Broadband arrayed waveguide gratings on InP

Broadband arrayed waveguide gratings on InP are presented using a novel S-shape design. This design was utilized to accommodate the large free spectral range required for broadband operation. Four and eight channel AWGs with a wavelength channel spacing of 18 nm are discussed. The output peaks of the AWGs have a wide FWHM of 11 nm which provides insensitive operation to polarization, temperature fluctuations, and chromatic dispersion.     

A mild synthesis of bis(indolyl)methanes using a deep eutectic solvent

Deep eutectic solvents (such as the combination of urea and choline chloride) are effective solvents/organocatalysts for the condensation of indole and aryl or alkenyl aldehydes to form bis(indolyl)methanes. The reaction conditions are quite mild and do not require additional Bronsted or Lewis acid catalyst, though they fail with ketones or aliphatic aldehydes. Given the inexpensive, non-toxic, and recyclable nature of the DES, these reaction conditions are simple and highly environmentally friendly.     

Partial information systems and the Smyth powerdomain

The dual of the join semilattice of proper compact Scott open subsets of a domain D is called the Smyth powerdomain of D. The Smyth powerdomain is used in programming semantics as a model for demonic nondeterminism. In this paper, we introduce the concept of partial information systems; and, as an application, show that the Smyth powerdomain of any domain can be realized in terms of the sub partial information systems of the domain??s corresponding information system.     

A note on class one graphs with maximum degree six

It is known that there are class two graphs with Δ=6 which can be embedded in a surface Σ with Euler characteristic χ(Σ)?0. However, it is unknown whether there are class two graphs on the projective plane or on the plane with Δ=6. In this paper, we prove that every graph with Δ=6 is class one if it can be embedded in a surface with Euler characteristic at least -3 and is C₃-free, or C₄-free or if it can be embedded in a surface with Euler characteristic at least -1 and is C₅-free. This generalizes Zhou's results in [G. Zhou, A note on graphs of class I, Discrete Math. 263 (2003) 339-345] on planar graphs.     

Observation of negative group delays within a coaxial photonic crystal using an impulse response method

An impulse response experiment is described which independently verifies the recently observed result of negative group delayed propagation within a coaxial photonic crystal. This result was unexpected when it was reported and has been the subject of enduring controversy because theoretical models predict that negative group delays should not be possible in passive linear media. The impulse response method allows for the determination of both the transmission amplitude and the group delay for narrow-band wave packets over a wide frequency range in one simple experiment. The impulse response results presented here confirm our earlier finding of negative group delays within the band gap region and are in excellent agreement with the more traditional approach of measuring transit times for wave packets. However, negative group delays are observed only over very narrow frequency ranges within the wider band gap suggesting that another interference mechanism is involved.     

On certain spanning subgraphs of embeddings with applications to domination

We prove the existence of certain spanning subgraphs of graphs embedded in the torus and the Klein bottle. Matheson and Tarjan proved that a triangulated disc with n vertices can be dominated by a set of no more than n/3 of its vertices and thus, so can any finite graph which triangulates the plane. We use our existence theorems to prove results closely allied to those of Matheson and Tarjan, but for the torus and the Klein bottle.     

Pricing and advertisement in a manufacturer–retailer supply chain

We use a game theoretical approach to study pricing and advertisement decisions in a manufacturer–retailer supply chain when price discounts are offered by both the manufacturer and retailer. When the manufacturer is the leader of the game, we obtained Stackelberg equilibrium with manufacturer’s local allowance, national brand name investment, manufacturer’s preferred price discount, retailer’s price discount, and local advertising expense. For the special case of two-stage equilibrium when the manufacturer’s price discount is exogenous, we found that the retailer is willing to increase local advertising expense if the manufacturer increases local advertising allowance and provides deeper price discount, or if the manufacturer decreases its brand name investment. When both the manufacturer and retailer have power, Nash equilibrium in a competition game is obtained. The comparison between the Nash equilibrium and Stackelberg equilibrium shows that the manufacturer always prefers Stackelberg equilibrium, but there is no definitive conclusion for the retailer. The bargaining power can be used to determine the profit sharing between the manufacturer and the retailer. Once the profit sharing is determined, we suggest a simple contract to help the manufacturer and retailer obtain their desired profit sharing.     

Isoperimetric Constants of Infinite Plane Graphs

   Abstract. Let G be an infinite locally finite plane graph with one end and let H be a finite plane subgraph of G . Denote by a(H) the number of finite faces of H and by l(H) the number of the edges of H that are on the boundary of the infinite face or a finite face not in H . Define the isoperimetric constant h (G) to be inf _H l(H) / a(H) and define the isoperimetric constant h (δ) to be inf _G h (G) where the infimum is taken over all infinite locally finite plane graphs G having minimum degree δ and exactly one end. We establish the following bounds on h (δ) for δ ≥ 7 :

Smoothing schemes for reaction-diffusion systems with nonsmooth data

Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002) 430–455] developed a class of Exponential Time Differencing Runge–Kutta schemes (ETDRK) for nonlinear parabolic equations; Kassam and Trefethen [A.K. Kassam, Ll. N. Trefethen, Fourth-order time stepping for stiff pdes, SIAM J. Sci. Comput. 26 (2005) 1214–1233] have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth-order (ETDRK4) scheme. They use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials, but this approach creates new difficulties in choosing and evaluating the contour for larger problems. Neither treatment addresses problems with nonsmooth data, where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully. Such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications, including computational chemistry and financial engineering. We introduce a new version of the fourth-order Cox–Matthews, Kassam–Trefethen ETDRK4 scheme designed to eliminate the remaining computational difficulties. This new scheme utilizes an exponential time differencing Runge–Kutta ETDRK scheme using a diagonal Padé approximation of matrix exponential functions, while to deal with the problem of nonsmooth data we use several steps of an ETDRK scheme using a sub-diagonal Padé formula. The new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices, having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel. In this approach it is only required that several backward Euler linear problems be solved, in serial or parallel. Numerical experiments are described to support the new scheme.