Duane W. DeTemple

Half integer approximations for the partial sums of the harmonic series

Abstract Let S n denote the n th partial sum of the harmonic series. An asymptotic approximation for S n is obtained in terms of the half integer variable n + 1 2 , including bounds for the remainder. Improved estimates of the Comtet function for the harmonic series are obtained as a consequence.

A Geometric Look at Sequences that Converge to Euler's Constant

In this article, we will generate and investigate sequences that converge to Eulers constant, y. What is novel h e is the elementary y t careful attention to the geomet ric descriptions of the t rms of the sequences, allowing us to obtain a converg nce ate of order 1/n2. G nerally, this improved rate can be ob ained only with a more cumbersom analytical analysis such as shown in [2], [3], and [4]. E lers c nstant is most often defined by c mpar g the natural logarithm, ln(n + 1), with the n h partial sum * 1 1 1 *. = ! + + + ... + -

A characterization and hereditary properties for partition graphs

Abstract A general partition graph is an intersection graph G on a set S so that for every maximal independent set M of vertices in G , the subsets assigned to the vertices in M partition S . It is shown that such graphs are characterized by there existing a clique cover of G so that every maximal independent set has a vertex from each clique in the cover. A process is described showing how to construct a partition graph with certain prescribed graph theoretic invariants starting with an arbitrary graph with similar invariants. Hereditary properties for various graph products are examined. It is also shown that partition graphs are preserved by removal of any vertex whose closed neighborhood properly contains the closed neighborhood of some other vertex.

Recent examples in the theory of partition graphs

Abstract A partition graph is an intersection graph for a collection of subsets of auniversal set S with the property that every maximal independent set of vertices corresponds to a partition of S . Two questions which arose in the study of partition graphs are answered by recently discovered examples. An enumeration of the partition graphs on ten or fewer vertices is provided.

Graphs associated with triangulations of lattice polygons

Two graphs, the edge crossing graph E and the triangle graph T are associated with a simple lattice polygon. The maximal independent sets of vertices of E and T correspond to the triangulations of the polygon into fundamental triangles. Properties of E and T are derived including a formula for the size of the maximal independent sets in E and T. It is shown that T is a factor graph of edge-disjoint 4-cycles, which gives corresponding geometric information, and is a partition graph as recently defined by the authors and F. Harary. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 05 C 99, 51 M 05, 52 A 43.

Squares Inscribed in Angles and Triangles

(1998). Squares Inscribed in Angles and Triangles. Mathematics Magazine: Vol. 71, No. 4, pp. 278-284.

On polygons circumscribing a closed convex curve

Length and area formulas for closed polygonal curves are derived, as functions of the vertex angles and the distances to the lines containing the sides. Applications of the formulas are made to the class of polygons which circumscribe a given convex curve and have a prescribed sequence of vertex angles. Geometric conditions are given for polygons in the class which have extremal perimeter or area.

A Geometric Method of Phase Plane Analysis

where the dot denotes the derivative with respect to the real variable t, and a, b, c, d are real constants. The usual method to study (1) is to introduce a real linear change of variables which transforms the coefficient matrix Q to a canonical (say, real Jordan) form. Our purpose here is to replace the algebraic approach by a geometric one. The development leads to a construction procedure by which the phase portrait can be quickly and accurately drawn in the original x, y-plane with no calculations whatever required. In Section 1 the real differential system is recast into an equivalent complex form involving both z = x + iy and the conjugate variable z = x -iy. Employing conjugate variable methods we derive in Section 2 some geometric results which are important in the sequel. In particular, the invariants of the coefficient matrix Q are identified geometrically in Section 3. In Section 4 the geometric description and construction procedures for phase portraits is illustrated. The concluding Section 5 shows the existence and construction of a homothetic family of conics isogonal to the trajectory system; the principal analytical tool in this section is the Schwarz function [2].

Permutations associated with billiard paths

Associated with a smooth closed convex curve C, a point P on C, and a natural number n>=3, is a billiard graph whose vertices are permutations on the set {1,2,...,n}. The graph is constructed and applied to billiard properties of C. Recursion properties of the graph as n increases are described with the aid of an appropriate generating function.

Colored Polygon Triangulations

(1998). Colored Polygon Triangulations. The College Mathematics Journal: Vol. 29, No. 1, pp. 43-47.