A semiconductor laser with a cavity normal to the wafer surface and emission through that surface.

The nonuniform spatial distribution of galaxies, quantified for example by a two-point correlation function or a matter power spectrum related through galaxy bias. Survey geometry and selection must be separated from physical clustering.

For a free ion with one $f$ electron in its weak-field ground multiplet, $L=3,S=1/2,J=5/2$ and $g_J=6/7$. The six magnetic substates have adjacent spacing $g_J{\mu }_{B}B$ and full spread $5g_J{\mu }_{B}B$. A crystal field can change the applicable effective low-energy description.

Disorder turns ideal sharp Landau levels into bands of states. In the integer quantum Hall effect, localized tails and extended states near level centres play different transport roles; a mobility gap need not be an empty spectral gap.

The two boundaries prefer the same block. Parallel lamellae then tend to favour matching terminations and integer-period commensurability.

The two boundaries prefer opposite blocks. In a simple parallel lamellar stack, opposite terminations favour half-integer repeat thicknesses.

A boundary has approximately equal interfacial free energies against the two blocks. One neutral boundary does not remove selectivity at the other boundary.

A constraint that depends explicitly on time can exert a normal force that does actual work, even when its force does no work on an allowed instantaneous virtual displacement. The surface motion gives the constrained particle a normal component of actual velocity.

A uniformly charged circular ring hasHere $k$ is the spring constant for extension of its full circumference and the symmetric gauge is used. The canonical angular momentum $J=m{r}^2\dot{\varphi }+qB{r}^2/2$ is conserved. Eliminating the angle gives $m\ddot{r}=J^2/(m{r}^3)-q^2{B}^{2}r/(4m)-4{\pi }^{2}k(r-r_0)$. This model neglects electric self-interaction and noncircular deformation.

A Lagrangian is degenerate when its Hessian with respect to velocities is singular. The momentum–velocity relation then cannot be inverted for every velocity. Some Euler–Lagrange equations can become algebraic constraints rather than independent second-order equations. A massless limit can produce this behaviour; disappearance of an equation at quadratic order does not establish an unconstrained mode of the exact nonlinear model.

Consider a cylinder of radius $r$, height $h\ge 0$, and a hemispherical cap of the same radius, enclosing fixed volume $V$. Let the cylindrical wall cost $w>0$ per area and the cap cost $\alpha w$ per area, with no floor cost. Then $V=\pi r^{2}h+2\pi r^3/3$ and $C=2\pi wrh+2\pi \alpha w{r}^2$. Eliminating $h$ gives $C(r)=2wV/r+2\pi w(\alpha -2/3)r^2$, on $0<r\le (3V/(2\pi ){)}^{1/3}$.

For $\alpha >1$, its interior minimum satisfies the displayed formulas; the positive second derivative proves uniqueness. For $0<\alpha \le 1$, cost decreases throughout the feasible interval up to its endpoint, so the minimum has $h=0$ and is a hemisphere alone. At cap-to-wall cost ratio two, the optimal cylinder height is its diameter, $h=2r$. This boundary check is part of the constrained optimization, since a formal critical point can otherwise require negative height.

For a closed right circular cylinder with radius $r>0$, height $h>0$, and total surface area $A=2\pi r(r+h)$, inscription in a sphere of radius $R$ gives $r^2+h^2/4=R^2$. The Lagrange multiplier equations imply $h^2+2rh-4r^2=0$, so the positive aspect ratio is $h/r=\sqrt{5}-1$. The dimensions are $r^2=R^2(5+\sqrt{5})/10$ and $h^2=2R^2(5-\sqrt{5})/5$.

There is also an eigenvalue proof of global maximality. Set $w=h/2$, so the constraint is $r^2+w^2=R^2$, and write $A/(2\pi )=r^2+2rw$. This quadratic form has symmetric matrix $\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)$ with largest eigenvalue $(1+\sqrt{5})/2$, the golden ratio. Its maximizing eigenvector lies in the positive quadrant, yielding the displayed area and aspect ratio.