Class ZMM.VJ]SMCS
KBPXB HXM@DB nx SLM@B TMXB HXM@DB
Vxmicmvlb7 Mp ms cnispxrcpb` ni pjb vxmicmvlb ne Tjbapspnib hxm`db, m)b) tjbi pjb hxm`db ms halaicb`, V Q
X ;
Z
, tjbxb V ai` Q axb ointi xbsmspaicbs,
X ms a uaxmahlb xbsmspaicb ai` Z ms pjb riointi xbsmspaicb) Cnispxrcpmni7 A cnispaipai xbsmspmub tmxb ne rimenxk `makbpbx ms knripb` ni a tnn`bi vlaio alnid tmpj a kbpbx xrlb) Cnvvbx spxmvs axb cniibcpb` ap pjb bi`s A ai` H ne pjb tmxb as sjnti mi pjb emdrxb) Pjb pbxkmials ni pjms spxmv axb cniibcpb` pn a happbx}) Ainpjbx cnvvbx spxmv ms emzb` hbptbbi pjbsb ptn spxmvs enxkmid ptn davs) Mi nib dav, a xbsmspaicb hnz X ms cniibcpb` ai` mi pjb npjbx dav, ai riointi xbsmspnx Z ms cniibcpb`) Nib bi` ne a daluainkbpbx ms cniibcpb` pn pjb km`+vnmip @ ne pjms spxmv ai` pjb npjbx bi` pn a gncob} tjmcj cai slm`b ni pjb tmxb AH) Enx a dmubi ualrb X, pjb gncob} ms slm`b` ni pjb tmxb mi srcj a ta} pjap daluainkbpbx sjnts wbxn `belbcpmni) Avvlmcapmnis ne kbpbx hxm`db7 1) Pn kbasrxb pjb riointi xbsmspaicb7 xbsmspai cb7 Pn emi` pjb ualrb ne riointi xbsmspaicb, pjb cmxcrmp ms axxaidb` as sjnti mi emdrxb) Pjb hxm`db cai hb halaicb` h} knumid pjb gncob} nubx pjb tmxb) Enx a dmubi ualrb X, pjb gncob} ms slm`b` ni pjb tmxb mi srcj a ta} pjap daluainkbpbx sjnts wbxn `belbcpmni) Me irll vnmip ms nhpamib` ap C srcj pjap AC ; l pjbi, CH ; 166+l) Pjb xbsmspaicb hbptbbi A ai` C ms paobi as V ai` pjap hbptbbi C ai` H ms paobi as Q) Smicb pjb hxm`db tmxb jas rimenxk axba ne cxnss+sbcpmni, cxnss+sbcpmni, V ∛ l ai` Q ∛166 ∘l ) V l ; Pjbi, Q 166 ∘l Exnk pjb vxmicmvlb ne Tjbapspnib hxm`db, Nx, Z
;
X
V
;
Q
166
∘
X Z
l
l Ointmid pjb ualrbs ne X ai` l, Z cai hb enri`)
⇘
l 166
; ∘
l
X Z
+++++++++++++++++++++++ +++++++++++++++++++++++ *1-
8) Pn cnkvaxb ptn riointi xbsmspaicbs7 Enx cnkvaxmid ptn riointi xbsmspaicb s X1 ai` X8, Z mi
pjb cmxcrmp ms xbvlacb` h} X 1) Pjb gncob} ms knub` nubx pjb hxm`db tmxb dbipl} ai` h} a`grspmid X, pjb halaicmid lbidpj l 1 ms nhpamib` tjbi pjb daluainkbpbx daluainkbpbx dmubs wbxn `belbcpmni) Mi srcj a casb, me AC; l 1, pjbi, CH;166+ l 1) Smicb pjb hxm`db tmxb jas rimenxk axba ne cxnss+sbcpmni, cxnss+sbcpmni, V ∛ l 1 ai` Q ∛166 ∘ l 1 ) Pjbxbenxb,
V Q
;
l 1 166 166
∘ l 1
Exnk pjb vxmicmvlb ne Tjbapspnib hxm`db, V X l 1 X
Q
;
X1
⇘
166
∘ l 1
;
X1
∰ X1 ; X 166 ∘ l 1 ∘ ∘ ∘ ∘*8l 1
Int, pjb bzvbxmkbip ms xbvbapb` tmpj pjb npjbx riointi xbsmspaicb X 8 mi pjb xmdjp dav) Me l 8 ms 166 ∘ l 8 pjb halaicmid lbidpj, pjbi tb dbp+ V ∛ l 8 ai` Q ∛ 166 Pjbxbenxb,
V Q
;
l 8 166 166
∘ l 8
Exnk pjb vxmicmvlb ne Tjbapspnib hxm`db,
+++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++ +++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++ ++ SXJSS.8616. CRXXBIP BLBCPXMCMP] Vadb 1 ne 2 Vxbvaxb` h}7 Sai`j}a)O
Class ZMM.VJ]SMCS
V Q
;
X X 8
⇘
∰ X 8 ; X
l 8 166 166 ∘ l 8
166 166 ∘ l 8
l 8
;
X X 8
∘ ∘ ∘ ∘*3-
@mum`mid bqrapmni *8- h} bqrapmni *3-, tb dbp+
∰
X1 X8
X1 X8
X
; X
166 166 ∘ l 1
l 1 166 166 ∘ l 8
;
166 166 ∘ l 1
l 1
⋂
l 8 166 166 ∘ l 8
l 8
l 166 166 ∘ l 1
; 8⋂
l 1 166 166 ∘ l 8
∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘*0-
Ointmid pjb ualrbs ne l 1 ai` l8 , pjb xapmn
X1 X 8
cai hb enri`)
VNPBIPMNKBPBX
A vnpbipmnkbpbx ms rsb` pn cnkvaxb b)k)es ne ptn cblls nx pn kbasrxb pjb mipbxial xbsmspaicb ne a cbll) Vxmicmvlb7 Pjb vxmicmvlb ne vnpbipmnkbpbx spapbs pjap tjbi a cnispaip crxxbip ms vassb` pjxnrdj a tmxb ne rimenxk axba ne cxnss+sbcpmni, pjb vnpbipmal `xnv acxnss ai} vnxpmni ne pjb tmxb ms `mxbcpl} vxnvnxpmnial pn pjb lbidpj ne pjap vnxpmni) Cnispxrcpmni7 A vnpbipmnkbpbx cnismsps ne a irkhbx ne sbdkbips ne tmxb ne rimenxk axba ne cxnss+ sbcpmni spxbpcjb` spxbpcjb` nubx a tnn`bi hnax` hbptbbi hbptbbi ptn cnvvbx spxmvs) spxmvs) Bacj sbdkbip ms 166ck lnid) A happbx} cniibcpb` hbptbbi pjb ptn pbxkmials sbi`s crxxbip pjxnrdj tmxb, tjmcj ms obvp cnispaip h} rsmid a xjbnspap) Pjbnx}7 Lbp U hb pjb vnpbipmal `meebxbicb acxnss cbxpami vnxpmni ne pjb tmxb, tjnsb xbsmspaicb ms X) Me M ms pjb crxxbip pjxnrdj pjb tmxb, pjbi U;MX) Tb oint, X
; ρ l
A
, tjbxb l, A ai` ρ axb lbidpj, axba ne
cxnss+sbcpmni ai` xbsmspmump} ne pjb kapbxmal ne pjb tmxb xbsvbcpmubl}) Pjbxbenxb, U ; M ρ
l A
Me cnispaip crxxbip ms vassb` pjxnrdj pjb tmxb ne rimenxk axba ne cxnss+sbcpmni, pjbi M ai` A axb cnispaips) Smicb enx a dmubi tmxb, ρ ms alsn cnispaip, tb jaub+
; a cnispaip Ô l Nx, U ∛ l U
Jbicb, me a cnispaip crxxbip elnts pjxnrdj a tmxb ne rimenxk axba ne cxnss+sbcpmni, pjb vnpbipmal `xnv acxnss ai} vnxpmni ne pjb tmxb ms `mxbcpl} vxnvnxpmnial vxnvnxpmnial pn pjb lbidpj ne pjap vnxpmni) Avvlmcapmnis ne vnpbipmnkbpbx 1) Pn cnkvaxb b)k)es ne ptn cblls7
Lbp B1 ai` B8 hb pjb b)k)es ne pjb ptn cblls) Pjb vnsmpmub vnlbs ne hnpj pjb cblls axb cniibcpb` pn pbxkmial A ne pjb vnpbipmnkbpbx) Pjb ibdapmub vnlbs ne pjb ptn cblls axb cniibcpb` pn pjb pbxkmials 1 ai` 8 ne a ptn ta} ob} tjmlb mps cnkkni pbxkmial ms cniibcpb` pn a gncob} G pjxnrdj pjb daluainkbpbx D) Mi pjb vxmkax} cmxcrmp, ai akkbpbx, a happbx} ne b)k)e B, a xjbnspap ai` a nib ta} ob} ms miclr`b`) +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++ +++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++ ++ SXJSS.8616. CRXXBIP BLBCPXMCMP] Vadb 8 ne 2 Vxbvaxb` h}7 Sai`j}a)O
Class ZMM.VJ]SMCS
Pn cnkvaxb pjb b)k)es ne ptn cblls, a cnispaip crxxbip ms vassb` pjxnrdj pjb vnpbipmnkbpbx tmxb hbptbbi A ai` H)Tjbi H)Tjbi pjb vlrd ms vrp hbptbbi hbptbbi 1 ai` 3 ne pjb ptn ta} ob}, pjb cbll cbll ne b)k)e B 1 ms miclr`b` mi pjb cmxcrmp) cmxcrmp) Lbp pjb halaicmid halaicmid lbidpj hb l 1) Me z ms pjb xbsmspaicb vbx rimp lbidpj ne pjb vnpbipmnkbpbx tmxb, ai` M, pjb cnispaip crxxbip crxxbip vassmid vassmid pjxnrdj mp, pjbi+ B1 ∛ * zl zl 1 - M +++++++++++++++++++++ +++++++++++++++++++++ *1Tjbi ob} ms vrp hbptbbi 8 ai` 3 ne pjb ptn+ta} ob} aepbx xbknumid mp exnk pjb dav 1 ai` 3, pjb cbll tmpj b)k)e B 8 ms miclr`b` mi pjb cmxcrmp) Lbp pjb halaicmid lbidpj hb l 8) Pjbi, B 8 ∛ * zl 8 - M +++++++++++++++ +++++++++++++++ *8@mum`mid bqrapmni *1- h} *8-,
B1 B8
;
l 1 l 8
B1 B8
∛
* zl 1 - M * zl 8 - M
++++++++++++++++ ++++++++++++++++ *3-
Kbasrxmid l 1ai` l8, pjb xapmn ne pjb b)k)es cai hb enri`)
8) Pn kbasrxb pjb mipbxial xbsmspaicb ne a cbll7
Pjb cmxcrmp ms axxaidb` as sjnti) Vlrd mi pjb ob} O ai` kamipami a cnispaip crxxbip M pjxnrdj pjb cmxcrmp) Emi` pjb halaicmid lbidpj l 1) zl 1 - M ++++++++++ *0 Pjbi, B 1 ∛ * zl Int, vlrd pjb ob} O 8 pn miclr`b S mi pjb cmxcrmp) Lbp pjb halaicmid lbidpj hb l 8) Pjbi, U ∛ * zl zl 8 - M +++++++++ *2@mum`mid bqrapmni *0- h} *2-, tb dbp+
B1
U B1
U
∛ ;
* zl 1 - M * zl 8 - M
l 1 l 8
++++++++++++++ *5-
Pjb mipbxial xbsmspaicb ne a cbll ms dmubi h}7
B ∘ 1 S U
x ;
Rsmid bqrapmni *5-, tb jaub+
l 1 ∘ 1 S l 8
x ;
Ointmid pjb ualrbs ne l 1, l8 ai` S, mipbxial xbsmspaicb cai hb calcrlapb`) calcrlapb`)
AKKBPBX Akkbpbx ms rsb` enx kbasrxmid crxxbip) Daluainkbpbx Daluainkbpbx ms a lnt xbsmspaicb `bumcb ai` caiinp hb rsb` pn kbasrxb crxxbip mi a cmxcrmp) Pn cniubxp a daluainkbpbx daluainkbpbx pn ai akkbpbx, a lnt ualrb xbsmspaicb xbsmspaicb callb` sjrip ms cniibcpb` mi vaxallbl pn pjb daluainkbpbx) As mps ualrb ms krcj skallbx pjai D, knsp ne pjb crxxbip elnts pjxnrdj mp ai` pjb daluainkbpbx daluainkbpbx `nbs inp dbp `akadb`) Knxbnubx, pjb bqrmualbip xbsmspaicb ne pjb kn`memb` daluainkbpbx ms lntbx pjai pjap ne pjb sjrip tjmcj tjbi cniibcpb` mi pjb cmxcrmp `nbs inp alpbx mps xbsmspaicb avvxbcmahl} avvxbcmahl} ai` jbicb pjb pxrb ualrb ne pjb crxxbip ms kbasrxb`)
+++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++ +++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++ ++ SXJSS.8616. CRXXBIP BLBCPXMCMP] Vadb 3 ne 2 Vxbvaxb` h}7 Sai`j}a)O
Class ZMM.VJ]SMCS
Enxkrla ne Sjrip Lbp MD ; kazmkrk crxxbip cavacmp} ne pjb daluainkbpbx, M ; `bsmxb` kazmkrk crxxbip mi pjb akkbpbx S ; ibcbssax} ualrb ne pjb sjrip Pjb ualrb ne pjb sjrip sjnrl` hb sn sblbcpb` pjap pjb crxxbip M D elnts pjxnrdj pjb daluainkbpbx ai` pjb halaicb M + M D ; Ms elnts pjxnrdj pjb sjrip as sjnti mi pjb emdrxb) Avvl}mid Omxcjjnee”s sbcni` xrlb pn pjb lnnv,
∘ M D D / * M ∘ M D - S ; 6 ∰ S ;
DM D M ∘ M D
UNLPKBPBX Unlpkbpbx ms rsb` pn kbasrxb pjb vnpbipmal `meebxbicb hbptbbi ai} ptn vnmips ne a cmxcrmp) Daluainkbpbx Daluainkbpbx tmpj srmpahlb kn`memcapmni ms rsb` enx pjms vrxvnsb) Pn cniubxp a daluainkbpbx pn a unlpkbpbx a jmdj ualrb xbsmspaicb xbsmspaicb callb` sbxmbs xbsmspaicb ms cniibcpb` mi sbxmbs tmpj pjb daluainkbpbx) @bxmumid enxkrla enx pjb Sbxmbs Xbsmspaicb Lbp MD ; kazmkrk crxxbip cavacmp} ne pjb daluainkbpbx, D ; xbsmspaicb ne daluainkbpbx, U ; `bsmxb` kazmkrk unlpadb mi pjb unlpkbpbx Xs ; ibcbssax} ualrb ne pjb sbxmbs xbsmspaicb Avvl}mid Njk”s lat pn pjb kn`memb` daluainkbpbx as sjnti mi pjb emdrxb, M D *D / X S - ; U M D X S X S
∰
;
;
U ∘ M D D U
∘
M D
D
DXNRVMID NE CAVACMPNXS
Cavacmpnxs cai hb cniibcpb` bmpjbx mi vaxallbl nx mi sbxmbs) Tjbi pjb cavacmpnxs axb cniibcpb` mi sbxmbs, pjb ibp cavacmpaicb @BCXBASBS) Tjbi pjb cavacmpnxs axb cniibcpb` mi sbxmbs, pjb ibp cavacmpaicb MICXBASBS)
CAVACMPNXS CNIIBCPB@ MI SBXMBS
Cavacmpnxs axb cniibcpb` mi sbxmbs me pjb vnsmpmub vlapb ne nib ms cniibcpb` pn pjb ibdapmub vlapb, as sjnti mi Emdrxb)
Lbp U1, U8 ai` U3 hb vnpbipmal `meebxbicbs acxnss pjb vlapbs ne pjxbb cavacmpnxs C 1, C8 ai` C3) Me U ms pjb vnpbipmal `meebxbicb acxnss pjb sbxmbs cnkhmiapmni, pjbi U ; U 1 /U 8 /U 3 +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++ +++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++ ++ SXJSS.8616. CRXXBIP BLBCPXMCMP] Vadb 0 ne 2 Vxbvaxb` h}7 Sai`j}a)O
Class ZMM.VJ]SMCS
Exnk pjb `bemimpmni ne cavacmpaicb, cavacmpaicb, U 1 Pjbxbenxb, U
;
q C 1
/
q C 8
/
q C 3
;
q C 1
? U 8
;
q C 8
? U 3
;
q C 3
; q 1 / 1 / 1 ++++++++++++++++ ++++++++++++++++ *1 C 1 C 8 C 3
Lbp C hb pjb bqrmualbip cavacmpaicb ne pjb sbxmbs cnkhmiapmni) Pjbxbenxb, tjbi a vnpbipmal `meebxbicb U ms avvlmb` acxnss pjb a cavacmpnx ne cavacmpaicb C, mp tmll spnxb sakb aknrip ne cjaxdb) q Jbicb, U ; +++++++++++++++++++++++ +++++++++++++++++++++++ *8C Exnk bqrapmnis *1- ai` *8-, tb dbp+
; q 1 / 1 / 1 C 1 C 8 C 3 1 1 ; / 1 / 1 ++++++++++++++++++++++++ ++++++++++++++++++++++++ *3C C 1 C 8 C 3
q C Nx,
Pjrs, pjb xbcmvxncal ne pjb xbsrlpaip *bqrmualbip- cavacmpaicb cavacmpaicb ne pjb sbxmbs cnkhmiapmni ne a irkhbx ne cavacmpnxs ms bqral pn pjb srk ne pjb xbcmvxncals ne pjb cavacmpaicbs ne mi`mum`ral cavacmpnxs)
CAVACMPNXS CNIIBCPB@ MI VAXALLBL
Ptn cavacmpnxs axb mi vaxallbl me pjb ibdapmub vlapbs axb cniibcpb` ai` pjb vnsmpmub vlapbs axb cniibcpb`, as sjnti mi Emdrxb
All pjb cavacmpnxs tmll jaub sakb vnpbipmal `meebxbicb acxnss pjbmx vlapbs) As pjb cavacmpnxs jaub `meebxbip cavacmpaicbs, cavacmpaicbs, pjb} tmll `xat `meebxbip aknrip ne cjaxdb exnk pjb snrxcb ne b)k)e) Me q 1, q8, q3 axb pjb ualrbs ne pjb cjaxdbs ni pjb cavacmpnxs C 1, C8 ai` C3 xbsvbcpmubl} xbsvbcpmubl} ai` U ms pjb vnpbipmal `meebxbicb acxnss pjb vaxallbl cnkhmiapmni, pjbi+ q1 ; UC 1 ? q8 ; UC 8 ? q3 ; UC 3
; q1 / q 8 / q3 q ; UC 1 / UC 8 / UC 3 ; U * C 1 / C 8 / C 3 - ++++++++++++++++++++++++++ ++++++++++++++++++++++++++ *0-
Pjb pnpal cjaxdb ni pjb vaxallbl cnkhmiapmni ne cavacmpnxs, q
Me C ms pjb bqrmualbip cavacmpaicb ne pjb vaxallbl cnkhmiapmni, cnkhmiapmni, pjbi+ q UC +++++++++++++++++++ +++++++++++++++++++ *2Exnk bqrapmnis *0- ai` *2-, tb dbp+ ;
U * C 1
/ C 8 / C 3 -
C ; * C 1 / C 8
UC
;
/ C 3 -
+++++++++++++++++ +++++++++++++++++ *5-
Pjrs, pjb xbsrlpaip *bqrmualbip*bqrmualbip- cavacmpaicb cavacmpaicb ne pjb vaxallbl cnkhmiapmni cnkhmiapmni ne a irkhbx ne cavacmpnxs ms bqral pn pjb srk ne pjb cavacmpaicbs ne mi`mum`ral cavacmpnxs)
+++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ +++++++++++++++++++++++++++ +++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++ ++++++++++++++++ ++ SXJSS.8616. CRXXBIP BLBCPXMCMP] Vadb 2 ne 2 Vxbvaxb` h}7 Sai`j}a)O
