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The Brownian transition semigroup identity proves that this positive density process has mean one for $t<1$. The backward heat equation and Itô's formula give $d{M}_t=M_t(y-B_t)(1-t{)}^{-1}d{B}_t$. On every fixed horizon $T<1$, Girsanov theorem under $dQ=M_{T}dP$ makes $B_t-{\int }_0^t(y-B_s)(1-s{)}^{-1}ds$ Brownian motion. Thus the original coordinate acquires the bridge drift coefficient toward $y$ at time one, without defining an equivalent endpoint density at the singular time.

Failure of boundedness: for every finite radius, some point lies outside the proposed bounding ball. A real-valued function is unbounded if its range is unbounded.

At a specified external pressure, the equilibrium temperature at which a pure liquid's saturation vapor pressure equals that external pressure. Bubble nucleation, hydrostatic head and capillary pressure can affect the observed onset of boiling without redefining the equilibrium property.

A relation on states of labelled transition systems matching every transition with a transition of the same label in either direction, with successor states again related. In particular, $\tau$ transitions are matched individually, rather than silently skipped. For CSP processes this preserves CSP traces, CSP stable failures and CSP divergences.

Use the zero-based geometric distribution, with $\mathbb{P}(N=k)=p(1-p{)}^k$ for $k\ge 0$ and $0<p<1$. After binomial thinning with retention probability $0\le r\le 1$, the retained count is again zero-based geometrically distributed, with the displayed success probability. Indeed, writing $q=1-p$, its probability generating function isFor $r=0$ this includes the degenerate success parameter $p^{\prime }=1$ and retained count zero.

A complete three-uniform hypergraph on four vertices has four edges. In the binomial random hypergraph $H^{(3)}(n,p)$, its count $X$ has mean $\mu =\left(\genfrac{}{}{0ex}{}{n}{4}\right)p^4$. If $np\to 0$, the first moment method gives $\mathbb{P}(X>0)\to 0$. Two distinct four-sets have independent clique indicators unless they share three vertices; in that case their combined edge set has seven triples. There are $O(n^5)$ such ordered pairs, so $\mathrm{Var}(X)\le \mu +O(n^5{p}^7)$. Consequently $\mathrm{Var}(X)/{\mu }^2=O((np{)}^{-4}+(n^{3}p{)}^{-1})\to 0$ if $np\to \infty$. The second moment method then gives $\mathbb{P}(X>0)\to 1$. These two implications establish the threshold function.

For a binomial distribution with parameters $n$ and $p$, the binomial theorem gives the displayed moment-generating function, finite for every real $t$. Products of these moment-generating functions show that the sum of independent binomial random variables with the same success probability again has a binomial distribution, with trial counts added.

A random variable with a binomial distribution.

An internal vertex that tests a condition and selects an outgoing branch. In a binary decision diagram, a decision node tests a Boolean variable and has one zero arrow and one one arrow.

A vertex that returns a final output rather than performing another test. In a binary decision diagram the two possible outputs are the Boolean constants zero and one.

A bilinear map between normed vector spaces is jointly continuous exactly when it satisfies the displayed product bound. Continuity at zero gives a bound on a product of balls; rescale each nonzero input separately to obtain the inequality. Conversely, expand the increment as $F(x+h,y+k)-F(x,y)=F(h,y)+F(x,k)+F(h,k)$ and apply the product bound to every term. Thus continuity at zero already implies joint continuity everywhere for a bilinear map.

Suppose two degree-$n$ projective plane curves meet at $n^2$ distinct points, and $nm$ of them lie on an irreducible degree-$m$ curve $R=0$, with $m<n$. Pick an extra point of $R=0$ outside the finite base locus. A nonzero linear combination $H=\lambda P+\mu Q$ vanishes at that point. If $R$ did not divide $H$, the Bézout theorem would bound their intersections by $nm$, contradicting the $nm$ original points and the extra point. Therefore $H=RS$ with $\mathrm{deg}S=n-m$. All remaining base points lie off $R=0$ and hence on $S=0$. This proves a residual incidence theorem by forcing one member of the curve pencil to acquire a prescribed component.

The Beth numbers ${\beth }_0={\aleph }_0$, ${\beth }_{n+1}=2^{{\beth }_n}$ form a strictly increasing countable sequence by Cantor's theorem. Their supremum ${\beth }_{\omega }$ has cofinality $\omega$ because no finite subset is cofinal. It exceeds $2^{{\aleph }_0}={\beth }_1$. It is also a strong limit cardinal: if $\rho <{\beth }_{\omega }$, some $n$ has $\rho <{\beth }_n$, so $2^{\rho }\le {\beth }_{n+1}<{\beth }_{\omega }$. Thus the singular cardinal example needs no Continuum hypothesis.

On a cyclic sequence of $n$ independent Bernoulli random variables with success probability $p$, count all runs of $k$ consecutive successes, allowing overlaps and taking $1\le k\le n/2$. The count has expected value $n{p}^k$. Its variance is the displayed expression: each starting indicator has variance $p^k(1-p^k)$; two starts at cyclic distance $j<k$ have joint probability $p^{k+j}$; nonoverlapping starts are independent. There are $2n$ ordered overlapping pairs at each such distance, because the two ranges do not meet when $k\le n/2$.

For $p\le 1/2$, summing a geometric series gives $\mathrm{Var}X\le n{p}^k(1+2p/(1-p))\le 3n{p}^k$. Markov's inequality therefore proves absence with high probability when $n{p}^k\to 0$, and Chebyshev's inequality proves existence and relative concentration when $n{p}^k\to \infty$. This remains valid for varying probabilities eventually at most one half.

For positive integers $i,k,r$, the floor function condition $i=\lfloor k\sqrt{r}\rfloor$ is equivalent to the displayed integer inequalities. Since $r\ge 1$, a witness satisfies $k\le i$. Finding the least $k\ge 1$ with $r{k}^2\ge i^2$ and checking the strict upper bound therefore decides membership by binary search. Scanning successive $i$ values with a persistent increasing witness index instead enumerates all values below a bound in linear time arithmetic-operation count. Exact integer comparison avoids floating-point errors near an integer boundary.

For a random sample $X_i\mid \theta \sim N(\theta ,{\sigma }^2)$ with known $\sigma >0$ and prior distribution $\theta \sim N(0,{\tau }^2)$ with $\tau >0$, compare $H_0:\theta <0$ to $H_1:\theta \ge 0$. The prior odds are one by symmetry. Normal-normal conjugacy gives a normal posterior distribution withIts negative and nonnegative masses are $\Phi (-z)$ and $\Phi (z)$, whereSince the Bayes factor is the posterior odds divided by the prior odds,Here $\Phi$ is the cumulative distribution function of the standard normal distribution. The ratio decreases with the sample mean. For any threshold $b>0$, the inequality $B_{01}<b$ is equivalent to $\Phi (z)>1/(1+b)$ and hence toFor $b=1/10$, the quantile is ${\Phi }^{-1}(10/11)\approx 1.33518$, and the equivalent posterior probability threshold for $H_0$ is $1/11$. The symmetry and proper normalization of the two hypothesis-conditional priors are essential to replacing the Bayes factor by the posterior odds directly.

The first barycentric subdivision of a delta-complex has distinct vertices within each simplex. Its vertices correspond to an increasing chain of original faces, and their strictly different dimensions prevent identifications within that chain. Distinct simplexes may still share the same global vertex set, as happens to the two halves of a subdivided loop edge.

For a delta-complex with distinct vertices within each simplex, a face inside a fixed simplex is uniquely determined by its vertex subset. In its barycentric subdivision, a vertex records an actual face, including its cell identity. A simplex records a chain of these faces; its vertex set recovers the unique order by dimension and each unique face inclusion. Thus no two simplexes have the same vertex set, and intersections are their common faces. Applying this to the first subdivision proves that the second subdivision is an abstract simplicial complex. The subdivision maps are affine on smaller simplexes and compatible on the glued faces, so they induce a homeomorphism of realizations.^[ref]