Extending Fisher’s inequality to gfarr/research/slides/Horsley-May2016Fisher... ·...
date post
02-Jan-2019Category
Documents
view
213download
0
Embed Size (px)
Transcript of Extending Fisher’s inequality to gfarr/research/slides/Horsley-May2016Fisher... ·...
Extending Fishers inequality to coverings
Daniel Horsley (Monash University, Australia)
Introduction 1Designs and Fishers inequality
(v, k, )-designs
A collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.
12
3
4
56
7
8
9
A (9,3,1)-design with 12 blocks
Necessary conditions for a design to exist
Obvious necessary conditions: If there exists a (v, k, )-design then
(1) r = (v 1)k 1 is an integer;
(2) b = rvk is an integer.
Necessary conditions for a design to exist
Obvious necessary conditions: If there exists a (v, k, )-design then
(1) r = (v 1)k 1 is an integer;
(2) b = rvk is an integer.
Necessary conditions for a design to exist
Obvious necessary conditions: If there exists a (v, k, )-design then
(1) r = (v 1)k 1 is an integer;
(2) b = rvk is an integer.
12
3
4
56
7
8
9
Possible (v,10,1)-designs
46 55 91 100 136 145 181 190v0
100
200
300
400
b
Eventually all exist
Possible (v,10,1)-designs
46 55 91 100 136 145 181 190v0
100
200
300
400
b
Eventually all exist
Possible (v,10,1)-designs
46 55 91 100 136 145 181 190v0
100
200
300
400
b
Eventually all exist
Necessary conditions for a design to exist
Obvious necessary conditions: If there exists a (v, k, )-design then
(1) r = (v 1)k 1 is an integer;
(2) b = rvk is an integer.
Fishers inequality (1940): There is no (v, k, )-design with v < k(k1) + 1.
Equivalently,I with b < v; orI with r < k.
Symmetric designs have v = k(k1) + 1 (or b = v or r = k).
Necessary conditions for a design to exist
Obvious necessary conditions: If there exists a (v, k, )-design then
(1) r = (v 1)k 1 is an integer;
(2) b = rvk is an integer.
Fishers inequality (1940): There is no (v, k, )-design with v < k(k1) + 1.
Equivalently,I with b < v; orI with r < k.
Symmetric designs have v = k(k1) + 1 (or b = v or r = k).
Necessary conditions for a design to exist
Obvious necessary conditions: If there exists a (v, k, )-design then
(1) r = (v 1)k 1 is an integer;
(2) b = rvk is an integer.
Fishers inequality (1940): There is no (v, k, )-design with v < k(k1) + 1.
Equivalently,I with b < v; orI with r < k.
Symmetric designs have v = k(k1) + 1 (or b = v or r = k).
Necessary conditions for a design to exist
Obvious necessary conditions: If there exists a (v, k, )-design then
(1) r = (v 1)k 1 is an integer;
(2) b = rvk is an integer.
Fishers inequality (1940): There is no (v, k, )-design with v < k(k1) + 1.
Equivalently,I with b < v; orI with r < k.
Symmetric designs have v = k(k1) + 1 (or b = v or r = k).
Possible (v,10,1)-designs
46 55 91 100 136 145 181 190v0
100
200
300
400
b
Possible (v,10,1)-designs
46 55 91 100 136 145 181 190v0
100
200
300
400
b
Possible (v,10,1)-designs
46 55 91 100 136 145 181 190v0
100
200
300
400
b
Possible (v,10,1)-designs
46 55 91 100 136 145 181 190v0
100
200
300
400
b
Incidence matrix arithmetic
Consider the incidence matrix of our (9,3,1)-design.
12 blocks
9po
ints
1 0 0 1 0 0 1 0 0 1 0 01 1 0 0 1 0 0 1 0 0 0 01 0 1 0 0 1 0 0 0 0 1 00 1 0 1 0 0 0 0 1 0 1 00 0 1 0 1 0 1 0 1 0 0 00 0 0 0 0 1 0 1 1 1 0 00 0 1 1 0 0 0 1 0 0 0 10 0 0 0 1 0 0 0 0 1 1 10 1 0 0 0 1 1 0 0 0 0 1
Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.
12 blocks
9po
ints
1 0 0 1 0 0 1 0 0 1 0 01 1 0 0 1 0 0 1 0 0 0 01 0 1 0 0 1 0 0 0 0 1 00 1 0 1 0 0 0 0 1 0 1 00 0 1 0 1 0 1 0 1 0 0 00 0 0 0 0 1 0 1 1 1 0 00 0 1 1 0 0 0 1 0 0 0 10 0 0 0 1 0 0 0 0 1 1 10 1 0 0 0 1 1 0 0 0 0 1
Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.
12 blocks
9po
ints
1 0 0 1 0 0 1 0 0 1 0 01 1 0 0 1 0 0 1 0 0 0 01 0 1 0 0 1 0 0 0 0 1 00 1 0 1 0 0 0 0 1 0 1 00 0 1 0 1 0 1 0 1 0 0 00 0 0 0 0 1 0 1 1 1 0 00 0 1 1 0 0 0 1 0 0 0 10 0 0 0 1 0 0 0 0 1 1 10 1 0 0 0 1 1 0 0 0 0 1
12
3
4
56
7
8
9
Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.
12 blocks
9po
ints
1 0 0 1 0 0 1 0 0 1 0 01 1 0 0 1 0 0 1 0 0 0 01 0 1 0 0 1 0 0 0 0 1 00 1 0 1 0 0 0 0 1 0 1 00 0 1 0 1 0 1 0 1 0 0 00 0 0 0 0 1 0 1 1 1 0 00 0 1 1 0 0 0 1 0 0 0 10 0 0 0 1 0 0 0 0 1 1 10 1 0 0 0 1 1 0 0 0 0 1
Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.
12 blocks
9po
ints
= A
1 0 0 1 0 0 1 0 0 1 0 01 1 0 0 1 0 0 1 0 0 0 01 0 1 0 0 1 0 0 0 0 1 00 1 0 1 0 0 0 0 1 0 1 00 0 1 0 1 0 1 0 1 0 0 00 0 0 0 0 1 0 1 1 1 0 00 0 1 1 0 0 0 1 0 0 0 10 0 0 0 1 0 0 0 0 1 1 10 1 0 0 0 1 1 0 0 0 0 1
Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.
12 blocks
9po
ints
= A
1 0 0 1 0 0 1 0 0 1 0 01 1 0 0 1 0 0 1 0 0 0 01 0 1 0 0 1 0 0 0 0 1 00 1 0 1 0 0 0 0 1 0 1 00 0 1 0 1 0 1 0 1 0 0 00 0 0 0 0 1 0 1 1 1 0 00 0 1 1 0 0 0 1 0 0 0 10 0 0 0 1 0 0 0 0 1 1 10 1 0 0 0 1 1 0 0 0 0 1
AAT =
4 1 1 1 1 1 1 1 11 4 1 1 1 1 1 1 11 1 4 1 1 1 1 1 11 1 1 4 1 1 1 1 11 1 1 1 4 1 1 1 11 1 1 1 1 4 1 1 11 1 1
