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IBM Journal of Research and Development

Issue 5 • Date Nov. 1960

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Displaying Results 1 - 15 of 15
  • Foreword [Combinatorial Problems]

    Publication Year: 1960, Page(s): 454
    IEEE is not the copyright holder of this material | PDF file iconPDF (234 KB)
    Freely Available from IEEE
  • Traces, Term Ranks, Widths and Heights

    Publication Year: 1960, Page(s):455 - 459
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (580 KB)

    The notions of widths and heights of (0, 1)-matrices are discussed in the general setting of known results concerning traces and term ranks. Proofs are omitted throughout. View full abstract»

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  • Automorphisms of Steiner Triple Systems

    Publication Year: 1960, Page(s):460 - 472
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (1145 KB)

    This paper treats the following problem in combinatorial analysis: Find an incomplete balanced block design D with parameters b, v, r, k, and λ = 1. possessing an automorphism group G which is doubly transitive on the elements of D and such that the subgroup H of G fixing all the elements of a block is transitive on the remaining elements. Also find transitive extensions of such groups G. I... View full abstract»

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  • The Enumeration of Trees by Height and Diameter

    Publication Year: 1960, Page(s):473 - 478
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (592 KB)

    The enumeration of trees which was begun by Harary and Prins, is simplified and elaborated in the interest of obtaining reliable numerical results. Height is a characteristic of a rooted tree, the length in lines of the longest path from the root, while the diameter of a (free) tree is the length of the longest path joining two endpoints. The most general enumerations given are for the case where ... View full abstract»

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  • Maximal Paths on Rectangular Boards

    Publication Year: 1960, Page(s):479 - 486
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (708 KB)

    A combinatorial approach is made to the problem of obtaining a path on a rectangular board of m by n squares with both terminals at the edges of the board. A square is said to be covered when the path enters one edge and leaves an adjacent edge. All other squares are said to be missed. Maximal paths are found, i.e., those which cover a maximum number of squares. For m = n, m − 2 squares are... View full abstract»

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  • On the Exceptional Case in a Characterization of the Arcs of a Complete Graph

    Publication Year: 1960, Page(s):487 - 496
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (723 KB)

    It is known that a certain simple set of properties characterize the relationship of adjacency of the n(n − 1)/2 arcs of the complete graph of order n, when n ≠ 8, and that these properties are not a sufficient characterization when n = 8 (see [1], [2], [3], [4], and [7]). The present paper describes a method for enumerating all counter-examples when n = 8. View full abstract»

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  • On Moore Graphs with Diameters 2 and 3

    Publication Year: 1960, Page(s):497 - 504
    Cited by:  Papers (27)
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (689 KB)

    This note treats the existence of connected, undirected graphs homogeneous of degree d and of diameter k, having a number of nodes which is maximal according to a certain definition. For k = 2 unique graphs exist for d = 2,3,7 and possibly for d = 57 (which is undecided), but for no other degree. For k = 3 a graph exists only for d = 2. The proof exploits the characteristic roots and vectors of th... View full abstract»

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  • Inductive Proof of the Simplex Method

    Publication Year: 1960, Page(s):505 - 506
    Cited by:  Papers (2)
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (360 KB)

    Instead of the customary proof of the existence of an optimal basis in the simplex method based on perturbation of the constant terms, this paper gives a new proof based on induction. From a pedagogical point of view it permits an earlier and more elementary proof of the fundamental duality theorem via the simplex method. Specifically we shall show that there exists a finite chain of feasible basi... View full abstract»

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  • Solving a Matrix Game by Linear Programming

    Publication Year: 1960, Page(s):507 - 517
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (800 KB)

    This paper presents (1) a new characterization, via linear programming, of extreme optimal strategies of a matrix game and (2) a simple direct procedure for computing them. The first pertains to the neat formulas of L. S. Shapley and R. N. Snow for a “basic solution”, and the second to the highly effective “simplex method” of G. B. Dantzig. Both are related to the autho... View full abstract»

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  • Some Combinatorial Lemmas in Topology

    Publication Year: 1960, Page(s):518 - 524
    Cited by:  Papers (15)
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (722 KB)

    For many years it has been known that a combinatorial result, called the Sperner Lemma, provides an elegant proof of the Brouwer Fixed Point Theorem. Although the proof is elementary, its complete formal exposition depends upon the somewhat complicated operation of subdividing a simplex. Also, the proof does not show whether the Sperner Lemma can be derived from the Brouwer Fixed Point Theorem. Th... View full abstract»

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  • Minimal Complete Relay Decoding Networks

    Publication Year: 1960, Page(s):525 - 531
    Cited by:  Papers (3)
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (843 KB)

    The standard relay-contact-tree network has been used extensively for many years. If n is the number of relays involved, it has always been assumed that the 2(2n−1) contacts used in the standard tree network is the smallest possible number of contacts with which such a network could be made. This paper proves that this is true, provided no sneak paths are allowed. This is in cont... View full abstract»

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  • A Bound for Error-Correcting Codes

    Publication Year: 1960, Page(s):532 - 542
    Cited by:  Papers (52)
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (840 KB)

    This paper gives two new bounds for the code word length n which is required to obtain a binary group code of order 2k with mutual distance d between code words. These bounds are compared with previously known bounds, and are shown to improve upon them for certain ranges of k and d. Values of k and d are given for which one of these bounds can actually be achieved; in such cases, the st... View full abstract»

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  • Minimization over Boolean Trees

    Publication Year: 1960, Page(s):543 - 558
    Cited by:  Papers (12)
    IEEE is not the copyright holder of this material | Click to expandAbstract | PDF file iconPDF (1431 KB)

    An algorithm is provided for what might be termed the general problem of logical design of circuits with one output and no feedback. Given a set B of logical building blocks, each with a positive cost, each with one output, and given a Boolean function f the problem is to prescribe a Boolean tree constructed from the available set of building blocks which realizes f and which has a minimum cost. A... View full abstract»

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  • Authors

    Publication Year: 1960, Page(s):559 - 560
    IEEE is not the copyright holder of this material | PDF file iconPDF (345 KB)
    Freely Available from IEEE
  • Contents of Previous Two Issues

    Publication Year: 1960, Page(s): 561
    IEEE is not the copyright holder of this material | PDF file iconPDF (258 KB)
    Freely Available from IEEE

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The IBM Journal of Research and Development is a peer-reviewed technical journal, published bimonthly, which features the work of authors in the science, technology and engineering of information systems.

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Editor-in-Chief
Clifford A. Pickover
IBM T. J. Watson Research Center