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u

u

v

u+v

v3.2

0.5

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1

2-

3

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u

u

v

v

v

u

w

w

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e

e

hh

eh

1

1

1

2

2

1(a)

(b)

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0 1

0 1

+ = 01

1(1/2)

(1/2)

(1/2)

(1/2)

/

/

/

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0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

A B

CD

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A B = A

C = AD = A

=

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A B

C = AD = B

=

A B = A

C = AD = A

=

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.P

(a) ar bit r ar y vector (b) arbi tr ary 1-form

at a poi n t P at a poi n t P

v

v

.P

x

x

1

2

e2

=2

P

. e1

=1

3x

= dx1 1

.

.

e3

=3 = dx

3 3

.

= dx2 2

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.

v

P. .

P

v

( ) = v1

( ) = v

v

u

u,

2( ) = 1 ( ) = 2 5

P

v

P

.

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U2

U1

U1 U2U

U1 U2U

A

B B

A

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(a) (b)

p

u =x

t

y t

x

yp

(p) t

u = x y-

x x+t

. .

.

.(p)

t

y x

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p

( v)(p)t*

v(p)

u(p)

v( (p))t

t(p)..

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2-simplex 3-simplex

. .

.

. .

.

.

.

.

.P P P

P P

P

P

P

0-simplex 1-simplex

P P

0 0

0

0

1

1

1

2

2

3

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+ _ =P P P P P P

P P P

0 0 01 1 1

222

. . .

.

.

.

. .

.

c

(2)

(0) (1)

1

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0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

P

PP0 1

2

(2)

(1)

(0)

1

1

1

2

t

t

1

2

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A BA

D = A

A B = AB = A

C = AD = AC = AC = A D = B

s

s

s

s s

s2

2

2 2

2

(2) (2)

(1) (1)

2

(1)

(2)

(c)(a) (b)

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.

x

x

x

1

2

3 j

surface

3

3

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3 3

t

t

t

t1

2

.

.

.

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x

x

x

1

2

3

x

x

x

1

2

3h

xh

.

n

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.

. h

h

0

33

3

3

2.

.

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0

(fluxon)

B

2

2-surface

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+external char ges+

++

+

+

+ + + +

+

.+++++

+++++

----

------

hand les

insulat ing

P

D=0

+

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3

x

3

1

22

1

x

x

S

2

1

P

h

-h

1

V2

V

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P..P

L

H

j~

H

ind

H

indH

A

(a) (b)

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HD

E B

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yl

jx

Bxy

x

zl

lx

y

z

IP

Q

HU

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Al AlSiO SiO SiO

Al

2 2 2

n n

p Si

+ +

z

y

x

II

UH

2DEG

draingate

source

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Bxy

H

U

x

jyjx

Uy

j

x

Q

P

I

y

xU

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B

R

R

R

2 4 60

0

0

2

4

6

8

10

H

L

k

T

0.2

0.8

0.6

0.4

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j

E B

el. current (twisted)

field strength (untw.)

electric

magnetic

2+1 decomposition

1+2 decomposition

2-forms1-

form

s

Lorentzforcecoupl.

Lorentzforcecoupl.

spacetimerelation s

pacetim

erela

tion

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D Hexcitation (twisted)

field strength (untw.)

electric

magnetic

3+1 decomposition

1+3 decomposition

2-forms1-

form

s

spacetimerelation

spacetimerelation

el./mag.reciprocity e

l./m

ag.r

ecipro

city

+ +

E B

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uR

uR||R

Q

Pu

v vP

P

QvQ||

T(u,v)

u,v[ ]

u

vu

v

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u

u(1)

(0)

.P

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x 2

x 1

pr

p

r +rp

r +p

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D

A

B

C

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D

D

A

A

B

BC

C

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=0K=0

(2)

=0(2)

no reciproc.

reciproc.

unknown condition

unknown condition

rank(b)=3

rank(b)=2

rank(b)=1

K=0

light cone

birefrigence

4th-order Fresnel equation

degenerated light cone

no apparent reduction

general 4th-order Fresnel eq.

4th-order Fresnel equation

no reciproc.

reciproc.

birefringence

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x

n

h

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1 2 3 4 x

0.2

0.40.6

0.8

1

1.2

1.4

y

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0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

B

E

1

2

ztorque

l

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0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

EB

I

force.z=0

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Jext

(1)=0

, = 1

(1)

matter

S

J(2)

=0

= =1

(2)

vacuum

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B(1)

v

E(2)

x

vx

x

x

(1)

(2)1

3

2

3

B

E(2)

(1)

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x

x

x

(1)

(2)

x

v

(2)B

E(1)

v2

3

1

1

B(2)

(1)E

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