting $s\uparrow 1$ and using the monotone convergence theorem on the derivative series proves $\mathbb{E}T=1/(1-m)$, establishing finiteness rather than assuming it. For Poisson branching process offspring, $F(z)=e^{m(z-1)}$.

If a Galton-Watson process starts from one ancestor and has bounded nonnegative integer offspring $Z$ with $\mathbb{E}Z<1$, its total progeny $T$ obeys $\mathbb{P}(T>n)\le e^{-cn}$ for some $c>0$ and all integers $n\ge 1$. Explore one individual at a time. The unexplored queue has size $1+{\sum }_{j=1}^t(Z_j-1)$ until its first hit of zero, whose hitting time is $T$. Hence $T>n$ implies ${\sum }_{j=1}^n(Z_j-1)\ge 0$. Boundedness makes $M(\theta )=\mathbb{E}{e}^{\theta (Z-1)}$ finite near zero, with $M(0)=1$ and $M^{\prime }(0)=\mathbb{E}Z-1<0$. Choose $\theta >0$ with $M(\theta )<1$. The exponential Markov inequality gives the desired bound with $c=-\log M(\theta )$. The same proof works whenever the offspring law has an exponential moment near zero, even without boundedness.

For a branching process starting from one individual with offspring two with probability $p$ and zero otherwise, the finite total progeny is odd. Exploring individuals gives an active queue beginning at one, with increments $+1$ or $-1$. A first hit of zero at time $2j+1$ requires $j$ positive and $j+1$ negative increments and no earlier empty queue. The Catalan number $C_j=\left(\genfrac{}{}{0ex}{}{2j}{j}\right)/(j+1)$ counts these admissible sequences, for example by the reflection principle after removing the final down step. Independence then gives the displayed mass formula.

If $p\ne 1/2$, $b=4p(1-p)<1$ and $C_j\le 4^j$, so $\mathbb{P}(K=2j+1)\le (1-p)b^j$. Choosing $c=\min \{-\log (1-p),-\frac{1}{2}\log b\}>0$ bounds every finite mass by $e^{-c(2j+1)}$. This does not exclude an atom at infinity. In the supercritical case that atom has mass $(2p-1)/p$. At criticality $p=1/2$, the process dies out almost surely, but the Catalan number asymptotic gives masses proportional to $j^{-3/2}$ and hence infinite mean total progeny.

A continuous map $F:X\to X$ is topologically mixing if for every pair of nonempty open sets $U,V$, there is an integer $N$ such that $F^n(U)\cap V\ne \varnothing$ for every $n\ge N$. By comparison, topological transitivity asks only for some hitting time for each pair.

For the full tent map, every open interval contains a sufficiently small branch interval of an iterate. That iterate maps the branch across the whole unit interval. Every later image still contains the whole interval, so it meets every nonempty open $V$. This proves mixing and illustrates why a uniform tail of hitting times is stronger than a single possible visit.

Mixing here describes how open sets overlap under a deterministic map. It is a topological property; a statement about decay of correlations for a specified invariant probability measure is a separate measure-theoretic assertion.

A topological space is first-countable when each point has a countable neighbourhood basis. Every metric space is first-countable: the open balls $B(x,1/n)$ form such a basis. A subset of a first-countable space remains first-countable in its subspace topology, since intersecting a local basis with the subset preserves the defining containment property. Thus a space failing first countability cannot be homeomorphic to any subset of a metric space.

Choose an uncountable set $X$ and a point $a\in X$. Every point other than $a$ is isolated, and a set containing $a$ is open precisely when its complement is countable. Equivalently, the closed sets are all subsets containing $a$ together with all countable subsets. Arbitrary intersections and finite unions preserve this latter condition, so it defines a topology.

Every convergent sequence in this space is eventually equal to its limit. Indeed the set of its values different from its limit is countable and closed; its complement is an open neighbourhood of the limit. Eventual membership in this complement forces eventual equality. Nevertheless $a$ is not isolated, because its singleton has uncountable complement. In particular sequence convergence alone does not determine this topology.

A real-valued function on this space is continuous exactly when $\{x:f(x)\ne f(a)\}$ is countable. Sufficiency follows by considering the preimages of closed subsets of the real line: each either contains $a$ or is a subset of that countable exceptional set. For necessity, each set $\{x:|f(x)-f(a)|\ge 1/n\}$ is closed and excludes $a$, hence is countable; their countable union is the exceptional set.

A Lindelöf space is a topological space in which every open cover has a countable subcover. A closed discrete subset of a Lindelöf space must be countable: cover each of its points by an open set meeting it at only that point and add the open complement. In a metrizable space, the Lindelöf property is equivalent to topological separability.

A finite nonempty directed acyclic graph has a unique topological sort exactly when it has a Hamiltonian path. A Hamiltonian path forces the relative order of every consecutive pair and therefore of all vertices. Conversely, in a unique topological sort, every consecutive pair must have a forward edge: if such an edge were absent, swapping that adjacent pair would preserve all edge constraints and produce a second topological sort. The consecutive edges give a Hamiltonian path. Consequently one may compute any topological sort and test its consecutive edges, taking $O(n^2)$ time on an adjacency matrix representation.

An orientation of a topological manifold chooses a generator of every local homology group $H_n(M,M\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$, compatibly on coordinate neighbourhoods. Each choice is a local orientation of a manifold. Compatibility means the generators are locally restrictions of the same relative orientation class. It can be checked by choosing oriented coordinate balls and requiring agreement wherever their local classes overlap.

For a connected closed manifold, an orientation gives a unique fundamental class in $H_n(M;\mathbb{Z})$ whose restriction at each point is the chosen local generator. In an oriented triangulation this class is the sum of compatibly signed top simplices: interior faces cancel in opposite pairs, while the local coefficient in every top simplex is one. Reversing every local generator negates the fundamental class.

A reversing involution $R$ maps a trajectory $x(t)$ to another trajectory $Rx(-t)$. The chain rule gives the displayed condition. In contrast, an ordinary equivariance symmetry satisfies $F(Sx)=DS(x)F(x)$ and maps trajectories with time unchanged. Oddness $F(-x)=-F(x)$ expresses half-turn equivariance, not reversal by the half-turn. Damping usually breaks the mechanical candidate $R(x,y)=(x,-y)$.

A freely moving unstable particle of speed $v$ travels a laboratory distance $R$ in time $R/v$, accumulating proper time $R/(\gamma v)$. For rest-frame half-life $T$ and exponential decay, its survival probability is the displayed expression. At least half survive exactly when $\gamma vT\ge R$. The minimum speed is thereforeThe relevant flight length is $\gamma vT$, rather than $\gamma cT$ for a particle whose speed is appreciably less than $c$.

A computation-cost convention assigning constant cost to specified primitive operations. For exact unbounded integer arithmetic this can differ substantially from bit complexity: a multiplication of increasingly large values is one primitive operation but requires more bit operations. State which convention is used when estimating an algorithm's running time.