The Cardinality of the Sets Involved in Lattice Games - Search Theory: A Game Theoretic Perspective - page 104

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Theorem 1. The cardinality of the set F

n , m is given by the expression

F

m =

m
k = 1 ( 1 − 2cosβ k )

m + k

1 1

+( − 1 )

tg 2 β k

0
n

n −

2 ,

(7.16)

,

( m + 1 )

where β k is given by ( 7.13 ).

n , m as

Proof. We can express the cardinality of F

m

h = 1 c hk 2

F

m = h , r a ( n )

m
k = 1 λ

m
k = 1 λ

0
n

k

h , r c hk c rk =

k

hr =

. (7.17)

,

First we will compute the sum

k 2
m

k 2
m

m
h = 1 c hk =( − 1 )

m
h = 1 ( − 1 )

h sin

( h β k )=( − 1 )

1 S

. (7.18)

+ 1 +

The value of S is obtained from the imaginary part of the expression

m
h = 0 ( − 1 )

h exp

( ih β k )

m e i ( m + 1 ) β k

+( − )

1 1

=

1 + e i β k

m e ik π m cos

1 +( − 1 ) 1 +( − 1 ) ( k π )

= =

e i β k e i β k

1 + 1 +

e − i β 2

m + k m + k

1 +( − 1 ) ( 1 +( − 1 ) )

= ,

2cos β k
2

e i β 2

e − i β k / 2 e i β k / 2

( + )

and the imaginary part of this value is equal to

( − β 2 )(

m + k m + k

sin 1 +( − 1 ) )

tg β k
2

1 +( − 1 )

= − = S . (7.19)

2cos β 2

2

Therefore,

k 2
m

1 2
m

m
h = 1 c hk =( − 1 )

m + k

1 tg β k

1 +( − 1 )

k +

1 S

=( − 1 )

+ + 2 2

and substituting the last value into ( 7.17 ) we conclude that