MMO / mmouchpos
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ISBN 966-7809-24-2
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J_p_ga_glb < F Ihjhogy ^hdlhj l_ogLqgbo gZmd ijhn_khj
E H =ZedLg ^hdlhj l_ogLqgbo gZmd ijhn_khj
Jha]eygmlh hkgh\g• f_lh^b c Ze]hjblfb qbk_evgh€ hilbf•aZp•€ Ijb\_^_gh hibk hjb]•gZevgbo Z\lhjkvdbo jhajh[hd • ^hke•^`_gv f_lh^•\ hilbf•aZp•€ Hkh[eb\m m\Z]m ijb^•e_gh aZklhkm\Zggx f_lh^•\ hilbf•aZp•€ \ aZ^ZqZo gZ\qZggy rlmqgbo g_cjhggbo f_j_`
IjbagZq_gh ^ey klm^_gl•\ ki_p•ZevghklL ©Ijh]jZfg_ aZ[_ai_q_ggy Z\lhfZlbah\Zgbo kbkl_fª Z lZdh` fh`_ \bdhjbklh\m\Zlbky •g`_g_jZfb Zki•jZglZfb • klm^_glZfb j•agbo ki_p•Zevghkl_c ^ey ijb^[Zggy ijZdlbqgbo gZ\bqhd \bj•r_ggy hilbf•aZp•cgbo aZ^Zq
ISBN 966-7809-24-2 |
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AF1KL
<klmi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Jha^•e E•g•cg_ ijh]jZfm\Zggy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
AZ^Zq• e•g•cgh]h ijh]jZfm\Zggy 7
<eZklb\hkl• hkgh\gh€ aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy 9
Kbfie_dkgbc f_lh^ j•r_ggy aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy 11
Jha^•e G_e•g•cg_ ijh]jZfm\Zggy. . . . . . . . . . . . . . . . . . . . . . . . . . 19
AZ]ZevgZ aZ^ZqZ g_e•g•cgh]h ijh]jZfm\Zggy 19
DeZkbn•dZp•y f_lh^•\ g_e•g•cgh]h ijh]jZfm\Zggy 21
Jha^•e H^gh\bf•jg• f_lh^b g_e•g•cgh€ hilbf•aZp•€. . . . . . . . . 24
F_lh^b ijyfh]h ihrmdm 24
F_lh^ jhaih^•em •gl_j\Zem gZ\i•e 24
F_lh^ N•[hgZqq• 25
Ihrmd f_lh^hf ³ahehlh]h i_j_lbgm´ 27
F_lh^ j•\ghf•jgh]h ihrmdm 28
F_lh^b ihrmdm a \bdhjbklZggyf ihe•ghf•Zevgh€ ZijhdkbfZp•€ 29
F_lh^ hp•gx\Zggy a \bdhjbklZggyf d\Z^jZlbqgh€ ZijhdkbfZp•€ 29
F_lh^ ihke•^h\gh]h hp•gx\Zggy a \bdhjbklZggyf d\Z^jZlbqgh€ ZijhdkbfZp•€ f_lh^ IZm_eeZ 31
F_lh^b ihrmdm a \bdhjbklZggyf iho•^gbo 32
F_lh^ GvxlhgZ JZnkhgZ 32
F_lh^ k_j_^gvh€ lhqdb f_lh^ ;hevpZgh 33
F_lh^ k•qgbo f_lh^ ohj^ 34
F_lh^ ihrmdm a \bdhjbklZggyf dm[•qgh€ ZijhdkbfZp•€ 34
4
Jha^•e ;Z]Zlh\bf•jg• [_a]jZ^•}glg• f_lh^b g_e•g•cgh€
hilbf•aZp•€ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
37 |
F_lh^ G_e^_jZ ± F•^Z 37
F_lh^ OmdZ >`b\kZ 41
F_lh^ kijy`_gbo gZijyfd•\ IZm_eeZ 43
Jha^•e ;Z]Zlh\bf•jg• ]jZ^•}glg• f_lh^b g_e•g•cgh€
hilbf•aZp•€ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
F_lh^ Dhr• 48
F_lh^ GvxlhgZ 48
:e]hjblfb kijy`_gbo ]jZ^•}gl•\ 49
IZjlZg ± f_lh^ 50
F_lh^ AZml_g^ZcdZ 51
;Z]ZlhiZjZf_ljbqgbc ihrmd 52
D\Za•gvxlhg•\kvd• f_lh^b 52
:e]hjblf E_\_g[_j]Z FZjd\Zj^lZ 54
MaZ]Zevg_gbc ]jZ^•}glgbc Ze]hjblf 56
Hdj_f• \biZ^db maZ]Zevg_gh]h ]jZ^•}glgh]h Ze]hjblfm 57
1gl_]jZevgbc f_lh^ hilbf•aZp•€ 58
Jha^•e F_lh^b [Z]Zlh\bf•jgh€ hilbf•aZp•€ a
h[f_`_ggyfb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
F_lh^ rljZngbo nmgdp•c 63
F_lh^ 6807 N•Zddh FZddhjf•dZ 65
Jha^•e F_lh^b hilbf•aZp•€ \ aZ^ZqZo ZijhdkbfZp•€ lZ
gZ\qZggy g_cjhggbo f_j_`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
NhjfZevgbc g_cjhg H^ghrZjh\bc i_jk_iljhg . . . . . . . . . . . . . . . . 69
;Z]ZlhrZjh\bc i_jk_iljhg 74
5
Ihj•\gyevgZ oZjZdl_jbklbdZ f_lh^•\ hilbf•aZp•€ ijb gZ\qZgg•
[Z]ZlhrZjh\bo g_cjhggbo f_j_` 76
Kihkl_j•]Zxqbc Ze]hjblf gZ\qZggy 78
JZ^•Zevgh [Zabkg• g_cjhgg• f_j_`• 85
F_lh^b hilbf•aZp•€ \ aZ^ZqZo ZijhdkbfZp•€ 87
:e]hjblf deZkl_j j_]j_kLcgh€ ZijhdkbfZpL€ lZ ch]h
g_cjhf_j_`_\Z Lgl_jij_lZpLy 92
Jha^•e Ijh]jZfg• aZkh[b qbk_evgh€ hilbf•aZp•€. . . . . . . . . . . . 107
Hilbf•aZp•y • g_cjhgg• f_j_`• \ iZd_l• 0DWODE 107
Hilbf•aZp•y • g_cjhgg• f_j_`• \ Z\lhfZlbah\Zg•c kbkl_f•
©>•Z]ghklbdZª 117
<bkgh\db. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
E•l_jZlmjZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
>h^Zlhd Iml•\gbd aZ kibkdhf e•l_jZlmjb . . . . . . . . . . . . . . . . . . 133
Summary. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6
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Kn_jZ aZklhkm\Zggy f_lh^•\ hilbf•aZp•€ ^m`_ rbjhdZ FZl_fZlbqg_ fh^_ex\Zggy ieZgm\Zggy • ijh]gham\Zggy \ _dhghf•p• lZ l_og•p• kihemq_g• a g_h[o•^g•klx ihrmdm gZcdjZsh]h a fh`eb\bo \Zj•Zgl•\ ihe•ir_ggy ydhkl• gZ^•cghkl• lZ ^hklh\•jghkl• fh^_e_c kdhjhq_ggy €ogvh€ jhaf•jghkl•
KZfZ aZ^ZqZ jhajZomgdm iZjZf_lj•\ qbk_evgh€ fZl_fZlbqgh€ fh^_e• \ [•evrhkl• \biZ^d•\ a\h^blvky ^h \bj•r_ggy ^_ydh€ hilbf•aZp•cgh€ aZ^Zq• H^gbf a lZdbo \biZ^d•\ } rlmqg• g_cjhgg• f_j_`• GF d_jm\Zevg• iZjZf_ljb ydbo ijb fh^_ex\Zgg• gZ ?HF jhajZoh\mxlv \bj•rmxqb hilbf•aZp•cgm aZ^Zqm f•g•f•aZp•€ ihfbedb fh^_e•
M p•c dgba• jha]eygml• f_lh^b qbk_evgh€ hilbf•aZp•€ sh gZc[•evr rbjhdh aZklhkh\mxlvky gZ ijZdlbp• ^_lZevgh hibkZg• ijbdeZ^b aZklhkm\Zggy f_lh^•\ hilbf•aZp•€ ijb \bj•r_gg• aZ^Zq gZ\qZggy rlmqgbo GF Z lZdh` \bj•r_ggy aZ^Zq [Z]Zlh\bf•jgh€ g_e•g•cgh€ ZijhdkbfZp•€
>ey ihe_]r_ggy ihrmdm \bagZq_gv l_jf•g•\ • ihgylv m l_dkl• dgb]b \hgb \b^•e_g• `bjgbf rjbnlhf M ^h^Zldm ijb\_^_gbc l_fZlbqgbc iml•\gbd aZ kibkdhf e•l_jZlmjb
:\lhjb \bkeh\exxlv sbjm ih^ydm k\h€f dhe_]Zf sh ijbcfZeb mqZklv \ hdj_fbo ^hke•^`_ggyo j_amevlZlb ydbo ijb\_^_g• \ p•c dgba• j_p_ga_glZf Z lZdh` lbf olh gZ^Z\Z\ ^jm`gx i•^ljbfdm Z\lhjZf ijb gZibkZgg• dgb]b
7
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f•g•fZevgh]h agZq_ggy nmgdp•€ |
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iSb mfh\Zo |
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L = n.
Kmdmig•klv qbk_e [ [1, x2, ..., xn sh aZ^h\hevgy} h[f_`_ggyf
gZab\Z}lvky ijbimklbfbf j•r_ggyf Z[h ieZghf .
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IeZg [* = (x1*, x2*, ..., xn* ijb ydhfm p•evh\Z nmgdp•y aZ^Zq•
ijbcfZ} fZdkbfZevg_ f•g•fZevg_ agZq_ggy gZab\Z}lvky hilbfZevgbf.
AgZq_ggy p•evh\h€ nmgdp•€ iSb ieZg• [ [m^_fh ihagZqZlb
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M lhfm \biZ^dm dheb ihlj•[gh agZclb f•g•fmf nmgdp•€
F = c1x1+c2x2+...+cnxn,
fh`gZ i_j_clb ^h agZoh^`_ggy fZdkbfmfm nmgdp•€
F1 = -F = -c1x1 - c2x2 - ... -cnxn,
hkd•evdb PLQ ) PD[ )
H[f_`_ggy g_j•\g•klv \bo•^gh€ aZ^Zq• EI sh fZ} \b]ey^ £ ",
fh`gZ i_j_l\hjblb \ h[f_`_ggy j•\g•klv ^h^Z\Zggyf g_\•^ }fgh€ af•ggh€ Z h[f_`_ggy g_j•\g•klv \b]ey^m ³ \ h[f_`_ggy j•\g•klv
\bjZom\Zggyf a ch]h e•\h€ qZklbgb ^h^Zldh\h€ g_\•^}fgh€ af•ggh€ LZdbf qbghf h[f_`_ggy g_j•\g•klv
ai1x1 + ai2x2 +...+ ainxn £ bi
i_j_l\hjx}lvky \ h[f_`_ggy j•\g•klv
ai1x1 + ai2x2 +...+ ainxn + xn+i = bi (xn+i ³ 0),
9
Z h[f_`_ggy g_j•\g•klv
ai1x1 + ai2x2 +...+ ainxn ³ bi
\ h[f_`_ggy j•\g•klv
ai1x1 + ai2x2 +...+ ainxn - xn+i = bi (xn+i ³ 0).
M lhc `_ qZk dh`g_ j•\gyggy kbkl_fb h[f_`_gv
ai1x1 + ai2x2 +...+ ainxn = bi
fh`gZ aZibkZlb m \b]ey^• g_j•\ghkl_c
ìai1x1 + ai2 x 2 |
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î ai1x1 |
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Qbkeh g_\•^ }fgbo af•ggbo sh \\h^ylvky |
iSb i_j_l\hj_gg• |
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h[f_`_gv g_j•\ghkl_c |
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h[f_`_ggy j•\ghkl• |
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qbkem |
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g_j•\ghkl_c sh i_j_l\hjxxlvky |
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<•^agZqbfh gZj_rl• |
sh |
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mfh\• |
g_\•^ }fghkl• lh €€ \Zjlh aZf•gblb ^\hfZ g_\•^}fgbfb af•ggbfb Xk lZ Yk ihdeZ\rb [k = uk - vk.
<eZklb\hkl• hkgh\gh€ aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy
AZibr_fh hkgh\gm aZ^Zqm e•g•cgh]h ijh]jZfm\Zggy m \_dlhjg•c nhjf• AgZclb fZdkbfmf nmgdp•€
F = cx |
(1.6) |
iSb mfh\Zo |
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x1p1+x2p2+...+xnpn = p0; |
(1.7) |
x ³ 0, |
(1.8) |