Formulas for Continued Fractions. An Automated Guess and Prove Approach

We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial conditions. This is used to generate the first few coefficients and from there a conjectured formula. This formula is then proved automatically thanks to a linear recurrence satisfied by some remainder terms. Extensive experiments show that this simple approach and its straightforward generalization to difference and $q$-difference equations capture a large part of the formulas in the literature on continued fractions.Comment: Maple worksheet attache

Complex sine-Gordon-2: a new algorithm for multivortex solutions on the plane

We present a new vorticity-raising transformation for the second integrable complexification of the sine-Gordon equation on the plane. The new transformation is a product of four Schlesinger maps of the Painlev\'{e}-V to itself, and allows a more efficient construction of the $n$-vortex solution than the previously reported transformation comprising a product of $2n$ maps.Comment: Part of a talk given at a conference on "Nonlinear Physics. Theory and Experiment", Gallipoli (Lecce), June-July 2004. To appear in a topical issue of "Theoretical and Mathematical Physics". 7 pages, 1 figur

Advanced inductively coupled plasma etching processes for fabrication of resonator-quantum well infrared photodetector

Resonator-quantum well infrared photodetectors (R-QWIPs) are the next generation of QWIP detectors that use resonances to increase the quantum efficiency (QE). To achieve the expected performance, the detector geometry must be produced in precise specification. In particular, the height of the diffractive elements (DE) and the thickness of the active resonator must be uniformly and accurately realized to within 0.05 lm accuracy and the substrates of the detectors have to be removed totally. To achieve these specifications, two optimized inductively coupled plasma (ICP) etching processes are developed. Using these etching techniques, we have fabricated a number of R-QWIP test detectors and FPAs with the required dimensions and completely removed the substrates of the test detectors and FPAs. Their QE spectra were tested to be in close agreement with the theoretical predictions. The operability and spectral non-uniformity of the FPA is about 99.57% and 3% respectively

Advanced inductively coupled plasma etching processes for fabrication of resonator-quantum well infrared photodetector

Resonator-quantum well infrared photodetectors (R-QWIPs) are the next generation of QWIP detectors that use resonances to increase the quantum efficiency (QE). To achieve the expected performance, the detector geometry must be produced in precise specification. In particular, the height of the diffractive elements (DE) and the thickness of the active resonator must be uniformly and accurately realized to within 0.05 lm accuracy and the substrates of the detectors have to be removed totally. To achieve these specifications, two optimized inductively coupled plasma (ICP) etching processes are developed. Using these etching techniques, we have fabricated a number of R-QWIP test detectors and FPAs with the required dimensions and completely removed the substrates of the test detectors and FPAs. Their QE spectra were tested to be in close agreement with the theoretical predictions. The operability and spectral non-uniformity of the FPA is about 99.57% and 3% respectively

Remarks on quantization of Pais-Uhlenbeck oscillators

This work is concerned with a quantization of the Pais-Uhlenbeck oscillators from the point of view of their multi-Hamiltonian structures. It is shown that the 2n-th order oscillator with a simple spectrum is equivalent to the usual anisotropic n - dimensional oscillator

Differential constraints and exact solutions of nonlinear diffusion equations

The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries

Nonlinear Dirac and diffusion equations in 1 + 1 dimensions from stochastic considerations

We generalize the method of obtaining the fundamental linear partial differential equations such as the diffusion and Schrodinger equation, Dirac and telegrapher's equation from a simple stochastic consideration to arrive at certain nonlinear form of these equations. The group classification through one parameter group of transformation for two of these equations is also carried out.Comment: 18 pages, Latex file, some equations corrected and group analysis in one more case adde

The Hunter-Saxton equation: remarkable structures of symmetries and conserved densities

In this paper, we present extraordinary algebraic and geometrical structures for the Hunter-Saxton equation: infinitely many commuting and non-commuting $x,t$-independent higher order symmetries and conserved densities. Using a recursive relation, we explicitly generate infinitely many higher order conserved densities dependent on arbitrary parameters. We find three Nijenhuis recursion operators resulting from Hamiltonian pairs, of which two are new. They generate three hierarchies of commuting local symmetries. Finally, we give a local recursion operator depending on an arbitrary parameter. As a by-product, we classify all anti-symmetric operators of a definite form that are compatible with the Hamiltonian operator $D_x^{-1}$

The Moyal bracket and the dispersionless limit of the KP hierarchy

A new Lax equation is introduced for the KP hierarchy which avoids the use of pseudo-differential operators, as used in the Sato approach. This Lax equation is closer to that used in the study of the dispersionless KP hierarchy, and is obtained by replacing the Poisson bracket with the Moyal bracket. The dispersionless limit, underwhich the Moyal bracket collapses to the Poisson bracket, is particularly simple.Comment: 9 pages, LaTe