The genus of the 2-amalgamations of graphs

A graph G is called the 2-amalgamation of subgraphs G₁ and G₂ if G = G₁ ∪ G₂ and G₁ ∩ G₂ = {x, y}, 2 distinct points. in this case we write G = G₁∪_{{x, y}} G₂. in this paper we show that the orientable genus, γ(G), satisfies the inequalities γ(G₁) + γ(G₂) ? 1 ≤ γ(G₁ ∪_{{x, y}} G₂) ≤ γ(G₁) + γ(G₂) + 1 and that this is the best possible result, i. e., the resulting three values for γ(G₁ ∪_{{x, y}} G₂) which are possible can actually be realized by appropriate choices for G₁ and G₂.     

The feasibility pump

In this paper we consider the NP-hard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{c^Tx:Ax≥b,x_j integer ∀j ∈ }. Trivially, a feasible solution can be defined as a point x* ∈ P:={x:Ax≥b} that is equal to its rounding , where the rounded point is defined by := x*_j if j ∈ and := x*_j otherwise, and [·] represents scalar rounding to the nearest integer. Replacing “equal” with “as close as possible” relative to a suitable distance function Δ(x*, ), suggests the following Feasibility Pump (FP) heuristic for finding a feasible solution of a given MIP.We start from any x* ∈ P, and define its rounding . At each FP iteration we look for a point x* ∈ P that is as close as possible to the current by solving the problem min {Δ(x, ): x ∈ P}. Assuming Δ(x, ) is chosen appropriately, this is an easily solvable LP problem. If Δ(x*, )=0, then x* is a feasible MIP solution and we are done. Otherwise, we replace by the rounding of x*, and repeat.We report computational results on a set of 83 difficult 0-1 MIPs, using the commercial software ILOG-Cplex 8.1 as a benchmark. The outcome is that FP, in spite of its simple foundation, proves competitive with ILOG-Cplex both in terms of speed and quality of the first solution delivered. Interestingly, ILOG-Cplex could not find any feasible solution at the root node for 19 problems in our test-bed, whereas FP was unsuccessful in just 3 cases.     

On the back-scatter cross-sections of ice spheres

Résumé La section droite de diffusion arrière de sphères de grand diamètre constituées par de la glace ou une matière (Stycast) ayant le même indice de réfraction est envisagée d'une manière à la fois théorique et expérimentale. La section droite est évaluée théoriquement par l'application d'une méthode d'optique géométrique simplifiée dans laquelle on considère seulement les réflexions sur les faces avant et arrière de la sphère. Dans le domaine du rapport diamètre/longueur d'onde de l'ordre de 1,3 à 6,7, les résultats obtenus par cette méthode s'accordent bien avec le calcul rigoureux de la section droite obtenu par la solution de l'équation demie. Pour des diamètres plus grands, la solution approchée sous-estime la section droite demie par un facteur qui est cependant plus petit que 3. La théorie simplifiée est étendue au cas de sphères diélectriques ayant leur surface partiellement couverte par une couche réfléchissante.Des données expérimentales sur la diffusion arrière, à une longuer d'onde de 3,22 cm, par des sphères en Stycast recouvertes d'une calotte sphérique de petite dimension, sont présentées dans le but d'illustrer la théorie précédente. En effet, les résultats obtenus, pour toutes les positions de la calotte sphérique au cours de sa rotation autour d'un axe vertical, sont en plein accord, dans la limite des erreurs expérimentales, avec la théorie faisant intervenir la composition de 3 parcours optiques principaux.

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A summary of this work was presented at the Tenth Weather Radar Conference, Washington, D. C., April 22–25, 1963.     

First facets of the octahedron

A procedure is given for identifying the facets of the octahedron that are first intersected upon extending the edge of a polyhedral cone. The information generated by this procedure can be exploited to advantage by cut-search procedures for zero-one integer programming. Results are given which make it possible to determine the first two facets (or sets of “tied facets”) following the innermost facet with less effort than required to determine the innermost facet itself. Depending on the orientation of the extended edge relative to the octahedron, a number of successive additional facets may be determined with comparable ease.     

Polyhedral convexity cuts and negative edge extensions

This note shows that convexity cuts defined relative to polyhedral convex sets can utilize negative as well as positive edge extensions under appropriate circumstances, yielding stronger cuts than customarily available. We also show how to partially collapse the polyhedron to further improve these cuts.     

Cut search methods in integer programming

Cut search is a new approach for solving integer programs based on extending edges of a cone to probe the solution space for sets of hyperplanes that are proxies for solution points in the space. Once all proxy hyperplanes associated with a given point have been intersected by at least one of the extended edges, this point is included in a set of points to be examined for feasibility (algorithmically or by inspection). Thereupon, all edges of the cone are extended an additional distance to create a cut by passing a hyperplane through the endpoints of these extended edges.The flexibility of the cut search procedure permits a variety of strategies for exploring and cutting into the solution space. One useful version arises by taking the proxy hyperplanes to be members of a positive or semipositive coordinate system. Relative to such a system the procedure can be organized to reduce the set of vectors to be examined for feasibility and also to generate deeper cuts at the end of the edge probe.     

Activation of prephenate dehydratase by adsorption to agarose derivatives

A series of potential affinity adsorbents for prephenate dehydratase were synthesized by coupling allosteric effectors to agarose with and without spacer arms. Only cationic ligands containing a hexamethylene spacer arm adsorbed the enzyme. Activity could be removed from aminohexyl- and glycyl-aminohexyl-agaroses with 0.5 M NaCl (with loss of activity), 0.4 M phosphate, or 140 mM hexanediamine, but was not removed from ala-aminohexyl-, phe-aminohexyl-, met-aminohexyl-, leu-aminohexyl-, or phenylpyruvyl-aminohexyl-agaroses or by Gly, Ala, Phe (inhibitor), Met or Leu (both activators). Urea (1.0 M), 1 mM phosphate, 50% glycerol in phosphate or prephenic acid (substrate) did not release enzymatic activity from any of the gels. The gel-bound enzyme was found to have enhanced activity comparable to that of the methionine-activated soluble enzyme. The site and mode of the enzyme immobilization are discussed in relation to the possible hydrophobic/ionic subunit-subunit interactions.