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MN∈NK∈N
C∶={1,...,K}
k∈C mk∈Nk
Gk=(V={v1,...,vN}, Ek={ek
1,...,ek
mk}) V=V(Gk)
Ek=E(Gk)
GkN
Ek
k∈C
k∈CM
N
̂
E∶=k∈CEkG∶=(V, ̂
E)
GkVP
v∈V w ∈V∖{v}k∈C X(k)
v,w ∼P
X(k)
v,w x(k)
v,w ∈Rx(k)
v,w v
w x(k)
v,w v x(k)
v,w w
{v, w}∈Ek
x(k)
v,w
Gk
x(k)
v,w Dk=(V , Ak={ak
1,...,ak
mk}) Dk
V Ak⊆(V×V)
h∶A→V t ∶A→V a ∈A
h(a)t(a)h(a)∈V
t(a)∈V a ∈AkDk=(V, Ak)
Gk=(V, E k) {v, w}∈Ek
(v, w) (w , v)DkGk
k
x(k)
v,w ∈RX(k)
v,w
GkX(k)
v,w Gk
̂
A∶=k∈CAkD∶=(V, ̂
A)
Dkk∈C V
Gk=(V, E k)
X(k)
v,w ∼P±
±X(k)
v,w ∶=
+X(k)
v,w ,(w, v)∈Ak
−X(k)
v,w ,(v, w)∈Ak
+X(k)
v,w =X(k)
v,w −X(k)
v,w
P X(k)
v,w
X(k)
v,w ±1+1
−1v∈V
±X(k)
v,w ∼±P
f X(k)
v,w ∼N(0,1)f+X(k)
v,w
2
πe−x2
2x>0 0 N(µ, σ)
µ σ X(k)
v,w
v w
X(k)
v,w Dk
Gk
M
M
v∈V
Lvv̂
A(v)∶={a∈Akv a;k∈C}
̂
A(v)=Λ∈LvΛ Λ ≠∅Λ∈Lv
v Gke∈Ekv∈e a ∈Ak
Dkv h(a)=v t(a)=v
vΛ∈Lv
v
C
Λ∈LvXΛ∶={±Xλ}λ∈Λ
v
{±Xλ}λ∈Λ
v λ v
X(k)
v,w
Λvv
v w v w
XLv∶=Λ∈LvXΛv
v
v1v2
v3
v4
a2
2
a1
1a2
1
a1
4
a1
2
a2
3a1
3
D=(V∶={v1, v2, v3, v4},̂
A)̂
A={a1
1, a1
2, a1
3, a1
4, a2
1, a2
2, a2
3}
̂
A(v1)
̂
A(v1)={a1
1, a1
3}⊍{a2
2, a2
3}Λv1={a2
2, a2
3}
XΛv1={X(2)
v4,v1,−X(2)
v1,v2}
Lv1={{a1
1, a1
3},{a2
2, a2
3}} v1
XLv1v1
v1
D P X∼±P
XLvv∈V±P
X∼P G
vΛvv
Λv
max
λ∈Λv
±Xλ; 0,
±XλXλ∼P
(Xλ)λ∈Λ
Y∶=∑λ∈ΛXλ
φY=
λ∈Λ
±φXλφY
φY(t)=η(t)+iν(t)Re(φY)=ηIm(φY)=ν
P
w v w
w v w
w
φmax[Y;0](t)=E(eitmax[Y;0])=1
2[1+φY(t)]+i
2[ {φY}(t)−{φY}(0)]
max[Y; 0]i{φY}
φY
{φY(t)}(ω)∶=1
πP V ∞
−∞
φY(t)dt
ω−t∶=lim
→0
1
πω−
−∞
φY(t)dt
ω−t+∞
ω+
φY(t)dt
ω−t
t, ω ∈RP V
t
Z
E(Z)=i−1∂t[φZ(t)](0),
Z∶=max[Y; 0]
MGk=(V, Ek)
k∈C∶={1,2}V∶={v1, v2, v3, v4}E1={e1
1, e1
2, e1
3, e1
4}
E2={e2
1, e2
2, e2
3}G1G2
v1v2
v3
v4
e2
2
e1
1e2
1
e1
4
e1
2
e2
3e1
3
D=(V, ̂
A)G=(V, ̂
E)
X∼L(0,1)L(µ, b)µ
b
Lv1={{e1
1, e1
3},{e2
2},{e2
3}} Lv2={{e1
4},{e2
1},{e2
2}}
Lv3={{e1
1, e1
2, e1
4},{e2
1}} Lv4={{e1
2, e1
3},{e2
3}}
Λ∈Lvii∈{1,2,3,4}
φY
φY
φmax(Y;0)
φX(t)=1
1+t2X∼L(0,1)Λ∈Lvi
Λ=1X=Y φY=φX
{φY}(t)=t
1+t2
φmax[Y;0](t)=1
2[1+φY(t)]+i
2[ {φY}(t)−{φY}(0)]=1
21+1
1+t2+i
2t
1+t2
E1E2
u v w
a1a2
D
E(0)=0w D E(max[Xa2; 0])=E(Xa2)=1
2
Y∶=Xa1−Xa2φU(0,1)(t)φU(−1,0)(t)=
(1−e−it)(−1+eit)
t2{φY}2(t−sin(t))
t2limt→0{φY}(t)
E(max[Y; 0]) =∂tφmax[Y;0](0)
i=1
6
E(max[Xa1−Xa2; 0])≠E(max[Xa1; 0])+E(max[−Xa2; 0])
Xa1−Xa2v
v
u v
w
e
γ(v)v∈V
Gk=(V, E k) E(v) v
Dk=(V, Ak)v γ+(v)
a∈Akh(a)=v a ∈Akt(a)=v
v γ−(v)γ+∶v↦γ+(v)
γ−∶v↦γ−(v)γ(v)∶=γ+(v)−γ−(v)
Dkγ γ(v)v
Dk=(V, Ak)X∼±PΛv
v∈V
E(∑λ∈Λv±Xλ)=0γ(v)=0
E(max[∑λ∈Λv±Xλ; 0])=1
2∂t[ {φY}](0)γ(v)=0
w∈UC
vUC
vv k ∈C
x(k)
v,w v
w∈V∖{v}k yv,w (C)∶=∑k∈Cx(k)
v,w
v w yv,w
yv,w (C)=−yw,v (C)
v w max[yv,w (C); 0]zb(D)∶=
∑v∈V∑w∈UC
vmax[yv,w (C); 0]D=(V, ̂
A)
M
MD=
(V, ̂
A)X∼±P
k∈C
E(max[Yv,w (C); 0]) v w
Yv,w (C)=∑k∈C±X(k)
v,w φYv,w φmax[Yv,w ;0]
Yv,w (C)max[Yv,w (C); 0]
E(max[Yv,w (C); 0])=∂t[φmax[Yv ,w;0](t)](0)
i.
E(Zb(D))=
v∈V
w∈UC
v
E(max[Yv,w (C); 0])=
v∈V
w∈UC
v
∂t[φmax[Yv,w ;0](t)](0)
i
D
X∼±L(0,1) N=6
C={1,2}
D=(V, ̂
A)̂
A=A1⊍A2
Yv,w (C)=X(1)
v,w −X(2)
v,w
φYYv,w (C)1
1+t2=i
i+t−i
−i+t
φmax[Y;0](t)=1
2[1+1
1+t2]+i
2[ { 1
1+t2}(t)−{1
1+t2}(0)]=
2i+t
2i+2t{1
1+t2}(t)=t
1+t2E(max[Yv,w (C); 0])=∂t[2i+t
2i+2t](0)
i=1
2
v
v1v2
w
w1w2
v w
E(Zb(D))=10
2=5
GDD
G∶=({v, w},̂
E)N=2X∼N(0, σ)̂
E=
E1⊍... ⊍EK
v w K
v w
⋮
X1
X2
XK
G K
X(k)
v,w ∼N(0, σ)k∈C
φYv,w (C)(t)=φYw,v (C)(t)=e−t2Kσ 2
2
Y∶=Yv,w (C)=∑k∈CX(k)
v,w N(0,√Kσ)
{φY}(0)=0
φmax[Y;0](t)=1
2[1+φY(t)]+i
2[ {φY}(t)−{φY}(0)]
=1
2+e−t2Kσ2
2
2+it√Kσ
√2
√π
1
1+t2
1
1+t2
{1
1+t2}(ω)=2i 1
(1+t2)(ω−t),i+i1
(1+t2)(ω−t), ω=
i
−ω2−1+1
ω−i=ω
1+ω2ω t
{φY}(t)=2
√πt√Kσ
√2 (t)∶=e−t2∫t
0es2ds
v w
E(max[Y; 0]) =∂t[φmax[Y;0]](0)
i=∂tit√Kσ
√2
√π(0)
i=σK
2π.
N>2v(N−1)
k∈C
(N−1)σK
2π
2N(N−1)
2K
D=(V, ̂
A)N≥2
K>1C={1,...,K}X∼±P
E[∑k∈C±X(k)
v,w ]=E[Yv,w (C)]=0{v, w}
E(max[Yv,w (C); 0])=1
2∂t[ {φYv,k (t)](0) {v, w }
v w
M
k∈C
v∈V U(k)
vv k ∈C
yv,k (U(k)
v)∶=∑w∈U(k)
vx(k)
v,w ∈Rv
k yv,k
v k
vmax[yv,k (U(k)
v); 0]
zm(Dk)∶=∑v∈Vyv,k (U(k)
v) Dk
∑v∈Vyv,k (U(k)
v)=0zm(Dk)
zm(Dk)=
v∈Vyv,k (U(k)
v)=2⋅
v∈V
max[yv,k (U(k)
v); 0].
K
k∈C(K−1)
zm(D)∶=zm(Dk)+zb(D∖Dk),
D∖Dk∶=(V, ̂
A∖Ak)
M
D=(V, A)X∼±P
Ak±±X(k)
v,w
±PYv,k (U(k)
v)max[Yv,k (U(k)
v); 0]φYv
φmax[Yv;0]v Dk
Emax[Yv,k (U(k)
v); 0] =∂t[φmax[Yv;0](t)](0)
i,
Yv,k (U(k)
v)=∑w∈U(k)
v±X(k)
v,w
E(Zm(Dk))=
v∈V
Emax[Yv,k (U(k)
v); 0]=
v∈V
∂t[φmax[Yv;0](t)](0)
i
Dk
k
Zm(D)∶=Zm(Dk)+Zb(D∖Dk)
GkDk
X∼N(0, σ)
G=(V, E )N
X∼N(0, σ)YvYv,1=∑w∈U(k)
vX(1)
v,w Xv,w
X(1)
v,w Yv(U(k)
v)=∑w∈U(k)
vXv,w φYv(t)=
e−t2
2(N−1)σ2{φYv}(t)=2
√πt√N−1σ
√2
Emax[Yv(U(k)
v); 0] =1
i∂t
1
21+e−t2
2(N−1)σ2+it√N−1σ
√2
√π(0)=N−1
2πσ
v N
G2N(N−1)
2
D=(V, ̂
A)N≥2
K C ={1,...,K}X∼±P
E[∑w∈U(k)
v±X(k)
v,w ]=E[Yv ,k(Uk
v)]=0γ(v)=0k
Emax[Yv,k (U(k)
v); 0]=1
2∂t[ {φYv,k (t)](0)γ(v)=0k
G=(V, E ={e1, e2, e3})
X∼L(0,1)l, m, n ∈{1,2,3}m≠n
v1v2
v3
e1
e2
e3
v1v2
v3
a1
a2
a3
G D
D=(V, {a1, a2, a3})
G
1
1+t2=i
i+t−i
−i+tYvl∶=Xm−XnXmXn
Emax[Yvl(U(1)
vl); 0] =1
2∂t[ {φYvl}(t)](0)=1
2∂tt
1+t2(0)=1
2
vll∈{1,2,3}D
E(Zm(D))=E(Zm(D1))=
∑3
l=1Emax[Yvl(U(1)
vl); 0]=3⋅1
2=3
2
5
2
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
8=23G
G=(V, E )
1
1+t2
1
1+t2
1
1+t2
1
1+t2+it
1+t2
1
1+t2
φYvl(t)=1
(1+t2)2=1
1+t2
1
1+t2Yvl∶=Xm+Xnvl
{φYvl}(t)l∈{1,2,3}t(3+t2)
2(1+t2)2
Emax[Yvl,1(U(1)
vl); 0] =1
2∂tt(3+t2)
2(1+t2)2(0)=3
4
GE(Zm(G))=9
4
1
8(2⋅3
2+6⋅5
2)=9
4
k∈C D K ∈N
X∼±P
E(Zm(D)) =E(Zm(Dk))+E(Zb(D∖Dk)) <E(Zb(D))
v∈V
∂t[N−1φv,k (t)](0)
i+
v∈V
w∈UC
v
∂t[K−1φv,w (t)](0)
i<
v∈V
w∈UC
v
∂t[Kφv,w (t)](0)
i,
Mφv,⋅(t)∶=1
21+φX(t)M+i
2{φX(t)M}(t)−{φX(t)M}(0)
max[Y, 0]Y∶=∑M
j=1±XjXj∼P M ∈N
N−1K K −1X Xj
v
X∼P
v
∂t[N−1φv,k (t)](0)
i+(N−1)∂t[K−1φv,w (t)](0)
i<(N−1)∂t[Kφv,w (t)](0)
i
⇔(N−1)(∂t[Kφv,w (t)](0)−∂t[K−1φv,w (t)](0)) >∂t[N−1φv,K (t)](0).
1
2∂t[ {φX(t)N−1}](0)<(N−1)
2∂t[ {φX(t)K}](0)−∂t[ {φX(t)K−1}](0)
P
G=(V, E )N
X∼N(0,1)
∂t{φX(t)M}(0)=√M2
π
1
2√N−12
π<(N−1)
2
√K2
π−√K−12
π
.
K<N2
4(N−1)N>2
G=(V, E )X∼L(0,1)
φX(t)1
1+t2
1
i∂t[Mφv,⋅(t)](0)=1
2∂t[ {φX(t)M}](0)=Γ(1
2+M)
√πΓ(M)=M
22M2M
M
R
iR
t=ω
a3
a2
a1
am
am−1
...
CR
C
−R R
C φY(t)1
ω−t
max[Y; 0]Y=∑λ∈Λv±Xλ
φY
C∋z↦φY(z)
ω−zC
a1,...,am∈Cn1,...,nm∈N
φY(z)→0z→∞
{φY(t)}(ω)=2i
m
j=1φY(z)
ω−z, aj+iφY(z)
ω−z, ω.
{φY(t)}(ω)=2i
m
j=11
(nj−1)!lim
z→aj
∂nj−1
∂znj−1[(z−aj)njφY(z)
ω−z]−i lim
z→ωφY(z)
]−R, R[C=CR⊍]−R, ω −[⊍C⊍]ω+, R[
R, ∈]0,∞[C
CRC
CφY(z)
ω−z
a1,...,am∈Cn1,...,nm∈NφY(z)→0z→∞
t=ω R
φYC
C
C
φY(z)
ω−zdz =2πi
m
j=1φY(z)
ω−z, aj.
C
φY(z)
ω−zdz =ω−
−R
φY(t)
ω−tdt +C
φY(z)
ω−zdz +R
ω+
φY(t)
ω−tdt +CR
φY(z)
ω−zdz.
R→∞→0
P V ∞
−∞
φY(t)
ω−tdt =lim
R→∞lim
→0ω−
−R
φY(t)
ω−tdt +R
ω+
φY(t)
ω−tdt.
Cω+eiθ−π≤θ≤0
C
φY(z)
ω−zdz =lim
→0i0
−π
φY(ω+eiθ)eiθ
ω−(ω+eiθ)dθ=πiφY(ω)=−πiφY(z)
ω−z, ω.
CRReiθ0≤θ≤π
φY(z)→0z→∞CR
φY(z)
ω−zdz =lim
R→∞iπ
0
φY(Reiθ)Reiθ
ω−Reiθdθ=0.
P V ∞
−∞
φY(t)
ω−tdt =2πi
m
j=1φY(z)
ω−z, aj+πiφY(z)
ω−z, ω,
{φY(t)}(ω)=2i
m
j=1φY(z)
ω−z, aj+iφY(z)
ω−z, ω.
φX(t)=1
1−2it=1
1+4t2+2it
1+4t2X
Γ(α, β)α∶=1β∶=2α>0
β>0
φX(z) z→∞
φ∣X∣(z)
ω−z−i
2
{φX(t)}(ω)=iφ∣X∣(z)
ω−z, ω=2ω
1+4ω2−i1
1+4ω2
−(X)=−Xφ−X(t)=
1
1+4t2−2it
1+4t2
i
2
{φ−X(t)}(ω)=2i φ−∣X∣(z)
ω−z,i
2+iφ−∣X∣(z)
ω−z, ω=2ω
1+4ω2+i1
1+4ω2
Λvm∈N
vXi∼Γ(α, β)i∈{1,2,...,m}
Xi (1−βit)−α
Y∶=∑m
i=1Xi (1−βit)−αm YΓ(mα, β)
−i(1−βit)−αm
αβm
v
Yv∶=∑λ∈ΛvXλ
λ∈ΛvΛ+
vΛ−
vh(λ)=v t(λ)=v
ΛvΛv=Λ+
v⊍Λ−
vΛ+
v=γ+(v)
Λ−
v=γ−(v)
0=E(Yv)=E∑λ∈Λv±Xλ0=∑λ∈Λ+
vE(Xλ)+∑λ′∈Λ−
vE(−Xλ′)
0
0=Λ+
v⋅E(Xλ)+Λ−
v⋅E(−Xλ′) E(Xλ)=−E(−Xλ′)
0=[γ+(v)−γ−(v)]⋅E(Xλ)
γ(v)=0E(Xλ)>0γ(v)=0
E(∑λ∈Λv±Xλ)=0
max[∑λ∈Λv±Xλ; 0]=max[Yv; 0]
Emax[∑λ∈Λv±Xλ; 0]=1
2E(Yv)+1
2∂t[ {φYv}(t)−
{φYv}(0)](0)E(Yv)=0γ(v)=0
γ(v)=0γ+(v)=Λ+
v=Λ−
v=γ−(v)
φYv=∏λ∈Λ+
vφXλ∏λ′∈Λ−
vφ−X′
λ