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β g h zNr[`M g NR g YH^eMHPQM g N α = β = γ = δ = λ MJN α2 = α + 1 n h P&PM ] h TrTrMHV?u hcgUq Y g NMHV_N] hcg VXNWY g NMON M k YH^ s ^`M|TWYHuUu h TWN q MHV q [fY k hcg Y ^`M VY j] h N M α √ YHj >gUh 0PzcTrM q h T n α = 21 (1 + 5) ´y¶! c¶ Y g V^eM]rYDV h j ^eM!uM g NrY k hcg M q h TM VXN]XT h [fV M shcg'h zNr[`M g N s uUYHT √ @ h [fTo^eMuM g NrY k hcg MMJ N h [f^ M ^eYvP M PM"MJ Nrj q M s 0 1 α = (1 − 5) = −1/α 2 ]_[e q MHV?V?jUV K n ACEBD qUhcg N^`M V] h N MH V=V hcg N^eMHV q [eY k hcg YH^eMHV q juUM g NWY k hcg M MO Nrj q [ M
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¦ NMHYD]?bcM TWNrM? h x NXb2MuUM g NWY k hcgts N{ hQq [eY k hcg YH^fV ad b P MHMJN n LHMONNXb2M&PQMHYHV?jUTrM h xoNXb2M'Y gUk ^eM&zUMON{MDM g NXb2M VmM e E c C c e N{ hQq [fY k hcg Y ^eVzUM q M gUh NrM q z l NXb2MV?YDPQM @ ^ h {M TX?]_YHVmM K b a ^`MONWN_M T|YDVNXb2MQcM TWNrM?^eYDzUMJ^ @ VMDMNXb2M q [eY k TWYDP K n7 [ g ]_M YH^\N_M T g YJN_M|[ g N_M Tr[ h T=Y gUk ^`M VYHTrMaM cjUY ^ s NXb2MiNXbcTrMHMY gUk ^eMHV da b d x h T_PQM q z l NXb2M q [fY k hcg Y ^eVoY NoM Y]?bcMHTN_MXvYHTrM gUh {)] h P c e u2^eMJNrM ^ lQq MJNrMHTrP&[ g M q s YHVVb h { g [ g NXb2M q [eY k TWYDP n A B tb2M k [ cM g TrMJ^eYJNr[ hcg ]rY g'gUh {¿zUMiTrM VXNWY NrM q YHV s x`T h P a +Y c + d = a + YHcTr+ e[eP'[f^fYDT t d b f [ i V e [ & P 2 u w ^ ` [ M V X N c b J Y N M X V {bd[`]?b{MbcYOcM n gUq 4BCD d = e 4BAE {b2M g ]_M s
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x NXb2MacMHTNr[`]_MHV h xY g YD]_jcN_MH?Y gUk `^ M q N_Tr[fY gUk ^eM n b2MQYH^\Nr[wNrj q M'x`T h P hcg M h y ABC PMHMJNWVaNXb2M h uUu h V?[\N_MVX[ q MYJN n ®T h P D s uUMHTruM gUq [e]_j2^eYHTWV DE Y gUq DF YDTrM D q TWYO{ g N h NXb2M h NXb2M TN{ h V?[ q M V n £UT h cMNXbcYJNaNXb2M^`M gUk NXb h x EF e[ VNXb2MVYHPM {bd[`]?b2MmcM TycMHTN_MX[eV]?b h VM gon
j
]?bcYH[fTWV n b2MHTrMYHTrM ] hcg xBMHTrM g ]XMbcY ^f^bcYHVYT h j gUq NrYDzc^eM{i[wNXb q MJ^`M k Y NrMHVN h n[fV Tb2MHTVmM Y NYDT_zc[\N_n B x H M r T M X ] M t 2 b M R W T _ V N M e ^ M J Y _ N | M ? ] b m V M V d b W T D Y TW[f^ lon b2MHTrMH hcg g n q k h2h h X N 2 b M M e ^ M J Y _ N | M X V w [ W N V c u e ^ D Y X ] M = V N X N 2 b | M r T [ b N x X N 2 b M s h k h q M ^eM k YJN_M s x h T q k (k + 1)@ uUYHTWNr[e]_j2^eYHT NXb2kMiVMD] kN NXb2M!RTWV_N K| J M ` ^ M Y r N i M X V w [ W N V ? M y N n }~g s hcgUq&q k g h n h 1 ]?bc≤ YH[fT=k]r≤ Y g nzUM −h 1]X]_jUuc[`M q z l P h TrMNXbcY ghcg M q M ^eM k YJN_M n x h T{bd[`]?baNXbd[fV[fVu h V?V?[ezc^eM n ®[ gUq NXb2M|VMJN h xUdY ^fjM V n n ¦ NXb2M] YHxfNrMHT
H
n £UT h cM!NXbcY NNXb2MiVmM cjM g ]XM q M R g M q z l y = 1 Y gUq 0 yn+1 =
1 2
] hcg VX[eV_NrV hcg ^ lQh x[ g NrM k M TWV n
3yn +
q
2 −4 5yn
@xh T K n≥0
YDTrM V?ub2M TrMHV h xUj g [\NTWY q [wjUV=YHT_TWY gUk M q [ g V?uUYD]XM s s n jUu2u h VMvNXbcYJN 1 N . .j.]?b2B NMXHY]rNr^N N{ V h NXbcY NMHYD]?bQVuUb2MHBTrMi M ! V h l hh NXb2M TWVvM? N_M T g Y ^f^ lon LHMJN P zUMYau h [ g N c j r N X V [ & M H Y w ^ ^ X N 2 b H M V & M V U u 2 b H M r T M V Y e ^ J M N X N 2 b M u h [ g NrV h x] hcg NrYD]rN!zUM h q s gUq s s n CK 1 [eV . . .M CN_N @ tb2M^`M gUk NXb h xtNXb2MNWY gUk M g Nx`T h P N h NXb2N MaV?ub2M TrM M q q U g h P B 1≤i≤N z l n £UT h cMNXb2Miu2T h2q j]rN h xNXb2M cjUY g Nr[wNr[eMHV t [eV gUh NiP h TrM!NXbcY g NXb2MiuUT h2q j]rN h x t i i NXb2M q [fVXNWY g ]_MHV n PC
_H
i
MXHN){M k [ cMNXb2M u2T h zc^eMHP&V¿x`T h P NXb2M
h TrMHY g Y NXb2M ^ l PQuc[eY q {|TW[wNWNrM gi¦ u2Tr[w^ A EdA I s HGG nt¦=k YH[ g P l NXbc15 Y g ZV k h N h&p [f^w^ Y U g q b M h xNXb2M }~g N_M T g YJNr[ hcg Y ^ ^ l P&u2[fY q h P&P&[\NWN_MHM h xNXb2M Y g Y q [eY g¿ Y NX2 _ ] ` [ O M N x T z r N Y [ [ X N 2 b H M V | M 2 u T c z e ^ H M & P V x T U j T U j V M Dh l s h h g gUk h h h n
15
Zu ` d W" ` " !>
= # K
` p k$ $'
PQYJNr[`]rYH^ V s bcYH[fT PQYJNr[`]rYH^
n ® h TY|uUTW[ePQM p h xNXb2Mx h T_P p
12k + 1
s
= {(a, b) | a b ∈
Y gUq p
s
p
s
s
s
s
= {0 1 2 . . . p − 1}
+H
p - (4a3 + 27b2 )}
s ^`MON
n
® h T s (a0 , b0 ) ∈ s {MvV?Y l XN bcYJN (a, b) Y gUq (a0 , b0 ) YHTrM!M dj2[\dY ^`M g N[wxcNXb2M TrM b) [eVY (a, ? VX j ]?bNXbcYJN n gUhcg M T h M ^eMHPQM g N pc ∈ 0 4 Y gUq p | (b0 − bc6 ) ®[ gUq NXb2M|P&YOH[eP&YH^ g jUPQzUM T h x p[ g M cj2[ dYH^eM g N=pMJ^`|M (a PM g −NrV=ac [ g ) n
H
p
C
j
n ®[ gUq Y ^f^xej g ]rNr[ hcg V f : → V?Y Nr[fV?x l [ gUk f (x − y) = f (x) + xy + f (y) x h T=MmcM T l Y gUq MmcMHT l s {b2M TrM e[ VNXb2MiVmMON h x YH^w^TrMHY ^ x∈ y ∈ {f (x) | x ∈ } g jUPz2MHTWV n
n tb2Mix h w^ ^ h {i[ gUk `x Y]rNWVYHTrMZ gUh { g [ g Y|PQYJNXb2MHP&Y Nr[e]_V] hcg N_M VXN * @ Y K b2M g jUPQU z M T h
x uUT h zd^`M PQVN_M VXNrM q {|YDV n n≥4 @ z K dY]?b'u2T h zc^eMHP¯{|YDVV h ^\cM q z l M?HYD]rNr^ l x h jUT=] hcg N_M VXNWY g NWV n @]K ® h T MHYD]?bu2YH[fT h x u2T h zc^eMHP&V s NXb2MHTrM[eV'M?HYD]rNr^ lhcg M] hcg rN MHV_NrY g Ni{b h z h NX' b uUT h zd^`M PQV n j Y ^N h ¦ VVXjUP&[ gUk NXb2M g jUPz2MHT h xo] hcg N_M VXNWY g NWV[eV k TrM Y NrMHTNXbcY gh T=M cU s R x h T={bd[`]?bNXb2M TrM&Y ^\{|Y l VM?H[eV_NrV!YQ] hcg NrHM V_NrY g N4n {b h P&[ g [eP'jUPdY ^fjM h x YH^w^2NXb2M|uUT h zd^`M PQV n n
H
$k $'
V h ^ cM q gUq XN b2M V h ^ cM q
Y gUq x h Tu h VX[wNr[ cMTrM YH^ n ® h T s ^eMJN a1 +a2 +· · ·+an {b2MHTTrM =NXb2bM 1 b2jU·PQ · zU · bM nTWV j Pz2MHTWnV ≥ s3 s S = YDTrM|uUY [eT¹{i[eVM g U s s s s s s g a a . . . an b1 b2 . . . b n bi q [fVXNr[ g ]rN n 1 2 @ Y K ®D[ gUq NXb2M g jUPQzUM T h x q [fVXNr[ g ]rNTrMHY ^ ?MHT h M V h xNXb2M|u h ^ lcgUh P&[fYH^ f (x) = (x − b1 )(x − b2 ) · · · (x − bn )
H
@z K U £ T h cMNXb2M[ g M cjUY ^f[\N l
l
1 n−1
n X
j=1
1−
aj S
bj
n X
j=1
aj x − bj
n
1 n−1 n X T aj n >
S
j=1
bj
z2M»Y g YD]_jcN_MN_TW[eY gUk ^`M s Y gUq ^`MON zUM[wNWV']_[fTr]_jUP]_[fTr]_^`M n LHMONiNXb2M n LHMJN u M T_uUM gUq ABC [`]_j2^fYDT^w[ g Mix`T h P Nh PQMDMON YJN O n LHMJN zUM|Yu h [ g N hcg s Y gUq P O£UT cM ^`MON z2M&NXb2MQx h2h N h xNXb2MAuM T_uUBC M gUq [`]_j2^fYDT!O ^f[ g MQx`D Th P N h NXb2MQ^w[ g M n h Q[fx P NXbcY N [eV hcg NXb2M h jcNrVX[ q M h x Y gUq NXb2M g AB YHTrM s s s O 2∠QP B = ∠P BC D P Q ] h ^w^f[ g MHYHQT n LHMON zUM!NXb2M 2 [ g [ g ]XTrMHYHV?[ gUk u Tr[fPM|] h j g Nr[ gUk x`T h PNXb2M|V?PQY f^ ^eMHV_Nu2Tr[fPM 2 M T n ®ph n T M?HYHPQucn ^`M s s s s n p1 = 2 p 2 = 3 p 3 = 5 . . . @ Y K ® h TY k [ cM g T V?Y Nr[fV?x l [ gUk s ^eMJN r z2MNXb2M|V?PQY ^f^eMHV_N[ g NrM k M n ≥ 10
n 6 h Tq
2 ≤ r ≤ n−2
s
n − 4 + 1 < pr
s
Y gUq»q M R g M xh T £ T h cMNXbcYJN s s s n U ) YJ− NXb2M TrMMXD[fVXNWV Nss = (sp1 p2 · s · V?· jp]?r−1 baNXbc N gU1hcg M h sx = 1s 2s . . . s pr q [\d[ q HM V n j 1 ≤ j ≤ pr p1 p2 . . . p n Nj @ z K VX[ gUk NXb2MiTrMHVXj2^wN h x @ Y K s R gUq YH^w^u h VX[wNr[ cM[ g NrM k M TWV x h T{d b [e]?b p2m+1
< p 1 p2 · · · p m
n
m
C G ½MabcYOcMTrMH]XMJ[\cM q a Y V h ^fjcNr[ hcg N h £UT h d z ^`M P h x NXb2M DGDGG t ¨ jUV?V?[fY g JY NXb2MD PQYJNr[`]rYH^ ^ l PQuc[eY q DGDGd * I {bd[e]?bYO h [ q VNXb2M!]_Y ^`]_j2^wjUVyjUVmM q [ g NXb2MxBM Y NrU j TrM q V h ^wjcNr[ hcg HGG * C EDCª n
l
£ T h cM!NXbcY N n U 1
° $ e¸
p
1 + x2
+p
1 1+
y2
≤ p
2 1 + xy
xh T s n 0<x y≤1
!² !²$ ° O¶ND² o°O² D ¸ ¶
;
t¶¶ D¸ ½M uUT h cM7NXb2M7[ g M djUYH^w[wN l j U V NXbcYJN g q MHTNXb2M)VX^f[ k bNr^ l {MHYDZDMHT] hcgUq [\Nr[ hcg Y gUq s n x y ≥® 0T xy ≤ 1 h x s y ≥ 0 s ^eMJN
+H
F (x, y) =
tb2M g F (x, y) = = =
{b2MHTrM G(x, y)
tbdjUV s
F (x, y) =
p
2
1 + xy
p
1 + xy
!
−p
1 1 + x2
−p
1 1 + y2
!
n
p
p p p 1 + x2 − 1 + xy 1 + y 2 − 1 + xy p p + 1 + y2 1 + x2 y(y − x) x(x − y) p p p + p 2 2 2 1 + x + 1 + xy 1 + x 1 + y + 1 + xy 1 + y 2 (x − y)G(x, y) s p p p p 2 2 2 2 1 + x + 1 + xy 1 + x 1 + y + 1 + xy 1 + y
p p = x 1 + y 2 + 1 + xy 1 + y 2 p p − y 1 + x2 + 1 + xy 1 + x2 p p p = x − y − xy(x − y) + 1 + xy x 1 + y 2 − y 1 + x2 p 1 + xy(x2 − y 2 ) p = x − y − xy(x − y) + p x 1 + y 2 + y 1 + x2 ! p 1 + xy(x + y) n p = (x − y) 1 − xy + p 2 2 x 1+y +y 1+x (x − y)
1+
x2
+
p
2
! p 1 + xy(x + y) p 1 − xy + p x 1 + y 2 + y 1 + x2
p p p 1 + xy 1 + x2 1 + y 2 + 1 + xy 1 + y 2
} N[fV gUh {-Mmd[ q M g NNXbcY N F (x, y) ≥ 0 w[ x xy ≤ 1 n b2M q M V?[fTrM q TrMHVXj2^wNx h ^w^ h {|V n
n
CA MXHN{MTrMONrjUT ^f[fVXNrM q x h TNXb2M HGGDG k [\cM g [ g HGG * A N h ÁM h T k MaDdY k M ^ h u
g N h TrMHY q M TWV V h ^wjcNr[ hcg V=N }~g _N M T g YJNr[ hcg Y ^ ^ l PQuc[eY q E HA ª Y gUq ] hcg Nr[ g jM q [ g h j2^ h V sH¦ NXb2M g V s ÁyTrMDMH]XMix h
u T h zd^`M PQVuUT h u h VmM q Y gUq Vb h TWNX h U [ g
h TrM Y n b2MHVM&u2T h zc^eMHP&Vv{MHTrM HGG * d CJE DG nvl NXbcY g ZV k h TuUT h c[ q [ gUk jUV{i[wNXbV h ^wjcNr[ hcg V n
n @ % K MON_M T_P'[ g M&Y ^f^[ g NrM k M TWV n ≥ 2 VXj]?bNXbcY Nx h TYH^w^o[ g N_M k MHTWV a Y gUq b TrM f^ Y Nr[ cM ^ l u2Tr[fPM!N h s [wxtY gUqQhcg ^ l [fx n n a ≡ b (mod n) ab ≡ 1 (mod n) &½b2M g NXbd[fVuUT h zd^`M P{|YDV k [\cM g [ g HGG * AHª s NXb2M] hcgUk TrjM g ]XM {|YHVVXNWY NrM q [ g ] h T_TrMH]rNr^ l YDV n
$
ab ≡ 1 (mod n)
° $ e¸
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ab ≡ −1 (mod n)
y¶
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K
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® h TY gl [ g NrM k M T s NXb2M k [ cM g ] hcgUq [\Nr[ hcg [fVM dj2[\dY ^`M g NN h n≥2 @ } K ® h TMmcMHT l [ g N_M k MHT TrMJ^eYJNr[\cMJ^ l u2Tr[fPMN h s 2 n a n a ≡ 1 (mod n) ~} gUq MHM q s NXb2M k [\cM g ] hcgUq [wNr[ hcg [eP&u2^w[`M V @ } K s Y gUq ] hcg cM TWVmMJ^ l s [fx @ } K b h ^ q V s NXb2M g x h T Y gl [ g N_M k MHTWV Y gUq TrM ^fY Nr[ cM ^ l uUTW[ePQMN h s {M|bcYJcM a b n a ≡ b (mod n)
Y gUq LHMJN
ab ≡ 1 (mod n)
8?Dj $ E hcgUq [\Nr[ hcg
H
@
==⇒ ==⇒ ==⇒
ab ≡ a2 ≡ 1 (mod n)
s
a2 ≡ ab (mod n) n a ≡ b (mod n)
K z2MNXb2Mix`YD]rN h WT [ XYJNr[ hcg&h x [ g N h u2Tr[fPM V n n
n = pe11 pe22 . . . pe` ` ei ≥ 1
@ } K [eVM cj2[ dYH^eM g NN h @ }W} K ® h ToMHYD]?b s x h TMmcMHT l [ g NrM k M T TrMJ^eYJNr[\cMJ^ l uUTW[ePQMN h s n i a pi a2 ≡ 1 (mod pei i ) t¶ $ J¸ *| jUu2u h VM&NXbcYJN @ } K b h ^ q V n ½MQuUT h cM @ }W} K n} N![eVM gUh j k bN h ] hcg VX[ q MHT n jUuUu h VmMQNXbcY N a [fVaTrM ^fY Nr[ cM ^ l u2Tr[fPMQN h p nÀ} xV h PQM h xNXb2M uUTW[ePQMHV i s = 1s s q [ c[ q M a s V?Y l p s p s . . . s 1p q [\d[ q M a Y gUq p2 p3 . . . p ` N [ c [ M s s q2h|gUh q q a s NXb2M g a + pe1 p 2 p 3 · · · p [fVkTrM ^fY Nr[ cM ^ l u2Tr[fPM p k+1 k+2 ` 1 K s `{M|bcYJcM N h k+1npc. .l . @ } p n
A
p jcN{MÀYH^fV h
(a + pe11 pk+1 pk+2 · · · p` )2 ≡ 1 (mod n)
bcYOcM
n
n
(a + pe11 pk+1 pk+2 · · · p` )2 ≡ a2 (mod pe11 )
e1 a2 ≡ 1 h (mod {¯VXjUuUpu 1h Vm)MiNXbcY N @ }W} K b h ^ q V s Y gUq ^eMJN
n
iM g _] M s
zUMaTrMJ^eYJNr[\cMJ^ l uUTW[ePQM|N h n b2M g aK {MbcYOcM n @ e[ VTrM f^ Y Nr[ cM ^ l u2Tr[fPMN h M Y]?b
d b U j V z xh T n s l }W} s a pi a2 ≡ 1 (mod pei i ) Y gl s e[ P&u2^ l [ gUk NXbcY N 2 n i a ≡ 1 (mod pe11 · · · pe` ` ) j UuUu h VmM gUh { NXbcY N ? V Y r N f [ V?RMHVNXb2M k [ cM g ] hcgUq [\Nr[ hcgon 1 e2 n @ =K pe1{ p · M · pe` ` 2bcYO· c X N 2 b M z Q M NXbcYJNv[eV s x} s gts l }W} s n 2 22[ ≡ 1M T (mod pei i ) ei ≤ 3 hcg cM TWVmpMJ^il = x T Y r N M s a2 ≡ 1 (mod 8) h gl&h2qUq g k n a } x p > 2 s NXb2M gts z l @ }W} K s {MibcYJcM 22 ≡ 1 (mod pei ) NXbcY N[eV s p = 3 Y gUq i n hcg cMHTWVM ^ lts a2 ≡ 1 (mod 3) x h TY gl [ g N_M k MHT ai TrMJ^eYJNr[\cMJ^ l u2Tri[fPMvN h 3 n ei = 1tbdjUV s n V?Y Nr[fV?RMHVNXb2M k [ cM g ] hcgUq [\Nr[ hcg [fx Y gUqQhcg ^ l [wx n q [ c[ q M V 23 · 3 n b2M c j _ N ] P M V H Y TrM h h s s s s s s n n ∈ {2 3 4 6 8 12 24}
C
H
n @ ´ d¸ K £UT h cM7NXbcY NNXb2MVmMON h x&u h VX[wNr[ cM)[ g N_M k MHTWV{bd[e]?b-]_Y g2gUh NzUM TrMHu2TrMHVM g NrM q YDVY|V?jUP h x q [eV_Nr[ g ]rNuUMHTWxBMD]rNV cjUYHTrMHV=[eV=R g [\N_M n ° $ e¸ %'%$w¸¹¶ y¶ (c¶U· (c¶ $ % %$ U² t¶ ?¶H¶ "_¶
K
u T h uUMHTNr[`M V jUuUu h VmM!{M|bcYOcM|Y|u h VX[wNr[ cM[ g rN M k M T N {i[wNXbaNXb2Mx h f^ ^ h {i[ gUk U Y gUq 2 2 2 2 2 2 s N = a 1 + a2 + · · · + a m
2N = b1 + b2 + · · · + bn
{b2MHTrM YHTrMu h VX[wNr[ cMQ[ g N_M s s s s s x`TWY]rNr[ hcg aV 1 . . . as m b1 s . . . bns [fVYau h {MHT β [\NXa δ M&VbcYH^w^oNXb2M YH^w^ s s aαs /a{i b α /bδ bγs /aβ bγ /b ½ n α β γ δ α 6= β γ 6= δ P >
4N P−2
(2kN +1)2
]rY g zUMTrM uUTrM VmM g N_M q YDVoYVXjUP
k HM TWVVXj]?bNXbcYJN gUhcg M h xNXb2M h x 2 @ [ g ]_^fj q [ gUk 20 = 1K x h T g u2T h cM'NXbcYJNMmcMHT l [ g NrM k M T
H
h x q [fVXNr[ g ]rNtuM TrxBMH]rNV cjUYHTrMHV n
Á k=0 y [ cM g Y gl VXj]?b|[ g NrM k M T s {M{|TW[wNrM [ g XN b2Mx h TrP s {b2M TrM P P P = 4N q + r r−1 n [ g ]XM r ≡ P (2kN + 1)2 (mod 4N ) Y gUq XN b2M|^fY NWNrMHTV?jUP 0 ≤ r ≤ 4N − 1 [eV=^eMHV?VNXbcY g s {MiPQY l {|TW[wNrM k=0 P
P =
r−1 X
(2kN + 1)2 + 4N t
s
k=0
{ M¥`jUVXNyNrYDZDM x h TV h Q P Miu h ?V [\Nr[\cM![ g N_M k MHT s w[ x n} x r = 0 s s {b2M TrM t r≥1 P = 4N t n t=q P 2ui P LHMJN z2MNXb2M|zc[ g YDT l XM u2Y g VX[ hcgQh x n b2M g t= 2 + 22vj +1 t i
j
r−1 P P P (2kN + 1)2 + (2ui +1 aα )2 + (2vj +1 bγ )2 k=0 j,γ P = P ui +1 Pi,α vj +1 2 2 (2 a ) + (2 b ) α γ i,α
H
j,γ
[wx [wx
r≥1 r=0
s n
tbdjUV s ]rY g zUMiTrMHu2TrMHVM g NrM q YDVY|VXjUP h x q [eV_Nr[ g ]rNuM rT xBMH]rN=V cjUYHTrMHV n P } NyrT M PQY [ g V hcg ^ l N h Vb h { NXbcY NNXb2MHTrM!M?H[eV_NrVY!u h V?\[ Nr[\cM[ g NrM k M T N YDVYz h cM n [fVV?j]?b'Y g jUPz2MHT s V?[ g ]XM p jcN 2 2 Y gUq 2 2n
k
29
29 = 2 + 5
58 = 3 + 7
n @ ´ ?¸ K }~g NXb2Muc^eY g M{MvbcYOcM n TrMH]rNrY gUk ^`M V{i[wNXbiu2YDTWY ^f^eM ^VX[ q MHV n tb2MV?[ q M V h x q [fVXNr[ g ]rNTrMH]rNrY gUk ^`M V!^w[`M hcg»q [fVXNr[ g ]rN^w[ g M V n tb2M'z h j gUq YDTW[`M V h xoNXb2MTrMD]rNWY gUk ^eMHV ]_jcNNXb2Muc^eY g Ma[ g N h ] hcg2g MD]rNrM q TrM k [ hcg V n¦ TrM k [ hcg [eVvVY [ q N h zUM d¸ "_¶i[fxo[\NvbcYHV!YJN ^`M YDV_N hcg M h xNXb2McMHTNr[`]_MHV h xoNXb2M TrMH]rNrY gUk ^`M V hcg [\NrViz h j gUq YHT lon tb2M TrMQ]rY g zUM n gUhcg ] hcg cM?TrM k [ hcg V s YHV={M ^w^tYHVTrM k [ hcg V={i[\NXbQP h TrMiNXbcY ghcg Maz h j gUq YHT l ]_jUT~cM n £UT h cMNXbcYJNNXb2MVXjUP h xoNXb2M g jUPQzUM TWV h xNXb2M|cMHTNr[`]_MHV h xyY ^f^ g [`]_MTrM k [ hcg Vv[fVv^eMHV?V NXbcY g n 40n
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X¶D¶ "X¶ D¸!¶
¶
cYD]?b!cMHTN_MX h xcY g [e]XMTrM k [ hcg [eVMJ[wNXb2M T] hcg c MX h T] hcg r] YJcM s DY Vy[ g XN b2MR k jUTrM V @ {b2MHTrM!NXb2M|VmbcY q M q u2YDTN[eVNXb2Mi[ g N_M Tr[ h T h x NXb2M|TrM k [ c h g ] hcg ?V [ q M rT M q K n
C
. .. . .. .. .. .. .. . .... ... .... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .
. . .... .... .... .... .
.................................................. .................................. ............................... ..................................... . .................................................................................................... ........................................................................................................................................... .................................
.................................................................................................................................... ....................... ...................... .......................... ....................... .........................
c hcg ]_YOcM h g cMX ® h TY'VX[eP&u2^eM]_^ h VM q ]_jUT~cM ] hcg NrY [ g M q [ g NXb2M'z h j gUq YHT lh xYTrM k [ hcg s {i[w^f^z2M]_Y ^f^eM q Y C % J¸ n ¦ z h j gUq YDT l uUY [eT [eVVY R[ q NXb2Mu2YH[fT @ TrM VuUMD]rNr[ cM ^ l'h jc(R, K n LHMON N h zUM (R, ¹¶ @ C) TrM VuUMD]rNr[ cM ^ l ¸ t¶ K [fx [eV[ g VX[ q M NrVX[ q C) M R C z2M!NXb2MVMJN h x z h j gUq YDT l u2YH[fTWV h x g [`]_MTrM k [ hcg V n LHMON z2M!NXb2MVMJN h x z h C j gUq YDT l B uUY [eTWV[ g {bd[e]?bYHTrM h jcNrMHT s Y gUq ^eMJN zUM|NXb2MVMJN{bdO [e]?bYDTrM[ g2g MHT n b2M g [fV B I NXb2M q [eVr¥ h [ g Nj g [ hcgh x Y gUq n h N_MNXbcY NNXb2MQVMJN [eV gUh N!MHP&ucN lts VX[ g ]_M'BNXb2M O I I hcg ^ l TrM k [ hcg {i[wNXbQ[ g R g [\N_MYHTrMHYa[eVvY g [e]XMTrM k [ hcg Y gUq Y ^f^ h xo[wNWVz h j gUq YHT l ]_jUT~cM V YDTrM[ g2g M T=z h j gUq YDTW[`M V n ® h T=Yz h j gUq YHT l u2YH[fT Y gUq z2M|NXb2M g jUPQzUM TWV h x s ^eMJN ] hcg cMX|cMHTNr[`]_MHVY gUq ] hcg ]_YOcbM= cMHT(R, Nr[`]_MHC) V h x ∈ Bs TrMHV?uMHv]rbNr[\cMJ^ l cs b{bd[`]?bQzUMJ^ hcgUk N h NXb2M R z h j gUq YDT l ]_jUT~cM n cYD]?bcMHTN_MX h xyNXb2M k [ cM g TrMH]rNrY gUk ^`M Va]XTrM Y NrMHVM?HYD]rNr^ lhcg M C ] hcg cMXcMHTN_MX'Y gUqhcg M] hcg ]_YOcMicMHTN_MX n hcg cM TWVmMJ^ l s M Y]?b] hcg ]_YOcMicMHTN_MX h xoY TrM k [ hcg ] h PM V=x`T h P%Y!cMHTN_MX h x hcg M h xNXb2M h Tr[ k [ g Y ^TrMD]rNWY gUk ^eMHV n b2MHTrMJx h TrM s
X
cb = 4n
n
+H
b∈B
® h TY&z h j U x NXb2Max h ^f^ h {i[ gUk M djUY Nr[ hcg g q YHT l u2YH[fT s z h NXbV?[ q M V h M?DuUTrM VVNXb2MV?jUP h xoNXb2MY gUk b^eMH= V!YJ(R, NNXb2C) M|cMHTNr[`]_MHV h x s { b2MHTrMNXb2M'Y gUk ^`MJY NvMHYD]?b C cMHTN_MX[eV]?b h VM g N h z2MNXb2M|Y gUk ^`M[ g NrMHTW[ h TyN h * R ◦
vb · 90 + cb · 270
◦
=
} Nx h ^w^ h {|VNXbcYJN
(
(vb + cb − 2) · 180◦ (vb + cb ) · 360◦ − (vb + cb − 2) · 180◦
vb − c b =
[ g ]_M I [eV gUhcg ?MHP&ucN lts {M|bcYJcM
{b2MHTrM
X
b∈B
(
4 −4
[wx s b∈O w[ x n b∈I
[fx s b∈O f[ x n b∈I
(vb − cb ) ≤ 4(k − 1) − 4 = 4k − 8
s
e[ VNXb2M g jUPz2MHT h xz h j gUq YHT l u2YH[fTWV[ g n ® kh TvM Y]?bz h j gUq YHT l uUY [eT xyY g [`B ]_M'TrM k [ hcg h s YJN^eMHYHVXN hcg M h xNXb2M C) cMHTNr[`]_MHV h x [fV|Y'cM TWNrM? h x hcg (R, M h xy NXb2M k [ cM g TrMH]rNrY gUk ^`M RV n» [ g ]XMMmcMHT l cMHTN_MX C h xNXb2M k [ cM g TrMH]rNrY gUk ^`M V|YDu2uM YDTWVi[ g MXHY]rNr^ l N{ h z h j gUq YHT l uUY [eTWV s {MQVMDM'NXbcYJN n k ≤ 8n ®[ g YH^w^ l s NXb2MiV?jUP h xNXb2M g jUPQzUM TWV h xNXb2M!cM TWNr[e]XM V h x YH^w^ g [`]_M|TrM k [ hcg V=[eV X
b∈B
(vb + cb ) =
X
b∈B
2cb + (vb − cb )
≤ 2 · 4n + 4k − 8 ≤ 40n − 8
n
C I
j
n o¶ $ ®D[ gUq HY ^w^DuUY [eTWV h xDxej g ]rNr[ hcg V f Y gUq g x`T h P4NXb2MVmMON h xTrMHY ^ g jUPz2MHTWV N h w[ NWVm(MJ^fxVXj]?b)NXbcY N x h TYH^w^UTrM YH^ g U j PQzUM TWV f x + g(y) = xf (y) − yf (x) + g(x) Y gUq n
x
y
° $ e ¸ ¶ D¸
%'%$w¸¹¶ y¶ (d¶
2·
5
(d¶&$ % $ D² t¶
;
X¶D¶ "X¶ D¸!¶
¶
+H
} N[fVMHYHV l N h VmMHMNXbcY N[wx f [fVi] hcg XV NWY g NNXb2M g f (x) = g(x) ≡ 0 n¦ V?V?jUPQM [eV gUhcg ?] hcg V_NrY g N n ¨tM u2^fY]_[ gUk z l [ g NXb2M k [\cM g M djUY rN [ c h gts {M|bcYOcM f x g(x) n f g(x) + g(y) = g(x)f (y) − yf g(x) + g g(x)
NXbcY N
[ g _] MNXb2M|^`MJxfNVX[ q M[fVV l P&PMON_Tr[e][ g x Y gUq y s NXb2MaTW[ k bNV?[ q MP&jUV_Nz2MaYH^fV h s Y U g q {M h zNrY [ g n @~A K
H
g(x)f (y) − yf g(x) + g g(x)
2YDZD[ gUk
= g(y)f (x) − xf g(y) + g g(y)
[ g NXb2M h TW[ k [ g YH^ M cjUYJNr[ hcg {M k MON x=0 f g(y)
*H
@ FK
= ay + b
{b2MHTrM Y gUq n MJNWNr[ gUk y = 0 [ g @ FK k [ cMHV f (b) = b n b { = g(0) @BA K −f djUY Nr[ hcag = {i[\NX(0) b [ c H M V gUh k y=0 @ OK n = ag(x) + bf (x) − bx + g(b)
g g(x)
}~g VmM TWNr[ gUk @ OK Y gUq @ FK [ g N h @BA K s {M h zdNWYH[ g
H
g(x)f (y) + ag(x) + bf (x) = g(y)f (x) + ag(y) + bf (y)
s
h T=M cj2[ dYH^eM g rN ^ t l s
g(x) − b
f (y) + a
=
g(y) − b
f (x) + a
@ IHK s
x h TY f^ ^ Y gUq n ¨txMDr] YH^w^f[ gUyk NXbcY N [fV gUhcg ] hcg XV NWY g N s {M]?b h2h VM VXj]?b!NXbcY N n y0 f (y0 )+a 6= 0 tb2M g {MiVmMHMx`T h P @ IDfK i { [\NXb XN bcY N y = y0
{b2MHTrM
A=
xh T Y f^ ^ x∈
g(y0 ) − b n f (y0 ) + a
g(x) − b = A f (x) + a
LHMJNWNr[ gUk
B = Aa + b
s
s {M|bcYJcM
H
g(x) = Af (x) + B
@ FK s
n}~g VmM TWNr[ gUk NXbd[eV=[ g N h NXb2M h WT [ k [ g YH^ M cjUYJNr[ hcgts {M h zdNWYH[ g n
= xf (y) − yf (x) + Af (x) + B £2jcNWNr[ gUk k [\cM V f x + g(A) = xf (A) + B s {bd[`]?b»Vb h {|VvNXbcYJN f [eVY y = A ^f[ g MHYHTxej g ]rNr[ hcgonpcl @ FK s [eVY ^eV h ^w[ g M YDT n h { s Y q [eTrMH]rN=] h P&u2jcNWY Nr[ hcgal [eM ^ q V g f x + g(y)
f (x) =
c (x − c)) c+1
xh T Y gl ] hcg VXNWY g N n c 6= −1
Y gUq
g(y) = c(y − c)
s
C
$
n ± D¸ ) LHMON O zUMaNXb2M&]_[fTr]_jUP]_M g N_TrM'Y gUq H NXb2M h TNXb h ]XM g N_TrM h xyY g Y]_jcNrM N_Tr[fY gUk( ^eM n £UT h cMNXbcY NNXb2M TrMiMXD[fVXNyu h [ g NrV D s E s Y gUq F hcg VX[ q MHV BC s CA s ABC Y gUq Y gUq TrM VuUMD]rNr[ cM ^ lts VXj]?baNXbcYJN s + DH = OE + EH = OF + F H NXb2Mi^fAB [ g MHV YDTrM|] OD hcg ]_jUTrTrM g N n s s Y gUq
AD BE
CF
; ) 180◦
Y gUq
∠XM N + ∠M N Y < 180◦
n
LHMJN
s s s Y gUq δ = ∠M N A n&}~g α =bcYJ∠N { M cM AB β = ∠ABM {bd[e]?b[fγPQuc=^f[eMH∠BM VNXbcYJN N s s s 4N AB AB < N B ∠AN B < ∠N AB = α Y gUq s NXbdjUV s {MbcYJcM n ~ } g s ∠ABV = ∠N AB + ∠AN B < 2a 4BM N s {bd[`?] bS[eP&u2^w[`M VaNXbcYJN ∠M BN > ∠BM N = γ s Y gUq s ] hcg VmMH M N > BN cjM g Nr^ l s n¦g Y ^ h2k h jUVX^ l s ∠U AB < 2β ∠M N Y = | ∠BM Y gUq M g ]XM s N + ∠M BN > 2γ n ∠XM N > 2δ
H
2α+2β > ∠ABV +∠U AB > 180◦ > ∠M N Y +∠XM N > 2γ +2δ
n
p jcN α + β = γ + δ = ∠AZM s {b2MHTrM Z [eVNXb2Mau h [ g N h x[ g NrMHTWVMD]rNr[ hcgh x AN Y U g q n ½MbcYJcMQY] hcg N_TWY q [e]rNr[ hcgon bdjUV s [ gUq MDM qts NXb2MHTrMM?H[eV_NrViY&NXbd[fT q TrM q u h [ g NBM n tb2M!x h ^f^ h {i[ gUk M?HYHPQuc^`MVb h {|VNXbcY NYix h jUTNXbTrM q u h [ g N g MDM q'gUh NMXD[fVXN s V h NXbcY N ?¶H¶=[eVoNXb2MvP'[ g [fP&jUP-V h j k bN n b2MN{ h NrY gUk M g Ny]_[eTr]_^eMHVy[ g NXb2MR k jUTrM q [f¬MHT V?^w[ k bNr^ l [ g V?[*?M n tb2M!YD]_jcN_M]_M g N_TWY ^2Y gUk ^eMHVYDTrM k TrMHYJN_M ToNXbcY g ◦ ny [ u h [ g NrV h x YDTrM¥`jUVXNYizd[wN h jcNrVX[ q MNXb2M!]_[eTr]_^eMHV s N{ h u h [ g NWVYDTrMNXb2M!]XM g N_TrMH60 V h xcNXb2M!]_[fTr]_^`M V shcg SM u h [ g N[fVNXb2Mu h [ g N h x2NrY gUk M g ] l&h x2NXb2Mi]_[eTr]_^eMHV s Y gUq'hcg Mu h [ g N[fV hcg NXb2M|] h PQP hcg NrY gUk M g NoN h NXb2M]_[fTr]_^`M VY NY q [eV_NrY g ]XM{bd[e]?b|[eVVX^f[ k bNr^ lak TrM Y NrMHTNXbcY g NXb2MvTWY q [fjUV h x NXb2M!^fYDT k M T]_[eTr]_^eM n b2M hcg ^ l NXbcTrMHMu h [ g NWV{bd[e]?bi{i[f^w^Uz2MP&YDT_ZDM q TrM q YDTrMNXb2MvN{ h ]XM g N_TrMHVY gUq NXb2Mu h [ g N h xyNrY gUk M g ] lon @ } x=V h PM h xNXb2M q [fVXNWY g ]_MHVbcYHuUuUM g N h zUM M djUYH^ sUhcg M]rY g V?^w[ k bNr^ l uUMHTNrjUTXzNXb2M|u h VX[wNr[ hcg V h xtY gl u h [ g NWV{i[\NXb h jcN=]?bcY gUk [ gUk NXb2M g MHYHTrMHV_N g M [ k b2z h jUTWV n K
+H
r
... . ............................... ............................................... . ........ ......... . ............ ..... ... ... ......... ...... ........ .... .... ..... .. .... ..... ........ ... .. .... .... .... ... .... . . . . .. ... .. .. ... ... . ... ... . . . . . ... ... ... .. . .. . . . . .. .... ....... ... . ... . . . . ... . . .. ........ ... .. ... .... . ................................................................................................................................................................................................................................ ... . . . . . . . ... ... . .... .. . . . . . . ... .. . .. ... .. ... .. ... .. .. .... . .. ... . . . . . ... ... .... .. .. .. .... ... ... .. .... .... .... .. .... ... .... .... .... ........ .... ..... ... ........... ........ ........ ............ . . . . . . . . . . . . . . . . . . ................. ................... . ....................... .......
r
r
l
r
r
r
r
r
r
r
n ±?¶ $ ) ¦ gUhcg ?MHP&ucN l VmMON A h xTrMHY ^ g jUPQzUM TWVv[eV!]rYH^w^`M q Y B E VmMON[fxNXb2M ] hcgUq [w(Nr[ hcg V s a s a s a s a s a ∈ A Y gUq a + a + a = a 3+ a + a a 1 5 6 1 2 [eP&u2^ l NXbcYJNvNXb2MQVm2M cjM 3g ]XM 4V Y gUq YDTr3M&[ q M g 4Nr[e]_Y ^yjU5uN h 6Y (a , a , a ) (a , a , a ) 1 2 3 4 5 6 uM T_P'jcNrYJNr[ hcgon ® h TY'VMJN h xTrMHY ^ g jUPz2MHTWV s ^`MON D(X) q M gUh NrMNXb2M q [w¬oM TrM g ]_M X VmMON £UT h cM!NXbcY N[fx Y gUq s n {|x − y| : x y ∈ X} YDTrMi[ g R Ag [\= N_M|{0 VM d= jM ag 0]_MH< V h ax1TrM K 0[fVTrMHuc^eYD]XM q z l n hcg VM djM g Nr^ lts F (3n) [fV h zNrY [ g M q 01 1 11 x`T h P z l TrM u2^fY]_[ gUk MHYD]?b [ g @BA K z l n g NXb2M h NXb2MHT fi+2 = fi+3 @~A K bcY gUq s F (n) [eV h zNrY [ g M q x`T h fP i+1 z l TrMHfuci+1 ^eYD]_[ + H M D Y ? ] b [ z l n gUk g F (n) fi+1 fi+3 tbdjUV s F (4n) n = F (4n) Fhc(3n) g cM TWVmMJ^ l s VXjUuUu h VmM.NXbcYJN n [fV-Y gUhcg g M k Y Nr[ cM%[ g NrM k M T VXj]?b NXbcYJN n }x h T s NXb2M g n ∈ E n h {¡YDV?V?jUPQMNXbcY N F (3n) =LHMJFN (4n)z2MQNXb2M n jU =Pz2 0MHT n = 1 x [ \ [ r N | V [ g NXb2M»zd[ g YHT l TrMHu2TrMHVM g NWY Nr[ hcgh x n n g h q k n 2MXHYDP'[ m n u ≥ [ X N 2 b M [ j r ] r N [ U u T x D Y z hcg g gUk gUq hcg h2h h cM s {M h z2VMHT~cMNXbcYJNiNXb2MM djUY Nr[ hcg h ]_]_jUTWV hcg ^ l [ g YDVM A Y gUq»hcg ^ l {b2M g F (3p) = F (4p) n ½M F (3n) = NXFbc(4n) _ N a M H Y f ^ V Y N [ gUh h s g YDVM A s {M'bcYOcM m−1 m−1 m−2 s {bd[e]?b [eP&u2^w[`M V=NXbcY NNXb2M|VmMH] hcgUq zc[ g YDT lQq [ k [wN p h < x 2 [eV n /3 tb2< M TrM 2x h TrM s /2 [ g =YHVm2M A s {MbcYOcM n 0 [wxY gUqhcg ^ l [wx n7pcl TrM R g [ gUk NXb2M[ gUq j]rNr[ hcg uUT h2h x=V?^w[ k bNr^ lts {M n ∈ E p ∈ E ] hcg ]_^wj q MNXbcYJNx h TYH^w^ gUhcg g M k Y Nr[ cMa[ g NrM k M TWV s NXb2M'M djUYH^w[wN l n F (3n) = F (4n) [eP&u2^w[`M VNXbcY N n n∈E
+H
+H
+H
DG ½M'bcYOcM'u2T h cM qh jUT]_^eY [eP%NXbcY N [wxyY gUq»hcg ^ l [fx n n z2∈ ®[ g YH^w^ l s {M!{i[f^w^ uUT h cMiNXbcY N s x h TM Y]?b gUFhcg (3n) g M k = Y Nr[ cFM|(4n) [ g NrM k M T NXb2M g jUP MHT Eh x s m VXj]?baNXbcY N [ g NrM k M TWV n [ g ]_M f m [fV m+1 s {M n ∈ E 0 ≤ n < 2 f ) m+2 m+2 = F (2 {i[f^w^ bcYJcMiuUT h cM q NXb2M q MHVX[eTrM q TrM V?j2^\N n ½Mu2T h ]XMHM q z l [ gUq j]rNr[ hcg&hcg n ® h T m = 0 s NXb2M g jUPQzUM T h x n ∈ E V?j]?b NXbcY N xh T m s NXb2M g jUPz2MHT h x n ∈ E VXj]?b»NXbcYJN m [eV 0 ≤ n < [e2 1 = f 2 {%Rd» mY =[ 1NrM M T n h g g k s Y gUq YDV?V?jUPQM s YDViNXb2M m V 0 ≤ n < 2 m ≥ 2 [ gUq j]rNr[ hcg b l u h NXb2M2V?[f= V s NXfbc3YJNNXb2M g jUPQzUM T h x VXj]?bNXbcY N [fV n P ∈ E NXb2M'[ N_M MH0TWV ≤ n < 2V?kj]?b x h TvY ^f^o[ g NrM k M TWV {i[\NXb n ¦ c h U g k g k f k 0 ≤ k < m n ∈ E NXbck+2 Y N m YDTrM|NXb h VMV?j]?b'NXbcY N m−1 s NXb2M g jUPQzUM T h x {bd[e]?b 0 ≤ n < 2 0 ≤ n < 2 @ jUV?[ gUk NXb2M'[ gUq j]rNr[ hcg b l u h NXb2MHVX[eV K n tb2M&TrMHP&YH[ g [ gUk VXj]?b»NXbcYJN [eV fm+1 nRU∈ EN{ H Y r T Q M X N b V M ? V j ? ] b X N c b J Y N
2 b M W T X V v N h n h zd[ g YHT l 0[ ≤ n x< 2nV?j]?b jUPz2MHT 2M m−1 Y ≤ nNXb2< 2MHm ' P U j X V N 2 z w [ W N V Y | M r T & P H Y [ [ [ \ [ r N V r T H M u2TrMHVM g N q k h gl g s gUq g gUkq k 10 tbdjUV NXb2M jUPQzUM T xVXj]?b Y g jUPz2MHT[ g {bd[e]?b[fV!^`M VnVNXbcY g m−2 [fV n s g h @ jUVX[ gUk NXb2Mv[ gUq E j]rNr[ hcg b l u h NXb2MHVX[eVY k YH2[ g K n tb2MN h NWYH^ g jUPz2MHT h x V?jn]?b|NXbcfYJnN n∈E [fV n tbd[fV] h PQuc^`MON_M VNXb2M|u2T h2h x n 0 ≤ n < 2n f +f =f
j
m+1
m
K
(
m+2
¸ ¹¶ ¸ ( c tb2MNWY gUk M g NrViY N Y gUq n Y]_jcNrM'N_Tr[fY gUk ^eM PQ)MDMONvNXb2MNrY gUk M g NYJN B Y N AY C T ^f[ g MHV Y gUq ABC PQMDMONYJN s Y gUq Q [eVNXb2MP&[ q Xu h AT BC P Y gUq PMHMJNYJN s Y gUq [eVNXb2MiP&[ q Xu h [ g N h x n CA R S BR @ Y K £UT h cMNXbcYJN n
N h XN b2MQ]_[fTr]_jUP]_[fTr]_^`M h x Y g r T H M ? V u H M r ] r N \ [ c J M ^ b2M gUq s ln
[g N U x X N 2 b a M f ^ [ H M V g h AP BU
∠ABQ = ∠BAS
@ z K OM N_M T_P'[ g M s [ g NrMHTrPQV h x TWY Nr[ h V h x VX[ q M!^`M gUk NXbcV s NXb2M!N_TW[eY gUk ^`M Vx h T{bd[e]?bNXbd[eV Y gUk ^`M[eVY|P&YOH[eP'jUP n
° $ e¸
%'%$w¸¹¶
y¶
(c¶U· (c¶ $ % %$ U² t¶
}~g 4ABC s ^`MON α = ∠A s β = ∠B s npcl NXb2MaWN Y gUk M g NX]?b h T q NXb2M h TrMHP s γ M = MJ∠C { k N ∠T BC = ∠T CB = ∠BAC = α n LHMJN q M gUh NrMNXb2M!]_[fTr]_jUPQTWY q [wjUV h x 4ABC n tb2MiR [fV h V]XMJ^`M VN_Tr[fY gUk ^eM k [\cM V BCT
BT =
BC 2 cos α
K
?¶H¶ "_¶
........................................... ....... .............. .............. ......... ........... ........ ....... ....... ..... . . . .... . .............. . ................ ....... ... . . . . . . ......... .... .... ............... .. . . . . . . ... ..... .. .. .. ... .... ..... ... .. ... .. ..... .. .. ..... .... .... .... .... ........ ... .. .. ..... ..... ... . ... .. .. ..... .... ..... ...... . . .. ... . .... . .. ....... .. .. .... . . . . . . . . . . . . . . . . . .... . . . . . . . ... . ....... .. . . ..... .. .................... ... ......... .... .. .......... ..... ..... .... .......... ..... .......................... .. .... ..... ... ..... ... ......... ..... .... ... .. .. ... ... ........... .......................... ..... . . . . . . .. .. . . . . . ... .. . . . ... .. ..... .......... ................... ... . . .... . . . ... ... . . . . . . . . . . ... .. .. . ... . .... ...... .......... ............ ...... ... ... ...... ..... .... .......... ............. .. ...... ... ...... ..... ...................... .. . .. .... .............................................. .................................................................................................................................................................... ... .. . . . . ..... . .. . . . . . . . ...... . ...... ........ .. ........ . .. ......... . . . . .... . ... ......... ... ....... .. ... .......... .. .......... .... ........... ... . ............................................................ ... ... . ... .. . . .. ... .. ... ... .. . . . . ... . . ... .. ... ... ... ... ... .. .. ... .. ..... . . ... ... ... ... ... ... ... ... .. .. ... .. .... . . ... .. ... ... ..... ......... .... .. .. .. ...
A
U
Q
R
= R tan α
pcl NXb2ML YJ{
n
h x [ g HM V|[ g 4ABT s {M bcYJcM AB sin ∠AT B n LHMJN n = ϕ = ∠BAT BT h Nr[ gUk NXbcYJN sin ∠BAT {s M k MJN ∠ABT = α + β sin(α + β + ϕ) AB = BT sin ϕ
n
S
B
P
C
T
V?[ gUk NXb2M|TrMJ^eYJNr[ hcg V s s Y gUq α + β + γ = 180◦ s AB = 2R sin γ BT = R tan α {M h zNrY [ g @BA K n cot ϕ = cot γ + 2 cot α
DGJI MXHN s {M'VMJN Y gUq YHuUuc^ l NXb2M»L YJ{ bd[eV k [\cM V ∠ABQ = ψ n 4BP Q BQ QA
=
Y gUq
sin ϕ
BQ
sin ψ
PQ
h x [ g MHVvN h 4BAQ Y gUq
sin ∠AP B
=
sin(β + ϕ) n sin(β − ψ)
=
sin(β − ψ)
[ g _] M Q f[ V'NXb2MSP'[ q ?u h [ g N h x AP s {MSbcYOcM P Q = QA n tbdjUV s NXb2MTWY Nr[ h V [ g NX2 b MN{ h M cjUYJNr[ hcg VaYz h c M YDTrMM cjUY ^ n tbd[fV k [ cMHV sin ϕ sin(β + ϕ) s=h T = sin ψ sin(β − ψ) sin(β + ϕ) sin(β − ψ) n iM g ]_M s s = sin β cot ϕ + cos β = sin β cot ψ − cos β sin ϕ sin ψ ^`M Y q [ gUk N h n}~g NXbd[eV=^fYDV_N=M cjUYJNr[ hcgts {MjUVmM @~A K N h'k MJN cot ψ = 2 cot β + cot ϕ @ FK n
H
H
H
cot ∠ABQ = cot ψ = 2(cot α + cot β) + cot γ
} N7x h f^ ^ h {|V s z l V l P&PMON_T lts NXbcYJN cot ∠BAS = cot ∠ABQ n @ V?[ g ]XMaNXb2M&] h NWY gUk M g Nxej g ]rNr[ hcg [fVV_N_Tr[e]rNr^ l»q MH]XTrMHYHV?[ gUk ∠ABQ = ∠BAS @Y K n u M [ _ N M ~ T d H Y ^ h g g ◦ ◦ K n ½M|bcYJcM|u2T h cM q (0 , 180 )M dj2[\dY ^`M NN P&[ [eP'[ _[ O Y H e [ ' P [ _ [ e [ & V gUk g h g gUk cot ψ s {bd[`]?b+]_Y g z2M jUV?[ gUk x h TrP&j2^fY @ OK nψ [ g ]XM
+H
cot α + cot β
= = ≥
{MbcYOcM cot ψ
≥
sin β cos α + sin α cos β
cos(α − β) − cos(α + β) 2 sin γ s 1 + cos γ 4 sin γ 1 + cos γ
= 4 tan
H
γ 2
+
=
sin α sin β 2 sin γ cos(α − β) + cos γ
+ cot γ 1 − tan2 2 tan
γ 2
γ 2
=
7 2
tan
γ 2
+
¦ uUuc^ l [ gUk NXb2M ¦ E Á ;}~g M cjUY ^f[\N l s {M h zdNWYH[ g
H
® h TM cjUY ^f[\N l s [wN[fV b2M VmM|] hcgUq [\Nr[ hcg V t NXb2ML YO{ h x [ g M V s h x 4ABC YHTrM
cot ψ ≥
q2hcg M
sin γ
=
sin α sin β 2 sin γ
tb2M g [ g XN b2M
s
7 tan
γ 2
·
1 tan
γ 2
=
1 2 tan
γ 2
n
√ n 7
g HM ]XM VV?YDT l XN bcYJN α = β = 90◦ − γ/2 Y gUq 7 tan γ = 1/ tan γ n 2 k [\cM sin γ = √7/4 Y gUq sin α = sin β = √14/4 n iM g ]_M s z 2l NXb2MY gUk ^eM f[ VYaPQYFD[fP&jUP{b2M g NXb2MaTWYJNr[ h V h xoV?[ q M|^eM gUk NXbcV ψ
a:b:c =
P
√ √ n 2: 2:1
tbcYJN=] h PQuc^`MON_M V NXb2M ¶ y x h TNXbd[eV=[fVVXjM ny M gUq PQM lch jUT g [`]_MiV h ^wjcNr[ hcg V s k M g M TWYH^w[ XYJNr[ hcg V s Y gUq ] hcg NrMHV_NP&Y rN MHTW[eY ^eV n
DG /
F
K6 P Mv6 : ° ¸ ¸ 2¸~¶ $
>
*
Kd
? > vG 0
B13, B"G n jUuUu h VmMNXbcYJN
f (x) = 1/x
A
b2M h TrM P s
s
s
z l
x1 x2 . . . x n
x1 + x 2 + · · · + x n =
tb2M g
1 x1
+
jUVX[ gUk
NXb2M] hcg cM?xej g ]rNr[ hcg
YHTrM|u h V?[\Nr[\cM g jUPQzUM TWVV?j]?baNXbcYJN 1
x2
+ ··· +
@ IHK
1 n xn
1 1 1 n + + ··· + ≥ 1 1 + (n − 1)x1 1 + (n − 1)x2 1 + (n − 1)xn
H
bd[eVo[ g M cjUY f^ [\N l PQY l z2M q M Tr[ cM q `x T h P NXbcY N 1 1 + (n − 1)x1
+
1 1 + (n − 1)x2
@ FK
@ FK [ g NXb2Mx h w^ ^ h {i[ gUk {|Y lon jUu2u h VM
+ ··· +
1 1 + (n − 1)xn
< 1
@Bª K
[ gQh NXb2M Ty{ h T q V s {MiV?jUu2u h VMNXbcYJN @ F K e[ Vx`YH^fVmM n t b2M g {M!{i[w^f^UVmb h { NXbcY N @ IDK YH^fV h xh T 1 q h M V U U g h Nb h ^ qon h XN bd[fVM gUq s ^`MON a = s s s n h N_M i = 1 2 ... n i 1 + (n − 1)xi NXbcY N Y gUq NXbcYJN 1 − ai x h TYH^w^ s s s n ½MY ^eV h bcYOcM ai > 0 xi = i = 1 2 ... n (n − 1)ai P z l @`ª K n iM g ]_M s ai < 1
1 − ai >
x h TY f^ ^ s s s n tbdjUV s i = 1 2 ... n x1 + · · · + x n =
n X
i=1
X j6=i
1 − ai (n − 1)ai
@ mK
aj = (n − 1)bi
> ≥ >
z l @ OK
n X ai
i=1 n X i=1
=
z l @ FK
n X bi a i=1 i
bi
z l @ FK
(n − 1)ai 1 − ai
1 1 1 n + + ··· + x1 x2 xn
|M g X] M s { MbcYOcMuUT h cM q XN bcY N|[fx @ OK qUh M V U g h Nab h ^ q s XN b2M g @ IDK U q h M V gUh Nab h ^ qn tb2M TrM x h TrM s @ IDK [fPQuc^f[eMHV @ OK s z l NXb2M|] hcg N_TWYHu h XV [wNr[ cM n nQ jzcVXNr[\NrjcNr[ gUk xh T [ g @ OK Y U V V_Nr[f^w^VYJNr[eVXR M q g qgUh Nr[ gUk XN bcY N @ HI K [fi 1/xi xi k [\cM VjUV
f $
x1 n − 1 + x1
+
x2 n − 1 + x2
+ ··· +
xn n − 1 + xn
≥ 1
n
A [ g ]_M xi
+H
n−1+x
= 1−
n−1
n−1+x
i i x h T=M Y]?b s s s s NXb2Mi[ g M djUYH^w[wN l ]_Y g zUMiTrMO{|TW[wNWNrM g [ g NXb2Mix h T_P i = 1 2 ... n
+H
1
n − 1 + x1
B13, B" #
+
1
+ ··· +
n − 1 + x2
1 n − 1 + xn
≤ 1
n
tbd[fV[fVY g [ g M djUYH^w[wN l [ g NXbcYJNy{|YDV q HM TW[\cM q U j V?[ gUk L Y k TWY gUk M j2^wNr[fu2^w[`M WT V n n} x x s x s . . . s x YHTrM|u h V?[\Nr[\cM g jUPQzUM TWV s NXb2M g 1 2 n
B
H
n
1
(n − 1)(x21 + x22 + · · · + x2n ) + n(x21 x22 · · · x2n ) n ≥ (x1 + x2 + · · · + xn )2
vtbd[fV[eViYZ gUh { g [ g M cjUY ^f[\N l ª {bd[`]?b»x h ^w^ h {|V!x`T h P tb2M h TrMHP jUVX[ gUk Y r T H M c u e ^ D Y _ ] [ i { w [ X N b NXb2M] hcg cMXxej g ]rNr[ hcg U g q U g k s s s s x f a1 a2 . . . a n 2 ln x1 TrM (x) V uUMD]r= Nr[ ceM ^ lon s s s 2 ln x . . . 2 ln x
/HTrM xB"M *TrMDB M VDjx %1T)b2 M ^fu2xe j2n^ VXtj b2MMHYHV_Nrjc[ NXb h V Ti{ h k k hcg n f!$3E 2
A I ª
n
j ^ q ^w[`ZDM»N h MXu2TrMHV?VQbd[eV k TWY Nr[\Nrj q MN h NXb2M h 2
+H
§!n TWN h JY ¥~M s g £ h u h d[`]_[wj V }~g M djUYH^w[wN l x h T c h g cMX4®j g ]rNr[ hcg V s F¶ ³ ¹¶! e ¸#"! ² s I @ HGGOK s JIU E n I d E I2 n $ A G @~HA CCDª K s 2 §!n TWN h YJ¥~M s £UT h zd^`M P A GdD s ²»¶ ³ ³ Un ÁyjM T hcgtsc jz2V_Nr[wNrjcNr[ hcg V sU}~g M cjUY ^f[\Nr[`M V s Y gUq ![eV_N h T l s ³ t¶! e¸#" @ HGGOK s EC G n ¤º=¸ »³
¶% e¸ " $³
¶% s )nwUn
^fYDPQZD[ gts g Y ! £UT h zd^`M P h xNXb2M hcg NXb #Hs S³ t¶! e¸#" º=¸ ³ t¶! e¸#" %$y³ t¶! s @ HGGOK s ªOE d n C s ³
¶% e¸ " -º=¸ ³
¶% e¸ " $ )nwUn
t^eYHPZH[ gts £UT h zc^eMHP ³
¶% s Ca@ DGDGFK s A n n
h z2MHT s g NXb2M ¦ TW[wNXbcPQMJNr[e]»Y gUq ÁM h PMON_Tr[e] MHY g Y gUq4hcg h ^ q MHT V }~g M djUYH^w[wN lts " ²»¶ =³ ° " Ca@BADC OK s I2 E I2 C n " ² s ¦!n LJjUuUYHV s g Y g}~g M cjUY ^f[\N l x h T hcg cM?®Dj g ]rNr[ hcg V s F¶ a³ ¹¶! e¸# @~AHC OK s AOE n n £ h u h c[e]_[fj sQ jUT]XM TWNWYH[ g MHVS[ g M k Y ^f[\N MH V cj2[']_YHTWY]rNM Tr[fVmM g N^eMHVSx hcg ]rNr[ hcg V ] hcg cMXDMHV s ² %$¹¶ $¹¶° e¸w¸ e¸ "X¶ $~¶ c¸~· ¤± ?¸ a°c¶ " e¸ À³ s AA p @~AHCª OK s A D EADª I n
+H
H
;
H
;
;
;
;H
G
§ YDVX[f^eM TWN h YJ¥~M g [\cM TWV?[\N l&h x£2^ h [`M VXNr[ ¨ h PQY g [eY
[email protected] AHC
|A(f )| x h TNXb2M&uM T_P'jcNrYJNr[ hcg g h zdNWYH[ g M q )| = n − 1 x`T h |Im(f P n f jUuUu VmMNXbcYJN x h TV h PQM h n bd[eVPM Y g VvNXbcY N ) = [n] − {d} x h Im(f TV h PQ M Y gUq s Y gUq TrMHdV_N_∈Tr[e]r[n] N_M q N h [eV!Y f f (b) = c a 6= b c f [n] − {b} zc[ (a) ¥~MD]rNr[= hcg»hcg N h [n] − {d} n tb2M gts ] hcg VXN_TWj]rNr[ gUk g : [n] → [n] YDV[ g @BA K s {M k MON Y gUq x h TvY ^f^ s Y gUq NXbcYJN g ∈ S [ g NXbd[eViVuUMD]_[wR ] g(b) f (x) x 6= b ]_YHVmM n =½M d g MDM q g(x) N h R gU= V Q P | M e x j r ] r N [ q h g hcg ϕ00 ∈ A(g) {bd[`]?b&[eV gUnh NPQYJN_]?b2M q {i[\NXb Y gl n ϕLH∈ MJN A(f ) Y gUq
s
s
[n] − {a b} s [n] − {c d}
s
s
= {v1 v2 . . . vn−2 } s s s = {w1 w2 . . . wn−2 }
V h XN bcY N x h TYH^w^ n= MJR g M 00 xh T g(vi ) = f (vi ) =Y wi YH^fV i ϕY (vi ) = wi+1 1 ≤ i ≤ n−3 Y gUq s U g q h U g q n b2M g ϕ00 [eVY 00 00 00 ϕ (v ) = w ϕ (a) = d ϕ (b) = c n−2 1 uM T_P'jcNrYJNr[ hcg Y gUq n ½MVmMHM&NXbcY N ϕ0 (e) = d x h TY ^f^ ϕ0 V?j]?bNXbcYJN ϕ00 ∈ A(g) [ ]_M x y T V Q P M Y gUq 00 h h n g s [\NyP&jUV_Nz2MNXb2M]_YHVmM ϕ → ϕ0 ϕ ∈ A(f ) a 6= e ϕ (a) = d NXbcY N x T H Y w ^ ^
c b Y N f [ V h n s n ϕ 6→ ϕ00 ϕ ∈ A(f ) |A(g)| > |A(f )| h {¡{M]_Y g bcY gUq ^`M ^fYH[fP Y gUq ] h PQuc^`MON_M h jUT&uUT h2h x h x!NXb2M YHT q 4 tb2M h TrMHP n * ½M|zUM k [ g z l YHVVXjUP&[ gUk NXbcY N uM T_P'jcNrYJNr[ hcg VXjUPQVYHTrMiVXN_TW[`]rNr^ l ^eMHV?V 12 K ]?b @ NXbcY g ® v T M Y ? ] » b r ] D Y T n h q C i = 1s 2s 3s 4 s h2h VmMY g M g N_T l e h ]X]_jUTrTr[ gUk Y i VM ^`M YDV_N 1/3 U j P 2 z H M T x r N f [ P M y V [ X N 2 b H M ? V U j & P V ½ v M r ] H Y w ^ ^ NXb2M '¸ c¸ $U¶ i x h To]_YHT q g h g n 12 e i n bdjUV s s s c s d } x h TyM Y]?b i s Y gUq e h ]_]_jUTWV[ g NXb2M 12 uM T_P'jcNrYJNr[ hcg C ∈NX{a i M i jUPQ V?jUiP&ViYJNvP eh iV_N b2M&iV?YDbPQ g i zUM T h xNr[fPM VYDViMHYDi ]?b h xNXb2M h NXb2MHTNXbcTrMDMQM g N_Tr[eMHV hcg ]rYDT q C n ½M{i[eVbN h ] h PQu2YDTrM!NXb2MiVXjUP e + e + e + e {i[wNXb'MHYD]?b h xNXb2M uM T_P'jcNri YJNr[ hcg VXjUPQV n h NrMNXbcY NY'uUMHTrP&jcNW1Y Nr[ hcg 2 V?jUP%3{i[w^f^o4bcYOcM gUh MJ^`M PM g NrV 12 [ g ] h P&P hcg {i[wNXb u2TrMD]_[fVmMJ^ l [fx NXb2MuM T_P'jcNrYJNr[ hcg @ ] hcg VX[ q MHTrM q eN1 +[\Nre e 3 + e4NXb2Mxej ]rNr[ 2^wx+ K YDVY|PQYHu&x`T h P V M J Y [ V h h q g hcg α : [4] → [4] q M R g M q z l [4]
1 2 α(i) = 3 4
bcYDu2uM g V N h z2M&YQuUMHTrP&jcNWY Nr[ hcgh }x {bdα[`]?b'YJ h [ q [eVV?[fPQuc^ l n αYO [ [ d(4) =P 9 V_N uM T_P'jcNrYJNr[ hcg V f [ V Y N h q gUk α h
8 3
[wx s ei = a i w[ x s ei = b i w[ x s ei = c i w[ x n e =d i
i
x
s XN b2M g NXb2M g jUPQzUM T h xuUMHTrP&jcNWY Nr[ hcg V NXb2[4] MHT¹{i[eVM s z l NXb2Mi^eMHP&PQY s NXb2M g jUPz2MHT h x n 8
n n ∈ /YOS tb2α MiP& H[e4P'jUP g jUPQzUM T h xuM T_P'jcNrYJNr[ hcg V?jUP&V{bd[e]?bYJ h [ q [ g NXbd[fV]_YHVmM [eV [\N'x h ^f^ h {|V&NXbcY N'[ g Y gl ?] b h [e]XM h x uUMHTrP&jcNWY Nr[ hcg V?jUP&V s αYJN&e^ MHYHVXNx h jUT 8 12
D P&jUV_NjUVMY N|^`M YDV_N hcg MP'[ g [fPQY ^M g N_T l n [ g ]XM e [eVNXb2MM g N_T l+hcg ]rYDT q C iV=NXbcY NMmcMHT {bd[`]?bQYHuUuUMHYHTWVNXb2M^eMHYHVXN g jUPz2MHT h x Nr[ePQeMHiV s [\Nx h ^f^ h {| l M g N_T lh xtNXbcYJNi TrMHV?uMH]rNr[\cM]rYDT q [eVY]_] h j g NrM q x h TY N^`M YDV_NYHVPQY gl Nr[ePQMHVYDVNXb2MP'[ g [fPQY ^tM g N_T l @ {i[\NXbd[ g NXb2M ]?b h VM g uM T_P'jcNrYJNr[ hcg V?jUP&V K n tbcYJN[fV s x h TvM Y]?b»P'[ g [fPQY ^yM g N_T lts {MQYHTrM k jUYDTW12 Y g N_MHM qhcg MQV?jUP ] hcg NWYH[ g M q [ g NXb2MQV?jUP h xNXb2M + ] ciT_+ ]?b h VM g uM T_P'jcNrYJNr[ hcg V s Y gUq aNXibd+ [fVV?bjUi P TrM Vdu ihcgUq VN h NXb2M]rYDT q N h {bd[`]?b'NXb2M h 12 P&[ g [eP&YH^tM g N_T l z2M ^ hcgUk V n [ g ]XMNXb2MHTrM|YHTrMYJN^`M YDV_N I P'[ g [fPQY ^tM g N_Tr[eMHV s [\Nx h ^w^ h {|V NXbcY NyNXb2M TrMiYHTrM!x h jUTVXjUPQV [ g NXb2MV?jUP h xYH^w^ uM T_P'jcNrYJNr[ hcg V n tb2M TrM x h TrM s NXb2M'VXjUP h xyY ^f^ ai +ubM iT_+ P'jccNri YJ+ Nr[ hcdg i V!P'jUVXNzUMY N^eMHYHVX12 N s {bd[`]?bPM Y g V uUMHTrP&jcNWY Nr[ hcg V12 ]_Y g2gUh NYH^w^ bcYJcMiV?jUP.VXN_TW[`]rNr^ l ^`M VVNXbc4Y g NXbcY NNXb2M n 12 1/3
8 3
n α∈S n 4 jUuUu h VmM s {i[\NXb h jcN^ h V?V h x k M g MHTWY ^f[\N l s NXbcYJN α(i) = i x h T i = 1 s 2 s 3 s 4 n tb2M TrMYDTrM uM T_P'jcNrYJNr[ hcg V?jUP&V={bd[e]?b&YDTrM] h P&u h VM q M g Nr[fTrM ^ lh x gUhcg XP&[ g [eP&YH^ M g N_TW[`M V sg YD9PQM ^ l NXb h VmM»] h TrTrMHV?u hcgUq [ gUk N h NXb2M q M TWY gUk M PM g NrV h x s s s 4} n tbdjUV s [ g Y gl VMJN h x uM T_P'jcNrYJNr[ hcg V?jUP&V s Y Nt^`M YDV_N h xNXb2M P-P'jUVXN{1 jUVM2P&[ 3g [eP& YH^ M g N_TW[`M V n ½M gUh N_M=NXbc12 Y N[fxdx h jUTouM T_P'jcNrYJNr[ hcg V?jUP&VyjUVm3MP'[ g [fPQY ^cM g N_Tr[eMHV s NXb2M g NXb2M uM T_P'jcNrYJNr[ hcg Yz h cM] hcg N_TWY q [e]rNr[ hcg b h ^ q V n tbdjUV s {M|YDV?V?jUPQM!{MbcYJcMM?HYD]rNr^ l V?jUP&Vv{bd[`]?b»jUVM&P&[ g [eP&YH^M g N_Tr[eMHV s Y gUq NXbcY N!Y ^f^ q MHTWY gUk MHPQM g NW3V|YDTrM'[ g ]_^wj q M q YDP hcgUk NXb2M uM T_P'jcNrYJNr[ hcg VXjUPQV ny [ g ]_M q M TWY gUk M 9PM g NrV]rY g2gUh Ny[ g ]_^fj q M s s 12 a b s2h T s z l V l P&PMON_T l NXb2MiV?jUP h xNXb2M q M TWY gUk M PM g NrV[eVM YDVX[f^ l VMDM g N h 1 z2M 2
c3
d4
9
3(b1 + c1 + d1 + a2 + c2 + d2 + a3 + b3 + d3 + a4 + b4 + c4 )
n
}~g NXb2MTrMHP&YH[ g [ gUk NXbcTrMHMauUMHTrP&jcNWY Nr[ hcg VXjUPQV s {MP&Y l YHVVXjUPMNXbcY N a YDu2uM YDTWV n p jcNyV?[ g ]XM[\NyYHuUuUMHYHTWV[ g YJNyP h VXNN{ h VXjUPQV gUh N] hcg NrY [ g [ gUk b s c sh T d1 @ g YDPQM ^ l K s {MP&Y l YHVVXjUPMNXbcYJN 2 3YHuUuUMHYHTW4VYHV {M ^w^ n Y gUq a +c +d +b a +d +b +c 1 2 3 4 1 2 3 4 h { VX[ g ]_M YDu2uM YDTWV[ g V h PMuM T_P'jcNrYJNr[ hcg V?jUP-^eMHV?VyNXbcY g b2 s {M!Z gUh {)NXbcYJN a 1/3 1 {bd[`]?b'PM Y g V n [eP'[f^fYDTW^ l s a + c + d > 2/3 n a1 < 1/3 b1 + c1 + d1 > 2/3 2 2 p jcN gUh {4NXb2M|VXjUP h xNXb2M g [ g M q M TWY gUk M PM g NrVP&jUV_N=zUMiY N^eMHYHVXN 2 3(b1 + c1 + d1 + a2 + c2 + d2 ) > 3
2 2 + 3 3
= 4
s
{bd[`]?b&] c h g N_TWY q [e]rNrVNXb2M|YHVVXjUPQudNr[ hcg NXbcY NNXb2M l M Y]?b&bcYOcM|VXjUP.^eMHV?VNXbcY g 1/3 n h NrMQNXbcY N!NXb2M»]rYDVM&{b2M TrM uM T_P'jcNrYJNr[ hcg V?jUP&VaYDTrMVXN_TW[`]rNr^ l ^eMHV?ViNXbcY g [eVTrM YH^w[ XYDzc^eM{i[\NXbiNXb2Mx h ^f^ h {i[ gU11 ]rYDT q ?VMJN s {bd[`]?bbcYDV uM T_P'jcNrYJNr[ hcg VXjUPQV k 1/3 11 M djUYH^UN h n 1/4
+H
+H
1/4 1 1/4 0 1/4 0 1/4 0
+H
0 0 1 0 0 1 0 0
xh T bcYHV V?jUP&V k TrM Y NrMHTNXbcY g ¦ e^ V
h s HHYHPQuc^`M nQ}~g x`YD]rN[\NvbcYDV 4/3dYDTW[eY N xtHHYHPQuc12 V?jUP&V=M djUYH^dN h Y gUnq = 4V?jUP&V=M 12 djUYH^dN h t 2 b ! M V D M ] ^`M n hcgUq g h 1/3 xh T bcYD5/3 VV?jUP&VM 12 cjUY ^UN h T t 2 b M m V | M ? M H H Y Q P c u ` ^ M = V N J M X N 2 b M T m V b 4 { X N c b J Y N h n h2k h NXb2Mi[ g nNrMH= T~dY 4^ [ gQh jUTNXb20M h TrMH4/3 P[eVz2MHV_Nu h V?V?[ezc^eM n [1/3, 4/3]
H
ª
H
h {MmcM T s P&Y gl cjM VXNr[ hcg VTrM Q P Y [ gon Y V h ^wjcNr[ hcg N h NXb2M ?]_YHT q uUT h zd^`M Px h T YH^w^ n} VNXb2M[ g N_M T~dHY ^ [1/2, 3/2] i n h2qUq n ≥ 3 HHYHPQuc^`M A Vb h {|VNXbcY N s [fxN_TWjM s NXbd[fVvTrMHVXj2^wN{ h j2^ q z2M'z2MHV_Nu h V?V?[ezc^eM n ® h T mM cM g s HHYHPQuc^`M Y gUq [wNWVN{ h dYHTr[fY g NWVVb h {4NXbcYJNNXb2Mi[ g N_M T~dYH^fV n
B1&D h
1 1 3 1 − , − 2 2n − 2 2 2n − 2
Y gUq
i
h
1 1 3 1 + , + 2 2n − 2 2 2n − 2
i
@ IHK
{ h 2 j ^ q z h NXb&z2MzUM VXNu h V?V?[ezc^eM[fxNXb2M l {MHTrMiV h ^wjcNr[ hcg V h xNXb2M ?]_YHT q uUT h zd^`M P n n n¦ TrMNXb2MN{ h [ g N_M T~dYH^fVv[ g @ IDK V h ^fjcNr[ hcg V h xNXb2M n ]rYDT q u2T h zc^eMHPx h T YH^w^MmcM g Y|V h ^fjcNr[ hcg'h xUNXb2M ]rYDT q uUT h zd^`M P &}~g u2YDTNr[`]_j2^fYDT s [eV 5/3] 4 V ^wjcNr[ £UT h nzc^e≥ MHP 4 VXj k k MHV_NrV!NXbcY NNXb2[2/3, M TrMQP& Y l zUM&P h TrMNXbcY ghcg M& h hcg N h NXb2M ]rYDT q uUT h zd^`M Px h TV h PM n}~g x`Y]rN s [\N[eV!M YDV l N h VmMHMNXbcYJN=NXb2Ma[ g NrMHT~dY ^ n n [0, 1] [eV|YV h ^fjcNr[ hcg x h T!Y gl n b2MTrMHYHV hcg [fV|V?[fPQuc^ l NXbcY NvNXb2MYOcMHTWY k M'dYH^wjM h xYH^w^ n uM T_P'jcNrYJNr[ hcg VXjUPQV[eV b2M g ]_M s NXb2MHTrM![fV=Y ^\{|Y l VV h PM!uM T_P'jcNrYJNr[ hcg {b h VmM!V?jUP [eVz2MHV_Nu h V?V?[ezc^eM n [eVYJNP h V_N n g ]_M|Y k 1YH[ gts2hcg M]rY g Vb h {4NXbcYJN ® h TvNX1b2M]_YHVmM h x [wN![eV gUh NibcYDT q [0, N h 1] Vb h {%NXbcYJN!NXb2M hcg ^ l P&[ g [eP&YH^ s V h ^wjcNr[ hcg VYDTrM|NXb2Ma[ g N_M T~ndYH= ^fV 3 s s Y gUq n tb2Mu2T h2h xNXbcYJN [1, 2] 2]N NXb2MiTrM Y MHT {i[\NXba { h TXZDV=[fV=V?[fP&[w^eYHTN h NXb2Miu2T h2h [0, x x h 1]T [1/2, 3/2]Y gUq [fV[1, ^`MJxfNy NXb2M h q s 3/2] bd[ g N * Vmb h { NXbcY NNXb2M TrMMXD[fVXNWVYJNP [1/2, _ V N | M U u H M r T & P c j W N Y r N [ i { w [ X N ' b X V U j P r T M Y r N H M y T X N c b Y g h hcg hcg k Y Y N P X V N U u H M r T & P c j W N Y r N [ V i { \ [ X N ' b ? V U j P ` ^ M V V X N c b Y s gUq h hcg g 1n 2 4 n)} V [1, 2] YH^fV h YV h ^wjcNr[ hcg x h T n = 4 ½bcYJNi[eVNXb2M»VX[wNrjUYJNr[ hcg x h T ^eYHT k MHT
B1&D
B1&D B1&D
n
G
n ®[ g YH^w^ l s [wNyVmMHMHP&VNXbcYJN s x h T s P&[ g [eP&YH^c[ g N_M ~T dYH^fVVYJNr[eVXx l [ gUk XN b2M T [fV ]rYDT q uUT h zd^`M P Y ^\{|Y l VbcYJcMi^eM gUk NXbQMXHnY]r>Nr^ l 2 n} V=NXbd[eV=N_TWjM } xV h s {b l n 1 [wNy¥`jUV_Ny{i[eVbdxej2^UNXbd[ g ZD[ gUk
f!$3 A
OO
O
O
KO
bNWN_u * O{!{!{ n PQYJNXb n j]_Y ^ k YHT lon ]rY DM q j]_YJNr[ hcg O?T YJNXb MXHYDP DGDGG ¥BTrPQYJNXb DGDGG n bN_P&^
OO
DO
bNWN_u * HPQYJNXbD{ h Tr^ qon { h ^wx`TWYDP n ] h P
cTW[`]!LHM g XY MHu2YDTN_PM g N h x JY NXb2MHP&Y Nr[e]_V Y gUqQ NrYJNr[eV_Nr[`]rV g [\cM TWV?[\N lQh x Y ^ k HY T l Y ^ k YHT ltsD¦ ^`z2MHTNrYi c A I Y g Y q Y eric
[email protected] O
M TWY gUk M PM g N n bN_P&^ p [w^f^ Y gUq V M uUYHTWN_PQM g N h x YJNXb2MHP&Y Nr[e]_V Y gUqQ NrYJNr[eV_Nr[`]rV g [ cMHTWVX[wN lQh x Y ^ k YHT l YH^ k YDT l s¦ ^ezUM TWNWYi d A I Y g Y q Y
[email protected] L
aZ*][hga5[i\e\]hm¥l$h\[l1]5a`9[Uhgu \cc,=Z hm\l1c},[v0z Z*ij]`^ hj,ij5`n5eo1cqpZ\[0[qqF`9c5^B]e^bNl$hg][ce0i*[=^¨B¦ ],dZ eeg]5p"b ][5\[ hg` [et1im]Z*a]5[q``1d5¦hjqi ij[`-p", [a e_ c,p5ij`Zgb d$a5hji\i\hm\c5l]`[Vhg^t[ ^Fn^Deol1hmclc"n]d,chrZmas^]F[]5egn^Ue|eoei\chg,pc [#Z\[`1p"q c[
]^rneo¥cl1n[vc^[,[,a5aNi\hgc5reri\hmhmllc"]5d$`h¦5^-](§,^[c,]5Zg`1d$hji\¨/c5`-]5eo
¥[e|ei\[e-]5`a¨/]5ehjij`s©fc,Zma^h\[ ij`chml[Lª-`i\[eg^oi\hjbLc ¨Vc5`hmeo[ ],Z1\c5e_hmeg]5`^Z*]hji\c5`^Hchml1[_neocpZ\[q^, /
7¢#
/
¡#
*
B0`
.k
n ± " c² W¸ n
@ Y K LHMON
@z K
@z K
1−x 1−y
%$
¶
s
s
x y z
0`
%$f¸ t¶ (
P
X¸
+
1−y
1−x
41
2 n 12 √ √ √ √ x y+ 1−x 1−y
≤
H
% J¶ n
%$
¶
P
X¸
T q f[ Vu2T h c MNXb2Mvx h ^f^ h {i[ gUkk M g M TWYH^w[ XYJNr[ hcg [ g h ^\d[ gUk u h V?[\Nr[\cMvTrM YH^ n {bd[e]?b'V?jUPN h * s s s x1 x2 . . . x n 1 n Y
xi
i=1
!
1+
D
n
.k
n X 1
x2i
i=1
!
≥
n3 + 1 n nn
2¸~¶
H¸ d¸
a3 + b 3 + c 3 a2 + b 2 + c 2 − abc ab + bc + ca
'³S¸ %$)
% O¶ n LHMJN q M gUh N_MSNXb2M7®D[`z hcg fn xh T f TrMHnP&= YH[ gUqfn−1 M Ty{b2+M g fn−2 [fV nq [ c≥[ q fkn−r @Y K @z K r=1
%$f¸ t¶ (
1 1 1 28 n xyz 1 + 2 + 2 + 2 ≥ x y z 27
(a + b + c)2 1 + a2 + b 2 + c 2 2
z2Mu h VX[wNr[ cMTrM YH^ g jUPQzUM TWVV?j]?b|NXbcY N £ T h cMvNXbcYJN n U x+y+z = 1
.k
41
% O¶
j PQzUM TWV s u2T h cM!NXbcY N } x a s b s c YHTrM|u h V?[\Nr[\cMiTrM YH^ g U
k M g M TWYH^w[ XYJNr[ hcgh x!NXb2M+YDz h cM[ g M cjUY ^f[\N l N h NXbcTrMDM h TQP h TrM
.kdj
?g jU£UPQT h zUcM M TWV h
0`
W¸ n
± " c²
@ Y K LHMON
Y gUq zUMu h V?[\Nr[\cM!TrM YH^ g jUPz2MHTWVx`T h P¯NXb2Mv[ g NrMHT~dY ^ 1 n U £ T h cMvNXbcYJN y 0, 2
x
} VNXb2MHTrMY ?g jUPQ zUM TWV
B0`
(?)
% J¶
2 ≤
+H
"
o¶" F¶;
c· ´
≥ 4
n
d¸ n
Y]_]_[VM djM g ]_M @ NXbcY N&[eV s Y gUq f0 = f 1 = 1 K n ® h TaYH^w^[ g NrM k M TWV O M _ N M _ T ' P [ M sq g NXb2M k[ ≥ ]_YH1VmM V M 2q z l 2 x h TyNXb2Mix h ^w^ h {i gUk * fn @]K r=2 ? r ∈ {3 s 4 s . . . s k − 1} n
n
0`
.kdl
)
0
% J¶ ?¸'$¹¶ X¶ c ¸~·H¶ ?¸ $ H ¸~¶ e ¸ n c¸ n LHMJN Y gUq zUMiu h VX[wNr[ cM![ g NrM k M TWV{i[\NXb s Y gUq ^eJM N a s a s . . . s a zUM k< 1 2 n TrMHY ^ g jUPQzUkM TWVV?jn]?baNXbcY N £UT n c! M NXbcY N n h a ≤ a ≤ ··· ≤ a
´
1
2
(a1 + a2 + · · · + an )
2
n
≥ n(a1 ak+1 + a2 ak+2 + · · · + an an+k )
@ {b2M rT MNXb2M|VXjz2V]_[eudNrVYDTrM!NrYDZDM g P h2q 2 j ^ h K [ g NXb2Mx h f^ ^ h {i[ gUk ]_YHVmM V * n @Y K @z K @]K n
0`
n = 2k
.k
n = 4k
H
0
$
k jDdfd
r ≥0 r ≥ −n
n
r
xr
x ∈ − 41 ,
Z\cl^.[mZ\[idn$dfpoMq(b!c*ZriZ s!d ["d.n^iZts!d b!fvuwd$fb.x
1 4
C
7 n ° $ e¸ A³¸#" t¶ $ ¸'$)$~¶y´ d¶
D ¸ t¶
"_¶
®[fTWVXN s ?V jUu2u h VMvNXbcYJN n b2M k [\cM g VXjUP[eVNXb2M g n≥0
r≥0
{i[f^w^UjUVmM!NXb2Mix h w^ ^ h {i[ gUk c Y l ^ h TVMHTW[`M VM?DuUY g ?V [ c h g* 1 (1 − z)m+1
=
;
t¶|¶ D¸
X
n + 2r r n ½M x r
∞ X m+k k n z k
k=0
|M TrM P&Y l z2MaY g lQgUhcg g M k Y Nr[ cM[ g N_M k HM T s Y gUq NXb2MMXu2Y g VX[ hcg [fV=dYH^w[ q x h T=YH^w^ ] h P&u2^em M? g jUPz2MHTWV ?V j]?baNXbcY N n z |z| < 1 z MRdDM qts Y gUq ^eMJN LHMJNY g [ g N_M k MHT 2 1 n tb2M gts x h T s {M|bcYJcM 0 < |z| < 1
r
f (z) =
f (z) =
∞ X n+r+k k
k=0
z r+1 (1 − z)n+r+1
z k−r−1
~} g NXbd[fVL HY jUTrM g NVmM Tr[eMHVx h T f (z) s XN b2Mv] h M ]_[eM g N h x z −1 [fV TrMHVX[ q jM h x M ¨tM ?V [ q jMi b2M h TrM P s YJN npcl NXb2& f (z) 0
{b2MHTrM
n + 2r r
1 = 2πi
I
f (z) dz
n
n + 2r n tbd[fVy[eVNXb2M r
s
C
q M U g h N_M V=NXb2M|]_[eTr]_^eM |z| = 1 _N TWYOcMHTWVM qQhcg ]_M] h j g N_M rT ]_^ h ]?ZJ{i[eVM n C z2M'RdDM qon ® 2h v LHMJN T Y ^f^ hcg s {MQbcYOcM |1 − z| ≥ x ∈ − 41 , 14 z C b2M g _] M s x z(1 − z) ≤
tb2M gts x h TYH^w^ hcg s z C ∞ X
r=0
x z(1 − z)
r
=
1 2
s Y gUq
|x| 1 = 4|x| < 1 n
1 2
2
1
−z(1 − z) s
= 2 x z −z+x 1− z(1 − z)
Y U { [wNXb TrMHV?uMH]rNN h g q NXb2M] hcg cM T k M g ]_M[fVSj g f[ x h rT P i h g c n tb2M j g [fx h TrP z { [\NXC ] hcg cM T k M g ]_MY ^f^ h {|VjUVaN h [ g _N M Tr?] bcY U bSNXb2MVXjUPQP&Y Nr[ hcg g k M»NXb2M[ g N_M k TWY Nr[ hcg i [ g NXb2Mix h ^w^ h {i[ gUk ]_Y ^`]_j2^fY Nr[ hcg* ∞ X n + 2r
r=0
r
x
r
I ∞ X 1 = f (z) dz xr 2πi C r=0 r I ∞ X 1 1 x = dz 2πi C z(1 − z)n+1 r=0 z(1 − z) I 1 −1 n = dz n 2 2πi C (1 − z) (z − z + x)
JIcG √
tb2M'u h ^ lcgUh P&[fYH^ bcYDVT h2h NWV 1 + 1 − 4x Y gUq p(z) = z 2 − z + x ζ+ = 2 √ √ Y gUq 1 − 1 − 4x n h NrM|NXbcY N n b2M ζ− = ζ+ + ζ − = 1 ζ+ − ζ− = 1 − 4x [ g NrM k TWY gUq YDz 2h cMabcYHVV?[fPQuc^`M|u h ^`M VY N Y gUq d z c j N ^ e [ = V [ X V [ M s hcg l ζ g q C npcl ζ+ ζ− − NXb2M&¨tMHVX[ q jMi b2M h TrM P s ∞ X n + 2r r
r=0
xr
= Res
−1
=
(1 −
z)n p(z)
, ζ−
=
−(z − ζ− )
lim
z→ζ− (1 − z)n p(z) 1 −1 = n n (1 − z) (z − ζ+ ) z=ζ− ζ+ (ζ+ − ζ− ) n @~A K 1 2 n √ √ 1 − 4x 1 + 1 − 4x
=
H
tbd[fV[fVNXb2M|TrM cj2[fTrM q ]_^ h VM q x h TrP.x h TyNXb2M]rYDVM n n ≥ 0Y h {-V?jUu2u h VM|NXbcYJN LHMONWNr[ gUk n gUq s = r − m s {MabcYJcM s n