Optimal node disjoint paths on partial 2-trees: A linear algorithm and polyhedral results

We present anO(p · n) algorithm for the problem of finding disjoint simple paths of minimum total length betweenp given pairs of terminals on oriented partial 2-trees withn nodes and positive or negative arc lengths. The algorithm is inO(n) if all terminals are distinct nodes. We characterize the convex hull of the feasible solution set for the casep=2.We gratefully acknowledge the referee's many helpful suggestions to improve the presentation of this paper.     

Adsorption of lignosulphonates on polystyrene latex particles from aqueous solution and flocculation of the dispersions by electrolytes

Summary The adsorption isotherms of lignosulphonates with various degrees of sulphonation (0.5, 1,0, 2.0 and 4.0 moles per 840 unit weight of lignin) were measured on polystyrene latex particles from aqueous solution at 25 °C. The results showed that the lignin with the lowest degree of sulphonation gave the highest adsorption. The lignin with 1.0 and 2.0 degrees of sulphonation gave similar adsorption which was lower than that obtained with the half sulphonated lignin. However, further increase in degree of sulphonation resulted in an increase in the adsorption.Adsorption layer thickness measurements showed that the lignosulphonate with half degree of sulphonation gave thicker layers than that with 2.0 moles sulphonation. Increasing electrolyte (NaCl and CaCl,) concentration resulted in a decrease in the adsorbed layer thickness. Microelectrophoresis at 25 °C, showed only small differences in the zeta potential as a function of degree of sulphonation. Addition of electrolytes (KCl, Na₂SO₄ and CaCl₂) showed a reduction in the zeta potential. Flocculation by salts of various valence types, namely KCl, Na₂SO₄ and CaCl₂ showed that the critical flocculation concentration (C.F.C.) decreases in the order KCl > Na₂SO₄ > CaCl₂. The change in degree of sulphonation had only a marginal effect on the C.F.C.

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Zusammenfassung Die Isothermen der Adsorption von Lignosulfonaten mit verschiedenen Sulfonierungsgraden (S = 0,5, 1,0, 2,0 und 4,0 Mole pro 840 Gewichtseinheit des Lignins) an PolystyrolLatex-Teilchen in wäßriger Lösung wurden bei 25 °C gemessen. Hiernach wird Lignin mit den niedrigsten S-Werten am stärksten adsorbiert. Lignine mitS = 1,0 und 2,0 zeigen ähnliche Adsorption; sie ist aber kleiner als die fürS = 0,5. Mit weiter steigendemS nimmt die Adsorption zu.Aus der Messung der Dicke der Adsorptionsschicht folgt, daß fürS = 0,5 größer als fürS = 2,0 ist. Bei wachsendem Zusatz von NaCl oder CaCl₂ fällt ab. Nach Mikroelektrophorese-Messungen bei 25 °C hängt das -Potential nur wenig vonS ab. fällt durch die Gegenwart von KCl, Na₂SO₄ oder CaCl₂ ab. Mit den gleichen Salzen erfolgt eine Ausflockung, wobei die kritische Flockungskonzentration CFC in der Reihenfolge KCl, Na₂SO₄, CaCl₂ abnimmt; CFC ändert sich nur wenig mitS.

     

The luminescence of Pr3+ in BaY4Si5O17

The cell constants of four new monoclinic compounds BaR₄X₅O₁₇ (R = Y, Gd; X = Si, Ge) are given. The luminescence of various RE activators in the silicates is reported. Pr³⁺-activated BaY₄Si₅O₁₇ shows efficient ultraviolet 5d → 4f emission and weak 4f → 4f emission (mainly red luminescence from the ¹D₂ level). The 5d → 4f emission is ascribed to Pr³⁺ on Y sites, the 4f → 4f emission to Pr³⁺ on Ba sites. Energy transfer from Pr³⁺ to Gd³⁺ has been observed. Gd³⁺ plays an intermediate role in the energy transfer from Pr³⁺ to Sm³⁺ and to Dy³⁺ in BaGd₄Si₅O₁₇. Upon activation with Tb³⁺ the silicates show characteristic green Tb³⁺ luminescence with a quantum efficiency of 75% for ultraviolet excitation.     

C-Nitroso compounds—XXXV : Reaction of organometallic compounds with 1-chloro-1-nitroso-2,2,6,6-tetramethylcyclohexane

Reaction of Grignard reagents and organolithium compounds (RM) with the congested 1-chloro-1-nitroso-2,2,6,6-tetramethylcyclohexane 1 leads to the formation of significant amounts of the reduction product 2,2,6,6-tetramethylcyclohexanone oxime 3 (61–90%) together with the corresponding oxime O-R ether 4 (0–11%). Attack on nitrogen is unimportant as shown by very low yields of nitrone. Formation of the products is rationalised with a pathway involving transfer of an electron from RM to 1. This leads—after separation of MCI—to a radical pair consisting of R and the relatively stable iminoxy radical 2 (Schemes 1 and 2). Combination of these radicals explains formation of oxime ether 4 and nitrone 5, while reaction of iminoxy radical 2 with excess of RM can give oxime 3. Reactive radicals R (i.e. Me, Ph, and to a minor extent n-Bu) are furthermore capable of abstracting hydrogen from the solvent (diethyl ether, toluene, or cumene), and the solvent derived radicals can also combine with 2 on oxygen, under formation of oxime ether (26% of 6a). The corresponding benzyl- and cumyl ethers 6b and 6c are only formed in trace amounts because dimerisation of benzyl radicals (7%) and cumyl radicals (22%) is favoured.     

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P₈-free bipartite graphs, but polynomial for P₅-free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6−ε, for any ε>0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.     

Number of connected components of groups of real points of adjoint groups

We present a unified approach to compute the number of connected components in the group of real points of adjoint almost simple real algebraic groups.     

Analytical Modeling of Nonaqueous Phase Liquid Dissolution with Green's Functions

Equilibrium and bicontinuum nonequilibrium formulations of the advection–dispersion equation (ADE) have been widely used to describe subsurface solute transport. The Green's Function Method (GFM) is particularly attractive to solve the ADE because of its flexibility to deal with arbitrary initial and boundary conditions, and its relative simplicity to formulate solutions for multidimensional problems. The Green's functions that are presented can be used for a wide range of problems involving equilibrium and nonequilibrium transport in semiinfinite and infinite media. The GFM is applied to analytically model multidimensional transport from persistent solute sources typical of nonaqueous phase liquids (NAPLs). Specific solutions are derived for transport from a rectangular source (parallel to the flow direction) of persistent contamination using first, second, or thirdtype boundary or source input conditions. Away from the source, the first and thirdtype condition cannot be expected to represent the exact surface condition. The secondtype condition has the disadvantage that the diffusive flux from the source needs to be specified a priori. Near the source, the thirdtype condition appears most suitable to model NAPL dissolution into the medium. The solute flux from the pool, and hence the concentration in the medium, depends strongly on the mass transfer coefficient. For all conditions, the concentration profiles indicate that nonequilibrium conditions tend to reduce the maximum solute concentration and the total amount of solute that enters the porous medium from the source. On the other hand, during nonequilibrium transport the solute may spread over a larger area of the medium compared to equilibrium transport.     

Minimal Degree and (k, m)-Pancyclic Ordered Graphs

Given positive integers k m n, a graph G of order n is (k, m)-pancyclic ordered if for any set of k vertices of G and any integer r with m r n, there is a cycle of length r encountering the k vertices in a specified order. Minimum degree conditions that imply a graph of sufficiently large order n is (k, m)-pancylic ordered are proved. Examples showing that these constraints are best possible are also provided.     

Local approximations for maximum partial subgraph problem

We deal with MAXH₀-FREE PARTIAL SUBGRAPH. We mainly prove that 3-locally optimum solutions achieve approximation ratio (δ₀+1)/(B+2+ν₀), where B=max_v∈Vd_G(v), δ₀=min_v∈V(H0)d_H0(v) and ν₀=(|V(H₀)|+1)/δ₀. Next, we show that this ratio rises up to 3/(B+1) when H₀=K₃. Finally, we provide hardness results for MAXK₃-FREE PARTIAL SUBGRAPH.