Think It Over
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THINK IT OVER
Think It Over
This section of Resonance presents thought-provoking questions, and discusses answers a few months later. Readers are invited to send new questions, solutions to old ones and comments, to Think It Over, Resonance, Indian Academy of Sciences, Bangalore 560 080. Items illustrating ideas and concepts will generally be chosen. Shailesh A Shirali Principal Amber Valley Residential School K M Road Mugthihalli Chickmagalur 577 101, India.
Email: [email protected]
On a Use of Normal Distribution
P r o b le m F o r p o sitiv e in teg ers n , co n sid er th e q u a n tities A B n d e¯ n ed b y Z1 ¼x An = dx ; co s2 n 2 ¡1 B
n
=
A
Z 1=pn
1 n
¡1=
p
n
co s2 n
¼x dx : 2
n
an d
(1 )
It is ea sy to a n ticip a te th a t A n ! 0 w h en n ! 1 ; so m e sa m p le va lu es a re d isp la y ed b elow , co m p u ted u sin g M athem atica.
Keywords Stirling approximation.
RESONANCE ç July 2005
A
100
¼ 0 :1 1 2 7 ;A
200
¼
0 :0 7 9 7 ;
A
400
¼ 0 :0 5 6 4 ;A
1000
¼ 0 :0 3 5 7 :
(2 )
S in ce th e ra n g e o f in teg ra tio n in th e in teg ra l d e¯ n in g B n stea d ily d im in ish es, w e m ig h t a lso g u ess th a t B n ! 0 a s n ! 1 . B u t th e ¯ g u res tell u s o th erw ise; in d eed w e 85
THINK IT OVER n ! a s n ! 1 . B u t th e ¯ g u res tell u s o th erw ise; in d eed w e ¯ n d th a t
B
100
¼ 0 :9 7 4 2 ;B
200
¼ 0 :9 7 3 9 ;
B
400
¼ 0 :9 7 3 8 ;B
1000
¼ 0 :9 7 3 7 :
(3 )
T h e ¯ g u res su g g est th a t B n co n v erg es to a n o n -zero va lu e (a p p rox im a tely 0 :9 7 4 ) a s n ! 1 . H ow m ay th is b e ex p la in ed ? W h a t is th e lim itin g va lu e, a n d ca n it b e ex p ressed in clo sed fo rm ?
Anagrams of Wiiliam Burnside: I WARN DUMB LILIES SIMILAR, BLUE WIND I WILL SUM IN BREAD MEDUSA ILL; WIN RIB I WILL DREAM IN BUS BILL; I MUSE INWARD INLAW IS MULE BIRD I DRAW IN SEMI-BULL I WILL DUB SEMINAR BILL; I MUSE DARWIN I AM ILL; WIN RED BUS ALL MINIBUS WIRED WELD RAIL, MINIBUS I, SIR, BAN WILD MULE DIAL; WE RUIN LIMBS REDIAL US; WIN LIMB IM BRAINWISE DULL ... Kanakku Puly
86
RESONANCE ç July 2005
Think It Over
This section of Resonance presents thought-provoking questions, and discusses answers a few months later. Readers are invited to send new questions, solutions to old ones and comments, to Think It Over, Resonance, Indian Academy of Sciences, Bangalore 560 080. Items illustrating ideas and concepts will generally be chosen. Shailesh A Shirali Principal Amber Valley Residential School K M Road Mugthihalli Chickmagalur 577 101, India.
Email: [email protected]
On a Use of Normal Distribution
P r o b le m F o r p o sitiv e in teg ers n , co n sid er th e q u a n tities A B n d e¯ n ed b y Z1 ¼x An = dx ; co s2 n 2 ¡1 B
n
=
A
Z 1=pn
1 n
¡1=
p
n
co s2 n
¼x dx : 2
n
an d
(1 )
It is ea sy to a n ticip a te th a t A n ! 0 w h en n ! 1 ; so m e sa m p le va lu es a re d isp la y ed b elow , co m p u ted u sin g M athem atica.
Keywords Stirling approximation.
RESONANCE ç July 2005
A
100
¼ 0 :1 1 2 7 ;A
200
¼
0 :0 7 9 7 ;
A
400
¼ 0 :0 5 6 4 ;A
1000
¼ 0 :0 3 5 7 :
(2 )
S in ce th e ra n g e o f in teg ra tio n in th e in teg ra l d e¯ n in g B n stea d ily d im in ish es, w e m ig h t a lso g u ess th a t B n ! 0 a s n ! 1 . B u t th e ¯ g u res tell u s o th erw ise; in d eed w e 85
THINK IT OVER n ! a s n ! 1 . B u t th e ¯ g u res tell u s o th erw ise; in d eed w e ¯ n d th a t
B
100
¼ 0 :9 7 4 2 ;B
200
¼ 0 :9 7 3 9 ;
B
400
¼ 0 :9 7 3 8 ;B
1000
¼ 0 :9 7 3 7 :
(3 )
T h e ¯ g u res su g g est th a t B n co n v erg es to a n o n -zero va lu e (a p p rox im a tely 0 :9 7 4 ) a s n ! 1 . H ow m ay th is b e ex p la in ed ? W h a t is th e lim itin g va lu e, a n d ca n it b e ex p ressed in clo sed fo rm ?
Anagrams of Wiiliam Burnside: I WARN DUMB LILIES SIMILAR, BLUE WIND I WILL SUM IN BREAD MEDUSA ILL; WIN RIB I WILL DREAM IN BUS BILL; I MUSE INWARD INLAW IS MULE BIRD I DRAW IN SEMI-BULL I WILL DUB SEMINAR BILL; I MUSE DARWIN I AM ILL; WIN RED BUS ALL MINIBUS WIRED WELD RAIL, MINIBUS I, SIR, BAN WILD MULE DIAL; WE RUIN LIMBS REDIAL US; WIN LIMB IM BRAINWISE DULL ... Kanakku Puly
86
RESONANCE ç July 2005
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