Copper—A “Modern” Bioelement

Copper is a bioessential element in biology with truly unique chemical characteristics in its two relevant oxidation states +I and +II. Significant progress has been made in recent years in the elucidation of the frequently surprising biochemistry of this trace element. Those advances were especially furthered through mutual stimulation involving results from biochemistry, molecular biology, and medicine on one hand and the synthesis as well as the structural and spectroscopic characterization of low molecular weight model complexes on the other. The most notable features of protein-bound active copper are its almost exclusive function in the metabolism of O₂ or N/O compounds (NO, N₂O) and its frequent association with oxidizing organic and inorganic radicals such as tyrosyl, semiquinones, superoxide, or nitrosyl. This unique biological role of copper can be rationalized given its chemical and assumed evolutionary background.     

Packing chromatic number, $$\mathbf (1, 1, 2, 2) $$ ( 1 , 1 , 2 , 2 ) -colorings,and characterizing the Petersen graph

The packing chromatic number $$\chi _{\rho }(G)$$     

Connected Domatic Number in Planar Graphs

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.     

Limited packings in graphs

We define a k-limited packing in a graph, which generalizes a 2-packing in a graph, and give several bounds on the size of a k-limited packing. One such bound involves the domination number of the graph, and here we show all trees attaining the bound can be built via a simple sequence of operations. We consider graphs where every maximal 2-limited packing is a maximum 2-limited packing, and characterize their structure in a number of cases.     

Convex labelings of trees

A convex labeling of a tree T of order n is a one-to-one function f from the vertex set of T into the nonnegative integers, so that f(y) ? (f(x) + f(z))/2 for every path x, y, z of length 2 in T. If, in addition, f(v) ? n ? 1 for every vertex v of T, then f is a perfect convex labeling and T is called a perfectly convex tree. Jamison introduced this concept and conjectured that every tree is perfectly convex. We show that there exists an infinite class of trees, none of which is perfectly convex, and in fact prove that for every n there exists a tree of order n which requires a convex labeling with maximum value at least 6n/5 – 22. We also prove that every tree of order n admits a convex labeling with maximum label no more than n²/8 + 2. In addition, we present some constructive methods for obtaining perfect convex labelings of large classes of trees.     

Convergence of the newton process to multiple solutions

In 1870, E.Schröder showed that the convergence of the Newton process of successive approximations to a multiple solution of a scalar equation was geometric in character, and that quadratic convergence could be restored by multiplying the ordinary corrections by a constant. Here, this result is extended to finite systems, and it is shown that there exist various subspaces of the given space in which the convergence is geometric with a rate characteristic of the given subspace. Quadratic convergence may be restored by applying a given fixed linear operator to the ordinary corrections. The conditions under which these results apply to equations in infinite-dimensional Banach spaces are given. Numerical examples involving scalar equations and a simple 2 × 2 system are presented.Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No. DA-11-022-ORD-2059.