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With the uniform prior, the binomial likelihood gives the beta distribution posterior $\mathrm{Beta}(s+1,n-s+1)$. Under squared-error loss, the Bayes estimator is the posterior mean:

The production manager's prior gives posterior density proportional to $p^{s+1}(1-p{)}^{n-s}$. Minimizing $\mathbb{E}[(1-p)(\hat{p}-p{)}^2\mid X=s]$ gives the weighted posterior meanusing the supplied beta function. Cross multiplication shows ${\hat{p}}_P>{\hat{p}}_G$ exactly when $n>2s$, so it is greater unless $s\ge n/2$.

Let $\varphi$ be the velocity potential. The linearized deep-water equations are ${\nabla }^2\varphi =0$ for $z<0$, ${\eta }_t={\varphi }_z$ and ${\varphi }_t+g\eta =0$ at $z=0$, with zero normal velocity at the four side walls. The normal modes arewhere $m,n\ge 0$ are not both zero. The free-surface conditions give the dispersion relation for a deep-water gravity wave:

The initial displacement is the $(m,n)=(3,4)$ mode, so $k=5\pi /a$. Release from rest selects a cosine in time:Velocity decays as $e^{kz}$, so it falls to $1/e$ of its surface value at depth $a/(5\pi )$.

Remove a small disc about $x$ and apply Green second identity to $u(x_0)$ and $G_0(x;x_0)$. Since ${\nabla }_{x_0}^2{G}_0=\delta (x_0-x)$ and ${\nabla }^{2}u=-\rho$, the shrinking inner boundary contributes $u(x)$ and gives

For the unit disc, the method of images gives the Dirichlet Green functionwhich vanishes on the boundary by the hinted identity. The boundary representation with $u(1,{\theta }_0)=\delta ({\theta }_0-\alpha )$ evaluates the normal derivative of $G_D$ at ${\theta }_0=\alpha$, giving the Poisson kernel

Differentiate the parametrized surface. The coefficients of the first fundamental form simplify toIf $c={\sigma }_u\times {\sigma }_v$, then $|c{|}^2=E$. MoreoverThus the second-fundamental-form coefficients satisfy $M=1/\sqrt{E}$ and $N=0$. The Gaussian curvature is consequently

A self-adjoint operator $T$ satisfies $⟨Tx,y⟩=⟨x,Ty⟩$. If $Tx=\lambda x$, then $\lambda ⟨x,x⟩=⟨Tx,x⟩=⟨x,Tx⟩=\overline{\lambda }⟨x,x⟩$, so $\lambda \in \mathbb{R}$. An eigenvector exists over $\mathbb{C}$; its orthogonal complement is $T$-invariant. Induction on dimension produces an orthonormal basis of eigenvectors, proving the finite-dimensional spectral theorem.

A self-adjoint $T$ is positive definite exactly when every eigenvalue is positive, since $⟨Tx,x⟩={\sum }_i{\lambda }_i|x_i{|}^2$ in an eigenbasis. The quadratic forms of two positive-definite operators add, so their sum is positive definite.

The same eigenbasis shows that $\mathrm{spec}T\subset [a,b]$ exactly when $T-\lambda I$ is positive definite for every $\lambda <a$ and negative definite for every $\lambda >b$. Finally, if $v$ is a unit eigenvector of $\alpha +\beta$ with eigenvalue $\gamma$, the Rayleigh quotient bounds giveThus every eigenvalue of $\alpha +\beta$ lies in $[a+c,b+d]$.

A Hermitian operator $A$ satisfies $⟨\phi ,A\psi ⟩=⟨A\phi ,\psi ⟩$ on its operator domain. HereTwo integration by parts operations, together with the boundary conditions in the Hamiltonian domain, show that the kinetic term is Hermitian; multiplication by the real function $V$ is Hermitian as well. Thus $H$ is Hermitian.

For a time-independent operator $A$, the Schrödinger equation and its adjoint give the Ehrenfest theoremUsing the canonical commutation relation $[x,p]=i\hslash$,Consequently