2,724 research outputs found

    Group divisible designs of four groups and block size five with configuration (1; 1; 1; 2)

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    We present constructions and results about GDDs with four groups and block size five in which each block has Configuration (1,1,1,2)(1, 1, 1, 2), that is, each block has exactly one point from three of the four groups and two points from the fourth group. We provide the necessary conditions of the existence of a GDD(n,4,5;λ1,λ2)(n, 4, 5; \lambda_1, \lambda_2) with Configuration (1,1,1,2)(1, 1, 1, 2), and show that the necessary conditions are sufficient for a GDD(n,4,5;λ1,(n, 4, 5; \lambda_1, λ2)\lambda_2) with Configuration (1,1,1,2)(1, 1, 1, 2) if n≢0(n \not \equiv 0 (mod 6)6), respectively. We also show that a GDD(n,4,5;2n,6(n1))(n, 4, 5; 2n, 6(n - 1)) with Configuration (1,1,1,2)(1, 1, 1, 2) exists, and provide constructions for a GDD(n=2t,4,5;n,3(n1))(n = 2t, 4, 5; n, 3(n - 1)) with Configuration (1,1,1,2)(1, 1, 1, 2) where n12n \not= 12, and a GDD(n=6t,4,5;4t,2(6t1))(n = 6t, 4, 5; 4t, 2(6t - 1)) with Configuration (1,1,1,2)(1, 1, 1, 2) where n6n \not= 6 and 1818, respectively

    Fixed block configuration group divisible designs with block size six

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    AbstractWe present constructions and results about GDDs with two groups and block size six. We study those GDDs in which each block has configuration (s,t), that is in which each block has exactly s points from one of the two groups and t points from the other. We show the necessary conditions are sufficient for the existence of GDD(n,2,6;λ1,λ2)s with fixed block configuration (3,3). For configuration (1,5), we give minimal or near-minimal index examples for all group sizes n≥5 except n=10,15,160, or 190. For configuration (2,4), we provide constructions for several families of GDD(n,2,6;λ1,λ2)s

    Fixed block configuration GDDs with block size 6 and (3, r)-regular graphs

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    Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Most of the definitions used in the thesis will be defined, and we provide a basic survey of topics in graph theory and design theory pertinent to the topics studied in this thesis. In Chapter 2, we are concerned with the study of fixed block configuration group divisible designs, GDD(n; m; k; λ1; λ2). We study those GDDs in which each block has configuration (s; t), that is, GDDs in which each block has exactly s points from one of the two groups and t points from the other. Chapter 2 begins with an overview of previous results and constructions for small group size and block sizes 3, 4 and 5. Chapter 2 is largely devoted to presenting constructions and results about GDDs with two groups and block size 6. We show the necessary conditions are sufficient for the existence of GDD(n, 2, 6; λ1, λ2) with fixed block configuration (3; 3). For configuration (1; 5), we give minimal or nearminimal index constructions for all group sizes n ≥ 5 except n = 10, 15, 160, or 190. For configuration (2, 4), we provide constructions for several families ofGDD(n, 2, 6; λ1, λ2)s. Chapter 3 addresses characterizing (3, r)-regular graphs. We begin with providing previous results on the well studied class of (2, r)-regular graphs and some results on the structure of large (t; r)-regular graphs. In Chapter 3, we completely characterize all (3, 1)-regular and (3, 2)-regular graphs, as well has sharpen existing bounds on the order of large (3, r)- regular graphs of a certain form for r ≥ 3. Finally, the appendix gives computational data resulting from Sage and C programs used to generate (3, 3)-regular graphs on less than 10 vertices

    Interlocking structure design and assembly

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    Many objects in our life are not manufactured as whole rigid pieces. Instead, smaller components are made to be later assembled into larger structures. Chairs are assembled from wooden pieces, cabins are made of logs, and buildings are constructed from bricks. These components are commonly designed by many iterations of human thinking. In this report, we will look at a few problems related to interlocking components design and assembly. Given an atomic object, how can we design a package that holds the object firmly without a gap in-between? How many pieces should the package be partitioned into? How can we assemble/extract each piece? We will attack this problem by first looking at the lower bound on the number of pieces, then at the upper bound. Afterwards, we will propose a practical algorithm for designing these packages. We also explore a special kind of interlocking structure which has only one or a small number of movable pieces. For example, a burr puzzle. We will design a few blocks with joints whose combination can be assembled into almost any voxelized 3D model. Our blocks require very simple motions to be assembled, enabling robotic assembly. As proof of concept, we also develop a robot system to assemble the blocks. In some extreme conditions where construction components are small, controlling each component individually is impossible. We will discuss an option using global controls. These global controls can be from gravity or magnetic fields. We show that in some special cases where the small units form a rectangular matrix, rearrangement can be done in a small space following a technique similar to bubble sort algorithm

    Designs and binary codes from maximal subgroups and conjugacy classes of ({rm M}_{11})

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    By using a method of construction of block-primitive and point-transitive 1-designs, in this paper we determine all block-primitive and point-transitive 1-((v, k, lambda))-designs from the maximal subgroups and the conjugacy classes of elements of the small Mathieu group ({rm M}_{11}). We examine the properties of the 1-((v, k, lambda))-designs and construct the codes defined by the binary row span of their incidence matrices. Furthermore, we present a number of interesting (Delta)-divisible binary codes invariant under ({rm M}_{11})

    Imprimitive flag-transitive symmetric designs

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    AbstractA recent paper of O'Reilly Regueiro obtained an explicit upper bound on the number of points of a flag-transitive, point-imprimitive, symmetric design in terms of the number of blocks containing two points. We improve that upper bound and give a complete list of feasible parameter sequences for such designs for which two points lie in at most ten blocks. Classifications are available for some of these parameter sequences
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