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1 S l u t s k y M a t r i x o g N e g a t i v e d e n i t e n e s s

2 . F . 1 0 o m t o o k F q n t d m n d u n t o n @ D A o m

t o o k P Q D = 1 n d w = 1 D l l X

I F g l u l t t l u t k m t S = Dpx(p,w) + Dwx(p,w)x(p,w)T

l u t t a @ I D I D I A

P F o t t @ D A d o n o t u l l ( l l t k o m F

n l u l t t l u t k w t n d t o n n o n u m n o m D

l l t n t n

w = 1 u m t o n F

1 . 1 S o l u t i o n

t t o u t l u l t n Dpx(p,w)X

Dpx(p,w) =

p2wp21P p2w

p1P2w

p1P p2w

p1P2 p2wp1P2

p3wp2P2

p3wp22P p3w

p2P2w

p2P p3w

p2P2

wp3P

p1wp3P2

p1wp3P2

p1wp23P p1w

p3P2

@ I A

P (p1+p2+p3)F x o t ' t o n d m n d o m n n F ' t n l u d n o m ' t D n d n o n l n t t o o n d

u t t u t o n ' t t o o m n t t o n u m F p t l l u l t

t ' t o m n n n o m

Dwx(p,w)X

Dwx(p,w) = p2p1P

p3p2P

p1p3P

@ P A

n l u l t t u l l n o m ' t

Dwx(p,w)x(p,w)T

D

3 3m t F

Dwx(p,w)x(p,w)T =

p2p1Pp3p2Pp1p3P

p2wp1P p3wp2P p1wp3P

@ Q A

Dwx(p,w)x(p,w)T =

p22w

p21P2

p3w

p1P2p2w

p3P2

p3w

p1P2p23w

p22P2

p1w

p2P2

p2w

p3P2p1w

p2P2p21w

p23P2

@ R A

I

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o t o l t d u t t u t o n ' t X

S = Dpx(p,w) + Dwx(p,w)x(p,w)T =

p2w(2p1+p3)p21P2

w(p1+2p3)p1P2

p2w(p1p3)p1p3P2

p3w(p2p1)p1p2P2

p3w(p1+2p2)p22P2

w(2p1+p2)p2P2

w(2p2+p3)p3P2

p1w(p2p3)p2p3P2

p1w(p2+2p3)p23P2

@ S A

n l u t n t l u t k m t n p = (1, 1, 1) n d w = 1 D o n t X

S(1, 1, 1) = 13

1 1 0

0 1 1

1 0 1

@ T A

w t d o n o t u l l n k F k n m l @ E o l u m n t o I E

o l u m n t o Q A a o l u m n t o P F e l o p Sp = 0D n a @ I D I D I A F f u t t t u l l l o l l F

n o m n n t m d ( n t v, v Sv 0 F n o n d t o n o @ D A u l l ( l l n t k o m F v t p = (1, 1, ), > 0 D n d n t t u t o t o n X

S(1, 1, ) =

12+1+2(2+)2

1(2+)2

0 3(2+)23

(2+)2

1(2+)

1(2+)2

1+2(2+)22

@ U A

o n d t o S(1, 1, ) n o t n t m d ( n t @ t t o t d ( n t A F n o l 2 . F . 3 D t t @ D A u l l ( l l v

n d o m o n o u o d o H D t n p S(p) = 0 n d S(p)p = 0 (p,w)Fp o m I D t P k n o t t @ D A u l l ( l l v n d o m o n o u o

d o H F o n l M . D . 4 W Q W F

T h e o r e m M . D . 4 X s

Sp = 0 p S = 0 n d d u d m t S n t d ( n t D t n n t d ( n t o l l t o n t u Tz = {z|zp = 0}F t o m n t d u d m t

S n m o n o n o n d o n

P

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o l u m n X

S(1, 1, ) =

12+

1+2(2+)2

0 3(2+)2

@ V A

o o n t t o

v R2 n d l u l t X

v Sv =1 + 2

(2 + )2v1v2

(2 + )

(2 + )2v21

(3)

(2 + )2v22 @ W A

i l l o t o n u v Sv > 0 F n > 0 n m o

(2 + )2 n d d u t o n X

v Sv > 0 (v1v2 2v21) > (v

21 + 3v

22 2v1v2) @ I H A

o n o n t t n d d n t t o

l l o t D n X

(v21 + 3v22 2v1v2) = (v1 v2)

2 + 2v22 > 0 @ I I A

o n o n t t o n t o X

(v1v2 2v21)

(v21 + 3v22 2v1v2)

> > 0 @ I P A

f l l o u o l

v2 > 2 v1 F u o o v = (1, 4) l l n u t t

v Sv > 0 o o m

F f n t o n o

v t X

(1 4 2 12)

(12 + 3 42 2 1 4)=

2

41> > 0

@ I Q A

u

S n o t n t d ( n t o l l t o D t u

S n n o t t F n m n

o u t o t o pD o u n d t t X

p R3; p = (1, 4, 0) p Sp > 0, when p = (1, 1, 2/41 ) @ I R A

u t t n p n d m k o t d ( n t D n d @ D A n n o t u l l ( l l t k o m F F i F h F

2 S t r a n g e d e m a n d c h a n g e s

s n P F p F I T o m t o o k n t o l l o n d m n d u n t o n X

x(p,w) =

p2p3

p1p3wp3

@ I S A

Q

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t D t o n u m d m n d o t l o o d I n d Q D u t d l o o d P F

e l o t d m n d o t ( t t o o o d d o n o t t t o n o t

o n u m o l l n o m F o u l d t n k o o o d I l o n u m t o n

o o d o u o n u m l l d m n d o d n t o l l D

n m l

p2 p3 F q o o d P t n l o u u l u l d o d n t o o n E u m p1 F y n l o o d Q m t o n o m l o o d D n d o u l d t o u t o o m k n d o n t m n t o o d o u t o m n F

d m n d o u l d l n d l k t X e u m t o k k n o D t

n u o o d I o n t n t o d D u t n o t t o k n l

F s t o u l d t t t o ' o m o k u n t n F r o D o o

l o u u l D o t t o n o o d I D t n l l o k m o F

n t o k o m l t d D t o j t o t o m p2 D n d l l o k n o m n d F f l l t t o n o l n n l t o t

o n u m D t u n n t D n t t n l n d t o

n t o n u m o m n t d t o u F s n t k d

t o X

I F o t t x(p,w) o m o n o u o d o H F n d u l l ( l l l v F

P F o t t

x(p,w)d o n o t u l l ( l l t k o m

Q F o t t t l u t k m t u l l ( l l v Sv = 0 v R3 F

2 . 1 P r o o f - H o m o g e n i t y o f d e g r e e 0 . a n d W a l r a s L a w .

p o m n t o n o t l n u m

o n t X

x(p,w) =

p2p3

p1p3wp3

= x(p,w) @ I T A

u @ D A o m o n o u o d H F e l o @ D A t ( l l o n X

p x(p,w) =p1p2p3

p2p1p3

+p3w

p3= w

@ I U A

f o t o t o l l o d t l F F i F h F

2 . 2 P r o o f - V i o l a t i o n o f t h e w e a k a x i o m

x t o t t @ D A d o n o t t t k o m F d u t o t

t t n o t u d n o o d I n d P F v t X

p = (1, 1, 1) w = 1 x = (1,1, 1) @ I V A

x o l t n o o o d I D n d o m n t o u o n u m D o t l l n

' o d n t n t u t o n X

p = (2, 1, 1) w = p x = 2 x = (1,2, 2) @ I W A

R

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o x l d d t o xD n d x o u l d n o t l d d xD l l t m

x n o t ' o d l n t u t o n I F n o t u n t l t t

n d d t X

p = (1, 1, 1) p x = 1 = w @ P H A

u t k o m n o t u l l ( l l d F o l m D t t u n

o o m n t n n D n n o t m d d t o u n d

n F F i F h F

2 . 3 T h e S l u t s k y M a t r i x

p n l l d d u t t l u t k m t F p t d m n d n o m n F

Dpx(p,w) =

0 1

p3p2p

2

3

1p3

0 p1p23

0 0 wp23

@ P I A

x t t o m n t d n o m ' t

Dwx(p,w) x =

0 0 00 0 0p2p23

p1p23

wp23

@ P P A

x o t t t o n l o o d Q ' t d o u n o m n F p n l l l u l t

t u t t u t o n ' t o n n t l u t k m t X

S = Dpx(p,w) + Dwx(p,w)x(x, p)T =

01

p3 p2

p23

1p3

0 p1p23

p2p23

p1p23

0

@ P Q A

g o o t o v R3 n d l u l t X

v Sv =v1v2p3

p2

p3

v3v1p3

+p1

p3

v3v2p3

v1v2p3

+p2

p3

v3v1p3

p1

p3

v3v2p3

= 0@ P R A

o n t u m n t o t F o I F t m n t R F t m n d o

o n F o

v Sv = 0 o t t o F F i F h F

f o t d m o n t t n l t o n o t n t d ( n t n o F

p t l D n n t m d ( n t D n n o n d t o n o t k

o m F y n t o t n d D n n t m d ( n t D n t d m n d

d o n o t u l l ( l l t k o m F u t n o m l t o n n p r o p o s i t i o n

2 . F . 2 F

S