Assume the Generalized continuum hypothesis. For necessity, König theorem for cardinal numbers gives ${\rho }^{\mathrm{cf}(\rho )}>\rho$ for every infinite cardinal. But $\lambda ={\kappa }^{\mu }$ implies ${\lambda }^{\mu }=\lambda$ by the currying law for cardinal exponentiation, so $\mathrm{cf}(\lambda )>\mu$. This direction uses only ZFC. Conversely, uncountable cofinality bounds the range of every countable sequence in $\lambda$. Under GCH, each smaller infinite cardinal $\nu$ satisfies ${\nu }^{{\aleph }_0}\le 2^{\nu }={\nu }^{+}\le \lambda$. There are only $\lambda$ ordinal bounds to consider, giving ${\lambda }^{{\aleph }_0}=\lambda$ and the representation with base $\lambda$ and exponent ${\aleph }_0$.

For each natural number $n$, the family $[A{]}^n$ of all $n$-element subsets is nonempty when $A$ is not finite. These families are pairwise distinct elements of $\mathcal{P}(\mathcal{P}(A))$, giving an explicit injection of the natural numbers. No representative from each family is selected. A power set cannot be countably infinite in ZF: if $\mathcal{P}(B)$ were countably infinite, singleton injection would make $B$ an infinite subset of a countable set and hence countably infinite; Cantor's theorem would then contradict countability of its power set. Apply this to $B=\mathcal{P}(A)$ to make the displayed lower bound strict.

The candidate key selected as the principal identifier for database tuples of a database relation.

A C-star isomorphism is a bijective C-star homomorphism: a complex-linear multiplicative map preserving the involution. Between unital C-star algebras it automatically preserves the unit. It preserves the Banach algebra spectrum, since it preserves invertibility in both directions. Applying the C-star identity and the spectral norm formula to $a^{\ast }a$ gives $\Vert \Phi (a){\Vert }^2=\Vert \Phi (a^{\ast }a)\Vert =\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$, so it is automatically isometric.

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 ordered binary decision diagram is a binary decision diagram using one fixed variable ordering of a binary decision diagram along every root-to-terminal branch. Tests may be omitted, but no branch may test a later variable before an earlier one. For the order $x<y<z$, a branch testing $x,z$ is permitted; a branch testing $z,x$ is not. Sharing equal residual Boolean functions can compress the diagram. Applying the reduction rules gives a reduced ordered binary decision diagram, canonical for the chosen order; a different order can have a dramatically different size.

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.