Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs

In this paper, we classify the reflexible regular orientable embeddings and the self-Petrie dual regular orientable embeddings of complete bipartite graphs. The classification shows that for any natural number n, say (p₁,p₂,…,p_k are distinct odd primes and a_i>0 for each i?1), there are t distinct reflexible regular embeddings of the complete bipartite graph K_n,n up to isomorphism, where t=1 if a=0, t=2^k if a=1, t=2^k+1 if a=2, and t=3·2^k+1 if a?3. And, there are s distinct self-Petrie dual regular embeddings of K_n,n up to isomorphism, where s=1 if a=0, s=2^k if a=1, s=2^k+1 if a=2, and s=2^k+2 if a?3.     

Reinvestigation of Ge4Se9 based on single‐crystal data

Tetra­germanium nona­selenide, Ge₄Se₉, adopts a two‐dimensional layered structure. The layer is made up of infinite chains of corner‐sharing GeSe₄ tetra­hedra and the chains are connected via the Ge₂Se₇ unit to form the two‐dimensional layer. These layers are stacked to form the three‐dimensional structure with a van der Waals gap. A previous structure report on Ge₄Se₉ based on powder diffraction data [Fjellvåg, Kongshaug & Stølen (2001). J. Chem. Soc. Dalton Trans. pp. 1043–1045] is comparable with our results except for the absolute structure determination.     

Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems

We construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order PDEs in the state variable (e.g., pressure) and its flux (e.g., Darcy velocity). An important point is that no staggered grids or covolumes are used to stabilize the system. Only a single primary grid system is adopted, and the degrees of freedom are imposed on the interfaces. The approximate flux is sought in the lowest-order Raviart-Thomas space and the pressure field in the rotated- nonconforming space. Furthermore, we demonstrate that the present finite volume method can be interpreted as a rotated- nonconforming finite element method for the pressure with a simple local recovery of flux. Numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method.

     

Circulant Double Coverings of a Circulant Graph of Valency Four

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors (see [3], [8]–[15]). A covering of G is called circulant if its covering graph is circulant. Recently, the authors [4] enumerated the isomorphism classes of circulant double coverings of a certain kind, called typical, and showed that no double covering of a circulant graph of valency 3 is circulant. In this paper, the isomorphism classes of connected circulant double coverings of a circulant graph of valency 4 are enumerated. As a consequence, it is shown that no double covering of a non-circulant graph G of valency 4 can be circulant if G is vertex-transitive or G has a prime power of vertices. The first author is supported by NSF of China (No. 60473019) and by NKBRPC (2004CB318000), and the second author is supported by Com²MaC-KOSEF (R11-1999-054) in Korea.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

Gas permeability and n‐butane solubility of poly(1‐trimethylgermyl‐1‐propyne)

The gas permeability and n‐butane solubility in glassy poly(1‐trimethylgermyl‐1‐propyne) (PTMGP) are reported. As synthesized, the PTMGP product contains two fractions: (1) one that is insoluble in toluene and soluble only in carbon disulfide (the toluene‐insoluble polymer) and (2) one that is soluble in both toluene and carbon disulfide (the toluene‐soluble polymer). In as‐cast films, the gas permeability and n‐butane solubility are higher in films prepared from the toluene‐soluble polymer (particularly in those films cast from toluene) than in films prepared from the toluene‐insoluble polymer and increase to a maximum in both fractions after methanol conditioning. For example, in as‐cast films prepared from carbon disulfide, the oxygen permeability at 35 °C is 330 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐soluble polymer and 73 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐insoluble polymer. After these films are conditioned in methanol, the oxygen permeability increases to 5200 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐soluble polymer and 6200 × 10^?10 cm³ (STP) cm/(cm² s cmHg) for the toluene‐insoluble polymer. The rankings of the fractional free volume and nonequilibrium excess free volume in the various PTMGP films are consistent with the measured gas permeability and n‐butane solubility values. Methanol conditioning increases gas permeability and n‐butane solubility of as‐cast PTMGP films, regardless of the polymer fraction type and casting solvent used, and minimizes the permeability and solubility differences between the various films (i.e., the permeability and solubility values of all conditioned PTMGP films are similar). © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2228–2236, 2002     

Comparison of V‐cycle multigrid method for cell‐centered finite difference on triangular meshes

We consider a multigrid algorithm (MG) for the cell centered finite difference scheme (CCFD) on general triangular meshes using a new prolongation operator. This prolongation is designed to solve the diffusion equation with strongly discontinuous coefficient as well as with smooth one. We compare our new prolongation with the natural injection and the weighted operator in Kwak, Kwon, and Lee ( 8 ) and the behaviors of these three prolongation are discussed. Numerical experiments show that (i) for smooth problems, the multigrid with our new prolongation is fastest, the next is the weighted prolongation, and the third is the natural injection; and (ii) for nonsmooth problems, our new prolongation is again fastest, the next is the natural injection, and the third is the weighted prolongation. In conclusion, our new prolongation works better than the natural injection and the weighted operator for both smooth and nonsmooth problems. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Synthesis and characterization of polyimides from bis(3‐aminophenyl)‐4‐(1‐adamantyl)phenoxyphenyl phosphine oxide

A novel diamine, bis(3‐aminophenyl)‐4‐(1‐adamantyl)phenoxyphenyl phosphine oxide (mDAATPPO), was synthesized via the Williamson ether reaction of 4‐(1‐adamantyl)phenol and bis(3‐nitrophenyl)‐4‐fluorophenyl phosphine oxide, followed by reduction. The phenol group was prepared by the Friedel–Crafts reaction of 1‐bromoadamantane and phenol, whereas the phosphine oxide group was synthesized by the Grignard reaction of 1‐bromo‐4‐fluorobezene and diphenyl phosphinic chloride, followed by nitration. The monomer and its intermediate compounds were characterized with Fourier transform infrared, NMR, and melting‐point apparatus. The monomer was then used to prepare polyimides with 2,2‐bis(3,4‐dicarboxyphenyl)hexafluoropropane dianhydride, 3,3′,4,4′‐benzophenonetetracarboxylic dianhydride, 4,4′‐oxydiphthalic dianhydride, and pyromellitic dianhydride by the conventional two‐step synthesis: the preparation of poly(amic acid) followed by solution imidization. The molecular weights of the polyimides were controlled to 20,000 g/mol by off‐stoichiometry, and the synthesized polyimides were characterized with Fourier transform infrared, NMR, gel permeation chromatography, thermogravimetric analysis, and differential scanning calorimetry. In addition, the solubility, intrinsic viscosity, dielectric constant, and birefringence of the polyimides were evaluated. Novel polyimides with mDAATPPO exhibited good solubility, high glass‐transition temperatures (290–330 °C), excellent thermal stability (>500 °C), low dielectric constants (2.77–3.01), low refractive indices, and low birefringence values (0.0019–0.0030). © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 2567–2578, 2006