Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering

We demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities. As an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium. We show that the conjectured fractal Weyl law for open chaotic systems [W. T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)] is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating Fresnel's laws.Comment: 8 pages, 12 figure

Eigenvectors in the Superintegrable Model II: Ground State Sector

In 1993, Baxter gave $2^{m_Q}$ eigenvalues of the transfer matrix of the $N$-state superintegrable chiral Potts model with spin-translation quantum number $Q$, where $m_Q=\lfloor(NL-L-Q)/N\rfloor$. In our previous paper we studied the Q=0 ground state sector, when the size $L$ of the transfer matrix is chosen to be a multiple of $N$. It was shown that the corresponding $\tau_2$ matrix has a degenerate eigenspace generated by the generators of $r=m_0$ simple $sl_2$ algebras. These results enable us to express the transfer matrix in the subspace in terms of these generators $E_m^{\pm}$ and $H_m$ for $m=1,...,r$. Moreover, the corresponding $2^r$ eigenvectors of the transfer matrix are expressed in terms of rotated eigenvectors of $H_m$.Comment: LaTeX 2E document, using iopart.cls with iopams packages. 17 pages, uses eufb10 and eurm10 fonts. Typeset twice! vs2: Many changes and additions, adding 7 pages. vs3: minor corrections. vs4 minor improvement

Development of the Bélanger Equation and Backwater Equation by Jean-Baptiste Bélanger (1828)

A hydraulic jump is the sudden transition from a high-velocity to a low-velocity open channel flow. The application of the momentum principle to the hydraulic jump is commonly called the Bélanger equation, but few know that Bélanger's (1828) treatise was focused on the study of gradually varied open channel flows. Further, although Bélanger understood the rapidly-varied nature of the jump flow, he applied incorrectly the Bernoulli principle in 1828, and corrected his approach 10 years later. In 1828, his true originality lay in the successful development of the backwater equation for steady, one-dimensional gradually-varied flows in an open channel, together with the introduction of the step method, distance calculated from depth, and the concept of critical flow conditions