Title: Fourier????????????,????????????????
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- Fourier?????????????????????????????????????
- ???????????,Fourier????????????????????????????
?,?sin(x)?cos(x)????????????
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9 Artificial Neural Network(ANN)
x1
w1
x2
w2
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y
w3
x3
wn
xn
10(No Transcript)
11?????
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12?????(?)
- ????????,??????f(x)?????????????????????????,?????
????,??????????????????(????????????)??????! - ???????,??????,???????????????????????,???????????
,????,???????????
13?????(?)
- ????????,??????????????!
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?,????????09????????100?,?1000?????
14?????(?)
- ?????,?????????????????100,????????????A??????,
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15? ?
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17????
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18????
- ??????????,Mandelbrot????????????????????
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20????(?)
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???,??????????? - ??????????????????,????,??????????????????????????
???????,???????????,????????????? - ????????,??????????
21????(?)
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22Koch ??
23Koch ??(?)
- Koch???????????????
- ??????????????????????
- ??,???????????????,???????????,??????????????????
?
24?????????
- ??????????????????????
- England??????????????????
25?????
- ????????B.B.Mandelbrot,??????????,??????????
????????????(????????????)?????????????????????--
--fractal? - fractal????Mandelbrot???,???????????fractus
- ?????????????????,????????????????????????????????
???????????????????1967???????????????????????
????,???????????????1973?,??????????,?????????????
?,????????????????????????
26?????(?)
- ??????????????F
- F??????,????????????????
- F?????,?????????????????
- F??????????,????????????????
- F????????????????F?????
- F???????????,???????
27Julia Set
- Julia Set
- Zn1 Zn2 C
- ??? C ????,? Z ?????????,? Z ?????????,??????,
????????????,?? Julia Set ???? - ??C?Z0?????????Julia??
28Mandelbrot Set
- ?????,M????????????
- Zn1Zn2C?
- ??,Z?c????,??????
- ?????
- Xn1iYn1 (XniYn)2CxiCy
29- ???
- Xn1 Xn 2 -Yn2Cx (??)
- Yn1 2XnYnCy (??)
- ????????????,?????,??Mandelbrot?,??M????????,Z????
?0,?C??????0??,?C????????????????,??????fc(Z)Z2
C???,??M?? - Mc?Cfck(0)/? 8 (k?8)?
30- ????C???????,?????Zk???????
- (1)Zk???????????????,???
- (2)Zk???????????????,????
- ??????????????,???????Zk????,?????????,?????M?????
??????????Zk????,?????????,????????????????,????M?
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31(No Transcript)
32???????
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- ???????????,Julia?Mandelbrot????????????????,?????
????(???????),????????????????????????????????????
- ????,?????????????????????????????????????????????
?????,?????????,?????????????????
39????????
- Mandelbrot?????????????
- ?Euchlid?????????????
- ?---0?
- ?---1?
- ?---2?
- ?---3??
40????????(?)
????1?????n??,????r,? n r 1
????1????n??,??????? ????r,? n r2 1
????1????n??,??????? ????r,? n r3 1
41????
- ???????????,r ????????????????,?????
- n rD 1
- ?????????
-
- ??,D???????.
42????(?)
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- ??n??,?????????4n,??????
- ?Koch??????
- ?????????D1.25 (Mandelbrot)
44????????(?)
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- ???????????????,?Dt??,?????,???????????????,0?
??????????????,????,??????????????????0,?????????
?????????1.??,???????????????Dtd.????????????????
????????.
45- Hausdorff??
- ???????????????,?????????"?r"???,??????????N
- ????????????,??????
- ????????????,????.???????
- N(r)r-Dh
- ?????????,?????
- DhlnN(r)/ln(1/r)
- ?
- DhlimlnN(s)/ln(1/s)
- ???Dh?????????,??????,??????.
46- ?????,??????,?D????.????????????????????,????Dh??
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49(No Transcript)
50?????
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- ??D?????????(??)???(??)???,?????????,?????????????
????????. - ???????????????????????????,???????????????(????).
- ???????,?????????????????????.????????????????????
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51??????????????
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???????????? (??,??,??,??,??) ?? ? ????? 0 ???? 1 ?2 ? 3
???????(?????) ?????? (???,???,???,??,???,??) ? ? ?????,?? ????? (?????)
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52IFS
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55?????(?)
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57? ?(turbulence)
- ???????????????????
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- ?????????(modes)?????
- ???????????,?????????????
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- f ???????????
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61- ????,?? ????????,???????
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62????????
- ??(mapping)????(iteration)
- xn12xn,?x13 ,?x26,x312?
- ?????????,?????(feedback),???????,??????????,?????
???? - ??????????????
- ????????,??????????????????????????,????????????
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63- ??????????
- ??a???,?????-2,4,???????0,4???,?????
-2,0??????x????0,1??????????????a?(1-
?)????,???_10,?_21-1/a????? f ??,f ?2?????f
2,3?????f 3,?? ???,????????????f ??????f ???1??f
2????????f ???2????f n?????f ???n??????
64- ??f ???m????????
- ???????f ????(?????,????1?)??????????
- ?lt1,??,???gt1,??,????1,???0,????
65- ???a??????x?????(stable steady states)??????,
??1??2??????????????????(??????????????),?????,???
???????????????????????????? ??????????
66(No Transcript)
67(No Transcript)
68(No Transcript)
69- ?????????????????????????????,?????????
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- d????????????
70The general concept of chaos
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78