|P in Prog : Geom Hue . - * * Y near-field . do b c - - PDF document

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|P in Prog : Geom Hue . - * * Y near-field . do b c - - PDF document

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Skew field : multiple . not . Commutative ' ' " ) Quaternions 4. din complex ( or skew field : field over ) ^ DESARGUES 's than |P in Prog : Geom Hue . - * * Y near-field . do b c y=ax+b Ga 's plane prog DESARGUES

slide-1
SLIDE 1

Skew field :

multiple . not

.

Commutative

Quaternions

(

' '
  • 4. din complex
")

: field

  • r skew field

DESARGUES 's than

^

Hue

in Prog:

Geom

.
  • ver)
  • |P

Y

near-field

¥¥÷÷*⇒*

. do b

c

Ga

y=ax+b

slide-2
SLIDE 2

prog

's plane

DESARGUES ← division ring

Finite division rig

are fields

(

EDDERBURN

finite

2→ Galois

  • F finite

non - DES . planes

  • rder
:

h

htt

= #pts on a line

117=111

n'tntl

  • f

W={ n ftp.p

. of order n}

N z { prime powers )

slide-3
SLIDE 3

✓ = {orders of finite pp

. }

N Z {prime powers }

all

known f.pp

. s have

prime

power order

  • BRUCK
  • RYSER

TAM

Tf

n =L

  • r 2

mod 4

  • N E N

⇒ (Fa

,b) ( n

= a't b

')

  • 2

3 45 6 7 89

10

11

12

✓ u

r -

X

u r - f

  • p
  • C. W
. H. LAM

175

1990

10 EIN

2

slide-4
SLIDE 4

LATIN

SQUARE

hxn

11

2-332

2231

13

33

12.21

ORTHOGONAL

I.Squares

EI

. If

n odd 23

⇒ 7- pair of orthog. L

.Squares

EX .

F

. .

4×4

i .

EI

.

F

"

axa

bxb

→ ab

x ab

missing

:

h

2141

EULER h=6

"

36 officers problem

"
slide-5
SLIDE 5
  • TERRY

1903

:

No

6×6

HE 2 (4)

1960 's

BOSE-SHRIKHANDE- PARKER

Hn E 2 (4)

,

n > G

F pair of ortho L Sgs

n = 6
  • nly exception
  • EI
.

If

n - pk

⇒ Fn- I

pairwise orth Lsgs

  • EI

Sf

L

, . . ( m

are

rainworth

.

nxn

L . Sgs

⇒

m En

  • l
  • Et
  • Fn - I

← Ffpp ardern .

slide-6
SLIDE 6

LATIN

RECTANGLE

kxn

11237

rows :

2312

perm's of

I?

Col's

:

all el's distinct

THI

Every Latin rectangle

can

completed to a L

. Sq .
slide-7
SLIDE 7

SYSTEMS

  • f

SDR

DISTINCT REPRESENTATIVES

H=(V. E)

E={ E ,

. . . Em }

ve

. EE,

all vi. distinct

  • No

SDR

if

  • A EE
.

←

m > n

ease

÷÷

I !f¥il4I

!

Hall

  • obstacle

KENG

  • HILL TAM

3- SDR ⇒ * Hale .gs#oSDRG6oDCHARACT

slide-8
SLIDE 8

DENYING

1916

kzl

k - uniform

, k-regular hyp .

EI

→ satisfies

Hall condita.

(§ obstacle)

ftp.iI-SDRE#kxnL.Recfayle

Ken

→ extend to

date) x n

L . Rect .

  • incidence matrix

m=n4

i

Et

.

?

k

bk4k=

M /¥¥/

k¥0

⑨ ④ n

k

÷

. .tn : Fwm =L

E

, Ej

Em Col

# SDRS

= # rook placements

  • n 17=1
slide-9
SLIDE 9

A = Cag

.)

nxn

Permanent

per CAKE

IT ai sci)

GE pen

  • #SDRs

= per M

I

incid . matrix

  • A : stochastic matrix
:

* row : prob

. distribution

aij to ¥

, ai;

=/

W

A

doubly stochastic

:

& same for columns

Iho

' . .? )

( i ! ! )

perm

. motor .
slide-10
SLIDE 10

A doubly Stoch:

aij 20

A Sow a l

tf

Solemn =L

EXAMPLES

permutation matrices

n !

! ! : it

a-

=

  • Per (pen
. matrix)

= 1

Der ( I J )

=

h '

Per (T )

= h !

hi

> In

Per A- § Tain!

slide-11
SLIDE 11

( ( a ) - ha

2

THM

Per CI) =L

  • Perl's 5) =n÷

are"

Vander Woerden's

permanent confect

.

µ±÷E¥%±r

PERMANENT IN EQ .

h

"

1976

EGTRYCHEVTIALIKMAN ER

per A

E t

=L ⇐ A permit

.

matrix

→ MANY

SDR

K- unit

, k -reg

→ MANY

Lat . Sg

Llnl

  • 7

"

STS