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A branch point of a multivalued analytic function is a point around which analytic continuation along a closed loop can return a different value. A branch cut is a curve or union of curves removed from the domain so that loops causing this monodromy are excluded and one single-valued branch can be chosen.

For $f\in C[0,1]$, define its Bernstein polynomialIf $X_{n,x}\sim \mathrm{Bin}(n,x)$, thenLet $M=\Vert f{\Vert }_{\infty }$. By uniform continuity, for any $\epsilon >0$ choose $\delta >0$ such that $|f(u)-f(v)|<\epsilon /2$ whenever $|u-v|<\delta$. Splitting the expectation according to that event and using Chebyshev's inequality gives, uniformly in $x$,For sufficiently large $n$ this is below $\epsilon$. Thus $B_{n}f\to f$ uniformly on $[0,1]$. An affine change of variable treats every compact interval, proving the Weierstrass approximation theorem.

The no-cloning theorem for two pure states states that no single unitary operator can clone two distinct nonorthogonal states. Suppose, to the contrary, thatPreservation of the inner product by $U$ givesWriting $z=⟨{\alpha }_0|{\alpha }_1⟩$, this requires $z=0$ or $z=1$. The first case is orthogonality; the second means that the normalized states represent the same ray. Both contradict the assumptions. Therefore distinct nonorthogonal quantum states cannot be cloned by a unitary process.

For $|\psi ⟩=a|0⟩+b|1⟩$, the Controlled-NOT gate givesA cloned product would instead beEquality up to a physically irrelevant global phase requires $ab=0$. ThusFor a genuine superposition it creates quantum entanglement instead of two independent copies.

Start from the product of two Gamma function integrals:Use the sum-and-ratio substitution for gamma integralswhose Jacobian is $r$. Absolute convergence for $\mathrm{Re}p,\mathrm{Re}q>0$ justifies the change of variables and separates the integral:Therefore

In the integral defining the Beta function, put $t=(1+x)/2$ and then $u=x^2$. Symmetry givesThis proves the identity for $\mathrm{Re}z>0$. Both sides extend by analytic continuation to the same meromorphic function, equivalently by the beta-gamma identity and the Gamma duplication formula. The identity holds meromorphically for every $z\in \mathbb{C}$; both sides are finite for $z\notin \{0,-1,-2,\dots \}$ and have matching poles at the excluded points.

The deterministic equation has solution $n(t)=n(0)e^{(B-D)t}$ and is identical in form to the equation for the stochastic expected value. It therefore identifies only the net rateThe variance equation also contains the total event rate $s=B+D$. Once $r$ and $s$ are inferred,The deterministic trajectory cannot distinguish frequent births and deaths that nearly cancel from infrequent events with the same net growth, whereas fluctuations can.

Multiply the birth-death master equation by $n$ and sum. Index shifts giveFor the second moment, a birth changes $n^2$ by $2n+1$ and a death changes it by $-2n+1$, soSince ${\sigma }^2=\mathbb{E}[N^2]-{\mu }^2$,Together these are the moment equations of a linear birth-death process.

With $p_{-1}=0$, the birth-death master equation for birth rate $b_n=Bn$ and death rate $d_n=Dn$ is

Take a context-free grammar for $L$ with start symbol $S$, rename nonterminals if necessary, and add a fresh start symbol $S_0$ withEvery derivation concatenates finitely many words of $L$, and every finite concatenation can be derived. The resulting grammar generates $L^{\ast }$, proving closure of context-free languages under Kleene star.