text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
and k denotes thermal
conductivity. Compared with mechanical refrigeration, thermoelectric cooling offers the
following advantages:
-No moving parts
-Environmentally friendly
-No loss of efficiency with size reduction
-Can be integrated with electronic circuits (e.g. CPU)
-Localized cooling with rapid response
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Ge
BiSb
PbTe
Bi2Te3
200
400
600
800
1000
1200
Temperature (K)
Room Temperature
Image by MIT OpenCourseWare.
Image by MIT OpenCourseWare.
In the current investigations, people have tried different compositions to improve ZT. At
different temperature ranges, in the right figure we have different best materia... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
exhibit low-dimensional behaviors. Recall the following cases
given in homework.
D. O. S.
D. O. S.
D. O. S.
D. O. S.
3D
E
2D
E
1D
E
E
0
D
2) Thermal conductivity can be significantly reduced by the scattering of phonons at the
interfaces. In the following case, the electrical conductivity is not strongly... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2.57 Fall 2004 – Lecture 20
Quantum Dots
145
We have discussed the seebeck coefficient enhancement in 2D and 1D structure.
Reducing the dimension to 0D may further increase S. However, difficulty exists in
making connections t... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
(assumed to be equal)
–Subband potential barrier: ∆Ec
–Effective masses: mA and mB
–Phonon mean free path: L
LA
LB
mA mB
Ec
A B
∆Ec = Band offset
Y-M. Lin and M.S. Dresselhaus., Phys. Rev. B 68, 0753045 (2003)
The utilized approaches are
–Determination of the (sub)band structure
–Derivation of the dispersi... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
The following figure demonstrate the idea of making standing nanowires.
Si (100)
predeposited layers
(adhesive / conductive / patterned)
thermal evaporation
Al
Si (100)
predeposited layers
barrier layer
electrochemical
polishing
Al
Si (100)
anodization
Si (100)
porous alumina
selective etch
1) patterni... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
004)
Carrier profiles and electronic
band energies across p-n
junction
9. Conclusions
1) Model systems show that:
ZT for 0D nanowire superlattice
> ZT for 1D quantum wires
> ZT for 2D quantum wells
> ZT for bulk for same material
2) New research directions now being pursued:
-Self assembled bulk composites of ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
��
= L T
12 ⎜
⎛ 1 dΤ ⎞
⎟
⎝ T dx ⎠
is similar to S
∆ =
dQ
T
and may be
compared with entropy flux.
The heat transferred is
J q = ∑∑∑ vx ( E − µ) f = L21 ⎜ −
2
V kx
k y
kz
⎛ dΦ ⎞
⎝ dx ⎠
⎟ + L22
dT
dx
For open circuits, Je=0. We obtain
− =
S
dΦ / dx
=
dT / dx T − T
c
V
h
L
12 ,
=
L
11
where S is ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
point, but closer to the average temperature from 1 to 3. This is different from
the normal thermocouples.
1
2
3
1
When dT/dx=0, we have
J = L −
21 ⎜
q
⎛ dΦ ⎞ L21
⎟
⎝ dx ⎠ L11
=
J = Π
e
J ,
e
where the Peltier coefficient Π = TS , L21=TL12. Note one thermoelectric coefficient (S
here) can be used to express... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
T
dS
dT
=
/
1 dq dx
J e dT / dx
= β .
Je
dQ/dx
The overall energy equation is
dT
dΦ
.
q =
e dx
dx
2
=σJ + k
e
dJ q
dx
+ J
+ Thomson term .
And thermal conductivity is
k = L L / L − L .
e
22
21
11
12
The Wiedemann-Franz law states
ke
σT
where the constant L is Lorenz number.
2
= const = L = 2... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
o
r o
r
2.57 Fall 2004 – Lecture 21
153
(3) We ignore
∂fo
∂t
on the LHS compared with
g
τ
on the RHS. This indicates
∂fo
g
∂t τ
⇒ t τ .
The theoretical solution of
Now consider the above transient process in which an infinite wall is heated sudd... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
condition that
(Λ/T dT/dx) cannot be satisfied. There is no convincing experimental data showing the
validity of the hyperbolic equation. In femto-laser heating, the temperature of electrons
is raised much higher than that of the phonons and after the relaxation time electrons
exchange energy with phonons (the foll... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
valid for rarefied gas flow, the drift-
diffusion equation is not applicable to electrons.
Chapter 7 Classical Size Effects
y
d
x
When the electron and/or phonon mean free paths are comparable to or larger than the
thin film thickness, they will collide more with the boundaries. The previous requirement
Λ / L 1... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
g( )r is changed by
scattering. For statistical calculation, caution needs to be taken when summation is
conducted.
2.57 Fall 2004 – Lecture 21
156
2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 22
We have talked about the heat flux as
qx =
1 ∑∑∑ fv x =ω.
V k
k
k
y
z
x
y
vy
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
0, f = f0, g = 0 for θ ∈⎜ 0,
⎪
⎪
⎨
⎪
y
⎪⎩
Finally we get
⎛ π⎞
⎟
⎝ 2 ⎠
⎛π ⎞
,π ⎟
⎝ 2
⎠
, f = f0, g = 0 for θ ∈⎜
d=
.
At y = 0,θ 0,
∈ ⎜
⎛ π⎞
, C = -S , g y,θ) = S 1− exp
⎟
⎝ 2 ⎠
⎛
0 ⎜
⎝
(
0
⎞ ⎞
y
⎛
⎜ −
,
⎟ ⎟
⎝ vτ cos θ ⎠ ⎠
2.57 Fall 2004 – Lecture 22
157
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
v sinθcos ϕ according to our spherical coordinate system, f=g+f0. Thus
q
x (
)
y =
ωmax
∫
0
2π
∫
0
π
∫ = (
0
dω dϕ[ 2 ω v cos ϕsin θ −τv cos ϕsinθ
⎛
)⎜
⎝
df0 dT ⎛
⎜
dT dx
⎝
⎛
exp −
⎜
⎝
y
⎞
⎟
vτcos θ
⎠
−1
⎞ ⎞ D(ω)
⎟ ⎟
4π
⎠ ⎠
sin d
θ θ +
∫π =ω(v cos ϕsin θ) −τv cos ϕsin θ
π
2
⎛
⎜
⎝
df0 dT ⎛
dT dx... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
∫0
2
cos ϕdϕ π
=
, the total heat
= −
π dT ωmax
4 dx
∫0
=
ω
df0
dT
2
τv D (
) d
ω ω ∫0
[
1
2
⎛
1( − µ dµ Λµ exp
⎜⎜
⎝
⎛
⎜
⎝
)
⎛ ξ⎞
⎜ −
⎟ −
µ
⎝
⎠
1
⎞
⎞
− d ⎟⎟
⎟
⎠
⎠
+
1
∫0
2
⎛
(1− µ )dµ⎜ −Λµ⎜1− exp
⎜
⎝
⎛
⎝
⎛ ξ⎞ ⎞
−
⎜
µ
⎝
⎞
⎟ ⎟ − d ⎟]
⎟
⎠ ⎠
⎠
= −kd
dT
dx
,
which yields (if Λ is indepen... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
ω instead of the
C v d
simplified k
=
CvΛ
3
, which gives an underestimation of Λ . This is because the Debye
approximation overestimates the velocity approach the edge of the first Brillouin zone,
where the group velocity should be zero.
ω
Optical
Acoustic
k
For partial specular (momentum conserved) and pa... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
C = ∫
y
y ⎞
⎟
⎟
y ⎠
τv
. In general, use C y to replace
(
)
g y( ) = (
C y 0
y
⎛
⎞
) exp ⎜ −
⎟ + ∫y
⎝ τv cosθ⎠
y
0
S
0 (
)'
y
'− ⎞
y
⎟
⎟
y ⎠
⎛
y
exp ⎜
⎜ τv
⎝
τvy
dy
' .
y =
( )
0, f =
, = 0, C 0 = 0 for θ∈ 0,
f g
0
Now the boundary condition is (elastic scattering on boundaries)
⎧
⎪
⎪
⎨
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
The temperature profiles for two extreme cases are drawn in the following figure. For
ξ→ 0 (note T1 ≠ T2 ), it is in nonequilibrium state but we define the equilibrium
conception, temperature, based on the average value.
θ
1
ξ→ ∞
0ξ→
y/d
1
T1
T1
Teq
Teq
T2
T2
2.57 Fall 2004 – Lecture 22
161
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
gives
g +τvy
g
∂
y
∂
= −τv
df0 dT
x dT
dx
= S0 ( ) .
x
We can solve g first and then substitute the expression f = g+f0 into any flux equation.
Under the diffuse assumption, we obtain the following figure for the conductivities of the
material.
σ
σ
bulk
=
k
k
bulk
In the y direction, we have
2.57 Fall 20... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
, first we discretize the equation with
2d T Ti+1 − 2Ti + T
dx2
2∆x2
=
.
i−1
0
ξ
Half trapezium at both ends
The integration is calculated by dividing the area into many trapezia. We have
ξ
∫θ(η')E (|η η ' |) dη θ 0
−
1
' = ( ) 1 (
)
E η
2
E1 (ξ η)
−
)
∆ + (
η θ ξ
2
η ∑ (
∆ + ∆η θ η i
) E (|η η |) ,
−... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
q
Te q
T2
T2
More generally, θ should be the internal energy of the carriers. For photons,
θ(
y
) =
) − 2
(
u y u T y T 4
u
T2
1
) − 2 =
u2
−
4 (
4 −
T
1
4
.
The interpretation of the temperature discontinuity is worthy of special attentions.
Sometimes, the jump is physical, while in other cases the jump is ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
T2 are only emitted phonon temperature
entering the thin film, not the local equilibrium temperature as we use in the Fourier law
or solved directly from the Boltzmann equation. However, if the interface reflectivity is
not zero, there exists a temperature jump just as in the case of thermal boundary
resistance tha... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
discussed above.
One example is in the experiment determining the quantized conductance of a nanowire.
The electrodes are large and the measured voltage drop should represent the differences
of electrons entering the channel.
Je
Chemical potential
Superlattice
Large electrodes to assure
uniform temperatures
2 ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
left figure, at the interface we have
2.57 Fall 2004 – Lecture 23
166
q = ∑ τ ωvx1 f1 + ∑ τ21 =ωvx2 f2 .
=
12
vx1 >0,vy1 ,v
z1
vx 2 >0, vy 2 ,v
z 2
Note in the right figure, only rightward arrows indicate transport to the second region
(V > 0 ). Define τ ' , τ ' as the average transmissivity in each region.... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2 ')
2
''
τ23
+
,
+ 3 /
4 Λd
where 3 /
d
4 Λ <<
1−
1 (τ12 '+τ21 ') 1−
2
+
τ21 '
1 (τ23 '+τ22 ')
2
τ23 '
in thin films, while in bulk materials
d3 /
4 Λ is dominant. We have similar relationship as previous cases.
k
kbulk
T1
1
T2
2
3
ξ=
d
Λ
1
Chapter 9 Liquids
For gases, after two particles c... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
168
2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 24
In the last lecture, we have talked about Einstein’s work on the Brownian motion.
P(x)
P(x+dx)
Stoke’s flow
3F
=
D uπ µ
In the left figure above, pressure difference exists in the fluid. For one particle, the
osmotic pressure is determined by... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
dissipation theory as viscosity is a measure of dissipative process and diffusivity is a
measure of random walk (fluctuation) process. The relationship between thermal
2.57 Fall 2004 – Lecture 24
169
diffusivity and viscosity is also called Einstein relation. In chapter 6, t... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
the Brownian particle can be written
as,
m
du
dt
= −mηu R ( ) ,
+
t
where η is the friction coefficient, and for Brownian particles in a fluid the Stokes law
gives F = 3πDµu , so that the random driving force R(t) has the following characteristics:
R ( )t = 0 (average of random driving force is zero)
R t • u t =... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
⎝ 2
⎡
⎛
⎢⎝
⎣
d
2
2
2
1/ 2
⎤
⎞
AC = ⎢⎜ r + cosθ⎟ + ⎜ sinθ⎟ ⎥
⎠ ⎥
⎦
⎛ d
⎝ 2
⎡
⎛
⎢⎝
⎣
d
2
⎞
⎠
2
2
1/ 2
,
.
Under the approximation r>>d, we obtain
φ(r,θ) = −
qβ cosθ
4πεor 2
,
where β=Qd is the dipole moment.
Note: (1) The superimposed two fields yield Φ ~ r
two dipoles, the potential becomes Φ = − Cr −... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
.2.3 Electric double layer potential
Bounded
Ions
-
-
-
-
-
-
Liquid
+
+
+
+
+
-
+
+
-
+
Counterions
O
Stern
Layer Double Layer
Electric
x
(Solid surface should not be separated from the negative ions.)
Surfaces immersed in liquids are usually charged due to the ionization or dissociation of
surface... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
equation.
JG
JG
The Maxwell equation determines the displacement D as
∇ ⋅ D = ρnet , D = εE = ε(−∇Ψ ) .
Thus we obtain the Poisson-Boltzmann equation
JG
JG
−εε
0 r
d 2Ψ
2
dx
= ρ =
net ∑ i
Z en exp
oi
i
⎛ Z eψ ⎞
i
.
⎜ −
⎟
κ T
⎝
⎠
B
Finally we obtain the Debye length,
= ∑
1
δ
2 2
Z e n
oi
i
ε ε κ ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
. Two main potentials interactions exist. One is the
van der Waals (usually attractive) and the other is double layer interaction (repulsive). In
2.57 Fall 2004 – Lecture 24
173
the above figure (a), a superposition of the double layer potential (repulsive electro
potential) and van der ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
My conclusion,
based on a simple orders of magnitude analysis, as given in an example in the book, says
not in practical cases. You can analyze this problem by considering how much bonding
force an atom experiences from the wall, and compare that to the shear stress generated in
practical situations.
U: Potential ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
can be approximately estimated from the work of cohesion,
12
1
2
2.57 Fall 2004 – Lecture 24
175
W12 = W11dW22d = 2 γ1dγ 2d .
Thus
γ = γ +γ − 2
1
1 γγ
d 2
12
2
.
d
Surface tension is very important for microsystems. There are two basic equations for... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
increases as the liquid droplet radius decreases.
For a given vapor pressure, smaller droplets tend to evaporate.
2.57 Fall 2004 – Lecture 24
176
MIT OpenCourseWare
http://ocw.mit.edu
2.57 / 2.570 Nano-to-Macro Transport Processes
Spring 2012
For information about citing these mat... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
18.413: ErrorCorrecting Codes Lab
February 10, 2004
Lecturer: Daniel A. Spielman
Lecture 3
3.1 Analysis of repetition code metachannel
When we specialize our interpretation of the output of a channel to the meta channel formed by
encoding using the repitition code and transmitting over another channel, we solve... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2e598567b712a0493fd5a634b3d1de2c_lect3.pdf |
chance 0 and half chance 1). I think of each channel
transmission as an experiment, and I now want to determine the probability that w was 1 given
the results of both experiments.
By the theorem from last class, we have
P [w = 1|y1 = b1 and y2 = b2] =
P [y1 = b1 and y2 = b2|w = 1]
P [y1 = b1 and y2 = b2|w = 1] + ... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2e598567b712a0493fd5a634b3d1de2c_lect3.pdf |
Lecture 3: February 10, 2004
32
Applying P [w = 0 y1 = b1] = 1 − P [w = 1 y1 = b1], we can also comput
|
|
P [y1 = b1 and y2 = b2|w = 0]
(1 − p1)P [y1 = b1] (1 − p2)P [y2 = b2]
P [w = 0]2
.
Combining these equations, and P [w = 0] = P [w = 1] = 1/2, we obtain
(3.1) =
p1p2
p1p2 + (1 − p1)(1 − p2)
.
In particu... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2e598567b712a0493fd5a634b3d1de2c_lect3.pdf |
= 1|(y1, y2) = (1, 1)]
P [w = 1]
�
,
�
�
2(1 − p)2
(1 − p)2 + p2
�
,
�
�
22p
(1 − p)2 + p2
2(1 − p)2
(1 − p)2 + p
2
�
,
�
,
�
22p
(1 − p)2 + p
�
.
2
We now compute I(w; y1, y2) by summing over all events:
I(w; y1, y2) =
�
a,b1,b2
P [w = a, y1 = b1, y2 = b2] i (w = a; y1 = b1, y2 = b2)
= �
(1 − ... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2e598567b712a0493fd5a634b3d1de2c_lect3.pdf |
usually written
Pprior [w = 1] .
In general, when w can take one of many values a1, . . . , am, the prior distribution is the vector of
prior probabilities
�
Pprior [w = a1]
, Pprior
[w = a2] , . . . , Pprior [w = am]
�
.
Our experiments reveal the extrinsic probability of w = 1 given the outcome of the experi... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2e598567b712a0493fd5a634b3d1de2c_lect3.pdf |
the calculation of the previous section, we obtain
Ppost [w = 1|y = b] =
Pext [w = 1|y = b] Pprior [w = 1]
Pext [w = 1|y = b] Pprior [w = 1] + Pext [w = 0 y = b] Pprior [w = 0]
|
.
A useful exercise would be to rederive the probability that w = 1 given y = b assuming that w is
not uniformly distributed, and to ob... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2e598567b712a0493fd5a634b3d1de2c_lect3.pdf |
6.825 Techniques in Artificial Intelligence
First-Order Logic
Lecture 5 • 1
At the end of the last lecture, I talked about doing deduction and propositional logic
in the natural deduction, high-school geometry style, and then I promised you
that we would look at resolution, which is a propositional-logic proof sys... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
3
In first-order logic, variables refer to things in the world and you can quantify over
them. That is, you can talk about all or some of them without having to name them
explicitly.
3
FOL motivation
• Statements that cannot be made in propositional
logic but can be made in FOL.
Lecture 5 • 4
The book has a ni... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
sterilize a jar. Well, it kills
all the bacteria in the jar. Now, you don't want to have to name all the bacteria; to
have to say, bacterium 57 is dead, and bacterium 93 is dead. Each one of those guys
is dead. All the bacteria are dead now. So you'd like to have a way not only to talk
about things in the world, bu... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
spend the first half of the
lecture doing the same thing we did with propositional logic and going over syntax
and semantics, and the second half practicing with the logic and, in particular, with
trying to write down statements in logic
8
FOL syntax
• Term
Lecture 5 • 9
The big difference between propositional... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
symbols, as well.
So another way to make a name for something is to say something like "F(X)". If F
is a function, you can give it a term and then F(X) names something. So, you
might have mother-of(John) or F(F(x)).
These three kinds of terms are our ways to name things in the world.
12
FOL syntax
• Term
• Cons... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
ane, Joan), sister(Mother-of(John), Jane),
its-raining()
• t1=t2
A sentence can also be T1 = T2. We’re going to have one special predicate called
equality. You can say this thing equals that thing, written term, equal-sign, term.
Lecture 5 • 14
14
FOL syntax
• Term
• Constant symbols: Fred, Japan, Bacterium39 ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
ecture 5 • 16
Finally we have closure under the sentential operators that we had before, so you
can do and, or, implies, not, parentheses, like we had before in propositional logic.
All that basic connective structure is still the same, but the things that we can say on
either side have gotten a little bit more com... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
symbol Fred could be that person.
19
FOL Interpretations
• Interpretation I
• U set of objects; domain of discourse; universe
• Maps constant symbols to elements of U
• Maps predicate symbols to relations on U (binary
relation is a set of pairs)
Lecture 5 • 20
The next mapping is from predicate symbols to rela... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
Now we can do the definition of what it means for something to be true, and then
we'll do examples. First we'll talk about terms. Terms name things, but we like
to be fancy so we say a term denotes something, so we can talk about the
denotation of a term, that is, the thing that a term names.
22
Basic FOL Semantic... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
, then given
undefined
I(F)(I(term))
² I P(t1, …, tn)
Lecture 5 • 26
In the context of propositional logic, we looked at the rules of semantics, which told
us how to determine whether a sentence was true in an interpretation. Now, in
first-order logic, we’ll add some semantic rules, for the new kinds of sentence... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
antics
Denotation of terms (naming)
• I(Fred)
• I(x)
• I(F(term))
if Fred is constant, then given
undefined
I(F)(I(term))
² I P(t1, …, tn) iff <I(t1), …, I(tn)>
brother(John, Joe)??
∈
I(P)
• I(John) =
[an element of U]
First, we look up the constant symbol “John” in the interpretation and find that it
na... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
complicated relation.
31
Basic FOL Semantics
Denotation of terms (naming)
• I(Fred)
• I(x)
• I(F(term))
if Fred is constant, then given
undefined
I(F)(I(term))
² I P(t1, …, tn) iff <I(t1), …, I(tn)>
brother(John, Joe)??
∈
I(P)
[an element of U]
[an element of U]
• I(John) =
• I(Joe) =
• I(brother) = {... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
symbol to I. It's kind of like
temporarily binding a variable in a programming language.
34
Semantics of Quantifiers
Extend an interpretation I to bind variable x to
element a
• ² I ∀
I
U: x/a
x.Φ iff ² Ix/a Φ for all a
∈
∈
U
Lecture 5 • 35
Now, how do we evaluate the truth under interpretation I, of the sta... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
• Quantifier applies to formula to right until an
enclosing right parenthesis:
Lecture 5 • 37
It’s hard to understand the precedence of these operators using the usual rules. A
quantifier is understood to apply to everything to its right in the formula,
stopping only when it reaches an enclosing close parenthesis.... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
have four predicates: above, circle, oval, square. The numbers above them
indicate their arity, or the number of arguments they take.
Now these particular predicate names suggest a particular interpretation. The fact
that I used this word, "circle", makes you guess that probably the interpretation
of circle is goin... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
45
Now, what kind of a thing is I(above)? Well, above is a predicate symbol, and the
interpretation of a predicate symbol is a relation, so I(above) is a relation.
Here’s the particular relation we define it to be; it’s a set of pairs, because
above has arity 2.
45
FOL Example Domain
,
,
,
}
• U = {
• Const... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
) = {<
• I(Hat) = {<
>}
>,<
,
>}
,
>,<
>}
>}
,
The Real
World
Lecture 5 • 48
And we’ll say that the hat of the triangle is the square and the hat of the oval is the
circle.
48
FOL Example Domain
,
,
,
}
• U = {
• Constants: Fred
• Preds: above2, circle1, oval1, square1
• Function: Hat
• I(Fred... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
}
• I(circle) = {<
• I(oval) = {< >,< >}
• I(Hat) = {< , >,< , >}
• I(square) = {< >}
>}
• ² I square(Fred)?
• ² I above(Fred, Hat(Fred))?
What about this one? Does the above relation hold true of Fred and the hat of Fred?
Lecture 5 • 51
51
FOL Example
• I(Fred) =
• I(above) = {< , >,< , >}
• I(circle) = ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
FOL Example
• I(Fred) =
• I(above) = {< , >,< , >}
• I(circle) = {<
• I(oval) = {< >,< >}
• I(Hat) = {< , >,< , >}
• I(square) = {< >}
>}
• ² I square(Fred)?
• ² I above(Fred, Hat(Fred))?
• I(Hat(Fred)) =
• ² I above( ,
) ?
Lecture 5 • 54
Now the question is: does the above relation hold of the triangle a... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
above(Fred, Hat(Fred))?
• I(Hat(Fred)) =
• ² I above( ,
) ?
• ² I ∃
x. oval(x)?
Okay. What about this sentence: there exists an x such that oval x. Is there a thing
that is an oval? Yes. So how do we show that carefully?
Lecture 5 • 56
56
FOL Example
• I(Fred) =
• I(above) = {< , >,< , >}
• I(circle) = {<
... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
² I ∀
x.
y. above(x,y) v above(y, x)
∃
Lecture 5 • 58
Here’s a more complicated question in the same domain and interpretation. Is the
sentence: “For all x there exists a y such that either x is above y or y is above x”
true in I?
58
FOL Example: Continued
• I(Fred) =
• I(above) = {< , >,< , >}
• I(circle) = ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
ask whether the sentence “There exists
a y such that either x is above y or y is above x” is true in the new interpretation.
Existentials are easier than universals; we just have to come up with one y that
makes the sentence true. And we can; if we bind y to the square, then that
makes above(y,x) true, which makes ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
= {< , >,< , >}
• I(square) = {< >}
>}
• ² I ∀
y. above(x,y) v above(y, x)
x.
∃
• ² Ix/ ∃
• ² Ix/ , y/ above(x,y) v above(y,x)
y. …
• ² I ∀
x.
∀
• ² Ix/
y. above(x,y) v above(y,x)
above(x,y) v above(y,x)
, y/
Lecture 5 • 62
If it’s going to be true, then it has to be true for every possible instantiation... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
) =
• I(above) = {< , >,< , >}
• I(circle) = {<
• I(oval) = {< >,< >}
• I(Hat) = {< , >,< , >}
• I(square) = {< >}
>}
• ² I ∀
y. above(x,y) v above(y, x)
x.
∃
• ² Ix/ ∃
• ² Ix/ , y/ above(x,y) v above(y,x)
y. …
• ² I ∀
x.
∀
• ² Ix/
y. above(x,y) v above(y,x)
above(x,y) v above(y,x)
, y/
And, therefore... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
similar type. Even in this first batch of problems, you should try to think of
the answer before you go on to see it.
66
Writing FOL
• Cats are mammals
[
cat1, mammal1]
How would you use first-order logic to say “Cats are mammals”? (You can
use a unary predicate “cat” and another unary predicate “mammal”).
Lect... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
or(Jane)
• A nephew is a sibling’s son [nephew2, sibling2, son2]
Lecture 5 • 71
A nephew is a sibling's son. Nephew, sibling, and son are all binary
relations.
I'll start you off and say for all X and Y, X is the nephew of Y if and only if
something. In English, what relationship has to hold between X and Y for X... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
Lecture 5 • 73
When you have relationships that are functional, like mother-of, and
maternal-grandmother-of, then you can write expressions using functions
and equality. So, what’s the logical way of saying that someone’s maternal
grandmother is their mother’s mother? Use mgm, standing for maternal
grandmother, an... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
z. x=mother-of(z) Æ z=mother-of(y)
xy. x=mgm(y)
•
↔ ∃
↔ ∃
• Everybody loves somebody [loves2]
∀
∀
∀
Using a binary predicate loves(x,y), how can you say that everybody loves
somebody?
Lecture 5 • 75
75
Writing FOL
• Cats are mammals
x. cat(x)
[
cat1, mammal1]
mammal(x)
•
→
• Jane is a tall surveyor [tal... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
ew2, sibling2, son2]
z . [sibling(y,z) Æ son(x,z)]]
xy. [nephew(x,y)
•
• A maternal grandmother … [functions: mgm, mother-of]
z. x=mother-of(z) Æ z=mother-of(y)
xy. x=mgm(y)
•
↔ ∃
↔ ∃
• Everybody loves somebody [loves2]
•
•
x.
∃
y.
∀
∀
∃
y. loves(x,y)
x. loves(x,y)
∀
∀
∀
There exists a y such that for al... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
∃
• Everybody has a father
Everybody has a father.
Lecture 5 • 81
81
Writing More FOL
• Nobody loves Jane
loves(x,Jane)
x. loves(x,Jane)
•
•
x.
¬
∀
¬∃
• Everybody has a father
•
x.
∀
y. father(y,x)
∃
Forall x Exists y F(y,x)
Lecture 5 • 82
82
Writing More FOL
• Nobody loves Jane
loves(x,Jane)
x... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
, no, that’s not enforced by the logic. For that matter, they could be the
same as x. Now, if you use the typical definitions of father and mother, they
won’t be the same, but that’s up to the interpretation.
85
Writing More FOL
• Nobody loves Jane
loves(x,Jane)
x. loves(x,Jane)
•
•
x.
¬
∀
¬∃
• Everybody ha... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
x.
∀
∃
yz. father(y,x) Æ mother(z,x)
• Whoever has a father, has a mother
x.[[
•
∀
∃
y. father(y,x)]
[
∃
→
y. mother(y,x)]]
Lecture 5 • 88
And we can describe x’s that have a mother by Exists y. mother (y,x).
Finally, we put these together using implication, just as we did with the “all
cats are mammals” ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
English sentences, write a
corresponding sentence in FOL
1. Somebody loves Jane.
2.
3.
4.
5.
6.
Please do these recitation problems before the next recitation. See you
then!
Lecture 5 • 90
90 | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/2e6d5a261d06ac45abdaa2de545b9205_Lecture5FinalPart1Save.pdf |
Topic 5 Notes
Jeremy Orloff
5 Introduction to harmonic functions
5.1 Introduction
Harmonic functions appear regularly and play a fundamental role in math, physics and engineering.
In this topic we’ll learn the definition, some key properties and their tight connection to complex
analysis. The key connection to 18.04 i... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/2e739bb156efb0bc7103fc43d0897dda_MIT18_04S18_topic5.pdf |
have harmonic pieces
The connection between analytic and harmonic functions is very strong. In many respects it mirrors
the connection between e and sine and cosine.
Let = + and write () = (, ) + (, ).
Theorem 5.2. If () = (, ) + (, ) is analytic on a region then both and are harmonic
functions on .
Proof. This is... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/2e739bb156efb0bc7103fc43d0897dda_MIT18_04S18_topic5.pdf |
Checking the Cauchy-Riemann equations we
[
have
]
[
]
=
− −
Since is harmonic we know = −, so = . It is clear that = −. Thus
satisfies the Cauchy-Riemann equations, so it is analytic.
3. Let be an antiderivative of :
5
INTRODUCTION TO HARMONIC FUNCTIONS
3
Since is simply connected our statement ... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/2e739bb156efb0bc7103fc43d0897dda_MIT18_04S18_topic5.pdf |
ic conjugates
Definition. If and are the real and imaginary parts of an analytic function, then we say and
are harmonic conjugates.
Note. If () = + is analytic then so is () = − + . So, if and are harmonic conjugates
and so are and −.
5.4 A second proof that and are harmonic
This fact is important enough that we w... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/2e739bb156efb0bc7103fc43d0897dda_MIT18_04S18_topic5.pdf |
and differentiation is not a problem.
(cid:240) = and is in the disk (cid:240) −
0
(cid:240) < .
0
5.5 Maximum principle and mean value property
These are similar to the corresponding properties of analytic functions. Indeed, we deduce them
from those corresponding properties.
Theorem. (Mean value property) If is ... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/2e739bb156efb0bc7103fc43d0897dda_MIT18_04S18_topic5.pdf |
ima
by using a minus sign.
5
INTRODUCTION TO HARMONIC FUNCTIONS
5
5.6 Orthogonality of curves
An important property of harmonic conjugates and is that their level curves are orthogonal. We
start by showing their gradients are orthogonal.
Lemma 5.4. Let = + and suppose that () = (, ) + (, ) is analytic. Then th... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/2e739bb156efb0bc7103fc43d0897dda_MIT18_04S18_topic5.pdf |
�() ≠ 0 we know that
= (, ) ≠ 0.
Likewise, ≠ 0. Thus, the gradients are not zero and the level curves must be smooth.
Example 5.5. The figures below show level curves of and for a number of functions. In all cases,
the level curves of are in orange and those of are in blue. For each case we show the level curves
s... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/2e739bb156efb0bc7103fc43d0897dda_MIT18_04S18_topic5.pdf |
At the origin this is not a smooth curve.
Look at the figures for 2 above. It does appear that away from the origin the level curves of
intersect the lines where = 0 at right angles. The same is true for the level curves of and the lines
where = 0. You can see the degeneracy forming at the origin: as the level curves... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/2e739bb156efb0bc7103fc43d0897dda_MIT18_04S18_topic5.pdf |
software
studio
asynchronous calls
Daniel Jackson
1 some history
in the 1990s
› most web sites issued a whole page at a time
› clunky for users, excessive bandwidth
idea
› update page incrementally
› do it asynchronously, so browser doesn’t freeze
in 1999, XMLHttpRequest arrives
› Microsoft invents XHR ide... | https://ocw.mit.edu/courses/6-170-software-studio-spring-2013/2e772463eebb035f1b5d38accee189ed_MIT6_170S13_47-asyn-intro.pdf |
using network inspector
› example from Safari
7 using network inspector
› can see here that request was get
8 encoding data for transit
XML
› parsing built into browser (XHR)
› comes back as DOM: not convenient
JSON
› Javascript object literals
› JQuery uses parser, not eval (why?)
<person>
<firstName>Jo... | https://ocw.mit.edu/courses/6-170-software-studio-spring-2013/2e772463eebb035f1b5d38accee189ed_MIT6_170S13_47-asyn-intro.pdf |
18.409 An Algorithmist’s Toolkit
September 10, 2009
Lecturer: Jonathan Kelner
Scribe: Jesse Geneson (2009)
Lecture 1
1 Overview
The class’s goals, requirements, and policies were introduced, and topics in the class were described. Every
thing in the overview should be in the course syllabus, so please consult th... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/2e812afd941cb290f887e3a0e53e51df_MIT18_409F09_scribe1.pdf |
v1,
corresponding eigenvalues of M as its diagonal entries. So M =
. . . , vn, and Λ is diagonal, with the
�
n
i=1 λivivT
i .
In Proposition 2, it was important that M was symmetric. No results stated there are necessarily true
in the case that M is not symmetric.
Definition 3 We call the span of the eigenvectors ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/2e812afd941cb290f887e3a0e53e51df_MIT18_409F09_scribe1.pdf |
is the degree of the ith vertex.
For unweighted G, the Laplacian matrix is clearly symmetric. An equivalent definition for the Laplacian
matrix is
LG = DG − AG,
where DG is the diagonal matrix with ith diagonal entry equal to the degree of vi, and AG is the adjacency
matrix.
4
Example Laplacians
Consider the gra... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/2e812afd941cb290f887e3a0e53e51df_MIT18_409F09_scribe1.pdf |
(1)
X(3)
The action of LG on v is then
1
LGv = −1
⎛
⎝
0
−1
2 −1
1
0 −1
⎞ ⎛
⎠ ⎝
X(1)
X(2)
X(3)
⎞ ⎛
⎠ = ⎝
X(1) − X(2)
2X(2) − X(1) − X(3)
X(3) − X(2)
⎞ ⎛
⎠ = ⎜
⎝ 2
⎞
� X(1) − X(2)
�
X(2) − [ X(1)+X(3) ⎟
] ⎠
2
X(3) − X(2)
For a general Laplacian, we will have
[LGv]i = [di ∗ (X(i) − average of X ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/2e812afd941cb290f887e3a0e53e51df_MIT18_409F09_scribe1.pdf |
5
Matlab Demonstration
As remarked before, vectors v ∈ Rn may be construed as maps Xv : V → R. Thus each eigenvector assigns a
real number to each vertex in G. A point in the plane is a pair of real numbers, so we can embed a connected
R2 . The following examples generated in Matlab show that
graph into the plane ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/2e812afd941cb290f887e3a0e53e51df_MIT18_409F09_scribe1.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.917 Topics in Algebraic Topology: The Sullivan Conjecture
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Generating Analytic Functors (Lecture 10)
Let Funan denote the category of analytic functors from Vectf to... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2ea8b21ce4498d757a261c7123784662_lecture10.pdf |
Vect. In particular, we have a surjection
in the category Fun. In particular, we can write F as a filtered colimit of subfunctors
FI0 = im(⊕i∈I0 PVi → F ) ⊆ F.
⊕i∈I PVi → F
where I0 ranges over finite subsets of I. Since the collection of good functors is stable under colimits, it
will suffice to show that each FI0 is... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2ea8b21ce4498d757a261c7123784662_lecture10.pdf |
Then there exists an injection
F �→ ⊕i∈I IVi
F �→ ⊕α∈A Symnα
where the set A is finite.
For each i ∈ I, let Fi denote the image of F in IVi . Then Fi is a quotient of F , and therefore a polynomial
functor. Moreover, we have an inclusion F �→ ⊕i∈I Fi. It will therefore suffice to prove Proposition 4 after
replacing ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2ea8b21ce4498d757a261c7123784662_lecture10.pdf |
of F in
(S∞)⊗nj . Then we have a monomorphism F → ⊕j∈J Fj , and it will suffice to prove the result after applying
F by Fj . In other words, we may reformulate Proposition 5 as follows:
Proposition 6. Let F be a polynomial functor, and suppose there exists an injection
Then there exists an injection
for some m ≥ 0. ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2ea8b21ce4498d757a261c7123784662_lecture10.pdf |
If equality holds, then dF � = dF , so that F � = F . Thus every chain of proper subfunctors of F has length at
most dF (0) + . . . + dF (m).
We can now Proposition 6 to the following:
Proposition 8. Let F be a functor of the form (Sk)⊗n, where k ≥ 1 and n ≥ 0. Then there exists a
monomorphism F �→ Symm for some m ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2ea8b21ce4498d757a261c7123784662_lecture10.pdf |
∈ V . We can therefore describe the space Sk+1(V ) as the
⊕1≤d≤k+1V ⊗d by the following relations:
Sym∗(V )
→
quotient of
(1) If σ ∈ Σd is a permutation, then
v1 ⊗ . . . ⊗ vd = vσ(1) ⊗ . . . ⊗ vσ(d)
in Sk+1(V ).
(2) If d < k, and v ∈ V , then
in Sk+1(V ).
v1 ⊗ . . . ⊗ vd ⊗ v = v1 ⊗ . . . ⊗ vd ⊗ v ⊗ v
We define a... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2ea8b21ce4498d757a261c7123784662_lecture10.pdf |
)
cancel if j =�
k, while the terms associated to the sequence (i1, . . . , id, j, j) appear on the left hand side as
associated to the sequence (i1, . . . , id, j + 1). To complete the proof, it will suffice to show that no other
terms appear on the left hand side. In other words, we must show that if 2i1 + . . . + 2... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2ea8b21ce4498d757a261c7123784662_lecture10.pdf |
. Let F and F � be nonzero subfunctors of S∞. Then F ∩ F � = 0
.
Proof. We compute that the endomorphism ring
R = HomFun(IF2 , IF2 ) � IF2 (F2)∨ � F2 ⊕ F2
has dimension 2 over the field F2. The endomorphism ring S∞ is properly contained in R, and therefore has
dimension 1 over F2. It follows that every nonzero endom... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2ea8b21ce4498d757a261c7123784662_lecture10.pdf |
18.413: ErrorCorrecting Codes Lab
March 4, 2004
Lecturer: Daniel A. Spielman
Lecture 9
9.1 Related Reading
• Fan, Chapter 2, Sections 3 and 4.
• Programming Tips #10: Working with GF (2) matrices.
9.2 LDPC Codes
We will examine the simplest and most natural lowdensity paritycheck (LDPC) codes. These are
spe... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2eb3152fd9bef9307e97d393de620d1b_lect9.pdf |
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