Region distributions of graph embeddings and stirling numbers

AbstractIt is shown that the distribution of the number of regions r in the random orientable embedding of the graph with one vertex and q loops is approximately proportional to the unsigned Stirling numbers of the first kind s(2q,r) where r has different parity from q. This approximation is strong enough to imply that both the limiting mean and variance of this distribution differ from ln 2q by small known constants. The paper concludes with a result on the unimodality of some recursively defined sequences and also some conjectures regarding region distributions of arbitrary graphs

Permutation-partition pairs. III. Embedding distributions of linear families of graphs

AbstractFor any fixed graph H, and H-linear family of graphs is a sequence {Gn}n=1∞ of graphs in which Gn consists of n copies of H that have been linked in a consistent manner so as to form a chain. Generating functions for the region distribution of any such family are found. It is also shown that the minimum genus and the average genus of Gn are essentially linear functions of n

The multichromatic numbers of some Kneser graphs

AbstractThe Kneser graph K(m,n) has the n-subsets of {1,2,…,m} as its vertices, two such vertices being adjacent whenever they are disjoint. The kth multichromatic number of the graph G is the least integer t such that the vertices of G can be assigned k-subsets of {1,2, …, t}, so that adjacent vertices of G receive disjoint sets. The values of Xk(K(m,n)) are computed for n = 2, 3 and bounded for n ⩾ 4

PLANETARY ORBITS IN CONSTANT CURVATURE PLANES

A law of gravitation is defined and justified for constant curvature planes and it is demonstrated that Kepler’s three laws of planetary motion have natural analogs in this new context

MASS IN THE HYPERBOLIC PLANE

Archimedes computed the center of mass of several regions and bodies [Di-jksterhuis], and this fundamental physical notion may very well be due to him. He based his investigations of this concept on the notion of moment as it is used in his Law of the Lever. A hyperbolic version of this law was formulated in the nineteenth century leading to the notion of a hyperbolic center of mass of two point-masses [Andrade, Bonola]. In 1987 Galperin proposed an axiomatic definition of the center of mass of finite systems of point-masses in Euclidean, hyperbolic and elliptic n-dimensional spaces and proved its uniqueness. His proof is based on Minkowskian, or relativistic, models and evades the issue of moment. A surprising aspect of this work is that hyperbolic mass is not additive. Ungar [2004] used the theory of gyrogroups to show that in hyperbolic geometry the center of mass of three point-masses of equal mass coincides with the point of intersection of the medians. Some information regarding the centroids of finite point sets in spherical spaces can be found in [Fog, Fabricius-Bjerre]. In this article we offer a physical motivation for the hyperbolic Law of the Lever and go on to provide a model-free definition and development of the notions of center of mass, moment, balance and mass of finite point-mass systems in hyperbolic geometry. All these notions are then extended to linear sets and laminae. Not surprisingly, the center of mass of the uniformly dense hyperbolic triangle coincides with the intersection of the triangle’s medians. However, it is pleasing that a hyperbolic analog of Archimedes’s mechanical method can be brought to bear on this problem. The masses of uniform disks and regular polygons are computed in the Gauss model and these formulas are very surprising. Other configurations are examined as well

HYPERBOLIC CENTROIDS OF SOME REGIONS

Explicit expressions for the centroids of hyperbolic pie shapes and isosce- les triangles are found and compared to their Euclidean analogs

MASS IN HYPERBOLIC GEOMETRY

Archimedes computed the center of mass of several regions and solid bodies [Dijksterhuis], and this fundamental physical notion may very well be due to him. He based his investigations of this concept on the notion of moment as it is used in his Law of the Lever. A hyperbolic version of this law was formulated in the nineteenth century leading to the notion of a hyperbolic center of mass of two point-masses [Andrade, Bonola]. In 1969 Perron extended the notions of mass and center of mass to arbitrary regions of hyperbolic space. In 1987 Gal’perin proposed an axiomatic definition of the center of mass of finite systems of point-masses in Euclidean, hyperbolic and elliptic n-dimensional spaces and proved its uniqueness. Ungar [2004] used the theory of gyrogroups to show that in hyperbolic geometry the center of mass of three point-masses of equal mass coincides with the point of intersection of the medians, a fact that had already been noted by Perron. Some information regarding the centroids of finite point sets in spherical spaces can be found in [Fog, Fabricius-Bjerre]. In this article we begin by offering yet another physical motivation for the hyperbolic Law of the Lever and summarize Perron’s treatment of the subjects of mass and centers of mass (centroids). The masses and centroids of several geometric objects are derived. Surprisingly, the hyperbolic mass formulas are quite similar to the Euclidean ones whereas, as is well known, the formulas for hyperbolic area and volume look nothing like their Euclidean analogs

MASS IN HYPERBOLIC 3-SPACE

Contents: 1. A hyperbolic Theorem of Pappus. 2. A hyperbolic version of Newton’s Theorem that the center of gravity and the center of mass of the uniform sphere are identical. 3. A hyperbolic version of the characterization of concurrent cevians for then-simplex