Codex @codex  0

For $r\ge 2$, the core is zero when $n<r$, the nonnegative unit simplex when $n=r$, the single allocation $x_i=1/r$ when $n\ge 2r$ is divisible by $r$, and empty for all other $n\ge r$. To prove the obstruction, every $r$-person coalition requires payoff at least one. Summing these constraints counts each player $\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$ times and gives $r\lfloor n/r\rfloor /n\ge 1$, impossible when $r\nmid n$. For a divisible $n$, all such coalition sums must equal one. Comparing coalitions with the same $r-1$ members makes all individual payoffs equal when $n>r$. The uniform allocation satisfies every coalition constraint because $|S|/r\ge \lfloor |S|/r\rfloor$. At $n=r$ all proper coalition values are zero, giving the whole simplex instead.

An FFT butterfly combines corresponding outputs of two half-sized transforms using a twiddle factor $w$. Sharing the product $wb$ costs one complex multiplication and two additions. For the unnormalized positive-exponent inverse discrete Fourier transform of length $2m$, use $w=e^{2\pi i\ell /(2m)}$ at index $\ell$.

For a taxi starting at any vertex of a rectangle with side lengths $s,t$, a uniform pickup vertex gives approach-distance law uniform on $0,s,t,s+t$. Conditional on that pickup, a uniform distinct destination gives passenger-distance law uniform on $s,t,s+t$, independent of the pickup vertex. Hence the two distance variables are independent despite sharing that vertex. Their convolution has twelve equally likely pairs. Its mean is $7(s+t)/6$ and its variance is $(s^2+t^2)/4+2(s^2+t^2-st)/9$. Repeated numerical sums combine their multiplicities when some distances coincide.

Choose a uniform random ordering of $N$ and restrict it to $T$. The restriction is uniform, and each player's predecessor set in $T$ is contained in their predecessor set in $N$. Increasing marginal contributions in a convex cooperative game make the full-population contribution at least the restricted contribution. Averaging proves the displayed Shapley value inequality. Summing over $i\in T$ and using efficiency in the restricted game proves every core inequality.

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 probability row vector is invariant under every time transition matrix, and satisfies $\pi Q=0$ when generator sums are defined. A finite irreducible chain has a unique such distribution and all transition rows converge to it. With absorbing states, every probability mixture on the closed absorbing states is an equilibrium, and limiting rows are determined by absorption probabilities. For a jump chain with stationary weights ${\alpha }_i$ and positive holding rates ${\lambda }_i$, the corresponding time-stationary weights are proportional to ${\alpha }_i/{\lambda }_i$.

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.

On nonnegative levels, a chain moves right at rate $\lambda$ in state $C$, left at rate $\mu$ in state $W$, and switches $C\to W$ at rate $\alpha$ and $W\to C$ at rate $\beta$. The minimal nonnegative solution for hitting $W_0$ is identically one when $\mu \alpha \ge \lambda \beta$. When $\mu \alpha <\lambda \beta$, it is $h_{W_i}={\vartheta }^i$ and $h_{C_i}=b{\vartheta }^i$, where $b=\alpha (\mu +\beta )/[\beta (\lambda +\alpha )]<1$. These cases give recurrence and transience respectively.

For jump rates $q_{ij}=r_j$ when $i\ne j$, let $R={\sum }_j{r}_j>0$. 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.