從需求到價格: Inter-sector input output analysis

從需求到價格(From demand to price): Inter-sector input output analysis

假定:
單位商品(半成品)有n種其總產出單位數 X1 X2 X3 ..... Xn
市場價格P1.....Pn
單位製程有m種其總使用單位數 Y1 Y2 Y3 ..... Ym
分別形成 (n x 1) 之 X及P矩陣 和 (m x 1) 之 Y矩陣

單位製程j 使用各種原料i之量Aij Xi 生產出各種i之量Bij
分別形成 (n x m) 之 Aij 和 Bij矩陣

A11 A12..A1j...A1m
A21 A22..A2j...A2m
.....
Ai1 Ai2..Aij...Aim
.....
An1 An2..Anj...Anm

可計算出總產出X=BY 總耗用 AY
得出(n x 1)之盈餘需求矩陣D = (B-A)Y = X-AY 供資本投入及終端消費使用
各單位製程之成本矩陣 (A')P 收入矩陣 (B')P
價差(附加價值value added)矩陣 V = (B'-A')P = ((B-A)')P皆為(m x 1)

關於D = (B-A)Y
由於不同製程可產生相同產品(不同飼料都可以養豬),
同一製程也可產生不同副商品(如一頭豬不同部位可以分售)
即使同樣之需求D矩陣,同樣之產出B及耗用A矩陣可能有不同之Y會符合
但生產者會傾向價差(附加價值)較低之製程
而同樣製程若價差越大,如同供應曲線一般.生產者也會傾向執行更多單位該製程
此彈性關聯會反映在製程使用矩陣Y與價差矩陣V之間
假定此彈性關聯如電路中之電阻倒數一般,係數分別為C1,C2.....,Cm
形成(m x m)之製程意願矩陣C  (若製程m耗用及產出皆放大3倍,Cm應變1/9Cm)
C1  0 .... 0
 0 C2 .... 0
....
 0  0 ....Cm
而使 Y = C V
代入前面2式
D = (B-A)Y = (B-A) C V = [(B-A) C ((B-A)')]P
價格矩陣 P = [(B-A) C ((B-A)')]^(-1) D
在已知各製程耗用及產出和生產者製程意願下,由市場需求量可以解出各商品價格

如同電路一般,並聯時電流會傾向走電阻較小的路徑,但電阻較大的路徑仍會有電流,
價格(Price)相當於電動勢(Voltage),貨物流量相當於電流(current),
而最終消費者的需求(demand)相當於電池(battery)賦予各種商品價格與電動勢


當然該關聯矩陣C可能會有不同模式如毛利率或總獲利......
特殊形態: 當每種商品只有唯一製程,製造過程亦不產生副商品時,
m=n使A和B成為(n x n)方陣

Fractional Flow Reserve (FFR)

FFR的生理意義是對於某特定流域之心肌,
目前(有大血管狹窄下)血流/最大可能(排除大血管阻力後)血流

Aorta-> coronary artery -> microvasculature -> cardiac vein -> right atrium

在沒有大血管(冠狀動脈)狹窄的情況下主要的阻力在microvasculature(包括arteriole,capilary)

假定分布於某特定流域之心肌,
其冠狀動脈有一段狹窄,該段阻力R,
而微小血管阻力最小值Rc(使用adenosine注射使微小血管充分擴張而達到),
靜脈壓力(Pv)相對極低接近於0

Pa -> R -> P1 -> Rc -> Pv(=0)
基於連通管任意斷面血流F相等 (Pa-P1)/R=(P1-Pv)/Rc=(Pa-Pv)/(R+Rc)=F
R=(Pa-P1)/F ;  Rc=(P1-Pv)/F

目前血流F = (Pa-0)/(R+Rc)
最大可能血流Fmax: 當R經處理成為0 => Fmax= Pa/Rc
FFR = F/Fmax= Rc/(R+Rc) = (P1-0)/(Pa-0) = P1/Pa
Pa 可直接由導管測量, P1則需使用特製導線深入冠狀動脈狹窄後方作測量

FFR 若小於 0.75-0.8 表示目前血流僅剩最大血流之 0.75-0.8倍
Rc/(R+Rc)< 0.75-0.8 意即 R/Rc > 0.33-0.25
此時對此狹窄做處理較有可見之效果

那如果有2段狹窄而中間無大分支的狀況
Pa -> R1 -> P1 -> R2 -> P2 -> Rc -> Pv(=0)
(Pa-P1)/R1=(P1-P2)/R2=(P2-Pv)/Rc=(Pa-Pv)/(R1+R2+Rc)=F
R1=(Pa-P1)/F ;R2=(P1-P2)/F ; Rc=(P2-Pv)/F

兩段狹窄總阻力R=R1+R2, 總合FFR為 P2/Pa
R1 FFR(排除R2後)= Rc/(R1+Rc) = (P2-Pv)/(Pa-P1+P2-Pv) = P2/(Pa-P1+P2)
R2 FFR(排除R1後)= Rc/(R2+Rc) = (P2-Pv)/(P1-P2+P2-Pv) = P2/P1

n objects within controlled bilateral vibration interconnected with (n+1) spring coils

n objects within controlled bilateral vibration interconnected with (n+1) spring coils
(all k/m substitutes with  k here)

Xi'' = k Xi-1 -2k Xi + k Xi+1
X0, X n+1 were controlled vibration of cos & sin wave
X0    = Gj cos Wj t
Xn+1= Hj cos Wj t

let Xi be linear,  Rij cos Wj t + Qij sin Wj t, composed of different j
Xi'' + 2k Xi =   (2k-Wj^2)(Rij cos Wj t + Qij sin Wj t) = k Xi-1  + k Xi+1
k R(i+1)j - (2k-Wj^2) Rij  + k R(i-1)j =0
consider characteristic kx^2 - (2k-Wj^2) x + k =0 , with x solution a, b

When Wj^2 > 4k => a,b belongs to Real number
Rij =  Gj  (a^(n+1-i) - b^(n+1-i)) / (a^(n+1) - b^(n+1)) + Hj  (a^i - b^i) / (a^(n+1) - b^(n+1))
When Wj^2 =0 => a=b=1
Rij =  Gj  (n+1-i) / (n+1)  + Hj  i / (n+1)


When 0< Wj^2 < 4k => a,b belongs to Imaginery number
(Wj^2 - 2k) / 2k +- Wj x (Wj^2-4k)^0.5 /2k
-1 <(Wj^2 - 2k) / 2k <1
 => let cos Y = (Wj^2 - 2k) / 2k, sin Y = Wj (4k - Wj^2)^0.5 /2k
Rij = Gj sin ((n+1-i)Y) /sin ((n+1)Y) + Hj sin( iY) /sin( (n+1)Y)

共振頻率為 W=2(sin (hπ/[2(n+1)])) k^0.5,
當n夠大時,最高頻趨近 2k^0.5
最低頻 2(sin (π/[2(n+1)])) k^0.5,剛好從0點到(n+1)點形成半波長駐波

other

lim k2->k1 (e^(-k2t)-e^(-k1t))/(k2-k1) =  - t e^(-kt)

(若E>0,則r1與r2有一個會是負數,只剩一個正數)

兩星體質量分別為M,m,距離r,以質心為圓心作圓周運動,
a=V^2/R , V=Rw -> a=Rw^2 => w^2=a/R
m之旋轉半徑R為 rM/(M+m), 受萬有引力產生之a為GM/r^2
w^2=a/R= GM/r^2 / (rM/(M+m)) =G(M+m)/r^3


三星體呈正三角,質量皆為m,距離r,以質心為圓心作圓周運動,
旋轉半徑R為 r/3^0.5, 受萬有引力產生之a為Gm/r^2 x 3^0.5/2 x 2 = 3^0.5 Gm/r^2w^2=a/R= G(3m)/r^3

角動量(向量) d [m (r0+r)x(v0+v)] /dt = m(v0+v) x (v0+v) + m(r0+r) x (a0+a) = (r0+r) x (ma0+ma)
當a0=0, F與r同軸 => d [m (r0+r)x(v0+v)] /dt = r0 x ma
兩星體質量分別為M,m,互相萬有引力而不受外界力量,其空間位置向量 r1,r2,質心位置r0
d [M (r0+r1)x(v0+v1) + m (r0+r2)x(v0+v2)] /dt = r0 x (Ma1 + ma2) = 0 (因互為反作用力相加為0)

formula

planck's law
2k^4/(h^3c^2)  T^4 x^3/(e^x-1) dx
x=h ν/(kT)

x             x/(1-e^(-x))
0            1
1.593    1.999534952
2.821    2.999616249  (max  ν)
3.920    3.999351512
4.965    4.999888979  (max λ)

Maxwell–Boltzmann distribution
2/pi^0.5 x^0.5 e^(-x) dx
x= E/(kT) = mv^2/(2kT)

multiple step decay

A -k1-> B -k2-> C -k3-> D -k4->E -k5-> F
A'= -k1A; d (lnA)= d(-k1t) ; A=A0 e^(-k1t)
B'=  k1A - k2B
 B'+k2B=k1A=A0 k1 e^(-k1t);令B(t)=x(t)e^(-k1t)+y(t)e^(-k2t);
> B'(t)+k2B(t) = (k2-k1)x(t)e^(-k1t) + x'(t)e^(-k1t) + y'(t)e^(-k2t) > (k2-k1)x(t)+x'(t) = A0k1 & y'(t)=0 ; y(t)=y0
if k2  = k1 then x'(t) =A0k1; x(t)= A0k1t + x0
>B(t)= (A0k1t + x0)e^(-k1t)+y0e^(-k1t)= A0k1t e^(-k1t)+ (x0+y0)e^(-k1t); B0= (x0+y0)
>B(t)= B0e^(-k1t) + A0k1t e^(-k1t)
if k2 != k1 then x(t) = A0 k1/(k2-k1)
>B(t)= A0 k1/(k2-k1) e^(-k1t)+y0e^(-k2t); B0=  A0 k1/(k2-k1) +y0
>B(t)= A0 k1/(k2-k1) e^(-k1t)+ (B0-A0 k1/(k2-k1))e^(-k2t)
>B(t)= B0e^(-k2t) + A0k1( e^(-k1t)/(k2-k1) + e^(-k2t)/(k1-k2) )

C'=  k2B - k3C

if k3 != k2 != k1
C(t) =  C0e^(-k3t) + B0k2( e^(-k2t)/(k3-k2) + e^(-k3t)/(k2-k3) ) + A0k1k2 ( e^(-k1t)/[(k3-k1)(k2-k1)] + e^(-k2t)/[(k3-k2)(k1-k2)] + e^(-k3t)/[(k1-k3)(k2-k3)] )



IF k1=k2=k3=.........=kn=k AND B0=C0=D0=E0=F0=.........=0 let A1=B, A2=C,.....,An
A=A0 e^(-kt)
A1=B=A0 kt e^(-kt)
A2=C=A0 [ (kt)^2 /2! ] e^(-kt)
A3=D=A0 [ (kt)^3 /3! ] e^(-kt)
A4=E=A0 [ (kt)^4 /4! ] e^(-kt)
A5=F=A0 [ (kt)^5 /5! ] e^(-kt)
An=..=A0 [ (kt)^n /n! ] e^(-kt)
All summation: A0  e^(kt) e^(-kt) = A0

for each An, peak An occured when (An)'=0; (kt/n)=1;  tmax = n/k
[n - (n-1)]/[(tmax)n-(tmax)n-1 ] = k ; So The peak wave moves at speed k(let x=kt),  (x^n/n!)e^(-x) dx = d[(-x^n/n!) e^(-x)] + (x^(n-1)/(n-1)!)e^(-x) dx,
(-x^n/n!) e^(-x) will be 0(when x=infinite), 0(when x=0,n>0), -1(when x=n=0)

intergration An from x=0 to infinite:
A0[ (kt)^n /n! ]e^(-kt) dt = A0/k[ (kt)^n /n! ] e^(-kt) d(kt) = A0/k (x^n/n!)e^(-x) dx =A0/k[0-(-1)]=A0/k

An= An-1 x ( kt / n )
for each time frame t (and kt), peak A occurred at nmax=kt (where kt/n=1;An=An-1)
at nmax, Anmax-2/Anmax-1= (kt-1)/kt; Anmax+1/Anmax= kt/(kt+1);when kt large enough ->

雜記

vector a x b : vector a sweeps in direction of vector b creates the area
0=(ai+bj)x(ai+bj)=ab (i x j + j x i)
i x j = - j x i


vector(a,b) moved as vector (c,d) painted green area
green area=blue-yellow=orange-pink=ad-bc


red area = green area



(a,b) & (-b,a) make ad-bc becomes a^2+b^2

(fgh)'= f'gh + fg'h + fgh'
(fg)''=f''g + 2f'g' + fg''
(fg)'''=f'''g + 3f''g' + 3f'g'' + fg'''

Z(x(t),y(t))  ; dZ = Zx dx + Zy dy  ; dZx = Zxx dx + Zxy dy :  dZy= Zxy dx + Zyy dy
則 ddZ = Zxx (dx)^2 + Zyy (dy)^2 + 2 Zxy dxdy + Zx ddx + Zy ddy (Zxx 表Z對x偏微分2次)


sin(x)=cos(x-pi/2)
cos(x)=sin(x+pi/2)

( f(t)e^(kt) ) '=k( f(t)e^(kt) )+ f'(t)e^(kt)
>  f(t)e^(kt) dt= d( f(t)e^(kt)/k )-  f'(t)e^(kt) /k dt>  f'(t)e^(kt) dt= d( f'(t)e^(kt)/k )-  f''(t)e^(kt) /k dt
f(t)e^(kt) dt
= d( f(t)e^(kt)/k-f'(t)e^(kt)/k^2 ) +  f''(t)e^(kt) /k^2 dt
= d [(f(t)/k-f'(t)/k^2+f''(t)/k^3+.......+fn-1(t)/k^n) e^kt] + (-1)^n fn(t)e^(kt) /k^n dt
當k=-1 => f(t)e^(-t) dt = d [ -(f(t)+f'(t)+f''(t)+............+fn-1(t)) e^(-t) ] + fn(t)e^(-t) dt

G(t)=f(t)e^(k1t)
> G'(t)--kG(t) = (k1-k) f(t)e^(k1t) + f'(t)e^(k1t)

( ln(1-e^(-x)) )'=  1/(e^x-1)

1/[(x-a)(x-b)]=[1/(x-a)-1/(x-b)]/(a-b)=1/(x-a)/(a-b)+1/(x-b)/(b-a)
1/[(x-a)(x-b)] dx = d ln(x-a)/(a-b)+ d ln(x-b)/(b-a) = d {ln [(x-a)/(x-b)]}/(a-b)

1/[(x-a)(x-b)(x-c)]=[1/(x-a)-1/(x-b)]/(a-b)/(x-c)=[1/(x-a)/(x-c)-1/(x-b)/(x-c)]/(a-b)
= {[1/(x-a)-1/(x-c)]/(a-c) - [1/(x-b)-1/(x-c)]/(b-c)}/(a-b)
=1/(x-a)/(a-c)/(a-b)+1/(x-b)/(b-c)/(b-a)+1/(x-c)/(c-a)/(c-b)

1/[(x-a)(x-b)(x-c)] dx = d ln {(x-a)^(1/(a-c)/(a-b)) (x-b)^(1/(b-c)/(b-a)) (x-c)^(1/(c-a)/(c-b))}

曲率半徑for U(x,y)=k
(UxUy)^2/ (Ux^2 +Uy^2)^(3/2) [Uxx/Ux^2 + Uyy/Uy^2 -2 Uxy/(UxUy)]
曲率半徑為正->凹向原點, 曲率半徑為負->凸向原點,