Combination discussion

timer Asked: Mar 20th, 2014

Question Description

i need this within 3 hours only (i need mentioned probs only).. 



i can provide notes as well..if anyone can help

201401 Math 322 Assignment 5 Due: Friday, March 21, 2014, before the lecture starts. Talking to each other about assigned work is a healthy practice that is encouraged. It is expected that, in the end, each person’s solution will be written in her own words and reflect her own understanding. There are 5 problems, worth a total of 30 marks, plus a bonus problem worth 4 bonus marks, on two pages. 1. For each of the following, either fill in the missing values to obtain the full collection of parameters (v, b, r, k, λ) of a design or demonstrate that no design with these parameters can exist. (a) [2] b = 35, k = 3, λ = 1. (b) [2] v = 14, b = 7, r = 4. (c) [2] r = 6, k = 4, λ = 2. (d) [2] v = 21, b = 28, k = 3. 2. [5] Let v, b, r, k, and λ be integers with k < v, Let A be a v × b 0-1 matrix. Prove that A is the incidence matrix of a (v, k, λ)-design if and only if AAt = (r − λ)I + λJv×v and Jv×v A = kJv×b , where Js×t is an s × t matrix with every entry equal to 1. 3. [3] Suppose (V, B) is a (v, k, λ)-design such that no two blocks are the same and not every k-subset of V is a block. Let B be the collection of k-subsets of V that do not belong to B. Show that (V, B) is a design and find its parameters. 4. A (v, k, λ)-design is called resolvable if its blocks can be partitioned into r groups (or parallel classes), each of which is a partition of the point set V. (a) [2] Show that the condition k | v is necessary for the existence of a resolvable (v, k, λ)-design and, if this condition and (k − 1) | λ(v − 1) are satisfied, then so is k(k − 1) | λv(v − 1) (so that the latter conditon is not needed). (b) [5] Show that if a (v, k, λ)-design is resolvable, then b ≥ v + r − 1. (Hint: in the incidence matrix A, the collections of columns corresponding to the parallel classes all have the same sum; now apply some results from linear algebra to reduce the upper bound on the rank of A .) 5. (a) [2] Make a formal definition of isomorphism of BIBDs D1 = (V1 , B1 ) and D2 = (V2 , B2 ). (b) [2] Prove that, up to isomorphism, there is only one (v, 2, 1)-design. (c) [3] Let D be the (15, 7, 3)-design with V = {0, 1, . . . , 9, A, B, C, D, E} whose blocks (with brackets and commas omitted) are: 0123456, 012789A, 012BCDE, 03478BC, 0349ADE, 05678DE, 0569ABC, 13579BD, 1367ACE, 1458ABE, 14689CD, 2358ACD, 23689BE, 24579CE, 2467ABD. Construct the dual design Dd and show that D and Dd are not isomorphic. (Hint: look at the number of triples of blocks that intersect in three elements.) 6. (Bonus Problem) Suppose a league schedule for 2n teams is to be constructed, with each pair of teams playing exactly once. (In such a schedule, n games are played simultaneously on each day the league operates.) It is desirable that each team should alternate home and away games as much as possible. Define a break in this pattern to be a repetition of being home, or of being away, in two consecutive games. (a) [2] Show that the league schedule must contain at least 2n − 2 breaks. (b) [2] Show how to construct a schedule with exactly 2n − 2 breaks. Page 2

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