Paul C. Kainen
Adjunct Associate Professor
Department of Mathematics and Statistics
Georgetown University

Last edited Dec. 20, 2013

For information on current courses and related material, please see my index page.

My research page needs updating.

The following papers are available here; see also "explore page" publications for some selected papers.

With Rachel Hunter: Quadrilateral embedding of G x Q_s, Bulletin of the Institute of Combinatorics and Its Applications, 52 (Sept.)(2007) 13--20. See a pdf copy of the paper .

Topological constancy in the perception of Lissajous figures. Draft paper of 10 pages, Jan. 24, 2005.

(pdf file) Replacing points by compacta in neural network approximation Journal of the Franklin Institute, Vol. 341, No. 4, pp. 391--399, July 2004. A subset M of a metric space X is called approximatively compact if for any x in X, any sequence in M which converges in distance to the infimum of the distances from x to m in M must contain a subsequence which is convergent to some element in M. In particular, the infimum is achieved. It is shown that for subsets A and B of a metric space X, A x B (the cartesian product) is approximatively compact (ac) when A is ac and B is compact. More briefly, the product of an ac and a compact set is ac. Since product with a point is the identity, this result embodies the title's description of a program proposed by Dugundji. It is also shown that A + B is ac when A is ac and B is compact, where A and B are subsets of an F-space and so of a normed linear space, and A + B {a + b: a in A and b in B}. The paper is dedicated to the memory of Hewitt Kenyon, who was the author's topology professor at George Washington University and supervised my honors' thesis.

(pdf file) On robust cycle bases , Proc. 9th Quadrennial Conf. on Graph Theory, Combinatorics, Algorithms and Applications, Ed. by Yousef Alavi, Dawn Jones, Don R. Lick and Jiuqiang Liu, Kalamazoo, MI, 4--9 June 2000; conference in honor of Yousef Alavi. Introduces cycle bases for graphs satisfying a recursive condition, applications to "forcing" commutativity of diagrams, especially cubes. This paper is available on the Elsevier site, Science Direct website (search for "robust cycle basis"); the citation is: "Electronic Notes in Discrete Mathematics," VOl. 11, (July 2002), article # 38, pp. 430--437.

(pdf file) Isolated squares in hypercubes and robustness of commutativity Cahiers de Topologie et Geom\'etrie Diff\'erentielle Cat\'egoriques, XLIII (2002) 213--220. On "blocking" the commutativity of cubical diagrams and statistical commutativity.

(pdf file) An octonion model for physics Fourth International Conference on Emergence, Coherence, Hierarchy, and Organization (ECHO IV), Odense, Denmark, July 31 -- Aug. 4, 2000. This paper contains some remarks on the octonions and their possible relevance for physics. There are also some connections with the 4-color theorem and quantum algebra.

With Shannon Overbay: Extension of a theorem of Whitney (pdf file) to appear in Applied Math Letters (AML 2315), Vol. 20, No. 7, July 2007. Here is an earlier version of this paper Book embeddings of graphs and a theorem of Whitney (Tech. Report GUGU-2/25/03) which has been cited in the literature.

Papy16 (with Vera Kurkova and Marcello Sanguineti, in a normed linear space, the error functional of a compact subset is well-posed in the generalized Hadamard sense, when restricted to an approx. compact subset)

(pdf file) Best approximation by linear combinations of characteristic functions of half-spaces, Journal of Approximation Theory, Volume 122, Number 2, June 2003, 151-159, with Vera Kurkova and Andrew Vogt, in L_p of the unit cube in d-dimensional space, p in [1,oo), for n a positive integer, the set of all n-fold linear combinations of half-space characteristic functions, restricted to the cube, is an approximatively compact set, so in particular best approx. exists using a fixed number of hidden units of Heaviside type in a feedforward neural net.

(pdf file) Best approximation by Heaviside perceptron networks Neural Networks 13(7) (2000) 695-697, with Vera Kurkova and Andrew Vogt. This was an announcement of the results proved in JAT 2003 above and considered the application to neural networks.

(pdf file) A graph-theoretic model for time, appeared in Computing Anticipatory Systems, Daniel M. Dubois, Ed., American Institute of Physics Conf. Proc. #573, 2001, pp. 490--495.) A graph model for time which extends the usual path-model by making two vertices adjacent if they have distance at most 3 along the path; by a result of Harary and Kainen, the model is maximal planar. The specific form of the model as ``cube of a path'' produces various combinatorial and analytic properties which are described here. This paper won a prize for best in its session at the conference.

A rather old resume is available.

List of all writings (reports, abstracts, preprints and papers): publications . Will be updated eventually.