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For $n$ points in a triangle and two fixed endpoint vertices, the longest convex chain length is the largest number of sampled points in a convex chain joining those endpoints. It is a certifiable function: a chain of length $a$ supplies a certificate of exactly $a$ point coordinates. Changing one point changes the maximum by at most one, since deleting that point from any optimal chain loses at most one vertex. For independent uniform points, its median is $O(n^{1/3})$ and its fluctuation scale about the median is at most $n^{1/6}$, as proved by convex-chain probability in a triangle and convex-chain median concentration.

For independent uniform points in $0\le y\le x\le 1$ with the two diagonal endpoints, the displayed formula is exact. Order the $k$ intermediate points so that both coordinates increase, and write the $k+1$ positive coordinate spacings as $(u_i)$ and $(v_i)$, with each sum equal to one. The two spacing vectors have uniform simplex measure, independently, and the full ordered-coordinate region has volume $1/(k!{)}^2$. Simultaneous permutation of the spacing pairs preserves that measure, so all $(k+1)!$ orders of the ratios $v_i/u_i$ have equal volume. Strictly increasing ratios characterize a convex chain and automatically place the intermediate points below the diagonal. Thus its ordered volume is $1/((k!{)}^2(k+1)!)$. There are $k!$ ways to assign labels, and uniform triangle density is $2^k$, giving the formula. Consequently the first moment method bounds the probability of any length-$k$ chain among $n$ points by $\left(\genfrac{}{}{0ex}{}{n}{k}\right)2^k/(k!(k+1)!)$.

For nonnegative integrable functions converging almost everywhere to an integrable $f$, convergence of their integrals implies L1 convergence. Indeed $\min (f_n,f)\to f$ almost everywhere and is dominated by $f$, so the dominated convergence theorem gives convergence of its integral. Use $|f_n-f|=f_n+f-2\min (f_n,f)$. This applies to probability densities, giving convergence in total variation distance.

For fixed $k$ independent random variables whose families have the same large deviation principle with good rate function $I$, the product large-deviation principle gives the sum of coordinate costs. Under the minimum map, one coordinate must equal $m$ and the others must be at least $m$, yielding the displayed rate function. If $I$ is convex with unique zero $\mu$, then $J(m)=I(m)$ for $m\le \mu$ and $J(m)=kI(m)$ for $m\ge \mu$.

For $a<b$, the contraction principle for large deviations under scalar clipping gives $K(z)=I(z)$ in $(a,b)$, the displayed endpoint costs, and infinite cost outside $[a,b]$. The endpoint costs can differ from $I(a)$ and $I(b)$ because clipping collects a whole input tail into a single output atom.

For a homologous stellar polytrope of index $3/2$, take $P=K{\rho }^{5/3}+\mathcal{R}\rho T/\mu$ and zero surface pressure. The stellar virial theorem, with $\Omega =-6G{M}^2/(7R)$, givesThe normalized density profile fixes $A$ independently of mass and radius at fixed composition. Writing ${\overline{T}}_M=\alpha /R-\delta /R^2$ shows that its maximum is ${\alpha }^2/(4\delta )=3\mu G^2{M}^{4/3}/(49\mathcal{R}A)$, at $R=7A{M}^{-1/3}/(3G)$. Below this radius contraction increases the fraction of support provided by electron degeneracy pressure and lowers the thermal temperature. A fixed normalized profile also fixes the ratio of central to mass-averaged temperature, giving a hydrogen-burning minimum mass.

For a sector of width $\varphi$ and an independent uniformly distributed central bearing, a specified bearing is covered with probability $\varphi /(2\pi )$. Two antipodal bearings cannot both be covered when $\varphi <\pi$. For $\varphi \ge \pi$, the allowable centre arcs overlap in two components of length $\varphi -\pi$, proving the displayed conditional probability. Averaging over the width's probability density function gives unconditional coverage probabilities.

A Q-matrix has nonnegative off-diagonal entries and row sums zero. Its diagonal entry is the negative total rate out of the state.

An Equilibrium distribution of a continuous-time Markov chain is a probability row vector $\pi$ invariant under its transition semigroup:It satisfies $\pi Q=0$ when the generator sums are defined. An invariant law is a property of the entire evolution, so formal generator equations must be interpreted with the process's nonexplosion and domain conditions.

For a finite irreducible chain, the equilibrium distribution is unique and every transition row converges to it. For a general finite chain, the limiting row from state $i$ is a mixture of the stationary distributions ${\pi }^C$ of its closed irreducible classes:The chain eventually enters one of these classes and then mixes within it. If every closed class is a single absorbing state, this reduces to absorption probabilities at those states. Merely having some absorbing states does not exclude other closed classes.

Holding times explain the link to an embedded jump chain. For a finite irreducible chain with strictly positive exit rates $q_i$ and jump-chain Stationary distribution $\alpha$, the time-stationary distribution isVisit frequency is multiplied by mean time $1/q_i$ per visit. In the countable setting, these time weights need a finite positive normalizer and a nonexplosive process before they define an equilibrium probability. For an alternating two-state jump chain, $\alpha =(1/2,1/2)$ and rates $a,b$ give $\pi =(b/(a+b),a/(a+b))$: slower holding rates increase time occupancy.

A continuous-time Markov chain with finite holding rates is nonexplosive when its jump times have no finite accumulation almost surely. It then remains in its specified state space at every finite time, rather than being lost to a cemetery state after infinitely many jumps. Bounded rates are sufficient for nonexplosion, but finite rates at individual states alone are not.

Consider states $C_i,W_i$ for integer levels $i\ge 0$ and strictly positive rates $\lambda ,\mu ,\alpha ,\beta$. From $C_i$ the chain moves to $C_{i+1}$ at rate $\lambda$ and to $W_i$ at rate $\alpha$. From $W_i$, $i\ge 1$, it moves to $W_{i-1}$ at rate $\mu$ or to $C_i$ at rate $\beta$. At the reflecting boundary $W_0$ there is only the switch to $C_0$; no negative level is allowed. All rates are bounded, so the chain is nonexplosive.

Let $h$ be the probability of hitting $W_0$, with $h_{W_0}=1$. First-step equations arewith the first equation also valid at $i=0$. Put $\vartheta =\mu (\lambda +\alpha )/[\lambda (\mu +\beta )]$. If $\mu \alpha <\lambda \beta$, then $\vartheta <1$ andSubstitution verifies both equations and the boundary. Eliminating the $C$ variables gives a second-order recurrence with modes $1,{\vartheta }^i$. The displayed decaying solution is the minimal nonnegative harmonic solution and hence the Hitting probability.

When $\mu \alpha \ge \lambda \beta$, the growing mode is excluded by boundedness; at equality the second mode is linear and is likewise excluded. The only bounded solution with the boundary value is one, proving almost-sure hitting. Since the specified reflected chain is irreducible, these cases classify it as recurrent or transient respectively. Specifying the boundary rule is essential for that recurrence statement; making $W_0$ absorbing would be a different chain.

For jump rates $q_{ij}=r_j$ when $i\ne j$, assume $r_j\ge 0$ and $0<R={\sum }_j{r}_j<\infty$. The positive-rate states form the unique Closed communicating class; every zero-rate state is an open singleton. The stationary law is ${\pi }_i=r_i/R$, with Detailed balance ${\pi }_i{q}_{ij}=r_i{r}_j/R$. When at least two rates are positive, the embedded jump-chain law is proportional to $r_i(R-r_i)$. With only one positive rate, that state is absorbing and the genuine sequence of jump times stops there; an extended discrete chain can use an absorbing self-loop.

A Markov jump process is a nonexplosive, piecewise-constant continuous-time Markov process. Transition rates $q(x,y)$ give exponential holding time of rate ${\sum }_{y\ne x}q(x,y)$ and next-state probabilities proportional to these rates. Being Markov relative to a specified filtration requires the conditional future law given that filtration to depend only on the present state.

// Target: probability-and-statistics.bigb

A continuous-time Markov chain is nonexplosive if it makes only finitely many jumps on every bounded time interval. A uniformly bounded total jump intensity permits domination by a Poisson process. More generally, if a population process stays in a finite random set of states on each finite interval and every individual state has finite exit rate, localizing to fixed finite sets gives the same conclusion. In an open migration process, external Poisson immigration bounds the total population while internal moves preserve it and exits reduce it, giving this localization.

A linear functional on a normed vector space is continuous exactly when its kernel is closed. The forward direction follows because zero is closed in the scalar field. Conversely, if $l\ne 0$, choose $v$ with $l(v)=1$ and write $M=\ker l$. Closedness gives $d=\mathrm{dist}(v,M)>0$. Since $x/l(x)-v\in M$ when $l(x)\ne 0$, $|l(x)|\le \Vert x\Vert /d$. The zero functional is continuous separately. Thus the criterion does not require completeness or the closed graph theorem.

For bounded self-adjoint $a_i,a$, no uniform bound on $\Vert a_i\Vert$ is needed. The resolvent identity gives $(a_i-i{)}^{-1}-(a-i{)}^{-1}=(a_i-i{)}^{-1}(a-a_i)(a-i{)}^{-1}$, and the first factor has norm at most one. Thus both resolvents at $i$ and $-i$ converge strongly. Products preserve strong convergence for uniformly bounded nets; uniform approximation by the algebra of these resolvent functions then gives the displayed statement for every continuous function vanishing at infinity.

For a Positive element of a C-star algebra $x$, the displayed improper integral converges in norm. Near zero its integrand has norm at most $t^{-1/2}$; near infinity it has norm at most $\Vert x\Vert t^{-3/2}$. The scalar identity follows by setting $t=s{u}^2$ for $s>0$ and using ${\int }_0^{\infty }2/(1+u^2)du=\pi$; for $s=0$ both sides vanish. The bounds are uniform on the compact spectrum of an element, so continuous functional calculus gives the operator identity.

For a normal C-star element $x$ in a nonunital algebra, take its spectrum of an element in the C-star unitization. Zero belongs to that spectrum. Continuous functions on this compact spectrum that vanish at zero form an ideal canonically identified with the displayed function algebra. Their values under the unital continuous functional calculus belong to the original algebra because the scalar quotient evaluates them at zero. Star-polynomial approximation with the constant term removed gives exactly the image $C^{\ast }(x)$.

The Takagi function is a continuous nowhere differentiable function. The triangular summands are continuous and the geometric tail is bounded uniformly. On dyadic intervals of length $2^{-N}$, all terms of index at least $N$ vanish at both endpoints, while every lower summand has slope $+1$ or $-1$. Each refinement therefore changes the secant slope by exactly one in absolute value. The secants straddling a differentiability point lemma rules out a derivative at every real point.

For $f:D\to \mathbb{R}$ and $a\in D$, continuity at $a$ is equivalent to the following condition: every sequence $(x_n)$ in $D$ with $x_n\to a$ satisfies $f(x_n)\to f(a)$. This is useful when a numerical method or a proof approaches a point through successively refined function inputs: applying a continuous function preserves the intended limit.

For the forward direction, choose $\delta$ from the epsilon-delta definition of continuity. A convergent sequence eventually lies in that input window, so its function outputs eventually lie in the requested output band. Conversely, if continuity fails, there is some ${\epsilon }_0>0$ for which every input window contains an allowed $x$ with $|f(x)-f(a)|\ge {\epsilon }_0$. Choosing such an $x_n$ within $1/n$ of $a$ gives $x_n\to a$ while $f(x_n)$ does not tend to $f(a)$. This converse relies on the countable shrinking windows available in metric spaces; a general topological space needs a different formulation.