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For birth rates $b_i$ and death rates $d_i$, a reversible measure satisfiesThus ${\pi }_{i+1}/{\pi }_i=b_i/d_{i+1}$, and it is an invariant distribution when the resulting measure is summable and the chain is nonexplosive.

For individual birth and death rates $\lambda ,\mu$ and one initial individual, the finite-time extinction probability satisfiesWhen $\lambda \ne \mu$,

The Yule process is the pure birth process with rate $n\lambda$ in population state $n$. Starting from one individual, it is nonexplosive because the independent holding times have divergent total almost surely.

On a finite measure space, the convergence in the Birkhoff ergodic theorem also holds in $L^1$. Prove it first for bounded truncations by the dominated convergence theorem, then use the $L^1$ contraction $\Vert A_{n}h{\Vert }_1\le \Vert h{\Vert }_1$ and Fatou lemma to remove the truncation.

The surface shear stress isThe torque exerted by the fluid on the disk per unit axial length is therefore

Let a discrete-time chain move up with probability $q$ and down with probability $p=1-q$ away from zero, while at zero it moves up with probability $q$ and stays with probability $p$. It is transient for $q>p$, null recurrent for $q=p$, and positive recurrent for $q<p$. In the positive-recurrent case its invariant distribution isand aperiodicity from the self-loop at zero implies convergence to this distribution.

For $s,t>0$, put $r=s+t$ and $u=t/(s+t)$. Then $(s,t)=(r(1-u),ru)$ and $dsdt=rdrdu$, separating radial gamma and ratio beta integrals.

At an interior maximum-likelihood fit of a logistic-regression model containing an intercept, the intercept score function isConsequently the sum of the fitted probabilities equals the number of observed successes exactly.

For $m=\mathbb{E}N$ and $v=\mathrm{var}N$, the Markov jump-process generator applied to $n$ and $n^2$ givesConsequently

The barycentric subdivision of a tetrahedron has f-vectorIts two-skeleton therefore has Euler characteristic $15-50+60=25$ and homology $H_0\cong \mathbb{Z}$, $H_1=0$, and $H_2\cong {\mathbb{Z}}^{24}$.

For $0<r<d$,Consequently

Let $f\in C^{\infty }(\mathbb{R})$ and suppose that for every $x$ there is an $n=n(x)\ge 0$ such that $f^{(n)}(x)=0$. Then $f$ is a polynomial.

In $C([0,1])$ with the uniform norm, the continuous functions that are monotone on some interval of positive length form a meagre set. Indeed, it is enough to use intervals with rational endpoints. For each such interval, the nondecreasing and nonincreasing functions form closed sets with empty interior: a sufficiently small local triangular perturbation breaks the relevant inequality. The Baire category theorem therefore shows that a dense set of continuous functions is monotone on no nontrivial interval.

A meagre, or first-category, set is a countable union of nowhere dense sets.

A subset $A$ of a topological space is nowhere dense when the interior of its closure is empty. Equivalently, every nonempty open set contains a nonempty open subset disjoint from $A$.

Every nonempty countable complete metric space has an isolated point. Otherwise it is the countable union of its singleton subsets, each of which is nowhere dense, contradicting the Baire category theorem.

If such a space has infinitely many points, it has infinitely many isolated points. Indeed, after deleting any finite collection of isolated points, the remaining set is closed, complete, countable, and nonempty. It therefore has a point isolated in the remaining space; because the deleted set is finite, that point is also isolated in the original space.

Let $S$ be a nonempty closed, convex, symmetric subset of a Banach space and suppose ${\bigcup }_{n\ge 1}nS=X$. Baire's theorem puts a ball $B(x,r)$ inside some $nS$. Symmetry also puts $B(-x,r)$ there, and convexity puts the midpoint of $x+z$ and $-x+z$, namely $z$, in $nS$ whenever $\Vert z\Vert <r$. Hence $B(0,r/n)\subset S$.

A Baire space is a topological space in which every countable intersection of open dense sets is dense, equivalently one in which no nonempty open set is a countable union of nowhere-dense sets. Every compact Hausdorff space and every complete metric space is a Baire space.

Magnetocentrifugal acceleration occurs when matter tied to a rotating inclined magnetic field moves outward under the effective centrifugal force. For a Keplerian disc, cold launching is possible when the field tilts by more than $30^{\circ }$ from the vertical.

An Alfvén surface is where the poloidal flow speed equals the poloidal Alfvén speed. Smooth passage through it imposes a regularity condition relating field-line angular velocity and angular momentum.