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Packing and covering in combinatorics
Book Reviews With regaLrdsto the contents; Chapters 3 - 7 (158 pages) deal with decision problems, the decisions being based upon concepts like the Neyman-Pearson criterion, the Bayes risk criterion and the minimax criterion. The sequential decision problem is treated in Chapter 6 whilst in Chapter 7, two nonparametric decision rules are mentioned (the sign test and the signed rank test). The last foui- chapters (73 pages) deal with estimation problems ,,~tartingwith basic concepts like the maximum likelihood estimator and the least-squares estimator. The most interesting chapters of the book are Chapters 9 and 11 that contain topics which are not standard in books about statistical inference. Chapter 9 is concerned with the estimation of a gaussian vector in the presence of gaussian noise; a linear, as well as a nonlinear model, is discussed. The last chapter deals with state estimation, in particular, with Kalman filters. Every chapter is concluded by a section in the form of exercises. To be honest, the book is rather disjointed in the sense that there is no covering theory and, personally, I prefer that electrical engineering students should receive a more general introduction to statistics.
P.C. SANDER Eindhoven University o f Technology Eindhoven, Netherlands A. SCHRIJVER (Ed.)
Packing and Covering in Combinatofies Mathematical Centre Tracts, 106 Mathematical Centre, Amsterdam, 1979, 313 pages, Dfl. 38.00 Both in combinatorial and m metricai ,~ontexts one is often interested in one of the following problems: (i) how many copies of a given set can be placed in the whole space without overlap but possibly with gaps in between? (packing problem); (ii) how many copies of a given set do you need at least in order to cover the whole space without gaps but possibly with overlap? (covering problem). From these kinds of problems both classical and modern resdts were presente6 during a study week 'Packing and covering' at the Mathematical Centre Amsterdam, 1978, and have been collected in this book.
73
A brief survey of its contents: (1) Some combinatorial co ncept s. (2) A. Sehrijver, Some background information from linear algebra. (3) I~ Haemers, Eigenvalue methods. (A packing in a graph can be seen as a set of mutually nonadjacent vertices. Bounding properties of packings can be . expressed in terms of the eigenvalues of the adjacency matrix.) (4) A.E. Brouwer and A. Schri]ver, Uniform hypergraphs. (Packing and covering problems of a fixed nset by a collection of k-subsets.) (5) A.E. Brouwer, Wilson's theory. (Asymptotic existence theorems for balanced incomplete block designs.) (6) A.E. Brouwer, Packing and covering of (~-sets. (Bounds on the number of k-sets that can be chosen in a given n-set that have intersection properties in terms of t-sets.) (7) A.E. Brouwer and M. Voorhoeve, Tur~in theory and the lotto problem. (How many lotto forms should be filled in at least, in order to be assured of winning a prize?) (8) H.M. Mulder, Ramsey theory. (Colouring graphs in an arbitrary way with a fixed number of colours, and looking for monochromatic subgraphs of a specific kind.) (9) M.R. Best, Optimal codes. (Construction of binary codes meeting given packing bounds, in particular the linear programming bound.) (1 O)J.H. van Lint, Sphere-packings, codes, lattices and theta-functions. (Relations between theta-functions and certain linear codes.) (11) A. Bos, Sphere-packings in Euclidean space. (Densest known packings of Euclidean space with congruent spheres. How many disjoint unit balls can touch a given unit ball?) (12) F. G6bel, Geometrical packing and covering problems. (Packing, covering and tiling of sets in the Euclidean plane.) (13) A. Schri/ver, Fractional packing and covering. (A weaker form of packing and covering of hypergraphs that has many applications.) (14) J.K. Lenstra and A.H.G. Rinnooy Kan, Complexity of packing, covering and partitioning problems. The bt-ok contains an extensive list of references "and is recommended by the reviewer.
K.A. POST Eindlzoven University o f Technology Eindlzoven, Netherlands.
P.C. SANDER Eindhoven University o f Technology Eindhoven, Netherlands A. SCHRIJVER (Ed.)
Packing and Covering in Combinatofies Mathematical Centre Tracts, 106 Mathematical Centre, Amsterdam, 1979, 313 pages, Dfl. 38.00 Both in combinatorial and m metricai ,~ontexts one is often interested in one of the following problems: (i) how many copies of a given set can be placed in the whole space without overlap but possibly with gaps in between? (packing problem); (ii) how many copies of a given set do you need at least in order to cover the whole space without gaps but possibly with overlap? (covering problem). From these kinds of problems both classical and modern resdts were presente6 during a study week 'Packing and covering' at the Mathematical Centre Amsterdam, 1978, and have been collected in this book.
73
A brief survey of its contents: (1) Some combinatorial co ncept s. (2) A. Sehrijver, Some background information from linear algebra. (3) I~ Haemers, Eigenvalue methods. (A packing in a graph can be seen as a set of mutually nonadjacent vertices. Bounding properties of packings can be . expressed in terms of the eigenvalues of the adjacency matrix.) (4) A.E. Brouwer and A. Schri]ver, Uniform hypergraphs. (Packing and covering problems of a fixed nset by a collection of k-subsets.) (5) A.E. Brouwer, Wilson's theory. (Asymptotic existence theorems for balanced incomplete block designs.) (6) A.E. Brouwer, Packing and covering of (~-sets. (Bounds on the number of k-sets that can be chosen in a given n-set that have intersection properties in terms of t-sets.) (7) A.E. Brouwer and M. Voorhoeve, Tur~in theory and the lotto problem. (How many lotto forms should be filled in at least, in order to be assured of winning a prize?) (8) H.M. Mulder, Ramsey theory. (Colouring graphs in an arbitrary way with a fixed number of colours, and looking for monochromatic subgraphs of a specific kind.) (9) M.R. Best, Optimal codes. (Construction of binary codes meeting given packing bounds, in particular the linear programming bound.) (1 O)J.H. van Lint, Sphere-packings, codes, lattices and theta-functions. (Relations between theta-functions and certain linear codes.) (11) A. Bos, Sphere-packings in Euclidean space. (Densest known packings of Euclidean space with congruent spheres. How many disjoint unit balls can touch a given unit ball?) (12) F. G6bel, Geometrical packing and covering problems. (Packing, covering and tiling of sets in the Euclidean plane.) (13) A. Schri/ver, Fractional packing and covering. (A weaker form of packing and covering of hypergraphs that has many applications.) (14) J.K. Lenstra and A.H.G. Rinnooy Kan, Complexity of packing, covering and partitioning problems. The bt-ok contains an extensive list of references "and is recommended by the reviewer.
K.A. POST Eindlzoven University o f Technology Eindlzoven, Netherlands.