COADJOINT ORBITS FOR THE CENTRAL EXTENSION OF Diff^＋（S^1） AND THEIR REPRESENTATIVES

According to Kirillov′s idea, the irreducible unitary representations of a Liegroup G roughly correspond to the coadjoint orbits O. In the forward direction one ap-plies the methods of geometric quantization to produce a representation, and in the reversedirection one computes a transform of the character of a representation, to obtain a coad-joint orbit. The method of orbits in the representations of Lie groups suggests the detailedstudy of coadjoint orbits of a Lie group G in the space g* dual to the Lie algebra g of G.In this paper, two primary goals are achieved: one is to completely classify the smoothcoadjoint orbits of Virasoro group for nonzero central charge c; the other is to find repre-sentatives for coadjoint orbits. These questions have been considered previously by Segal,Kirillov, and Witten, but their results are not quite complete. To accomplish this, theauthors start by describing the coadjoint action of D-the Lie group of all orientation pre-serving diffeomorphisms on the circle S^1, and its central extension D~, then the authors willgive a complete classification of smooth coadjoint orbits. In fact, they can be parameterizedby a subspace Of conjugacy classes of PSU~(1,1). Finally, the authors will show how to findrepresentatives of coadjoint orbits by analyzing the vector fields stabilizing the orbits, anddescribe the amazing connection between the characteristic (trace) of conjugacy classes of PSU~(1, 1) and that of vector fields stabilizing orbits.     

Core-suction technique for the fabrication of optical fiber preforms

A novel technique, named "core suction," for fabricating optical fiber preforms has been devised. The technique involves drawing the molten nonconventional core glass material into the cladding tube to form the preform. The developed technique is simple, inexpensive, and shows great potential for fabricating preforms of highly nonlinear nonconventional glasses as the core material. Preforms were made with Schott SF6 and a lead-tellurium-germanate glass in silica cladding tubes, and these preforms were then pulled into fibers.     

Signal-processing algorithm for white-light optical fiber extrinsic Fabry-Perot interferometric sensors

We present a novel signal-processing algorithm for single-mode optical fiber extrinsic Fabry-Perot interferometric sensors that can achieve both high-resolution, absolute measurement of the cavity length and a large dynamic measurement range simultaneously. The algorithm is based on an accurate model of the characteristics of a fiber-optic sensor that takes into account the phase shift that is due to the coupling of light reflected at the second surface to the lead-in fiber end.     

Solitary wave solutions to nonlinear evolution equations in mathematical physics

This paper obtains solitons as well as other solutions to a few nonlinear evolution equations that appear in various areas of mathematical physics. The two analytical integrators that are applied to extract solutions are tan–cot method and functional variable approaches. The soliton solutions can be used in the further study of shallow water waves in (1+1) as well as (2+1) dimensions.     

The Diagonal Distribution for the Invariant Measure of a Unitary Type Symmetric Space

Let Θ denote an involution for a simply connected compact Lie group U, let K denote the fixed point set, and let μ denote the U-invariant probability measure on U/K. Consider the geodesic embedding of Cartan. In this paper we compute the Fourier transform of the diagonal distribution for relative to a compatible triangular decomposition of G, the complexification of U. This boils down to a Duistermaat-Heckman exact stationary phase calculation, involving a Poisson structure on the dual symmetric space G₀/K discovered by Evens and Lu.     

Random-hole optical fiber evanescent-wave gas sensing

Research on development of optical gas sensors based on evanescent-wave absorption in random-hole optical fibers is described. A process to produce random-hole optical fibers was recently developed that uses a novel in situ bubble formation technique. Gas molecules that exhibit characteristic vibrational absorption lines in the near-IR region that correspond to the transmission window for silica optical fiber have been detected through the evanescent field of the guided mode in the pore region. The presence of the gas molecules in the holes of the fiber appears as a loss at wavelengths that are characteristic of the particular gas species present in the holes. An experimental setup was constructed with these holey fibers for detection of acetylene gas. The results clearly demonstrate the characteristic absorptions in the optical spectra that correspond to the narrow-line absorptions of the acetylene gas, and this represents what is to our knowledge the first report of random-hole fiber gas sensing in the literature.     

Tunable-optical-filter-based white-light interferometry for sensing

We describe tunable-optical-filter-based white-light interferometry for sensor interrogation. By introducing a tunable optical filter into a white-light interferometry system, one can interrogate an interferometer with either quadrature demodulation or spectral-domain detection at low cost. To demonstrate the feasibility of effectively demodulating various types of interferometric sensor, experiments have been performed using an extrinsic Fabry-Perot tunable filter to interrogate two extrinsic Fabry-Perot interferometric temperature sensors and a diaphragm-based pressure sensor.