Multinomial permutations on a circle

Multinomial permutations on a circle are considered in the framework of combinatorics. Different cases are presented and shown to agree with previously derived formula for the number of cyclic necklaces. Two applied examples are discussed with a view to illustrate the implications of derived formulas. Copyright © 2006 John Wiley & Sons, Ltd.     

Gromov's K-area and symplectic rigidity

Supported by the United States-Israel Binational Science Foundation grant 94-00302     

Enumerating disjunctions and conjunctions of paths and cuts in reliability theory

Let G=(V,E) be a (directed) graph with vertex set V and edge (arc) set E. Given a set P of source-sink pairs of vertices of G, an important problem that arises in the computation of network reliability is the enumeration of minimal subsets of edges (arcs) that connect/disconnect all/at least one of the given source-sink pairs of P. For undirected graphs, we show that the enumeration problems for conjunctions of paths and disjunctions of cuts can be solved in incremental polynomial time. Furthermore, under the assumption that P consists of all pairs within a given vertex set, we also give incremental polynomial time algorithm for enumerating all minimal path disjunctions and cut conjunctions. For directed graphs, the enumeration problem for cut disjunction is known to be NP-complete. We extend this result to path conjunctions and path disjunctions, leaving open the complexity of the enumeration of cut conjunctions. Finally, we give a polynomial delay algorithm for enumerating all minimal sets of arcs connecting two given nodes s₁ and s₂ to, respectively, a given vertex t₁, and each vertex of a given subset of vertices T₂.     

Body waves in fractured porous media saturated by two immiscible newtonian fluids

A study of body waves in fractured porous media saturated by two fluids is presented. We show the existence of four compressional and one rotational waves. The first and third compressional waves are analogous to the fast and slow compressional waves in Biot's theory. The second compressional wave arises because of fractures, whereas the fourth compressional wave is associated with the pressure difference between the fluid phases in the porous blocks. The effects of fractures on the phase velocity and attenuation coefficient of body waves are numerically investigated for a fractured sandstone saturated by air and water phases. All compressional waves except the first compressional wave are diffusive-type waves, i.e., highly attenuated and do not exist at low frequencies.Now at Izmir Institute of Technology, Faculty of Engineering, Gaziosmanpasa Bulvari, No.16, Cankaya, Izmir, Turkey.     

3, 3, 3-Trifluoropropyl(methyl)cyclosiloxanes

The influence of the number of 3, 3, 3-trifluoropropyl(methyl)siloxane links (/) in the cyclotetrasiloxanes _mD_4-m, where D represents the dimethylsiloxane link and m=0–4, on the rearrangement of these compounds in acetone solution under the action of sodium siloxanolate has been studied. The rearrangement takes place with the formation of a linear polysiloxane the degradation of which yields, in addition to the initial ring, cyclosiloxanes with a different structure. The rate of rearrangement of _mD_4-m and of the formation of a linear polysiloxane rises with an increase in m from 0 to 3. The equilibrium concentration of the linear polysiloxane formed from _mD_4-m is inversely proportional to m. Results have been obtained on the kinetics of the formation of the cyclosiloxanes _mD_n, where m=0–5, n=0–5, and m+n=3–6, in the rearrangement of the rings D₃, ₂D₂, ₃D, and ₄. The reactivity of the siloxane links rises in the sequence (CH₃)₂Si-O-Si(CH₃)₂ < (CF₃CH₂CH₂)-(CH₃) Si-O-Si(CH₃)₂ <(CF₃CH₂CH₂) (CH₃)Si-O-Si(CH₃) (CH₂CH₂CF₃) . Because of the negative inductive effect transferred through the siloxane links, the 3, 3, 3-trifluoropropyl groups strongly activate the siloxane ring with respect to nucleophiiic reagents.For part I, see [3].     

Artificial multiple criticality and phase equilibria: an investigation of the PC-SAFT approach

The perturbed-chain statistical associating fluid theory (PC-SAFT) is studied for a wide range of temperature, T, pressure, p, and (effective) chain length, m, to establish the generic phase diagram of polymers according to this theory. In addition to the expected gas-liquid coexistence, two additional phase separations are found, termed "gas-gas" equilibrium (at very low densities) and "liquid-liquid" equilibrium (at densities where the system is expected to be solid already). These phase separations imply that in one-component polymer systems three critical points occur, as well as equilibria of three fluid phases at triple points. However, Monte Carlo simulations of the corresponding system yield no trace of the gas-gas and liquid-liquid equilibria, and we conclude that the latter are just artefacts of the PC-SAFT approach. Using PC-SAFT to correlate data for polybutadiene melts, we suggest that discrepancies in modelling the polymer density at ambient temperature and high pressure can be related to the presumably artificial liquid-liquid phase separation at lower temperatures. Thus, particular care is needed in engineering applications of the PC-SAFT theory that aims at predicting properties of macromolecular materials.