Plural bookkeep using Exchange Algebra Yuji Onuki (University of tsukuba) Key Words:,,, 1 Deguchi(2004) 2 (1984) Staszkiewicz(2011) SNA n n r 1,1 < Name 1,1, > +r 1,2 < Name 1,2, > + + r 1,k1 < Name 1,k1, > +r 1,k1+1ˆ< Name 1,k1+1, > + + r 1,m1ˆ< Name 1,m1, > (1) r 1,i (i = 1, 2,, m1) Name 1,i (i = 1, 2,, m1) r 1,1 < Name 1,1,, base 1 > +r 1,2 < Name 1,2,, bas 1 > + + r 1,k1 < Name 1,k1,, base 1 > +r 1,k1+1ˆ< Name 1,k1+1,, base 1 > + + r 1,m1ˆ< Name 1,m1,, base 1 > (2) r 2,1 < Name 2,1,, base 2 > +r 2,2 < Name 2,2,, base 2 > + + r 2,k2 < Name 2,k2,, base 2 > +r 2,k2+1ˆ< Name 2,k2+1,, base 2 > + + r 2,m2ˆ< Name 2,m2,, base 2 > (3) r 2,i (i = 1, 2,, m2) Name 2,i (i = 1, 2,, m2)
r 1,1 < Name 1,1,, base 1 > +r 1,2 < Name 1,2,, bas 1 > + + r 1,k1 < Name 1,k1,, base 1 > +r 1,k1+1ˆ< Name 1,k1+1,, base 1 > + + r 1,m1ˆ< Name 1,m1,, base 1 > +r 2,1 < Name 2,1,, base 2 > +r 2,2 < Name 2,2,, base 2 > + + r 2,k2 < Name 2,k2,, base 2 > +r 2,k2+1ˆ< Name 2,k2+1,, base 2 > + + r 2,m2ˆ< Name 2,m2,, base 2 > (4) x = r 1,1 < Name 1,1,, base 1 > +r 1,2 < Name 1,2,, bas 1 > + + r 1,k1 < Name 1,k1,, base 1 > +r 1,k1+1ˆ< Name 1,k1+1,, base 1 > + + r 1,m1ˆ< Name 1,m1,, base 1 > +r 2,1 < Name 2,1,, base 2 > +r 2,2 < Name 2,2,, base 2 > + + r 2,k2 < Name 2,k2,, base 2 > +r 2,k2+1ˆ< Name 2,k2+1,, base 2 > + + r 2,m2ˆ< Name 2,m2,, base 2 > + +r n,1 < Name n,1,, base n > +r n,2 < Name n,2,, base n > + + r n,kn < Name n,kn,, base n > +r n,kn+1ˆ< Name n,kn+1,, base n > + + r n,mn < Name n,mn,, base n > (5) j Pro j[base j ](x) = r j,1 < Name j,1,, base j > +r j,2 < Name j,2,, base j > + +r j,k j < Name j,k j,, base j > +r j,k j+1ˆ< Name j,k j+1,, base j > + + r j,m ĵ < Name j,m j,, base j > (6) 3 2n (1) Staszkiewicz(2011) 4 2n (2) (1988)
5 2008 2n 2n 2n (2010) Q P i Q i P ti P ti t i Q 1 P t1 + Q 2 P t2 + Q n P n (2010) t i 6
2n i ri 2i-1 t 2i 1 = 1ˆ<,, > +1 <,, > +r 2i 1ˆ<,, > +r 2i 1 <,, > (7) 2i 2i-1 t 2i = 1ˆ<,, > +1 <,, > +2 <,, > +r 2i 1 <,, > +(2r 2i r 2i 1 ) <,, > +2r 2i <,, > (8) 2i-1 2i (t 2i 1 + t 2i ) = (1ˆ<,, > +1 <,, > +r 2i 1ˆ<,, > +r 2i 1 <,, > +1ˆ<,, > +1 <,, > +2 <,, > +r 2i 1 <,, > +(2r 2i r 2i 1 ) <,, > +2r 2i <,, >) = 1 <,, > +1 <,, > +(2r 2i r 2i 1 ) <,, > +(2r 2i r 2i 1 ) <,, > (9) (2r 2i r 2i 1 ) (2r 2i r 2i 1 ) i = 1 i = n (t 2i 1 + t 2i ) = n <,, > +n <,, > + (2r 2i r 2i 1 ) <,, > + (2r 2i r 2i 1 ) <,, > (10)
1 re n <,, > +n <,, > (11) (2r 2i r 2i 1 )/re <,, > + (2r 2i r 2i 1 )/re <,, > (12) 2i t 2i = 1ˆ<,, > +1 <,, > +2 <,, > +(r 2i r 2i 1 ) <,, > +(r 2i r 2i 1 ) <,, > +2r 2i <,, > +r 2î <,, > +r 2i <,, > (13) 2i-1 2i (t 2i 1 + t 2i ) = (1ˆ<,, > +1 <,, > +r 2i 1ˆ<,, > +r 2i 1 <,, > +1ˆ<,, > +1 <,, > +2 <,, > +(r 2i r 2i 1 ) <,, > +(r 2i r 2i 1 ) <,, > +2r 2i <,, > +r 2î <,, > = 1 <,, > +1 <,, > +r 2i <,, > +(r 2i r 2i 1 ) <,, > +(2r 2i r 2i 1 ) <,, > (14) i=n x1 = (t 2i 1 + t 2i ) = n <,, > +n <,, > + r 2i <,, > + (r 2i r 2i 1 ) <,, > + (2r 2i r 2i 1 ) <,, > (15) 2i ť 2i = (r e r 2i )ˆ<,, > +(r e r 2i ) <,, > (16) i=n x2 = ť 2i = (r e r 2i )ˆ<,, > + (r e r 2i ) <,, > (17)
x1 x2 + (x1 + x2) = n <,, > +n <,, > + r 2i <,, > + (r 2i r 2i 1 ) <,, > (2r 2i r 2i 1 ) <,, > + (r e r 2i )ˆ<,, > + (r e r 2i ) <,, > = n <,, > +n <,, > +nr e <,, > + (2r 2i r 2i 1 r e ) <,, > + (2r 2i r 2i 1 ) <,, > (18) 1 r e n <,, > +n <,, > n <,, > + (2r 2i r 2i 1 r e ) <,, > + (2r 2i r 2i 1 ) <,, > t k k 50 10 50 <, > +2 50 <, > +3 50 <, > +4 50 <, > +5 50 <, > +6 50 <, > +7 50 <, > +8 50 <, > +9 50 <, > +10 50 <, > = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) 50 <, > = 55 50 <, >= 2750 <, > (19) t 10 50 500 2750 500 5.5 500 t k k x1 = 50 <, 1 > +2 50 <, 2 > +3 50 <, 3 > +4 50 <, 4 > +5 50 <, 5 > +6 50 <, 6 > +7 50 <, 7 > +8 50 <, 8 > +9 50 <, 9 > +10 50 <, 10 > (20)
k k = 1 1, (k = 1, 2,, 10) x = 500 10 <, 10 >= 5000 <, 10 > (21) 9 50 <, 10 > +8 50 <, 10 > +7 50 <, 10 > +6 50 <, 10 > +5 50 <, 10 > +4 50 <, 10 > +3 50 <, 10 > +2 50 <, 10 > +1 50 <, 10 > = (9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1) 50 <, 10 > = 45 50 <, 10 >= 2250 <, 10 > (22) y = 5000 <, 10 > +2250 <, 10 > (23) t = 2250ˆ<, 10 > +2250ˆ<, 10 > (24) (y + t) = 2750 <, 10 > (25) t 10
(1998) Beaver and Wolfson(1982) Goldberg and Godwin(1994) FASB 7 Deguchi(2004) (1988) 8 10 200 x1 = 10 <, > +200ˆ<, > 10 1000 x2 = 10 <, > +1000ˆ<, > t1 = 10ˆ<, > +200 <, > t2 = 10ˆ<, > +1000 <, > (x1 + x2 + t1 + t2) = 200 <, > +1000 <, > +1200ˆ<, > 10 2 x3 = 10ˆ<, > +2 <, > 10 2 2
s1 = 10 <, > +10 <, > x3 = 10ˆ<, > +2 <, > s2 = (s1 + x3) = 2 <, > +10 <, > s2 = 2 <, > +2 <, > s2 s2 s2 = s2 10 <, >= 2 <, > name 10 < name, >= 2 < name, > 9 A A A 50 A 100 A 50 t1 = 50ˆ<, > +50 < A, > t2 = 100 <, > +50ˆ< A, > +50 <, > A t1 = 50ˆ<, > +50 < A, >
t2 = 200 <, > +50ˆ< A, > +150 <, > 150 3 75, t1 = 50ˆ<, > +50 < A, > t2 = 50 < A, > +50 <, > t2 = 200 <, > +100ˆ< A, > +100 <, > A t1 = 1ˆ<, A > +1 < A, A > t2 = 2ˆ<, A > +1 < A, A > +1 <, A > 1 <, A >= 50 <, > 1 <, A >= 100 <, > A A 2 A 10 Point of Event
1kWh=18.89 PM11-AM7 1kWh=11.82 2 AM1-AM6 1kWh=10.88 k h 2 100 100kWh x = 100 <, > +100 <, kwh > t1 = 100ˆ<, kwh > +1889 <, > (x + t1) = 100 <, > +1889 <, > PM11-AM1,AM6-AM7 t2 = 100ˆ<, kwh > +1182 <, > (x + t2) = 100 <, > +1182 <, > AM1-AM6 t3 = 100ˆ<, kwh > +1088 <, > (x + t3) = 100 <, > +1088 <, > 2500 2000 1500 Ç 1000 500 0 0 3 6 9 12151821 0 3 6 9 12151821 0 3 6 9 12151821 1 2 3 ì 1:
. x = 100 <, > +100 <, kwh > AM12 t1 = 100ˆ<, kwh > +1889 <, > (x + t1) = 100 <, > +1889 <, >= 1989 <, > AM1 t2 = 1088ˆ<, > +100 <, kwh > (x + t1 + t2) = 1989 <, > +1088ˆ<, > +100 <, kwh > AM12 = 901 <, > +100 <, kwh > t1 = 100ˆ<, kwh > +1889 <, > (x + t1 + t2 + t1) = 901 <, > +1889 <, >= 2790 <, >
Ç 4000 3500 3000 2500 2000 1500 1000 500 0 r(ø ' (Ø 0 3 6 9 12151821 0 3 6 9 12151821 0 3 6 9 12151821 1 2 3 ì 2: AM12 AM1.2 100 50 x = 100 <, > +100 <, kwh > AM12 t1 = 100ˆ<, kwh > +1889 <, > (x + t1) = 100 <, > +1889 <, >= 1989 <, > AM1 t2 = 1088ˆ<, > +100 <, kwh > (x + t1 + t2) = 1989 <, > +1088ˆ<, > +100 <, kwh > = 901 <, > +100 <, kwh > t3 = 50ˆ<, kwh > +50 <, kwh > (x + t1 + t2 + t3) = 901 <, > +50 <, kwh > +50 <, kwh > AM12 t4 = 50ˆ<, kwh > +945 <, >
(x + t1 + t2 + t3 + t4) = 901 <, > +945 <, > +50 <, kwh > = 1846 <, > +50 <, kwh > AM1 t2 = 1088ˆ<, > +100 <, kwh > (x + t1 + t2 + t3 + t4 + t2) = 758 <, > +100 <, kwh > +50 <, kwh > t3 = 50ˆ<, kwh > +50 <, kwh > (x + t1 + t2 + t3 + t4 + t2 + t3) = 758 <, > +50 <, kwh > +100 <, kwh > AM12 t4 = 50ˆ<, kwh > +945 <, > (x + t1 + t2 + t3 + t4 + t2 + t3 + t4) = 1703 <, > +100 <, kwh > 3000 2500 2000 r(ø ' (Ø Ç 1500 1000 500 0 0 3 6 9 12151821 0 3 6 9 12151821 0 3 6 9 12151821 1 2 3 ì 3: 50%.3 143 50 Point of Event
Point of Event 11 2n [1] Beaver, W. H. and Wolfson, M. A.(1982) Foreign Currency Translation and Changing Prices in Perfect and Complete Markets, Journal of Accounting Research, vol.20, No.2, pp.528-549. [2] Deguchi, H.(2004) Economics as an agent-based complex system. Springer-Verlag. pp.67-93. [3] Goldberg, S. R. and Godwin, J. H.(1994) Foreign Currency Translation Under Two Cases- Integrated and Isolated Economies, Journal of International Financial Management and Accounting, vol.5, No.2, pp.97-119.
[4] Staszkiewicz, P.(2011) Multi Entry Framework for Financial and risk reporting. [5] (1984), [6] (2010), [7] (1998) [ ], [8] (1999) [9] (1988) : http://ir.lib.fukushimau.ac.jp/dspace/bitstream/10270/1072/1/3-730.pdf (2012-12-30 ) [10] (2008),