The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).     

Optical-induced spectral tuning in grating-assisted nonlinearcouplers

The spectral (wavelength) properties of the grating-assisted nonlinear coupler are investigated theoretically. The interaction of linear and nonlinear wavelength-dependent phase detuning leads to a rich display of new and useful characteristics. Among these are bandpass tuning and spectral shaping. The grating frequency is viewed as a `free design parameter' which may be judiciously chosen to bring about an optimum behavior for a given application. As a wavelength processing element, the coupler behaves as an extended and continuously tunable wavelength filter, with tuning proportional to the input power. As an an-optical switch, a broadened and virtually transparent passband is easily achieved, which has desirable consequences for ultra-fast (large bandwidth) switching. The coupler may be operated in two complimentary modes: The active mode, where the power level of an input pulse or signal interacts in a nonlinear fashion with the passband of the coupler; and the passive mode, where a separate idler input is used to bias the device for a desired response to a smaller injected signal     

Very high-order microring resonator filters for WDM applications

High-order microring resonators having from 1 to 11 coupled cavities are demonstrated. These filters exhibit low loss, flat tops, and out-of-band rejection ratios that can exceed 80 dB. They achieve performance that is suitable for commercial applications.     

Untersuchung von Textilien und Textilrohstoffen

Ohne Zusammenfassung     

Correction to: On Webs of Maximum Rank (Geom. Dedicata 31 (1989), 19–35)

We correct an error in Example (3.4) in Geom. Dedicata 31 (1989), 19–35.     

Microring resonator channel dropping filters

Microring resonators side coupled to signal waveguides provide compact, narrow band, and large free spectral range optical channel dropping filters. Higher order filters with improved passband characteristics and larger out-of-band signal rejection are realized through the coupling of multiple rings. The analysis of these devices is approached by the novel method of coupling of modes in time. The response of filters comprised of an arbitrarily large dumber of resonators may be written down by inspection, as a continued fraction. This approach simplifies both the analysis and filter synthesis aspects of these devices     

The mixed-aldehyde synthesis of difunctional tetraarylporphyrins

The synthesis is reported of nine unsymmetrical, meso-substituted porphyrins. Among the compounds prepared are the following 5-(R)-10,15,20-tri-p-tolylporphyrins; R = 2,6-dinitrophenyl, 4-hydroxy-3-ethoxy-phenyl, 4-hydroxy-3-methoxy-5-nitrophenyl, 5-hydroxy-2-nitrophenyl and 4-hydroxy-3-nitrophenyl. Other porphyrins reported include 5-(2-(1-butoxy)phenyl)-15-(2-nitrophenyl)-10-15-di-p-tolylporphyrin and the two 5-(R)-10-15,20-tripropylporphyrins in which R = 2-nitrophenyl and 2-hydroxyphenyl. The disubstituted porphyrins offer a rational route to the synthesis of difunctional “tailed-porphyrins”.     

Mechanistic Aspects of Carbon Nanotube Nucleation and Growth

The discovery, synthesis, characterization, and applicability of carbon nanotubes have produced tremendous excitement and interest among scientists and engineers. In particular, the use of these unique tubular nanostructures for new strong lightweight materials, nanoelectronics, fuel storage and cells, electron emitters and bio, scanning probe microscopy, and chemical sensing devices has created an intense effort to advance the synthesis so as to mass produce carbon nanotubes with control over diameter and helicity. The massive and controlled synthesis of this heralded nanostructure has been a great challenge. Although significant progress has advanced the preparation, more synthetic development is required. The syntheses have so far involved three main approaches: arc discharge vaporization, laser vaporization, and catalytic chemical vapor deposition. The synthetic trend has progressed to a point where further advancement with these techniques will require a better understanding of the mechanism of nucleation and growth. The mechanics of carbon nanotube nucleation and growth involve very complex and diverse phenomena occurring under extreme conditions and on the mesoscopic scale. As yet the detail mechanism is unknown. Difficulties with experimental probing and computational simulation have increased the mystery of this mechanism. This review presents an account of research on the synthesis of carbon nanotubes and the mechanism of formation. This overview includes all three mentioned synthetic approaches and hybrids thereof. On the basis of this broad account a comprehensive mechanism for carbon nanotube nucleation and growth naturally arises. This mechanism is qualitative and it hopes to inspire more quantitative exploration and synthetic advancement.