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- Frank Bernhart, Paul C. Kainen
- J. Comb. Theory, Ser. B
- 1979

There are several geometric invariants which have been studied extensively for graphs-among them, genus and thickness. In this paper we introduce a new invariant defined by considering embeddings ofâ€¦ (More)

- Paul C. Kainen
- Electronic Notes in Discrete Mathematics
- 2002

Two types of robust cycle bases are defined via recursively nice arrangements; complete and bipartite complete graphs are shown to have such bases. It is shown that a diagram in a groupoid isâ€¦ (More)

- Paul C. Kainen
- 2003

It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which everyâ€¦ (More)

- Paul C. Kainen
- 1972

Very few results are known which yield the crossing number of an infinite class of graphs on some surface. In this paper it is shown that by taking the class of graphs to be Â¿-dimensional cubes Q(d)â€¦ (More)

- Vera KurkovÃ¡, Paul C. Kainen
- Neural Computation
- 1994

For a feedforward perceptron type architecture with a single hidden layer but with a quite general activation function, we characterize the relation between pairs of weight vectors determiningâ€¦ (More)

- Paul C. Kainen, Vera KurkovÃ¡, Andrew Vogt
- Journal of Approximation Theory
- 2003

It is shown that for any positive integer n and any function in Lp([0, 1]) with p âˆˆ [1,âˆž) there exists a best approximation by linear combinations of n characteristic functions of half-spaces.â€¦ (More)

- Paul C. Kainen, Arthur T. White
- Journal of Graph Theory
- 1978

- Paul C. Kainen, Vera KurkovÃ¡, Marcello Sanguineti
- IEEE Transactions on Information Theory
- 2012

The role of input dimension <i>d</i> is studied in approximating, in various norms, target sets of <i>d</i>-variable functions using linear combinations of adjustable computational units. Resultsâ€¦ (More)

- Paul C. Kainen
- 2004

A connection is investigated between integral formulas and neural networks based on the Heaviside function. The integral formula developed by KÅ¯rkovÃ¡, Kainen and Kreinovich is derived in a new wayâ€¦ (More)

- Vera KurkovÃ¡, Paul C. Kainen
- Neural Networks
- 2014

The role of width of Gaussians in two types of computational models is investigated: Gaussian radial-basis-functions (RBFs) where both widths and centers vary and Gaussian kernel networks which haveâ€¦ (More)