Search

The article explores the so-called λ-permutations shown by Steven Krantz and Jeffery McNeal in their paper "Creating more convergent series." They specified that a permutation σ of the positive integers is a λ-permutation if it has two characteristics. The first property is that for every convergent series, the summation of ak from k =1 to infinity, the summation of aσ(k) from k =1 to infinity also converges. The second property is that there is at least one divergent series which is the summation of ak from k =1 to infinity such that the summation of aσ(k) from k =1 to infinity converges.

The article provides two short and direct proofs of J. S. Pym's result in his 1969 paper "The linking of sets in graphs," regarding the Cantor-Bernstein theorem for paths in graphs. Both proofs are elementary, and they can be examined independently. The first proof employs transfinite induction. This proof starts from the first set of paths which is then transformed step by step into the wanted bijective set by integrating path segments from the second set. The choices made during this inductive process have been made arbitrary. For the second proof, these choices are made explicitly.

The article reports on the k-dependence and domination in king graphs. It states that the article concerns the struggle to support the conjecture presented by good upper bounds and as the variety of rival techniques used for different values of k. The author also presents the use of graph theory, number theory, group theory, real analysis, and integer linear programming. He also notes on the adoption of notation and terminology from graph theory by referring to board squares as vertices. Several images of the graph are presented.

The article discusses the combinatorial topology of barycentric subdivision and the characters of simplical complexes. Barycentric subdivision and triangulations are mathematical ideas that have originated from the root of combinatorial topology. The homology groups of simplical complex are invariant under subdivision and represent an important fact in the development of simplical homology theory. Investigations are being conducted to study the geometric realization of the complex in Euclidean space and analyze the dynamics of the features of successive triangulations.

The article examines the so-called mean paradox which was discovered during a departmental research project in which certain statistics related to the average length of the cycles in a random permutation had to be computed. When each of the n! permutations on n symbols is presented as a product of disjoint cycles, the mean cycle length, averaged over all these cycles, is asymptotic to n/log n. However, a computer simulation showed that the mean cycle length was substantially bigger than the n/log n expected. It is hereby demonstrated that there is a difference in averaging over all the cycles compared to determining the mean cycle lengths of the individual permutations and then calculating the average over all the n! permutations.

Discusses the approaches taken by mathematicians to see the same problem from four different viewpoints. Algebraic solution; Hyperplane solution; Zonotope solution; Combinatorial solution.