text stringlengths 16 3.88k | source stringlengths 60 201 |
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Bass Guitar
Human Voice
Snare Drum
Guitar
Synthesizer
Piano
13.75 Hz-
27.5 Hz
27.5 Hz-
55 Hz
55 Hz-
110 Hz
110 Hz-
220 Hz
220 Hz-
440 Hz
440 Hz-
880 Hz
880 Hz-
1,760 Hz
1,760 Hz-
3,520 Hz
3,520 Hz-
7,040 Hz
7,040 Hz-
14,080 Hz
14,080 Hz-
28,160 Hz
Image by MIT OpenCourseWare.
6.02 Fall 2012
Lecture 14 Slide #12
Co... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/3d71e96b6e7942d74c15fb2f99244985_MIT6_02F12_lec14.pdf |
(ΩΩ)
6.02 Fall 2012
Lecture 14 Slide #15
X(Ω) and x[n]
6.02 Fall 2012
Lecture 14 Slide #16
Fast Fourier Transform (FFT) to compute
samples of the DTFT for
signals of finite duration
P−1
k ) = ∑ x[m]e
− jΩ
km
,
X( Ω
m=0
(P/2)−1
x[n] = 1 ∑ X(Ω
P
k=−P/2
jΩ
kn
k )e
where Ω
k = k(2π/P), P is some integer... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/3d71e96b6e7942d74c15fb2f99244985_MIT6_02F12_lec14.pdf |
exp(jΩ1)(cid:3)
exp(jΩ0)(cid:3)
1
exp(jΩ
2)(cid:3)
–j
exp(jΩ
1)(cid:3)
6.02 Fall 2012
Lecture 14 Slide #18
Spectrum of Digital Transmissions
(scaled version of DTFT samples)
6.02 Fall 2012
6.02 Fall 2012
Lecture 14 Slide #19
Lecture 14 Slide #19
Spectrum of Digital Transmissions
6.02 F... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/3d71e96b6e7942d74c15fb2f99244985_MIT6_02F12_lec14.pdf |
Slide #23
MIT OpenCourseWare
http://ocw.mit.edu
6.02 Introduction to EECS II: Digital Communication Systems
Fall 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/3d71e96b6e7942d74c15fb2f99244985_MIT6_02F12_lec14.pdf |
MIT OpenCourseWare
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18.727 Topics in Algebraic Geometry: Algebraic Surfaces
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
ALGEBRAIC SURFACES, LECTURE 10
LECTURES: ABHINAV KUMAR
Recall that we had left to show that there are no... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/3d867f3ce99fa44cc12a6e39cd48177a_lect10.pdf |
the blowup of X at x,
x ∈ Y , then for D ∈ L a curve singular
˜
˜
then ˜D is a nonsingular rational curve
→ P1 . After
d a fiber of φ = φ ◦ π : X
an
→ P1 is a morphism and
further (at most K 2) blowups, we get an X � s.t. φ� : X �
one fiber of φ� is a smooth, rational curve. So X � is geometrically ruled over P1 ,... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/3d867f3ce99fa44cc12a6e39cd48177a_lect10.pdf |
2
LECTURES: ABHINAV KUMAR
Proof. ∃ a nonsingular elliptic curve D ∈
exact sequence
|−K| by the above lemma. The short
(1) 0 → OX ((n − 1)D) ⊗ TX → OX (nD) ⊗ TX → OX (nD) ⊗ TX ⊗ OD → 0
gives the long exact sequence in cohomology
2
X (nD) ⊗ X ⊗ OD) → H (X, OX ((n − 1)D) ⊗ TX )
→ H 2(D, OX (nD) ⊗ TX ) = 0
H 1(X, O... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/3d867f3ce99fa44cc12a6e39cd48177a_lect10.pdf |
� TX ⊗ OD) → H 1(OX (nD) ⊗ N )
1
D2 > 0, so we get H 1(OX (nD) ⊗ OD) = 0 = H 1(OX (nD) ⊗ N ) as desired, since
OX (nD) ⊗ OD and OX (nD) ⊗ N are sheaves of large positive degree on D for
�
n >> 0.
We now return to the proof of the proposition for p = char(k) > 0. Let A = W (k)
be the Witt vectors of k. A is a comp... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/3d867f3ce99fa44cc12a6e39cd48177a_lect10.pdf |
. Thus, H 1(X �, OX � ) = H 2(X �, OX � ) = 0. See Mumford’s Abelian
Varieties or Chapters on Algebraic Surfaces for more details.
Now, let K �� be an algebraic closure of K � and Ki
of K � inside K ��. Let X �� = X � ×K� k��, Xi = X � ×K� Ki
� the family of finite extensions
�. Let A�� be the integral
AL... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/3d867f3ce99fa44cc12a6e39cd48177a_lect10.pdf |
isomorphism b : Pic Xi
following: for L an
Li|Xi = L, and we set b([L]) = [Li|X ].
invertible OXi -module, ∃ an invertible OU
→
∼
Pic X defined by the
i -module Li s.t.
Proof. Omitted.
�
Proof of theorem: So we get a canonical isomorphism between Pic X and Pic X ��
which takes ωX to ωX �� . Since Pic X = ZωX , we ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/3d867f3ce99fa44cc12a6e39cd48177a_lect10.pdf |
rational, we have q(Y ) = p2(Y ) = 0. f
is separable, so it induces an injective map H 0(X, ω⊗n) H 0(Y, ω⊗n). Thus,
�
p2(X) = 0 and similarly q(X) = 0. By the above, it is rational.
→
→
X
Y
Remark. In Castelnuovo’s theorem, it is not enough to take q = pg = 0: coun
terexamples include the Godeaux surfaces, e.g. a q... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/3d867f3ce99fa44cc12a6e39cd48177a_lect10.pdf |
LX t2
1
⊗
OX T -module L s.t. L|X t1
}
is algebraically equivalent to zero. The Picard functor from schemes
over k to
sets is that which maps a scheme T to the set Pic X (T ) of all T -isomorphism
classes of invertible OX T -modules, where L1, L2 are T -isomorphic if
∃ and in
∼= L2 ⊗ p2
∗ (M ). This is a contrav... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/3d867f3ce99fa44cc12a6e39cd48177a_lect10.pdf |
(idX × f )∗(L
× Pic X called the universal bundle: the map f : T
) on X × T . Similarly for P ic0
→
X . | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/3d867f3ce99fa44cc12a6e39cd48177a_lect10.pdf |
Mechanical Issues
January 4th, 2005
Aaron Sokoloski
Agenda
The Maslab Workshop
Raw Materials
Other Materials
Fasteners
Tools
Safety & Maintenance
Mechanical issues
Motors
Techniques
Design Principles
Other resources
The Maslab Workshop
Goal: Be able to build a
simple robot with the tool... | https://ocw.mit.edu/courses/6-186-mobile-autonomous-systems-laboratory-january-iap-2005/3d97f0b560e89204cc83709e49835a4e_mechanical.pdf |
bulky parts
Cuts easily with hot knife
Also can be sculpted with hot knife for
interesting / irregular shapes
Other materials
Wooden dowels
Hollow metal tubing
Springs
PVC pipe
Foam pipe insulation
Gears
Others…
Fasteners
Bolts and machine screws
sizes from ¼” down
Wood screws
Gl... | https://ocw.mit.edu/courses/6-186-mobile-autonomous-systems-laboratory-january-iap-2005/3d97f0b560e89204cc83709e49835a4e_mechanical.pdf |
turn and tighten gradually
Rotary cutting tool
Quick, but inaccurate
Tools
Mitre saw
More accurate wood cuts, any angle
Use clamps for best result
Drill press
Wood, plastic, metal (carefully)
Clamp small or light pieces
Punch is preferable for sheet metal – if you have to
drill, make sure the p... | https://ocw.mit.edu/courses/6-186-mobile-autonomous-systems-laboratory-january-iap-2005/3d97f0b560e89204cc83709e49835a4e_mechanical.pdf |
: Metal bending
To bend without the brake, make guide cuts using
snips
(and holes along bend line for wide pieces)
This makes it bend where you want it to
Design Principles
• Rule of 3-5
(Saint Venant’s principle)
• Applies to shafts (rotary and
linear motion) wheel hubs,
others
• Anytime something should
move ... | https://ocw.mit.edu/courses/6-186-mobile-autonomous-systems-laboratory-january-iap-2005/3d97f0b560e89204cc83709e49835a4e_mechanical.pdf |
Lecture 3
Fast Fourier Transform
6.046J
Spring 2015
Lecture 3: Divide and Conquer:
Fast Fourier Transform
• Polynomial Operations vs. Representations
• Divide and Conquer Algorithm
• Collapsing Samples / Roots of Unity
• FFT, IFFT, and Polynomial Multiplication
Polynomial operations and representation
A poly... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/3dc84e78914f226665357234d6f4e25c_MIT6_046JS15_lec03.pdf |
for 0 ≤ k ≤ 2(n−1), because the degree
of the resulting polynomial is twice that of A or B. This multiplication is then
equivalent to a convolution of the vectors A and reverse(B). The convolution
is the inner product of all relative shifts, an operation also useful for smoothing
etc. in digital signal processing. ... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/3dc84e78914f226665357234d6f4e25c_MIT6_046JS15_lec03.pdf |
of Algebra. Addition
and multiplication can be computed by adding and multiplying the yi terms,
assuming that the xi’s match. However, evaluation requires interpolation.
The runtimes for the representations and the operations is described in the table
below, with algorithms for the operations versus the representat... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/3dc84e78914f226665357234d6f4e25c_MIT6_046JS15_lec03.pdf |
�
⎥
⎢
⎥
⎢
⎥ ⎢
=
⎥ ⎢
⎢
⎥
⎦ ⎣
⎤
1
y0
⎥
y
⎥
⎥
⎥
y
⎥
2
⎥
⎥
.
.
⎦
.
yn−1
where V is the Vandermonde matrix with entries vjk = xk
j .
Then we can convert between coefficients and samples using the matrix vector
product V · A, which is equivalent to evaluation. This takes O(n2).
Similarly, we can samples to coeffi... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/3dc84e78914f226665357234d6f4e25c_MIT6_046JS15_lec03.pdf |
, a3, a5, . . .
k
)
3
Lecture 3
Fast Fourier Transform
6.046J
Spring 2015
2. Recursively conquer Aeven(y) for y ∈ X 2 and Aodd(y) for y ∈ X 2, where X 2 =
{x2 | x ∈ X}.
3. Combine the terms. A(x) = Aeven(x... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/3dc84e78914f226665357234d6f4e25c_MIT6_046JS15_lec03.pdf |
the complex plane (including 1). These points are of the
form (cos θ, sin θ) = cos θ +i sin θ = eiθ by Euler’s Formula, for θ = 0, 1 τ, 2 τ, . . . , n−1 τ
(where τ = 2π).
n
n
n
The nth roots of unity where n = 2£ form a collapsing set, because (e
iθ)2
=
n
. Therefore the even nth roots of unity are equivalent t... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/3dc84e78914f226665357234d6f4e25c_MIT6_046JS15_lec03.pdf |
implementation of FFT is FFTW, which was described by Frigo and
Johnson at MIT. The algorithm is often implemented directly in hardware, for fixed
n.
Inverse Discrete Fourier Transform
The Inverse Discrete Fourier Transform is an algorithm to return the coefficients
of a polynomial from the multiplied samples. The tr... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/3dc84e78914f226665357234d6f4e25c_MIT6_046JS15_lec03.pdf |
because V · V¯ = nI.
This claim says that the Inverse Discrete Fourier Transform is equivalent to the
Discrete Fourier Transform, but changing xk from e
to its complex conjugate
e−ikτ /n, and dividing the resulting vector by n. The algorithm for IFFT is analogous
to that for FFT, and the result is an O(n lg n) algo... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/3dc84e78914f226665357234d6f4e25c_MIT6_046JS15_lec03.pdf |
, etc.
7
MIT OpenCourseWare
http://ocw.mit.edu
6.046J / 18.410J Design and Analysis of Algorithms
Spring 2015
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2015/3dc84e78914f226665357234d6f4e25c_MIT6_046JS15_lec03.pdf |
Lecture 1
The Hamiltonian approach to classical mechanics.
Analysis of a simple oscillator.
Program:
1. Hamiltonian approach to classical mechanics.
2. Vibrations of an electron in a lattice: simple oscillator
Questions you should be able to answer by the end of today’s lecture:
1. How to construct a Hamiltonian... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3dc854a84076e1a55e576a95413f1744_MIT3_024S13_2012lec1.pdf |
Why is this function important? – Let’s take a look how the Hamiltonian changes with respect to
momentum and position:
v
dx
dt
H
p
p m
H
x
dV x
dx
F x
m
d 2
x
dt 2
d
dt
m
dx
dt
dp
dt
The resulting pair of equations is referred to as Hamilton’s equations, ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3dc854a84076e1a55e576a95413f1744_MIT3_024S13_2012lec1.pdf |
to our initial example.
Example I: Particle falling under gravity:
The Hamiltonian in this case is:
H x, p
p2
2m
V x
2
p
2m
mgx
Then the Hamilton’s equations are:
H dx
p
H
x
dx
p
dt m
dp
dt
p
dt m
dp
dt
mg
mg
d 2 x 1 dp
dt ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3dc854a84076e1a55e576a95413f1744_MIT3_024S13_2012lec1.pdf |
simplistic case we can model this system as a ball (electron) on a spring attached to a wall
(massive ion).
I. The system:
Electron = ball of mass m is bound by a
stretchable spring of length l and stiffness K.
II. The Hamiltonian:
The energy for this system is:
x l2
2
p2 K
2m 2
mv2
2
K
E
2
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3dc854a84076e1a55e576a95413f1744_MIT3_024S13_2012lec1.pdf |
� x l 0,
K
m
The last equation is simply an equation for a simple harmonic oscillator. The solution for this
equation is:
x l Aeit Beit
We can find coefficients A and B from the initial conditions.
Let’s say at time t = 0, we have stretched the spring to the length of 2l and released with initial... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3dc854a84076e1a55e576a95413f1744_MIT3_024S13_2012lec1.pdf |
Key Concepts for section IV (Electrokinetics and Forces)
1: Debye layer, Zeta potential, Electrokinetics
2: Electrophoresis, Electroosmosis
3: Dielectrophoresis
4: Inter-Debye layer force, Van-Der Waals force
5: Coupled systems, Scaling, Dimensionless Numbers
Goals of Part IV:
(1) Understand electrokinetic phenomena... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3de45552ba2bb507838ab3c714017a90_electrokin_lec5.pdf |
i
E0
+
+
+
+
+
+
+
-
-
-
-
-
-
-
E0
-
-
-
-
-
-
+
+
+
+
+
+
-
+
Positive DEP
)σ σ>
(
0
Negative DEP
)σ σ<
(
0
Positive / negative DEP, etc.
pr
pr
+
-
-
-
+
+
+
+
+
-
-
-
-
+
+
+
-
-
-
+
r
E
Particle moves toward
the high field region.
r
E
Particle moves away from
the high field region.
DEP force is independent of th... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3de45552ba2bb507838ab3c714017a90_electrokin_lec5.pdf |
ves.
at Membrane Surfaces through Colloid Phase Transitions." (cid:10)
"Detection of Molecular Interactions
Nature 427 (January 8, 2004): 139-141.
Measurement of W
in different salt concentration
(inter-bilayer distance)
140
Hydrophobic tail
Polar head
(phosphate for the
lecithin layers
used above)
W
100
W (A)
... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3de45552ba2bb507838ab3c714017a90_electrokin_lec5.pdf |
)
Figure by MIT OCW.
Electrostatic interaction within electrolyte solution
Interactions and forces in micro / nanoscale
+
+
+
-
-
-
-
+
+
--
+
-
-
+
-
-
-
-
+
- -
+
--
+
+
1κ−
h
+
+
+
-
-
-
-
+
+
--
+
-
-
- -
+
--
+
+
-
-
-
-
+
+
weak or no interaction
+
+
+
-
-
-
-
+
+
--
+
-
-
- -
+
--
+
+
-
-
-
-
+
+
+
+
+
-
-
-
-
... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/3de45552ba2bb507838ab3c714017a90_electrokin_lec5.pdf |
Probabilistic models,
random variables
6.011, Spring 2018
Lec 12
1
Sample space and events
Sample space °
A specific
outcome c
Collection of
outcomes (event)
2 Random variable
°
c
Real line
X(c)
3 Joint pdf
© The MathWorks, Inc. All rights reserved. This content is excluded from our Creative Commo... | https://ocw.mit.edu/courses/6-011-signals-systems-and-inference-spring-2018/3e0a5c25cd7b5e1e0b1a328e13c82025_MIT6_011S18lec12.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.641 Electromagnetic Fields, Forces, and Motion
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 3: Electroquasistatic and Magnetoquasist... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/3e1182a94ee3f63c9407789f288af2c6_MIT6_641s09_lec03.pdf |
0
K 2 b π + π b2 dσsu = 0 ⇒ K = −
dt
r
r
b dσsu = −
2 dt
b ε
2
dE
0
dt
(cid:118)∫ i
H ds =
C
∂ ( E) i da ⇒ H 2 r
∫ ∂t
φ π = π 2
r ε
ε
S
dE
0
dt
⇒ H
φ =
r dE
0
ε
dt
2
(cid:118)∫ i
E ds = − µ
∫ ∂
H
t
∂
C
S
i da
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
Courtesy of H... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/3e1182a94ee3f63c9407789f288af2c6_MIT6_641s09_lec03.pdf |
λ = c ⇒ ω =
2ε 2
2 c ω µ b
4
π
λ
⇒
=
2
π
λ2
2
b (cid:19) 1 ⇒ b (cid:19)
λ
π
f=1 MHz in free space ⇒ λ =
×
3 10 8
6
10
= 300 m
If b (cid:19) 100 m
EQS approximation is valid.
II. Conditions for Magnetoquasistatic Fields
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
6.641, El... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/3e1182a94ee3f63c9407789f288af2c6_MIT6_641s09_lec03.pdf |
E1n
) dS = σsdS
S
S
ε (E - E ) = σ ⇒ n i ε (E - E ) = σ
1n
2n
2
0
s
s
⎤
1 ⎦
⎡
⎣ 0
2. Continuity of Tangential E
(cid:118)∫ E i ds = (E1t - E2t ) dl = 0 ⇒ E1t - E2t = 0
C
n× E 1 - E2 = 0
)
(
Equivalent to Φ1 = Φ2 along boundary
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 3
... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/3e1182a94ee3f63c9407789f288af2c6_MIT6_641s09_lec03.pdf |
Zahn
Lecture 3
Page 6 of 12
6. Electric Field from a Sheet of Surface Charge
a. Electric Field from a Line Charge
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 3
Page 7 of 12
dE =
r
dq
4πε0 (r + z )
2
2
cos θ =
λ0rdz
4πε0 (r2 + z2 )
3
2
E =
r
+∞
∫
z = −∞
dE
... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/3e1182a94ee3f63c9407789f288af2c6_MIT6_641s09_lec03.pdf |
.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 3
Page 9 of 12
⎧ σ0
⎪
⎪2ε0
= ⎨
σ0
⎪−
⎪ 2ε
⎩
0
y > 0
y < 0
Checking Boundary condition at y=0
E yy
( = 0 ) − E (y = 0 ) =
y
+
−
σ0
ε0
σ0
2ε0
σ0 ⎞
⎛
⎟ =
− −⎜
⎟
⎜
⎝ 2ε0 ⎠
σ0
ε0
c. Two sheets of Surface Charge (Capacitor)
E
1... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/3e1182a94ee3f63c9407789f288af2c6_MIT6_641s09_lec03.pdf |
2
2
)1
(− sin φ)
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 3
Page 11 of 12
= −
K dx y
0
2π x2 + y2
H = −
x
K0
2π
y
+∞
dx
∫ x2 + y2
x = −∞
= − 0
K y 1
2π y
tan −1
+∞
x
y
x = −∞
= ⎨
⎧ K
⎪− 0
⎪ 2
K0
⎪+
⎪⎩ 2
y > 0
y < 0
Check boundary condition at y=0:
Hx (y = 0+... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/3e1182a94ee3f63c9407789f288af2c6_MIT6_641s09_lec03.pdf |
MIT Quantum Theory Notes
Supplementary Notes for MIT’s Quantum
Theory Sequence
(cid:13)c R. L. Jaffe
1992-2006
February 8, 2007
Canonical Quantization and Application to
the Quantum Mechanics of a Charged
Particle in a Magnetic Field
(cid:13)c R. L. Jaffe MIT Quantum Theory Notes
Contents
1 Introduction
2
3
2 Canonical ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
. . 17
. . . . . . . . . 17
3.4.1 The location and size of Landau levels
3.4.2 A more careful look at translation invariance . . . . . . 19
4 The Aharonov Bohm Effect
23
5 Integer Quantum Hall Effect
28
5.1 The ordinary Hall effect and the relevant variables . . . . . . . 28
5.2 Electrons in crossed electric and magnetic ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
”
i.e. Hamiltonian, form of classical mechanics. Though it is very useful and
quite powerful, it is important to remember that it provides only the first
guess at the quantum formulation. The only way to figure out the complete
quantum mechanical description of a system is through experiment. Also,
recall from 8.05 that ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
8
2 Canonical Quantization
2.1 The canonical method
There is a haunting similarity between the equations of motion for operators
in the Heisenberg picture and the classical Hamilton equations of motion in
Poisson bracket form.
First let’s summarize the quantum equations of motion. Consider a sys-
tem with N degrees of ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
very different in
quantum mechanics than in classical mechanics: operator matrix elements
between states are the observables, and the states cannot have sharp values of
both x and p. Nevertheless (4) are identical in form to Hamilton’s equations
and the similarity has useful consequences.
In fact, it is the form of (2) ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
�xj
x˙ j +
∂
A
∂pj
p˙j
(cid:27)
j=1
X
(cid:26)
= {A, H}PB
(7)
where the second line follows from the first by substituting from (6) for x˙
and p˙.
Finally, to complete the analogy, note that the Poisson Brackets of the
x’s and the p’s themselves are remarkably simple,
{xj, xk} = 0
{pj, pk} = 0
{xj, pk} = δjk
(8)
because... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
Theory Notes
7
of) the simple steps necessary to find the quantum equivalent of a classical
Hamiltonian system —
• Set up the classical Hamiltonian dynamics in terms of canonical coor-
dinates {xj} and momenta {pj}, with a Hamiltonian H.
• Write the equations of motion in Poisson Bracket form.
• Reinterpret the classica... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
= −gyˆ. Now there is a potential
energy V (s) = mgy(s). The canonical operators are still sˆ and pˆ, but now
the Hamiltonian is H = pˆ2/2m + mgy(sˆ).
(cid:13)c R. L. Jaffe MIT Quantum Theory Notes
8
This is actually so oversimplified a problem that interesting physics has
been lost. A real bead is held on a real wire by... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
r2)
2
~
The radius of curvature at some point X(s) is the radius of a circular disk that is
~
adjusted to best approximate the wire at the point X.
3This problem has attracted attention recently in connection with the propagation of
electrons in “quantum wires”. If you’re interested I can supply references.
(12)
(cid:... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
a derivation of quantum mechanics
from classical mechanics. The substitution (9) cannot be motivated within
It represents a guess, or a leap of the imagination,
classical mechanics.
forced on us by the bizarre phenomena that were observed by the early
atomic physicists and that were inexplicable within the confines of c... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
ures from classical dynamics.
2.3.2 Operator ordering ambiguities
Classical dynamical variables commute with one another, so the order in
which they are written does not affect the dynamics. Not so in quantum
2 ˙2
mechanics. Suppose, our Lagrangian was m ξ ξ
2 . Then the classical Hamil-
R
tonian would have been H = 1 p... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
familiar example is spherical polar coordinates (r, θ, and φ). The origin,
r = 0, is a singular point for spherical polar coordinates — for example, θ
and φ are not defined at r = 0. If you follow the canonical formalism through
from Lagrangian to canonical momenta (pr, pθ, and pφ), to Hamiltonian, to
canonical commutat... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
the associated Hamilton’s equation,
∂H/∂qk = 0 is not an equation of motion. Instead it is a constraint that
must be satisfied by the canonical coordinates and momenta at each time.
The constraint may not be consistent with canonical commutation relations.
A simple, but not particularly interesting example, would be the... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
, constant in magnitude, direction, and time
(cid:13)c R. L. Jaffe MIT Quantum Theory Notes
13
fills a region of space. For definiteness we assume B0 points in the eˆ3 direction.
All points in the xy-plane are equivalent — a simple example of translation
invariance. The classical motion of a charged particle in a constan... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
this symmetry, however, because
the vector potential depends on x and y in an asymmetric manner. In the
end the physics must be translation invariant but it will take some work to
demonstrate this.
(cid:13)c R. L. Jaffe MIT Quantum Theory Notes
14
3.2 Lagrangian, Hamiltonian and Canonical Quantiza-
tion
A velocity-depe... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
˙
Next form the Hamiltonian, H(~x, p~) = ~x · ~p − L, with the result,
˙
H
1
˙
= m~x2
2
=
1
2m
e
(p~ − ~A)2.
c
(21)
Notice that the energy is merely 1
2 mv2 — since the magnetic field acts at
right angles to the particle’s velocity it does not contribute to the energy.
However the Hamiltonian must be regarded as a funct... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
0
/2mc, and L = x1p2 − x2p1 is the angular momentum
2
in the x1 − x2 plane. So the system looks like a particle in a two-dimensional
harmonic oscillator with an additional potential −ωL3. Having solved the
harmonic oscillator before, we can easily construct the energy eigenstates for
this problem.
2) − ωL3
(24)
3
3.3 A... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
with the usual commutation relations,
= δjk,
aj, a†k
h
[aj, ak] = [a , a ] = [a , a ] = 0
± ±
∓ ±
i
= 1,
a , a†
± ±
h
i
Substituting into H and L3 we obtain
L3 = ~
(cid:16)
H = ~ω
a†+a+ − a† a
,
− −
a†+a+ + a† a + 1
− ~ω
a†+a+ − a† a
− −
(cid:16)
= ~ωL
a† a +
− −
(cid:18)
(cid:17)
(cid:16)
,
(cid:17)
− −
1
2
(cid:19)
(... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
degenerate.
−
• For energy (n + 1
2)~ωL the tower of degenerate states begins at angular
momentum −n and grows in steps of ~ to infinity.
−
−
3.4 Physical Interpretation of Landau Levels
The results of the last section are as puzzling as they are enlightening. Clas-
sically a particle moving in the x1-x2y plane under th... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
to show
that
−
−
hx1i = `0 Re α
hx2i = `0 Im α,
(31)
p
where `0 =
(x1 − ix2)/`0 we can center a state with energy E0 = 1 ~
~/mω, so by choosing the real and imaginary parts of α =
2 ωL wherever we wish.
Momentum and velocity are not directly proportional in this problem.
The state |αi provides a graphic example: The qu... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
ΦL(n ) ≡ πr2B
and substitute for ωL,
L. Now the quantum theory
2 . This becomes more transparent if
0,
(cid:1)
hc
1
ΦL(n ) = (n + )
e
2
It appears that the flux through the particle’s orbit comes in units of a fun-
damental quantum unit of flux , Φ0 ≡ hc/e! In the next section, we shall see
that something very close to t... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
Thus the symmetry associated with the constants of the motion
(cid:13)c R. L. Jaffe MIT Quantum Theory Notes
20
(and the degeneracy of the Landau levels) is translation invariance. This is
a welcome result. However, we cannot simply diagonalize P1 and P2 along
with H because they do not commute with each other ,
[P1, P... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
2
2
1
2ib
1b2mω/
~
2ib
1b2mω/
~
T2(b2),
or
T1(b1)T2(b2).
(42)
(43)
(cid:13)c R. L. Jaffe MIT Quantum Theory Notes
21
So the finite translations fail to commute only by virtue of this multiplicative
factor of unit magnitude. If (and only if) we choose the parameters b1 and
b2 so that the phase is a multiple of 2π then th... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
1, so they represent the smallest trans-
lation that commutes with the Hamiltonian,
~
−
|~b, 0, 0i ≡ T1(b1)T2(b2)|0, 0i.
(46)
This state is normalized to unity because the operators T are unitary. It has
~= b. What does it look like in terms of the original
energy E = 1~
basis |n+, n i? To answer this we must express t... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
throughout the plane.
Although these states are not orthonormal, they they give us a qualitatively
correct picture of the solutions of the Landau problem as towers of nearly
2)~ωL situated in unit cells on a
localized energy eigenstates with E = (n + 1
grid labeled by any pair of distances b1 and b2 satisfying (45).
−
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
MIT Quantum Theory Notes
23
just like a plane wave, exp i(k1x1 + k2x2), would behave. The difference, of
course is that ψ has this simple behavior only for the special translations we
have discovered, not for an arbitrary translation, and as a consequence, the
“momenta” (k1, k2) are not conserved.
States like these aris... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
be regarded as merely useful, but inessential, abstractions.
~
~
~
~
In the quantum theory H rather than mx¨ is fundamental, so the possibil-
ity exists that physics depends on A. For the case we have studied in detail
— motion in a constant magnetic field — A = 1
2~x × B so we cannot even
~
define dependence on A indepe... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
, B = ∇ × A, give
~ ~
~
~
~ · ~
dl A =
d2S eˆ3 · × ~A
∇~
C
I
=
Z Z
Z Z
d2S eˆ3 · B = πr2B0,
~
(53)
~
so A cannot vanish everywhere on the circle C. In fact symmetry requires
that A point in the azimuthal, φ, direction, so an elementary calculation
gives,
ˆ
~
~
A =
ˆ
φ for
r > R
(54)
Φ
2πr
Of course the existence of a v... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
Now factor the phase g out of the wavefunction,
ψ(~r, t) = exp(ig) χ(~r, t),
(58)
(55)
(56)
(57)
(cid:13)c R. L. Jaffe MIT Quantum Theory Notes
26
and substitute into eq. (55). The result is that χ obeys the free Schr¨odinger
equation,
− ∇~ 2χ = i~χ˙ .
~2
2m
(59)
Thus all information about the vector potential is conta... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
l · ~
d A
χ1(~r, t2) + exp
ie
~c
(cid:20)
C1
Z
(cid:21) (cid:26)
ie
~
c C¯
I
(cid:20)
~dl · ~A
(cid:21)
χ2(~r, t2)
.
(cid:27)
(61)
Note that the relative phase is given by the loop integral over the closed path
C¯ = C2 − C1.
The relative phase in eq. (61) is measureable, for example by watching
the interference pattern... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
with
¯
), where C 0 = C20 − C1
g ¯
a
~
~
0
~A0
~= A − ∇~ Λ,
then
g C¯
(
) → g0(
C¯
) = g ¯(C) −
e
~c
IC¯
~
dl · ∇~
¯
Λ = g(C)
(63)
because the integral of the gradient of any continuous function around a
closed path is zero.
So Aharonov and Bohm have shown in this simple example, that the vec-
tor potential has physica... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
strip carrying a current along its length develops a current across its
width when placed in a magnetic field. The direction of the induced current
is sensitive to the sign of the electric charge of mobile species in the material
and can be used to show that conventional currents are carried by electrons
(negative charg... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
. A strip of conductor lies in the
xy-plane. A constant and uniform electric field, E, points in the y-direction.
~
A constant and uniform magnetic field, B, is oriented normal to the xy-plane.
First consider the case where B = 0. Mobile charge carriers7 with charge
~
~
7Electrons, for our case, but to keep track of sign... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
qE + q × B.
~
~v
c
(66)
The current comes from the charge carriers drift in response to Eeff and is
therefore given by ~j = σ0 ~Eeff. Also, ~v = ~j/nq, so (66) can be rewritten as
~
~
~
j = σ E
0 +
σ
0
nqc
~
~j × B.
(67)
The current is no longer only parallel to E: Because of the second term in
(67) it develops a compone... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
Hall effect system
In this section we ignore all the complexities of a physical conductor —
electron ion interactions, thermal effects, impurities, and so forth — and
(cid:13)c R. L. Jaffe MIT Quantum Theory Notes
31
~
~
consider the idealized problem of a gas of electrons moving in the xy-plane
subject to an electric fie... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
2
∂ξ
(cid:20)
∂2
∂η
− 2 + 2αη
,
(cid:21)
ξ = x/`0
η = y/`0
α = eE0`0/~ωL.
(71)
(72)
ξ and η are scaled coordinates and α measures the energy scale of electric
relative to magnetic effects.
We now take (ξ, η, pξ, pη) to be our canonical variables, so
H
=
L
~ω
2
(pξ − η)2 + p2
η + 2αη ,
(73)
where pξ = −i∂/∂ξ and pη = −i∂... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
associated wavefunction, ϕn(η, k) is a gaussian (multiplying a Hermite
polynomial) strongly localized around η = k − α,
ϕn(η, k) = exp − |η − k + α|2Hn(η − k + α).
1
2
(76)
The continuous variable k labels the degeneracy of the Landau levels if E = 0.
When E = 0, the highly degenerate Landau level spreads out into a ba... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
-range given by (77) —
∆k =
L
`0N
.
If we substitute N = Φ/Φ0, we obtain,
∆k =
2π`
W
0
.
The allowed values of k can therefore be labelled by an integer, p,
kp =
2π`0
W
p + ζ
(78)
(79)
(80)
0. The constant ζ cannot be determined by
0 ≤ p ≤ LW/4π`2
for −LW/4π`2
what we have done so far.
~
To summarize: For E = 0 we have... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
�, η) =
`
0
W
r
eikξ exp − |η − k + α|2Hn(η − k + α)
1
2
34
(82)
where we have introduced a factor
is normalized to unity over the rectangle of area LW .
p
exp(−η2/2)Hn(η) is normalized in η.).
`0/W so that the wavefunction
(We assume
• The eigenenergies are
E (k) = ~ω (n + ) + ~
L
n
1
2
ωL(αk − ).
α2
2
• The quantum n... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
e
mc
~
Aψ∗ψ
(85)
(cid:13)c R. L. Jaffe MIT Quantum Theory Notes
35
In the gauge we are using, Ay = 0. Given this, and given the fact that the
harmonic oscillator wavefunctions, Hn, are real, we conclude that there is no
current in the y (eˆ2) direction. This is remarkable since eˆ2 is the direction
of the external elec... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
of
the sample (L and W) and n — the label of the Landau band. If N Landau
bands are filled then the conductance is −Ne2/h.
In the matrix notation
of the previous section, we have found that the idealized quantum problem
leads to a purely off-diagonal conductance matrix
σ =
0
+ N e2
h
− N e2
h ,
0
(89)
The resistance matr... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
effects stand out: (1) the Hall conductance (1/RH) is
quantized in integer multiples of e2/h over ranges of magnetic field strength.
The plateaus in 1/RH are very flat and separated by steep increases as 1/RH
grows from one integer value to another. (2) The longitudinal resistance,
on the other hand, vanishes when RH is c... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
essentially) a continuum. For B0 = 0 the continuum breaks
up into Landau levels separated by gaps. Each Landau level can hold one
electron in an area corresponding to a single flux quantum. That area is
given by hc/eB0. The number of electrons that can be accomodated in a
6
(cid:13)c R. L. Jaffe MIT Quantum Theory Not... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
not
conduct current. A picture of the energy levels in a realistic two-dimensional
electron gas is shown in Fig. 5: localized states in between bands (in this case
Landau levels) of delocalized states. The striking appearance of the integer
quantum Hall effect can be understood by considering the sequential filling
of lo... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
lling the situation remains unchanged — the
localized electrons do not participate. When a Landau level again begins to
fill, scattering again generates a longitudinal resistance.
We have only scratched the surface of the myriad of phenomena asso-
ciated with the behavior of materials in the presence of magnetic fields.
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.438 Algorithms for Inference
Fall 2014
4 Factor graphs and Comparing Graphical Model
Types
We now introduce a third type of graphical model. Beforehand, let us summarize
some key perspectives on our first two.
Fir... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/3e3e9934d12e3537b4e9b46b53cd5bf1_MIT6_438F14_Lec4.pdf |
are expressed by the removal of edges, and which similarly reduces the complexity of
inference.
4.1 Factor graphs
Factor graphs are capable of capturing structure that the traditional directed and
undirected graphical models above are not capable of capturing. A factor graph
consists of a vector of random variable... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/3e3e9934d12e3537b4e9b46b53cd5bf1_MIT6_438F14_Lec4.pdf |
x2 ≤
x3 ≤
x4 ≤
and finally we need to decrease spending by 1, so x1 + x2 + x3 + x4 ≥
1. If we were
interested in picking uniformly among the assignments that satisfy the constraints,
we could encode this distribution conveniently with a factor graph in Figure 2.
The resulting distribution is given by
px(x)
∝
1x1≤... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/3e3e9934d12e3537b4e9b46b53cd5bf1_MIT6_438F14_Lec4.pdf |
irected graph (See Problem Set 2).
4.2.2 Converting Factor Graphs to Directed Models
Take a topological ordering of the nodes say x1, . . . , xn. For each node in turn, find
a minimal set U
U is satisfied
such that xi ⊥⊥ {
and set xi’s parents to be the nodes in U . This amounts to reducing in turn each
p(xi
x1, .... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/3e3e9934d12e3537b4e9b46b53cd5bf1_MIT6_438F14_Lec4.pdf |
Circuits Maximus v1.6
Documentation
MIT 6.01 Introduction to EECS I
Fall 2011
Contents
1 Getting Started
1.1 What Is CMax? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Running CMax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
. . . . . . . . . . . . . . . .
2.3 File Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Circuit Simulator
3.1 Common Simulation Error Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Keyboard Shortcuts Reference
2
2
2... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
1.6; if you do not see a window, or you see a different version number, please make sure you have
opened CMax using the instructions above.
CMax consists of two main components, the circuit editor and the circuit simulator (described in sections 2 and 3,
respectively).
2
2 Circuit Editor
CMax’s main window consis... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
ors
When a new resistor is added to the protoboard, its value is determined by the value of the prototype resistor (2).
There are two ways to change the value of the prototype resistor, both of which will immediately be reflected in the
prototype resistor’s button:
1. Clicking on one of the colored bands of the prot... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
just like using a multimeter to measure
voltages in a physical circuit. Clicking on the voltage probe button will place + and - probes on the lower rails.
Clicking on the voltage probe button with Shift held down will place + and - probes on the upper rails.
2.2 Modifying a Circuit
Elements can be edited once they ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
from the menu (keyboard shortcut
Ctrl-Shift-i); the currently-loaded simulation file is shown in the status bar at the bottom of the window. You
will never need to write your own simulation files; we will always provide them.
→
In order to get meaningful output from the simulator, your circuit must contain some means ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
You know that there should be something else plugged
into the bottom section of column 41, in this case.
• Could not solve for voltage at node j42
This usually occurs when you have a probe placed at node j42, but nothing else is connected there (there is
no voltage to measure).
•
Element not in valid holes
This ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.014 Calculus with Theory
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-014-calculus-with-theory-fall-2010/3e9a6928277e7349ac5a8029711dc870_MIT18_014F10_ChFnotes.pdf |
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