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HN VIỆT DỊCH S LƯỢC

GS Nguyễn Hữu Quang

Nguyn Giảng Vin Vật L Chuyn về Cơ Học Định Đề
(Axiomatic Mechanics, a branch of Theoretical Physics)
tại Đại Học Khoa Học Si Gn trước năm 1975

 

 

 

CHƯƠNG 08

 

TAM-THNH DỊCH

 

 

 

(Tiếp theo Kỳ 104)

 

TẠP-QUI-TRUYỆN

 

Cấu-trc cuả Tạp-Qui-Truyện đ được duyệt kỹ trong hai bi 96 v 97. Hay ho nhất l cắt nghi tại sao tm quẻ cuối truyện (Đại-qu, Cấu, Tiệm, Di, K-tế, Quy-muội, Vi-tế, Quyết) lại khng phản-đối từng đi một. Dưới đy l bổ-tc bằng Tn-ton-học. 

          GS Daniel Goldenberg đ chứng-minh rằng nhm 64 biệt-qui với php ton 2-chu-kỳ (ho chi giữa hai ho ln-cận = 2-cycle), l một nhm phi-abel (non-abelian group). Tnh phi-abel l hệ-quả hiển-nhin cuả kết-hợp giữa 2-chu-kỳ. Tuy nhin, nếu ta dng liệt-k mật-m Gray (Xem bn dưới), nhm phi-Abel ny sẽ trở thnh nhm Abel, bởi v với bất kỳ ba 2-chu-kỳ lin-tiếp no, php cộng đẳng-thặng 2 sẽ giao-hon. 

Nhờ Định-l khổng-lồ cuả ton-thuyết về nhm, nhm phi-Abel ny nghiễm-nhin l nhm giản-dị hữu-hạn  xưa nhất thế-giới. Định-l căn-bản Đại-số Chu-dịch (Định-l 7) bảo rằng: "Với bất kỳ một cặp biệt-qui no ta cũng tm ra được một biệt-qui thứ ba duy nhất, mệnh-danh l biệt-qui trung-gian, biến đổi một biệt-qu trong cặp thnh biệt-qui kia trong cặp qua php cộng hay php trừ đẳng-thặng 2". Tỷ như trong th-dụ sau đy:

                        001011   (Tiệm u)                 100001     (Di v[)                 

        + 100001   (Di v[)                  + 110100     (Quy-muội v)            

                       101010   (K-tế )             010101     (Vị-tế )

On the abstract level, professor Goldenberg showed that the group of 64 hexagrams of the Chouyi (R34, p. 149-51), under the binary operation of 2-cycle (permutation of 2 contiguous hsiaos or hsiao swap or bits swap) is a non-abelian group, i.e. satisfying closure, associativity, identity element, and inverse. The non-abelian property is obvious because of the order of composition of the 2-cycles.

However, if we use the Gray Code Sequencing (cf. Chapter Five, 2.7), the group becomes abelian, because for any three successive 2-cycles the addition or the substraction modulo 2 is commutative. Check!

Thanks to the Enormous Theorem in Group Theory (R36), this non-abelian group constitutes the oldest simple finite group of the World. The Goldenberg’s fundamental theorem of the Algebra of the Chouyi (R34, pp.163-4) reads: “For any hexagram-pair, there exists a third, unique, mediating hexagram which transforms either member of the pair into the other under addition or substraction (modulo-difference which is knowingly also the modulo-sum alias XOR of Computerese). E.g.,

 001011   Chien        Tiệm     (H53)       100001  I         Di    (H27)              

+ 100001   I        Di      (H27)         + 110100  Kuei-Mei   Quy-muội    (H54

   101010   Chi Chi   K-tế   (H63)        010101  Wei Chi     Vị-tế      (H64)
 

R34 Goldenberg, D.S, The Algebra of the Chouyi and its Philosophical Implications, Journal of Chinese Philosophy 2 (1975), 149-76. 

R36 Gorenstein, D., The Enormous Theorem, Sci. Am., Vol. 253, Nr. 6, Dec. 1985. 

 

The GRAY CODE ARRANGEMENT

 

The Gray code is an encoding of unsigned binaries so that adjacent binaries have a single bit different by 1. Usually it is called binary reflected Gray code for it can be generated as follows. Take the gray code 0, 1. Write it forwards then backwards 0, 1, 1, 0. Then prepend 0s to the first half and 1s to the second half: 00, 01, 11, 10, 10, 11, 01, 00 and so on. Each iteration doubles the number of codes. From the five successive iterations we get the 2, 3, 4, 5, 6-bit Gray code which represent successively the Di-, Tri-, Tetra-, Penta- and Hexagrams. 

The following tables are sort of bootstrap to 2.7 below. 

 Table 2.5.1                 Table 2.5.2                Table 2.5.3            Table 2.5.4

2-bit Gray code           3-bit Gray code            4-bit Gray code          5-bit Gray code

0          00                     00        000                 0000     0000               00000 

1          01                     01        001                 0001     0001               00001

1          11                     11        011                 0011     0011               00011

0          10                     10        010                 0010     0010               00010

                                    10        110                 0110     0110               00110  

                                    11        111                 0111     0111               00111 

                                    01        101                 0101     0101               00101

                                    00        100                 0100     0100               00100

                                                                      1100     1100               01100

                                                                      1101     1101               01101

                                                                      1111     1111               01111

                                                                      1110     1110               01110

                                                                      1010     1010               01010

                                                                      1011     1011               01011

                                                                      1001     1001               01001

                                                                      1000     1000               01000

                                                                                                        11000

                                                                                                        11001

                                                                                                        11011

                                                                                                        11010

                                                                                                        11110

                                                                                                        11111

                                                                                                        11101

                                                                                                        11100

                                                                                                        10100

                                                                                                        10101

                                                                                                        10111

                                                                                                        10110

                                                                                                        10010

                                                                                                        10011

                                                                                                        10001

                                                                                                        10000

Table 2.3 The 2, 3, 4, 5-bit Gray Code’s Arrangement

  

Table 2.5.4                   Table 2.5.5

5-bit Gray code              6-bit Gray code

 

00000                           000000              Khn                 K’un

00001                           000001              Bc                   Po

00011                           000011              Quan                Kuan

00010                           000010              Tỷ                    Pi

00110                           000110              Tụy                  Ts’ui

00111                           000111              Bĩ                     P’i

00101                           000101              Tấn                  Chin

00100                           000100              Dự                   Y

01100                           001100              Tiểu-qu           Hsiao Kuo

01101                           001101              Lữ                    L

01111                           001111              Độn                  Tun

01110                           001110              Hm                 Hsien

01010                           001010              Kiển                  Chien

01011                           001011              Tiệm                Chien

01001                           001001              Cấn                  Kn

01000                           001000              Khim               Ch’ien

11000                           011000              Thăng               Shng

11001                           011001              Cổ                    Ku

11011                           011011              Tốn                  Sun

11010                           011010              Tỉnh                 Ching

11110                           011110              Đại-qu            Ta Kuo

11111                           011111              Cấu                   Kou

11101                           011101              Đỉnh                 Ting

11100                           011100              Hằng                 Hng

10100                           010100              Giải                   Hsieh

10101                           010101              Vị-tế                 Wei Chi

10111                           010111              Tụng                 Sung

10110                           010110              Khốn                 K’un

10010                           010010              Khảm                K’an

10011                           010011              Hon                 Huan

10001                           010001              Mng                Mng

10000                           010000              Sư                    Shih

                                    110000              Lm                  Lin

                                    110001              Tổn                   Sun

                                    110011              Trung-phu         Chung Fu

                                    110010              Tiết                   Chieh

                                    110110              Đoi                  Tui

                                    110111              L                      L

                                    110101              Khuể                  K’uei

                                    110100              Quy-muội           Kuei Mei

                                    111100              Đại-trng           Ta Chuang

                                    111101              Đại-hữu             Ta Yu

                                    111111              Kiền                   Ch’ien

                                    111110              Quyết                 Kuai

                                    111010              Nhu                   Hs

                                    111011              Tiểu-sc             Hsiao Ch’u

                                    111001              Đại-sc              Ta Ch’u

                                    111000              Thi                   T’ai

                                    101000              Minh-di              Minh I

                                    101001              B                      Pi

                                    101011              Gia-nhn            Chia Jn

                                    101010              K-tế                 Chi Chi

                                    101110              Cch                  Ko

                                    101111              Đồng-nhn        T’ung Jn

                                    101101              Ly                     Li

                                    101100              Phong               Fng

                                    100100              Chấn                 Chn

                                    100101              Phệ-hạp             Shih Ho

                                    100111              V-vng             Wu Wang

                                    100110              Ty                   Sui

                                    100010              Trun                Chun

                                    100011              ch                    I (I4)

                                    100001              Di                     I (Yi1)

                                    100000              Phục                 Fu


Table 2.4 The 5-bit and 6-bit Gray Code’s Arrangement 

 

 

 

 

Xem Kỳ 106

 

 

 

 

 

 

GS Nguyễn Hu Quang
Nguyn Giảng Vin Vật L Chuyn về Cơ Học Định Đề
(Axiomatic Mechanics, a branch of Theoretical Physics)
tại Đại Học Khoa Học Si Gn trước năm 1975

 

  

 

 

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