M.Sc. Dissertation Presentation … · M.Sc. Dissertation Presentation EXISTENCE OF GROUP DIVISIBLE DESIGNS ON TWO GROUPS,BLOCK SIZE FOUR WITH EQUAL NUMBER OF EVEN AND ODD BLOCKS

M.Sc. Dissertation Presentation

EXISTENCE OF GROUP DIVISIBLE DESIGNS ON TWO GROUPS,BLOCK SIZE FOUR WITH EQUAL

NUMBER OF EVEN AND ODD BLOCKS.

Issa Ndungo - 2013/MSc/001/PS. Department of Mathematics

Mbarara University of Sci. & Tech.

January, 2016.

Supervisor

Prof. Dinesh G. Sarvate Ph.D.

Presentation outline

Definition of terms (02 minutes)

Statement of the problem (03 minutes) Necessary conditions (02 minutes) GDD(n,2,4;3,4) (04 minutes) GDD(n,2,4; 1, 2) with equal number of Even & odd blocks (12 minutes) A new application of GDDs (02 minutes) Conclusion and further research. (02 minutes) Acknowledgements (01 minute)

1.1 Definition of terms

1. A block design D consists of a point set V and a collection of blocks b where each block is a subset of V.

2. A group divisible design GDD(n, m, k; 1, 2) is a collection of k-element

subsets, called blocks, of an nm-set X Where the elements of X are partitioned

into m subsets (called groups) of size n each;

1.1 Definition of terms…

Pairs of distinct elements within the same group are called first associates of each other and appear

together in 1 blocks. While any two distinct elements not in the same group are called second

associates and appear together in 2 blocks

3. A block is even if it intersects each group in two points. While it is said to be odd if it intersects one group in one point and the other in three points.

A GDD whose blocks are all even is an even GDD and has block configuration (2, 2) . While that with all its block odd is an odd GDD and has block configuration (1, 3).

Example 1.1

A GDD(3, 2, 4; 3, 2) with two groups {1,2,3} and {4,5,6} has blocks:

{1,2,3,4}, {1,2,3,5}, {1,2,3,6}, {4,5,6,1}, {4,5,6,2}, {4,5,6,3}.

Every element occur in exactly 4 blocks.

In any GDD every element occurs a fixed number of times (replication No. r).

Also there are only two groups of the same size, each block intersects each group in exactly three points or in exactly one point. This is an odd GDD.

5

2.1 Statement of the problem

Several mathematical studies have been carried out to settle specific problems pertaining GDDs; all of which center at variation in number of groups, block sizes and

the indices 1, 2. For example problems with two groups & block size four , three groups & block size four and even GDDs and odd GDDs were studied by Hurd & Sarvate, mixed GDDs with three groups and block size four were settled by Zhu &Ge. However little is known about the equality of even and odd blocks.

Here, the existence GDD(n, 2, 4; 1, 2) has been studied with an extra property “Equal number of even & odd blocks”.

3. Methodology Use of the existing definitions, theorems designs

and mathematical relations.

Combinatorial tools such as 1-factorisation, Latin squares, and resolvable BIBDs.

Problem solving techniques by Po`lya & Kilipatrick

(1) Is it possible to satisfy the conditions?

(2) Are the conditions sufficient to determine the unknown or more conditions are needed?

(3) Is a figure necessary and are the notation used suitable for the problem?

4.1 Necessary conditions

If b is the number of blocks of the GDD(n, 2, 4; 1, 2) then we want the number of even blocks and odd blocks to be .

Counting the number of first associate pairs in the GDD if it exists, we have 2* +3* = first associate pairs. Hence,

=n(n-1)1……….….(i)

Similarly, there are 3* +4* = second associate pairs in the GDD if it exists and hence

= n2 2 …….……….(ii)

2

b

2

b

2

b

2

5b

2

b

2

b

2

7b

2

7b

2

5b

Necessary conditions cont…..

From (i) and (ii) we have

Or

Hence 7 divides n2 , i.e., 7| n or 7| 2.

Our aim was to obtain minimum values of

indices of 1 and 2 such that other values of indices are multiples of the minimum values.

)1(7

5 21

n

n

n

n

5

)1(7 12

Necessary conditions cont…..

Hence there are four cases to consider. (1) gcd(n-1, 5)=1 and 7 does not divide n then

GDD(n,2,4;1 =5n, 2 =7(n-1)).

(2) gcd(n-1,5)=1 and 7|n, if n=7t,

GDD(n,2,4; 1 =5t, 2 =7t-1)

(3) gcd(n-1,5)=5 and 7does not divide n, if n=5t+1, GDD(n,2,4; 1 =5t+1, 2 =7t)

(4) gcd(n-1,5)=5 and 7|n. Here n21(mod 35), so if n=35s+21,

GDD(n,2,4; 1 =n/7=5s+3, 2 =7s+4)

Necessary conditions …….

Also we note that for GDD with two groups and block size four we have the following necessary conditions (from literature)

and

Since r and b must be integers then,

1.

2.

3

)1( 21 nnr

)3(mod0)1( 21 nn

6

)1( 2

2

1 nnnb

)6(mod0)1( 2

2

1 nnn

4.2 GDD(n,2,4;3,4)

Necessary conditions

From cond. (1) above,

Hence

Note that: In this design, the number of even blocks need not to be equal to number of odd blocks.

)3(mod0)1( 21 nn

)3(mod04)1(3 nn

)3(mod0n

GDD(n,2,4;3,4) cont. …

Example 4.2.1

For n=3, a GDD(3,2,4;3,4) with groups {1,2,3} and {4,5,6} exists and the blocks as illustrated below in columns

GDD(n,2,4;3,4) cont… Example 4.2.2

For n=6, a GDD(6,2,4;3,4) with groups {1,2,3,4,5,6} and {7,8,9,10,11,12} exists and the blocks are illustrated below in columns.

Note the total number of blocks are

396/4*6*65*6*3

6

)1( 2

2

1

nnn

b

Example 4.2.2 GDD(6,2,4;3,4) cont…..

GDD(n,2,4;3,4) cont… Example 4.2.3 For n=9, a GDD(9,2,4;3,4)

the blocks are as below

Each of these blocks in column is taken twice

Example 4.2.3 cont…

In addition we have the following blocks in columns to make a total of 90 blocks

GDD(n,2,4;3,4) cont…

Theorem #1: For a GDD(n, 2, 4; 3, 4) exists except

possibly at n=18. Hence necessary conditions are

sufficient for the existence of GDD(n, 2, 4; 3, 4).

Proof:

This construction does not work for small

parameters

So we assume that n= 3s, s 4 in view of the above

three examples on n=3,6 and 9.

)3(mod0n


We partition the elements of each group into

3 subgroups each with s points. I.e

G1= {{A=a1,a2,…,as},

{B= b1,b2,...,bs},

{C= c1,c2,,…cs}} and

G2={{D=d1,d2,…,ds},

{E=e1,e2,…,es},

{F=f1,f2,…,fs}}.

ai, di; bi,ei; and ci,fi will

be referred to as

corresponding

elements and the order

of the elements is fixed


We construct a pair of IMOLSs of order s and label the

rows, columns and entries as follows:

Form blocks {i, j, io1j, io2j} for each case. We get 3(s)(s-1)

odd blocks with 3 elements form G1. Repeat the same to get

3(s)(s-1) blocks with 3 elements form G2

ROWS AND COLUMNS

OF (L1 , o1) AND (L2, o2)

ENTRIES OF (L1 , o1) ENTRIES OF (L2 , o2)

A B E

B C F

C A D


Now apart from the corresponding elements and the

pairs (ai, bi), (ai, ci), (ai, ei), (ai, fi), (bi, ci), (bi, di),

(bi, fi), (ci, di), (ci, ei), (di, ei), (di, fi), (ei, fi)

which do not occur at all, all first and second

associates occur twice.

Now use the triples {ai, bi, ci}and {di, ei, fi} to create

3 even blocks {ai,bi,di, ei}, {ai,ci,di, fi} and {bi,ci,ei, fi}

each taken twice for i=1,2,…,s. Here all pairs of the

corresponding pairs occur 4 times and other pairs

occur twice.


We now construct a self orthogonal Latin square of order

3s. Use the block {i, j, ioj, joi} while relabeling i, j by

elements of G1 and ioj, joi by elements of G2 1 i j.

Here first associate pairs occur once and second associate

pairs occur twice except pairs of the corresponding

elements which do not occur.

Thus first associate pairs occur 3 times and second

associate pairs occur 4 times altogether and number of

blocks is 3s(3s-1)/2+6s+6s(s-1)= . #

2

)17(3 ss


Here the number of even blocks =

and number of odd blocks =

If we wish to have equal number of even and

odd blocks, we need = .

So s=7 and hence n=3*7=21.

Thus, GDD(21, 2, 4; 3, 4) with equal number

of even and odd blocks exists.

2

)17(9 ss

)1(6 ss

2

)17(9 ss )1(6 ss

4.3.1 GDD(n,2,4; 5t, 7t-1)

This is the case where gcd(n-1,5)=1 and 7|n, so n=7t a) For odd t

Theorem 2(a): The necessary conditions are sufficient for the existence of GDD(7t, 2, 4; 5t, 7t-1) for all odd t.

Proof.

Let t=2s+1. Let G1= {a1,…,a7t}& G2= {b1,…,b7t}.

(1) We construct BIBD(7t,3,3) on G1 constructed

using ISLS of order 7t. We construct odd blocks by

replacing the block {ai,aj,aioaj=ak} by {ai, aj, ak, bk }.

Repeat the same for G2 to get blocks of the form

{bi, bj, bk, ak }. This gives 7t(7t-1) odd blocks with

occurrence of pairs (ai, aj) and (bi, bj)=3, (ai, bk) =2,

(ak, bk)=7t-1 times

Theorem 2(a) cont….. Existence of GDD(n,2,4; 5t, 7t-1) for odd t


(2) Now take s copies of BIBD(7t,3,6) on G1 to

get 7st near 3-resolvable classes and for each

class missing ai, construct odd blocks by adding

bi . Eg if (a1,a2,a3) is a block from a classing

missing ai then construct block (a1,a2,a3,bi ).

Repeat the same for BIBD(7t,3,6) on G2. This

yields 2.7ts(7t-1) odd blocks with occurrence of

pairs as (ai, aj), (ai, bk) and (bi, bj)=6s


(3) For even blocks we use 2 IMOLSs of order

7t label the rows and columns by G1 and entries

by G2 & take t copies of {ai, aj, ai o1 aj, ai o2 aj }. This yields 7t(7t-1).t even blocks with

occurrence of pairs as (ai, aj) & (bi, bj)=2t and

(ai, bk)=4t

Thus in all first associates pairs occur 5t times

and second associates occur 7t-1 times #

4.3.2 GDD(n,2,4; 5s, 7s-1)

This is the case when gcd(n-1,5)=1 and 7|n, so n=7s b) For even s (We use s in order to make a distinction between variables of Theorem 2(a) and 2(b))

Theorem 2(b): The necessary conditions are sufficient for the existence of GDD(7s, 2, 4; 5s, 7s-1) for all even s.

Theorem 2(b) cont….. Existence of GDD(7s, 2, 4; 5s, 7s-1) for even s

Proof. Let s=2t; G1= {a1,a2,…,a14t}and

G2= {b1,b2,…,b14t}

(1) We take a near 3-resolvable BIBD(14t,3,6t) on

G1 to get 14t2 near 3-resolvable classes.

For t near 3-resolvable classes with ai missing, create

a block by adding in each triple a corresponding

element bi from G2.Repeat the same BIBD(14t,3,6t)

on G2 to get a total of 2t.14t(14t-1) odd blocks with

1st and 2nd ass. occurirng 6t times except the pair

(ai, bi) which do not occur.


(2) Take 4t copies of complete graph K14t on

G1 and G2 which has 2t.14t.(14t-1) pairs. Use

one copy of K14t on G1& G2 & for each edge

(ai, aj) of K14t on G1 Create a block (ai, aj, bi, bj)

to get even blocks in which 1st and

2nd ass. pairs occur once except the pair (ai,

bj) which occur 14t-1 times.

2

)114(14 tt


(3) To use the 14t-1 copies K14t left we take a

SOLS of size 14t, label the rows and columns

by a1,a2,…,a14t and the entries by b1,b2,…,b14t

and construct blocks (ai, aj, aio1aj, ajo2ai), 1<i<j,

take 14t-1 copies of the resulting blocks to get

blocks with 1st associate pairs

occur 14t-1 times and 2nd associate pairs occur

2(14t-1) times.

2

)114)(114(14 ttt

Theorem 2(a) cont….. Existence of GDD(7s, 2, 4; 5s, 7s-1) for even s

All together we have

2t(147)(14t-1) + +

=4t(14t)(14t-1) blocks.

With 1st associate pairs occurring

6t+1+4t-1=10t=5s times

2nd associate pairs occurring

6t+1+2(4t-1)=14t-1=7s-1 times #

2

)114(14 tt

2

)114)(114(14 ttt

Theorems 2(a) & 2(b) taken

together prove the main

theorem below:

Theorem #2

“The necessary conditions are

sufficient for the existence of

GDD(7t, 2, 4; 5t, 7t-1) in which

number of even and odd blocks

is equal for all t 1.

4.4.1 GDD(n, 2,4; 5n,7(n-1))

We know the following results (Hurd and Sarvate):

(a) Even GDD(n,2,4;ns,2(n-1)s) exist for n odd and even GDD(n,2,4; ,(n-1)s) exist for n even, . (b) The necessary conditions are sufficient for the existence of odd GDD(n,2,4;n,n-1) for 0, 1, 2, 3, 4(mod 6). (c) The necessary conditions are sufficient for the existence of odd GDD(n,2,4;3n, 3(n-1)) when (mod 6).

2

ns

1s

n

5,2n

GDD(n, 2,4; 5n,7(n-1)) cont..

Taking the union of blocks of an even GDD(n,2,4;2n,4(n-1)) and an odd GDD(n,2,4;3n,3(n-1)) we obtain the collections of blocks for GDD(n,2,4;5n,7(n-1)). Note the number of blocks in even GDD is equal to those in the odd GDD =n2(n-1). (this completes the case 1) Theorem #3 Necessary conditions are sufficient for the existence of GDD(n,2,4;5n,7(n-1)) for all n.

4.5. GDD(5t+1,2,4; 5t+1, 7t)

Theorem #4: The necessary conditions are sufficient for the existence of GDD(5t+1,2,4; 5t+1, 7t) for all even t. Theorem #5: The necessary conditions are sufficient for the existence of GDD(5t+1,2,4; 2(5t+1), 14t) for all t.

5.0 Simple application of designs in weather prediction

Here we suggest a method where by it is clear that full data set may not be necessary for good prediction. It is known that outliers can be and should be deleted from the data set but which are the outliers in weather prediction? Our data is a first step in deciding on this.

6.1 Conclusion

In this dissertation four families of GDDs

were realized and the following new

theorems have been proved (and a paper

which includes these results has been

accepted for publication with discrete math.

Journal-339(2016)1344-1354.

Also, there was an attempt to establish a new

application of a special case of GDDs called

BIBDs

Theorem #1: The necessary conditions are


GDD(7t,2,4;5t,7t-1) in which number of even

blocks and odd blocks is equal for all Theorem #2: The necessary conditions are

sufficient for the existence of:

a) GDD(5t+1, 2,4;2(5t+1),14t) for all t, and

b) GDD(5t+1, 2,4;5t+1,7t) for all even t with equal

number of even and odd blocks.

1t

Conclusion cont….

Theorem #3: Necessary conditions are


GDD(n,2,4;5n,7(n-1)) for all n, with equal

number of even and odd blocks.

Theorem #4: Necessary conditions are


GDD(n,2,4;3t,4t) for except at n=18

and for n=21, the GDD has equal number of

even and odd blocks.

Conclusion cont….

1t

6.2 Further research

To complete the problem on the existence of GDD(n,

2, 4; 1, 2) with equal number of even and odd

blocks two families are to be worked on:

1.GDD(5t+1,2,4;5t+1,7t) for odd t

2. The case of gcd(n-1,5)=5and 7|n that is

GDD(35s+21,2,4;5s+3,7s+4), for .

3. Also, there is a need to complete the work on the

application of GDDs in finding outliers in data

predictions.

2t

The following organisations supported this research in distinguished ways; and are

therefore are acknowledged

END **I THANK YOU ALL for listening ** & ** GOD BLESS YOU**

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M.Sc. Dissertation Presentation … · M.Sc. Dissertation Presentation EXISTENCE OF GROUP DIVISIBLE DESIGNS ON TWO GROUPS,BLOCK SIZE FOUR WITH EQUAL NUMBER OF EVEN AND ODD BLOCKS