An Extremal Problem on v-Partite Graphs

Let r, t, and v be positive integers with 2 ? t ? v. For a fixed graph G with t vertices, denote by Г (r, v, G) the class of all v-partite graphs H with vertex classes V_i, |V_i| = r (i =1, 2, ... , v) satisfying the following condition: For every t-subset {i₁, i₂, ... , i_t} of {1, 2, ... , v} there exists a subgraph of H isomorphic to G having exactly one vertex in each of the classes V_iτ, τ =1, 2, ... , t. Let cl(H) denote the clique number of H, i.e. the maximum order of a complete subgraph of H. We are interested in determining the value of the class parameter

$D(r,v,G)=\underset{H\in \Gamma (r,v,G)}{\min }cl(H)$

The main result is given the form of a Ramsey-type theorem (see Theorem 2.2). We shall show that for fixed r and G, D (r, v, G) tends to infinity when v tends to infinity if and only if χ(G) (the chromatic number of G) is greater than r. Some results concerning D(r, v, K_t), where K_t, is the complete graph on t vertices, are also given.     

Surface adsorption effects in the inkjet printing of an aqueous polymer solution on a porous oxide ceramic substrate

The inkjet printing of a polymeric solution into a porous substrate was studied, with the focus on phenomena occurring within the pore space during infiltration. Lines of aqueous polyacrylic acid (PAA) solution were printed onto the surface of porous, high-green-density ceramic powder beds. The PAA is a binder for the ceramic particles, allowing removal of the printed line structure ("primitive") and characterization of the extent of polymer penetration. Large differences in cross section of the retrieved printed structure were observed between ceramic systems and for different specific surface area powders. A mechanism for "filtration" of the polymer by adsorption onto the ceramic particle surfaces during infiltration was proposed. The adsorption of PAA onto Al2O3, SiO2, and TiO2 was characterized via adsorption isotherms, and the trend of primitive cross section with PAA adsorption was consistent with the filtration hypothesis, as was the variation with powder-specific surface area. These results can be generalized to other systems where a solution is inkjet printed onto a porous substrate (e.g., inks on plain paper, porous coated papers, etc.) Utilization of the adsorption effects may allow confinement of the solute molecules (e.g., colorant) to a small region near the substrate surface.     

Die Chlorierung desp-Jod-dimethylanilins

Ohne Zusammenfassung     

Anwendung der Weierstraßschen Methode zur Umkehrung der elliptischen Integrale auf die Bestimmung konjugierter Punkte im Sinne der Variationsrechnung

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Blutnachweis

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