text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
of n bit nodes and a set of n/2
check nodes, in which each bit node has degree 3 and each check node has degree 6. To choose a
random graph with these parameters, consider the sockets on each side, where a socket is a place
where an edge is attached. So, each bit node has 3 sockets and each check node has 6. You can... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2eb3152fd9bef9307e97d393de620d1b_lect9.pdf |
node, it has an extra edge with just one endpoint
• Each bit node is called an equallity constraint (which corresponds to a repetition code)
• Each check node is called a parity constraint
• There is a variable on each edge.
At the beginning of the algorithm, each of the interal edges (those with two endpoints) is ... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2eb3152fd9bef9307e97d393de620d1b_lect9.pdf |
the phases alternate, the extrinsic probabilities output by one phase become the intrinsic prob
abilities input to the other.
While the treatment of intrinsic probabilities is identical to that of prior probabilities, we call them
intrisic instead of prior because they might not be the actual priors.
By the rule ab... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2eb3152fd9bef9307e97d393de620d1b_lect9.pdf |
that it is one.
Lecture 9: March 4, 2004
93
9.5 SNR, dB
The standard way to report the standard deviation of the Gaussian channel is through the signal
tonoise ratio (SNR). Symbolically, this is written Eb/N0, and if we are using ±1 signalling, it is
defined to be
Eb
N0
where R is the rate of the code you are... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/2eb3152fd9bef9307e97d393de620d1b_lect9.pdf |
Design of an ESD
Design of an ESD
Core Methodology
Core Methodology
Subject
Subject
February 7, 2007
Dick Larson, Dan Frey, with Roy Welsch
Engineering Systems: At the intersection
of
Engineering, Management & Social Sciences
Management
Social
Sciences
ESD
Engineering
2
For the new Methods
subject
We want to ... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2ebe106407339f82b451d497f44c9098_lec1_intro.pdf |
byproduct would be continued ‘class bonding’ of the first year
doctoral students, the primary focus is on intellectual content.
6
This is a
Knowledge
Requirement
> For students whose academic plan is to take MIT
subjects that go much deeper than this subject (in
statistics, probability, quantitative research meth... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2ebe106407339f82b451d497f44c9098_lec1_intro.pdf |
.
> If they want a true stats course, that would follow this course.
11
Want students to be able to work with ‘blank
sheets of paper.’
They know fundamentals and can derive results.
They are not just users of computer routines.
12
Go Deep,
Use all Available
Subjects
> We cannot think that ESD is so unique that no... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2ebe106407339f82b451d497f44c9098_lec1_intro.pdf |
$36,000
<19%
$30,000
Harvest
$34,200
$34,200
Figure by MIT OCW. After example by Akinc.
15
Linkages to ‘ilities…”
> Reliability
– Measures of..
– Systems designs with
redundancy
> Robustness
> Predictability
> Stability
0,n+1
0
0,1
1,0
2,0
0,n
0,2
0,3
3,0
n,0
1
2
3
:
n
1n+1
2n+1
3,n+1
4,n+1
n+1
n+1,0
Figure by MIT ... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2ebe106407339f82b451d497f44c9098_lec1_intro.pdf |
19 9 4
Figure by MIT OCW.
18
In a 162 game season,
we should not be surprised to see
> At least one 7 game loosing streak. :(
> At least one 7 game winning streak. :)
> All within the null hypothesis that each game is an
independent fair coin flip.
> But imagine the press coverage of these two events.
> Generalize t... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2ebe106407339f82b451d497f44c9098_lec1_intro.pdf |
Lecture 5
8.321 Quantum Theory I, Fall 2017
22
Lecture 5 (Sep. 20, 2017)
5.1 The Position Operator
In the last class, we talked about operators with a continuous spectrum. A prime example is the
position operator. Let’s first consider a particle in d = 1. We define x as the position operator,
with corresponding eigenstat... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
postulate from earlier in the course. After measurement,
the state of the particle will be such that a second measurement of x will yield a value between
x(cid:48) − ∆
2 and x(cid:48) + ∆ . Mathematically, this means that an initial state
2
ˆ
is sent by the measurement to a state
|ψ(cid:105) =
|ψ(cid:105) → |ψ (cid:105... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
exists is
(5.7)
ˆ
Prob(particle is somewhere) =
∞
dx(cid:48) |(cid:104)x(cid:48)|ψ(cid:105)|2 = 1 ,
which is true if (cid:104)ψ|ψ(cid:105) = 1. The position-space wavefunction is defined as
−∞
(cid:104)x(cid:48)|ψ(cid:105) := ψ(x(cid:48)) .
(5.8)
(5.9)
Lecture 5
8.321 Quantum Theory I, Fall 2017
23
5.1.2 Hilbert Spaces... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
the same
time. (It is interesting to consider what happens if we relax this assumption, for example, if we
have a particle in two dimensions where the x and y operators do not commute. We will discuss
such a system later in the course.)
We can then find a complete set of states that are simultaneous eigenstates of each ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
set of eigenstates of the
momentum operator. The momentum eigenstates |p(cid:48)(cid:105) satisfy
p(cid:12)
(cid:12)p(cid:48)
(cid:11) = p(cid:48)
(cid:12)
(cid:12)p(cid:48)
(cid:11)
,
and they form an orthonormal basis with
(cid:10)p(cid:48)(cid:48)
(cid:12)
(cid:12)p(cid:48)
(cid:11) = δ(cid:0)p(cid:48)(cid:48) − p(c... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
ψ := ψ p(cid:48)
(cid:11)
(cid:0)
(cid:12)
(cid:12)
(cid:1)
.
(5.22)
(5.23)
Although we will rarely use this notation,
space wavefunction as
it
is
worth
noting
that Sakurai denotes the momentum-
(cid:10) (cid:12)
p(cid:48) ψ := φ p(cid:48)
(cid:12)
(cid:11)
(cid:0)
(cid:1)
.
(5.24)
How are |p(cid:48)(cid:105) and |x(ci... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
(cid:48) − x(cid:48)(cid:48)
(cid:1) ,
(5.29)
2π(cid:126)|N |2δ(cid:0)x(cid:48) − x(cid:48)(cid:48)
(cid:1) = δ(cid:0)x(cid:48) − x(cid:48)(cid:48)
(cid:1) .
(5.30)
This only fixes the modulus of N , but by convention
we have
we choose N to be real and positive, so that
N = √
1
2π(cid:126)
.
This finally gives us
(cid:10... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
)
(5.33)
Thus, we see that the momentum-space wavefunction is related to the p
by the Fourier transform. We similarly have
osition
space wavefunction
(cid:104)p |ψ(cid:105) =
(cid:48)
ˆ
√
dx
(cid:48)
2π(cid:126)
(cid:48)
e ip(cid:48)x /(cid:126)(cid:104)x(cid:48)|ψ(cid:105) .
−
5.3 Normalization of Position and Momentu... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
that
eip(cid:48)(x(cid:48)+L)/(cid:126) = eip(cid:48)x(cid:48)/(cid:126) .
p(cid:48)L = 2πn , n ∈ Z .
(5.36)
(5.37)
Lecture 5
8.321 Quantum Theory I, Fall 2017
26
We now have discrete momenta, and a countable basis of momentum eigenstates. For finite L, we
require that
ˆ
0
L
dx(cid:48)
(cid:12)
(cid:12)
(cid:10)
x(cid:... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
.
You will prove this in the homework.
5.5 Momentum and Translation
We will now use the momentum operator p to define a unitary operator
T (a) = e−
(cid:126)
iap/ ,
a ∈ R .
This operator is unitary because p is Hermitian:
(T (a))† = eiap/ = (T (a))−
(cid:126)
1 .
(5.40)
(5.41)
(5.42)
ψ(x0)φ(p0)‘¯h/‘ Lecture 5
8.321 Quan... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
ipa/(cid:126)
(5.45)
dF
da
(cid:16)
=
=
i
(cid:126)
i
(cid:126) e−
i
(cid:126)
= −1 .
e−
=
(cid:126)
ipa/ i(cid:126)eipa/(cid:126)
We can then integrate this equation to find
F (a) = F (0) − a .
(5.46)
Because F (0) = x, this completes the proof.
For this reason, we will call T (a) the translation operator. What happens... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
)
1(a) x(cid:48) = x(cid:48) − a .
(cid:12)
(cid:12) (cid:11)
(cid:12)
The effect on the wavefunction is thus
(cid:10)x(cid:48)
(cid:12)
(cid:12)T (a)(cid:12)
(cid:12)ψ(cid:11) = (T −
1(a)(cid:12)
(cid:12)x(cid:48)
(cid:11), |ψ(cid:105)) = (cid:10)x(cid:48) − a(cid:12)
(cid:12)ψ(cid:11) = ψ(cid:0)x(cid:48) − a(cid:1) .
... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
I
Fall 2017
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/2ecc07c095d8ae56600f7145c6fb4a6a_MIT8_321F17_lec5.pdf |
Lecture 7
8.251 Spring 2007
Lecture 7 - Topics
• Area formula for spacial surfaces
Area formula for spatial surfaces
(“spatial” as opposed to “space-time”)
Consider 2D surface in 3D space
3D Space
�x = (x 1 , x 2 , x 3)
Parameter Space: ξ1 , ξ2 (directions along grid lines. Purely arbitrary. No con
nection to... | https://ocw.mit.edu/courses/8-251-string-theory-for-undergraduates-spring-2007/2ed11482f84c747194ee6a8bf3a226ce_lec7.pdf |
−
∂�x
∂ξ1
�2
∂�x
∂ξ2
·
2
Lecture 7
8.251 Spring 2007
�
A =
dA
Important that this formula is reparameterization-invariant.
Reparam. Invariance
Choose another coordinate par. (ξ�1 , ξ�2). Can write as functions of our (ξ1, ξ2)
coordinates. Must have:
dξ�1dξ�2
��
∂�x
∂ξ�1
·
∂�x
∂ξ�2
��
∂�x
∂ξ�2... | https://ocw.mit.edu/courses/8-251-string-theory-for-undergraduates-spring-2007/2ed11482f84c747194ee6a8bf3a226ce_lec7.pdf |
18.034, Honors Differential Equations
Prof. Jason Starr
Lecture 5
2/13/04
1. Quickly reviewed the proof of existence/uniqueness on a small interval, [t0, t0+C]
2. Explained how to do the same for [t0-c, t0], and then patch the 2 solutions. Checked the
solution is diff. at to.
3. Explained how uniqueness on small... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/2ee7d40005a4ab250cea7594569c523c_lec5.pdf |
Lecture 5
Quantum Mechanical Systems and Measurements
Today’s Program:
1. Wavefunctions in QM
2. Schrodinger’s Equation
3. Representing physical quantities: Observables
4. Hermitian operators
5. Principle of spectral decomposition.
6. Predicting the results of measurements, fourth postulate discrete and continu... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
infinite (at infinity as well as elsewhere)
(d) Continuous
(e) Piecewise continuous first derivative
Now let’s consider the simplest wavefunction for a particle. In a free space such as vacuum with
no external forces the particle can be approximated as a plane wave: eikxit , where k
p
is the
wavevector. Sinc... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
it
e
ikxit Eeikxit
i
e
t
ikxit
e
This exercise results in a curious statement about the propagation of particle ~ plane wave in free
space:
ikxit
e
2 2
2m x 2
ikxit i
e
t
In general in the presence of forces in 3D space energy of a particle would be:
2
E
V... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
Sixths Postulate of Quantum Mechanics: The time evolution of the wavefunction r, t
is
governed by the Schrodinger’s equation.
______________________________________________________________________________
Let’s take a closer look at the Schrodinger’s equation. During our derivation of the equation we
hav... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
ˆ r x, y, z
x xˆ x
Second Postulate of Quantum Mechanics: Every measurable physical quantity a is described
by an operator Aˆ acting on the wavefunction space. This operator is Hermitian and is called an
observable.
3
Properties of Hermitian operators:
1. Aˆ Aˆ (For for m... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
and obtain the operator fˆ
corresponding to the physical quantity f: f f r,
fˆ f
p, t
rˆ, pˆ, t
Example: Hamiltonian
In the first two lectures we have learned how to solve mechanical problems using Hamiltonian
approach. Remembering the general form of a Hamiltonian and using our recipe for c... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
�
t
Akin to a Classical system in Quantum Mechanics knowing the Hamiltonian defines the
system and it’s evolution in time and space.
A special and very important case: Time-independent Hamiltonian.
If there are no time dependent fields in the system, no time dependent forces, then Hamiltonian
does not dep... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
V
t
r
r
i
r
t
t
2
V
2m
2
V
2m
r r
r
r
r i
t
t
1
t
i
t
t
r and the right side of the
Note that the left side of the equation only depends on p... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
the units of energy.
kg
m
2s
2
i
I
t
E t
d
dt
II
2
2m
2 V r
Er
r
The equation (I) has a solution in a form: t e
i
E
t
.
The equation (II) is an eigenvalue/eigenfunction problem for the Hamiltonian:
Hˆ
r E
r
Fro... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
�
r, t
E
E
r E
t E
i
r e
t
E
In general, since the Hamiltonian may have many eigenvalues and corresponding eigenfunctions,
the solution for this system is a linear combination of all the possible solutions corresponding to
different energies:
5
r
, t
r
i... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
does that work? The answer lies in
one the most classic examples of Quantum Mechanics – particle in an infinite potential well. The
simplest approximation for quantum dot is an infinite potential well where all it’s electrons are
confined.
I The system: A particle of mass m in a potential well with infinitely tall ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
��
Hˆ xˆ, pˆ
u x
Eu x
2
2
2
2m x
u x Eu x
2
u x 2m
2
x
2
Eu x
0
uk x
ae
ikx beikx
,
k
2mE
2
Note: The undetermined a and b coefficients imply that there are an inifinite number of allowed
eigenfunctions corresponding to every eigenvalue (i.e. determini... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
bcos
kd
2
i sin
0
kd
2
ik
d
u ae
2
a b cos
d
2 be
ik
d
2 0 acos
i a bsin
0
kd
2
kd
2
kd
n
2
2
a b,
, n 1,3,5... or a b,
kd
n
2
2
, n 2,4,6...
Then the solution has the following form:
uE x
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
ˆ xˆ, pˆ x
2
2m x
Hˆ xˆ, pˆun x Enun x
a cos k x
n
n
2k 2
n
2m
a cos k x E a cos k x E x
n n
n
n
n
n
7
Because Hˆ is a linear operator any superposition of solutions is also a solution.
The boundary conditions have lead to a quantization of the energy levels... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
dn
d 2
2 sin2
2k
x
d
dx 1
cn
Discussion:
2 d
2
sin2 ky dy 1, y
2 x
d
d
n
2
d
Comparison between the eigenvalues and eigenvectors of the free particle and particle in a box.
One has continuous spectrum the other is discrete. The discrete character was a result of the
boundary co... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
�� En
2
2
2md 2
n 12 n2
2
2
2md 2
2n 1
8
What if we make the well size larger: d ' d , then the spacing between the energy levels
1
d 2
decreases quadratically:
~ En En ~
1
d '2
'
How is this relevant to quantum dots spectra?
Later in the cour... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
�� cos
3 x
d
has 1 node between the walls of the well.
has 2 nodes between the walls of the well…
The general trait is that the un(x) has n-1 nodes between the walls of the well.
4. Solutions are either odd or even:
The solutions are either symmetric=even ( cos
n x
d
) or anitsymmetric=odd ( sin
) around the
n... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/2eeeaab73afed1cd3969c79dd22c2a4f_MIT3_024S13_2012lec5.pdf |
15.081J/6.251J Introduction to Mathematical
Programming
Lecture 6: The Simplex Method II
1 Outline
• Revised Simplex method
• The full tableau implementation
• Anticycling
2 Revised Simplex
Initial data: A, b, c
1. Start with basis B = [AB(1), . . . , AB(m)]
and B−1 .
2. Compute p ′ = c ′
cj = cj − p ′ Aj ... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/2ef2f1dd7045b5f29e5faea299fb0798_MIT6_251JF09_lec06.pdf |
0 0 −1
1
1 1 0
1
1 0 1
1 0 −1 0
1 0
0 0
1 0
−1 1
0 0 −1 1
2.2 Practical issues
• Numerical Stability
B−1 needs to be computed from scratch once in a while, as errors accu
mulate
• Sparsity
B−1 is represented in terms of sparse triangular matrices
3 Full tableau implementation
−c ′ B−... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/2ef2f1dd7045b5f29e5faea299fb0798_MIT6_251JF09_lec06.pdf |
*
0
1 −1.5
3
0
0
0
1
0
0
0
1
Slide 13
Slide 14
x1 x2 x3
x4
3.6
0.4
x5
1.6
x6
1.6
0.4 −0.6
0
1
0 −0.6
0.4
0
0.4 −0.6
0.4
0.4
Slide 15
136
4
4
4
x3 =
x1 =
x2 =
0
0
1
0
0
0
0
1
x
3
.
B = (
)
A = (
)
E = (
)
.
.
.
D = (
)
x
1
C = (
.
)
x
2
4 Comparison of implementa... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/2ef2f1dd7045b5f29e5faea299fb0798_MIT6_251JF09_lec06.pdf |
bases.
5.2 Perturbations
(P ) min c x
′
(Pǫ) min c x
′
s.t. Ax = b
s.t. Ax = b +
x ≥ 0
x ≥ 0.
ǫ
ǫ2
.
.
.
ǫm
5.2.1 Theorem
∃ ǫ1 > 0: for all 0 < ǫ < ǫ1
ǫ
.
..
ǫm
Ax = b +
x ≥ 0
is non-degenerate.
5.2.2 Proof
Let B1, . . . , Br be all the bases.
B−1
r ... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/2ef2f1dd7045b5f29e5faea299fb0798_MIT6_251JF09_lec06.pdf |
��
, B−1
r b =
r
b
1
.
. .
r
bm
+ Br + · · · + Br
i1θ
im θm
is a polynomial in θ
•
r
Roots θi,
r
1, θi,
2, . . . , θr
i,m
•
r
If ǫ = θi,
1, . . . , θr
r
i,m ⇒ bi
r
+ Bi
1ǫ + · · · + Br ǫm = 0.
im
•
Let ǫ1 the smallest positive root ⇒ 0 < ǫ < ǫ1 all RHS are
non-degen... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/2ef2f1dd7045b5f29e5faea299fb0798_MIT6_251JF09_lec06.pdf |
= b, x ≥ 0. Then B is feasible for Ax = b +
(ǫ, . . . , ǫm) , x ≥ 0 for sufficiently small ǫ if and only if
′
ui = (bi, Bi1, . . . , Bim) > 0, ∀ i
L
B−1 = (Bij)
(B−1b)i = (bi)
5.4.2 Proof
B is feasible for peturbed problem “⇔” B−1 (b + (ǫ, . . . , ǫm) ′ ) ≥ 0 ⇔
bi + Bi1ǫ + · · · + Bimǫm ≥ 0 ∀ i
⇔ First non-zero... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/2ef2f1dd7045b5f29e5faea299fb0798_MIT6_251JF09_lec06.pdf |
rule
1. Choose an entering column Aj arbitrarily, as long as cj < 0; u = B−1Aj.
2. For each i with ui > 0, divide the ith row of the tableau (including the
entry in the zeroth column) by ui and choose the lexicographically smallest
row. If row l is lexicographically smallest, then the lth basic variable xB(l)
exit... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/2ef2f1dd7045b5f29e5faea299fb0798_MIT6_251JF09_lec06.pdf |
iting variable?
•
Otherwise, two rows in tableau proportional ⇒ rank(B−1A) < m ⇒
rank(A) < m
5.7 Theorem
If simplex starts with all the rows in the simplex tableau, other than the zeroth
row, lexicographically positive and the lexicographic pivoting rule is followed,
then
(a) Every row of the simplex tableau, oth... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/2ef2f1dd7045b5f29e5faea299fb0798_MIT6_251JF09_lec06.pdf |
Massachusetts Institute of Technology
6.042J/18.062J, Fall ’05: Mathematics for Computer Science
Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld
September 7
revised August 30, 2005, 956 minutes
InClass Problems Week 1, Wed.
Problem 1. Identify exactly where the bugs are in each of the following bogus proofs.1
... | https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2005/2f40fa3ccb1dfac791414d27201383a6_cp1w.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.013/ESD.013J Electromagnetics and Applications, Fall 2005
Please use the following citation format:
Markus Zahn, 6.013/ESD.013J Electromagnetics and Applications, Fall
2005. (Massachusetts Institute of Technology: MIT OpenCourseWare).
http://ocw.mit.edu (accessed MM DD, YYYY... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/2f455332ba1548a8db83ef1d78592088_lec8.pdf |
1
1
µ H¯ 2 = −E ¯ J ¯
|
2
∂t 2
�
� E¯ 2 +
|
|
| −
·
|
= −
− E ¯ J ¯
·
dV = E ¯ × H ¯ · dS ¯
�
V
�
�
E ¯ × H ¯ +
� ·
E ¯ × H ¯ �
� · �
�
¯ H ¯
E × · d¯
a +
S
S
d
dt
� �
V
1
� E 2 + µ ¯
| ¯ |
2
1
2
|H|2
�
�
¯
= − E · J dV
¯
dV
V
S ¯ = E ¯ × H ¯
W = �
� 1
2 �|E¯ |2 + 1
V
�
¯
¯
·
Pd = V E... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/2f455332ba1548a8db83ef1d78592088_lec8.pdf |
Approach, by Markus Zahn, 1987. Used with permission.
Outside circuit elements
�
¯ E · d ¯ l ≈ 0,
C
� × ¯ E = 0 ⇒ ¯ E = −�Φ (Kirchoff’s Voltage Law
�
k vk = 0
(Kirchoff’s current law
�
k
ik = 0
� × H ¯ = J ¯ ⇒ � · J ¯ = 0,
�
S
J ¯ · dS ¯ = 0
�
Pin = − E ¯ × H ¯ dS ¯
S
¯ �
�
¯
= − � · E × H dV
�
·
�
E ¯... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/2f455332ba1548a8db83ef1d78592088_lec8.pdf |
J
�� �
−ik
S
�
C. Complex Poynting’s Theorem (Sinusoidal Steady State, ejωt)
E¯ˆ(¯r)ejωt �
�
E¯(¯r, t) = Re
H¯ˆ (¯r)ejωt �
�
H¯ (¯r, t) = Re
=
=
E¯ˆ(¯r)ejωt + Eˆ¯∗(¯r)e−jωt �
1 �
2
Hˆ¯ (¯r)ejωt + Hˆ¯ ∗(¯r)e−jωt �
1 �
2
�
��
�
The real part of a complex number is
one-half of the sum of the number and... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/2f455332ba1548a8db83ef1d78592088_lec8.pdf |
¯r)e−2jωt �
4
1 �
�
E∗(¯r) × H(¯r) + Eˆ(¯r) × H ∗(¯r)
ˆ
¯
¯
4
�
�
1
Eˆ¯(¯r) × Hˆ¯ ∗(¯r)
Re
2
ˆ
¯
�
�
Eˆ¯∗(¯r) × Hˆ¯ (¯r)
Re
� �
¯
S =
=
=
1
2
=
ˆ
¯
(A complex number plus its complex conjugate is twice the real part of that number.)
�
� · Sˆ = � ·
¯
Sˆ¯ =
�
1
Eˆ(¯r) × Hˆ ∗(¯r)
¯
2
=
¯
=
=
1
E¯... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/2f455332ba1548a8db83ef1d78592088_lec8.pdf |
ˆd =
¯ � · Sˆ + 2jω [�wm� − �we�] = −Pˆd
III. Transverse Electromagnetic Waves (ρf = 0, J ¯ = 0)
A. Wave equation
H ¯
∂
∂t
∂E ¯
∂t
� × ¯
E = −µ
¯
� × H = �
� · ¯
E = 0
� · H ¯ = 0
3
�
�
� × � × ¯
∂ �
� ×
E = −µ
∂t
�
������0
E
�
E
�
� × � × ¯ = � � · ¯ − �
2 ¯
E = −�µ
¯ �
H = −µ
∂ �
�
∂t
∂E ¯... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/2f455332ba1548a8db83ef1d78592088_lec8.pdf |
, c = √
�µ
B. Plane waves
�
Ex(z, t) = Re ˆE
x(z)ejωt �
Image by MIT OpenCourseWare.
ω2
c2
ˆEx
d2 ˆEx
dz2 = −
+ k2Eˆx = 0
d2Eˆx
dz2
4
Ex- = Re[Ex-(z)exp ](j t)Ex+ = Re[Ex+ (z)exp ](j t)Hy+ = Re[Hy+ (z)exp ](j t)Kx = Re[K0 exp ](j t)Hy- = Re[Hy-(z)exp ](j t)e, me, mxyzwwwww where we have
is the wavenumber,... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/2f455332ba1548a8db83ef1d78592088_lec8.pdf |
t) = Ex,i(z = 0, t) + Ex,r(z = 0, t) = 0
Eˆi + Eˆr = 0 ⇒ Eˆr = −Eˆi
6
From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
For Eˆi = Ei real we have:
�
Ex(z, t) = Ex,i(z, t) + Ex,r(z, t) = Re Eˆi e−jkz − e +jkz ejωt
� �
�
Hy(z, t) = Hy,i(z, t) + Hy,r(z, t... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/2f455332ba1548a8db83ef1d78592088_lec8.pdf |
− Eˆ
i
∗e−jk1z ��
r
�Sz,i� =
=
=
=
=
�Sz,t� =
1
2
1
2
η1
1
η1
2
+
1
2
η1
|Eˆi|2
2η1
1
2η2
Re
|Eˆi|2 − | Eˆr|2 �
�
�
1
EˆrEˆ
Re
2η1
�
|Eˆi|2 − | Eˆr|2 �
�
1 − R2�
�
∗e 2jk1z − Eˆ
i
��
pure imaginary
r
∗Eˆie−2jk1z �
�
|Eˆt|2 =
ˆ 2T 2
|E
i|
η2
2
=
| ˆ |
Ei
2(1 − R2)
2η1
= �Sz,i�
V.... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/2f455332ba1548a8db83ef1d78592088_lec8.pdf |
Wave-particle Duality: Electrons are not just particles
• Compton, Planck, Einstein
– light (xrays) can be ‘particle-like’
• DeBroglie
– matter can act like it has a ‘wave-nature’
• Schrodinger, Born
– Unification of wave-particle duality, Schrodinger
Equation
©1999 E.A. Fitzgerald
1
Light has momentum: Compt... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/2f6b82d24ba7b65e1fa265e8dec88989_lecture_3.pdf |
tot
oi
E-field inside metal wall
is zero (due to high
conductivity)
0
L
Therefore, sinkz must equal zero at z=0 and z=L
sin kL = 0; kL =πn; k =
n
π
L
Also, since k=2π/λ,
n =
L
2
λ
or λ=
L
2
n
In 3-D, λ=
2L
2 + n 2
nx
2
y + nz
Note that the wavelength for E-M
waves is ‘quantized’ classically jus... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/2f6b82d24ba7b65e1fa265e8dec88989_lecture_3.pdf |
determined for deriving ρ (ν )
ρν
( ) =
N( ) E
ν wave
volume
=
kT
8πν 2 L3
c 3
L3
=
2
8πν kT
c 3
The classical assumption was used, i.e. Ewave=kbT
This results in a ρ (ν ) that goes as ν 2
At higher frequencies, blackbody radiation deviates substantially from this dependence
©1999 E.A. Fitzgerald
6
80... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/2f6b82d24ba7b65e1fa265e8dec88989_lecture_3.pdf |
)
(
EP E dE
∞
∫
= ∞
0
E
∫ P(E)dE
0
= if P(E) is normalized = ∫
)
(
EP E dE
= kbT
∞
0
©1999 E.A. Fitzgerald
8
Light is Quantized: Planck
•If P(E) were to decrease at higher E, than ρ (ν ) would not have ν 2 dependence at higher ν
•P(E) will decrease at higher E if E is a function of ν
•Experimental fit... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/2f6b82d24ba7b65e1fa265e8dec88989_lecture_3.pdf |
999 E.A. Fitzgerald
10
Light is always quantized: Photoelectric
effect (Einstein)
• Planck (and others) really doubted fit, and didn’t initially believe h was
a universal constant
• Photoelectric effect shows that E=hν even outside the box
I,E,λ
e-
metal
block
Maximum
electron
energy,
Emax
Emax=h(ν-νc)
!... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/2f6b82d24ba7b65e1fa265e8dec88989_lecture_3.pdf |
transitions in atoms
– distinct energies: E=hc/λ; E~ 10keV or so (core e- binding
energies)
Collimator crystal (decreases spread in θ and λ)
Thermionic
emission
e-
λCu
Cooled Cu
target
‘single-crystal’
diffraction
©1999 E.A. Fitzgerald
detector
θ
θ
sample
sample
‘double crystal’,
‘double axis’ diffrac... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/2f6b82d24ba7b65e1fa265e8dec88989_lecture_3.pdf |
999 E.A. Fitzgerald
18
Imaging Defects in TEM utilizing Diffraction
• The change in θ of the planes around a defect changes the Bragg condition
• Aperture after sample can be used to filter out beams deflected by defect
planes: defect contrast
Image removed due to copyright restrictions.
Please see any explanation... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/2f6b82d24ba7b65e1fa265e8dec88989_lecture_3.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
(cid:10) 6.642 Continuum Electromechanics
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
(cid:13)
6.642, Continuum Electromechanics, Fall 2004
Prof. Markus Zahn
Lecture 9: Plasma Stability (z-θ pinch)
Continuum E... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2f958b16182841cc15aec7ee419806bc_lec09_f08.pdf |
, a G a,R
m
⎢
G R, a F a,R
⎢
⎣
m
m
m
)
)
(
(
(
(
)
)
b
⎡
(cid:3)
h
⎤
⎢
r
⎥ ⎢
(cid:3)
⎥
⎦
h
⎢
⎣
r
c
⎤
⎥
⎥
⎥
⎦
⎡
(cid:3)
P
⎢
⎢
(cid:3)
P
⎣
α
β
⎤
⎥
⎥
⎦
= j
(
ω
- kU
)
⎡
F
m
ρ ⎢
G
⎢
⎣
m
(
,
β α
(
)
G
)
,
F
β α
m
m
(
(
,
α β
,
α β
(cid:3)
v
(cid:3)
v
)
)
⎡
⎤
⎢
⎥ ⎢
⎥
⎦ ⎢
⎣
α
r
β
r
⎤
⎥
⎥
⎥
⎦
F
m
(
)
x, y =
'
I
m
⎡
⎢
1
⎣
'
k... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2f958b16182841cc15aec7ee419806bc_lec09_f08.pdf |
⎡
2
1 - m - kR
⎣
R
(
2
)2
⎤
⎦
P n = T n -
γ∇ i
n n
ij
j
i
i = r, n = n = 1
i
r
P = T n + T n + T n + T
r
θ
rz
rr
z
θ
r
sr
T =
rr
1
2
μ
0
⎡
⎢
h - H 1 -
⎢
⎣
⎛
⎜
⎝
⎛
⎜
⎝
2
r
t
ξ
R
⎞
⎟
⎠
+ h
θ
2
⎞
⎟
⎠
- H + h
a
(
z
2
)
⎤
⎥
⎥
⎦
≈
1
2
⎡
μ ⎢
⎣
0
-H - 2H h -
t
θ
2
t
⎛
⎜
⎝
H
ξ
t
R
⎞
⎟
⎠
- H - 2H h
a z
2
a
⎤
⎥
⎦
'T =
rr
−μ
0
⎡... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2f958b16182841cc15aec7ee419806bc_lec09_f08.pdf |
m m
⎡
(cid:3)
ξ ⎢
R
R
⎣
H + kH
t
a
⎤
⎥
⎦
(cid:3)
h = jk
zc
(cid:108)
Ψ
c
= kF
m
(
)
a, R
⎡
(cid:3)
ξ ⎢
⎣
m
R
H + kH
t
a
⎤
⎥
⎦
IV. Dispersion Relation
2
ω ρ
(cid:3)
)
F 0, R = - H F
ξ
μ
(
m
0
t m
(
a, R
)
m m
(cid:3)
ξ
R
R
⎡
⎢
⎣
+
(cid:3)
γξ ⎡
2
1 - m - kR
⎣
R
(
2
2
)
⎤
⎦
H + kH +
a
t
⎤
⎥
⎦
μ
2
H
0
t
R
(cid:3)
ξ μ
- ... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2f958b16182841cc15aec7ee419806bc_lec09_f08.pdf |
2
kR + m - 1 + F
0 m
μ
2
)
⎤
⎦
(
a, R
)
m
R
⎡
⎢
⎣
H + kH
t
a
2
⎤
⎥
⎦
-
μ
2
H
0
t
R
Stabilizing
Destabilizing
6.642, Continuum Electromechanics Lecture 9
Prof. Markus Zahn Page 4 of 5
V. Stability
Surface tension: stabilizing for m≥1
destabilizing fo... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2f958b16182841cc15aec7ee419806bc_lec09_f08.pdf |
�gwëq
∼
9A_ío AíôàwfAí8 9v† ^_wífë¢â ô_ífA9_ío Aí 9†¢Uo _m 9v† B†à_ôA9â õ_9†í9Aëà φ f†Üí†f ^â
ï}I UI àa F
φ;
∇
MIT OpenCourseWare
https://ocw.mit.edu
2.062J / 1.138J / 18.376J Wave Propagation
Spring 2017
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2f9f71f7f386b36c9c42f48c03aecfe6_MIT2_062J_S17_Chap1.pdf |
6.891: Lecture 4 (September 20, 2005)
Parsing and Syntax II
Overview
• Weaknesses of PCFGs
• Heads in context-free rules
• Dependency representations of parse trees
• Two models making use of dependencies
Weaknesses of PCFGs
• Lack of sensitivity to lexical information
• Lack of sensitivity to structural frequen... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
VP � VP PP | VP) then (b) is
more probable, else (a) is more probable.
Attachment decision is completely independent of the words
A Case of Coordination Ambiguity
NP
CC
and
NP
NNS
cats
(a)
NP
NP
PP
NNS
IN
NP
dogs
in
NNS
houses
(b)
NP
NP
NNS
dogs
IN
in
PP
NP
NP
CC
NP
NNS
and NNS
houses ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
analysis (Bill does the believing),
so the two analyses receive same probability.
Heads in Context-Free Rules
Add annotations specifying the “head” of each rule:
S ∈ NP VP
VP ∈ Vi
VP ∈ Vt NP
VP ∈ VP PP
NP ∈ DT NN
NP ∈ NP PP
IN NP
PP ∈
Vi ∈ sleeps
Vt ∈ saw
NN ∈ man
NN ∈ woman
NN ∈
DT ∈
IN ∈ with
IN ∈ ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
� DT
NP � DT
NNP
Rules which Recover Heads:
An Example of rules for VPs
If the rule contains Vi or Vt: Choose the leftmost Vi or Vt
Else If the rule contains an VP: Choose the leftmost VP
Else Choose the leftmost child
e.g.,
VP ∈ Vt NP
VP ∈ VP PP
Adding Headwords to Trees
S
NP
VP
DT
the
NN
lawyer
Vt
... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
can find the highest scoring parse under a PCFG in this
form, in O(n3|R|) time where n is the length of the string being
parsed, and |R| is the number of rules in the grammar (see the
dynamic programming algorithm in the previous notes)
A New Form of Grammar
We define the following type of “lexicalized” grammar:
• ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
yer, NN
)
DT
NN
the
lawyer
VP(
questioned, Vt
)
Vt
NP(
witness, NN
)
questioned
DT
NN
the witness
• Also propagate part-of-speech tags up the trees
(We’ll see soon why this is useful!)
Heads and Semantics
S
∈ like(Bill, Clinton)
NP
VP
Bill
Vt
NP
likes Clinton
Syntactic structure ∈
Semantics/Logi... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
Vt(questioned,Vt) NP(lawyer,NN)
≈
(questioned, Vt, lawyer, NN, VP, Vt, NP, RIGHT)
Headwords and Dependencies
VP(told,V[6])
V[6](told,V[6])
NP(Clinton,NNP) SBAR(that,COMP)
�
(told, V[6], Clinton, NNP, VP, V[6], NP, RIGHT)
(told, V[6], that, COMP, VP, V[6], SBAR, RIGHT)
Headwords and Dependencies
S(told,V[6])
... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
6]
SBAR COMP S
VP
S
Vt
VP
SPECIAL)
LEFT)
NP
NP
RIGHT)
SBAR RIGHT)
RIGHT)
LEFT)
RIGHT)
NP
NP
A Model from Charniak (1997)
S(questioned,Vt)
≈
P (NP( ,NN) VP | S(questioned,Vt))
S(questioned,Vt)
NP( ,NN) VP(questioned,Vt)
≈
P (lawyer | S,VP,NP,NN, questioned,Vt))
S(questioned,Vt)
NP(
lawyer
,NN) ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
�2 ×
Count(S( ,Vt)�NP( ,NN) VP) )
Count(S( ,Vt))
× ( �1 ×
Count(lawyer | S,VP,NP,NN,questioned,Vt)
Count(S,VP,NP,NN,questioned,Vt)
+�2 ×
Count(lawyer | S,VP,NP,NN,Vt)
Count(S,VP,NP,NN,Vt)
+�3 ×
Count(lawyer | NN) )
Count(NN)
Motivation for Breaking Down Rules
• First step of decomposition of (Charniak 1997):
S(... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
6 ... 10
11 ... 20
21 ... 50
51 ... 100
> 100
by Type
6765
1688
695
457
329
835
496
501
204
439
Statistics for rules taken from sections 2-21 of the treebank
(Table taken from my PhD thesis).
Modeling Rule Productions as Markov Processes
• Step 1: generate category of head child
S(told,V[6])
≈
S(t... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
,NN) NP(Hillary,NNP)
VP(told,V[6])
�
S(told,V[6])
STOP
NP(yesterday,NN) NP(Hillary,NNP)
VP(told,V[6])
Ph(VP | S, told, V[6]) × Pd(NP(Hillary,NNP) | S,VP,told,V[6],LEFT)×
Pd(NP(yesterday,NN) | S,VP,told,V[6],LEFT) × Pd(STOP | S,VP,told,V[6],LEFT)
Modeling Rule Productions as Markov Processes
• Step 3: generate r... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
head.
S(told,V[6])
??
NP(Hillary,NNP)
VP(told,V[6])
∈
S(told,V[6])
NP(yesterday,NN) NP(Hillary,NNP)
VP(told,V[6])
Ph(VP | S, told, V[6]) × Pd(NP(Hillary,NNP) | S,VP,told,V[6],LEFT)×
Pd(NP(yesterday,NN) | S,VP,told,V[6],LEFT,� = 0)
The Final Probabilities
S(told,V[6])
STOP
NP(yesterday,NN) NP(Hillary,NNP)
... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
• yesterday is a temporal modifier
• But nothing to distinguish them.
Complements vs. Adjuncts
• Complements are closely related to the head they modify,
adjuncts are more indirectly related
• Complements are usually arguments of the thing they modify
yesterday Hillary told . . . ∈ Hillary is doing the telling
• ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
])
VP(told,V[6])
{NP-C}
Ph(VP | S, told, V[6]) × Plc({NP-C} | S, VP, told, V[6])
• Step 3: generate left modifiers in a Markov chain
S(told,V[6])
?? VP(told,V[6])
{NP-C}
≈
S(told,V[6])
NP-C(Hillary,NNP)
VP(told,V[6])
{}
Ph(VP | S, told, V[6]) × Plc({NP-C} | S, VP, told, V[6])×
Pd(NP-C(Hillary,NNP) | S,VP,to... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
esterday,NN) | S,VP,told,V[6],LEFT,{})×
Pd(STOP | S,VP,told,V[6],LEFT,{})
The Final Probabilities
S(told,V[6])
STOP
NP(yesterday,NN) NP-C(Hillary,NNP)
VP(told,V[6])
STOP
Ph(VP | S, told, V[6])×
Plc({NP-C} | S, VP, told, V[6])×
Pd(NP-C(Hillary,NNP) | S,VP,told,V[6],LEFT,� = 1,{NP-C})×
Pd(NP(yesterday,NN) | S,VP,... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
0,{SBAR-C})×
Pd(STOP | VP,V[6],told,V[6],RIGHT,� = 0,{})
Summary
• Identify heads of rules ∈ dependency representations
• Presented
two variants of PCFG methods applied
to
lexicalized grammars.
– Break generation of rule down into small (markov
process) steps
– Build dependencies back up (distance, subcategoriz... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
)
86.7%
Generative Lexicalized Model (Charniak 97)
87.5%
Model 1 (no subcategorization)
88.1%
Model 2 (subcategorization)
74.8%
84.3%
85.7%
87.5%
86.6%
87.7%
88.3%
Effect of the Different Features
P
R
A
V
MODEL
Model 1 NO NO
75.0% 76.5%
86.6% 86.7%
Model 1 YES NO
Model 1 YES YES 87.8% 88.2%
85.... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
(S (NP The men) (VP dumped (NP sacks) (PP of (NP the substance))))
S(told,V[6])
NP-C(Hillary,NNP)
VP(told,V[6])
NNP
Hillary
V[6](told,V[6])
NP-C(Clinton,NNP)
SBAR-C(that,COMP)
V[6]
told
NNP
Clinton
COMP
that
S-C
NP-C(she,PRP)
VP(was,Vt)
PRP
she
Vt
NP-C(president,NN)
was
NN
president
TOP
V[6]
... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
0
11
12
13
14
15
16
17
CP
29.65
40.55
48.72
54.03
59.30
64.18
68.71
73.13
74.53
75.83
77.08
78.28
79.48
80.40
81.30
82.18
82.97
P
29.65
10.90
8.17
5.31
5.27
4.88
4.53
4.42
1.40
1.30
1.25
1.20
1.20
0.92
0.90
0.88
0.79
Count Relation
11786 NPB TAG TAG L
PP TAG NP-C R
4335
S... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
49
74.34
94.55
79.20
74.93
97.49
90.54
92.41
Prec
93.46
94.04
95.11
84.35
92.15
97.98
81.14
96.85
93.93
86.65
75.72
92.04
79.54
78.57
92.82
93.49
88.22
Accuracy of the 17 most frequent dependency types in section 0 of the treebank,
as recovered by model 2. R = rank; CP = cumulative percentage... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
51
42
23
18
16
15
15
95.75
92.41
94.27
92.41
74.67
93.27
78.57
93.76
94.72
97.42
94.55
90.56
94.40
97.59
84.31
66.67
69.57
38.89
100.00
46.67
100.00
95.11
92.15
93.93
88.22
78.32
78.86
68.75
92.96
94.04
97.98
92.04
90.56
89.39
98.78
70.49
84.85
69.57
63.64
100.00
46.67
88... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
35
23
19
12
4473
289
174
129
28
25
25
19
14
84.99
83.62
90.24
75.56
68.57
0.00
21.05
50.00
82.29
55.71
74.14
72.09
71.43
60.00
12.00
78.95
85.71
84.35
81.14
81.96
78.16
52.17
0.00
26.67
100.00
81.51
53.31
72.47
69.92
66.67
71.43
75.00
83.33
63.16
763
61.47
62.20
Type
... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
0
29
27
15
12742
495
476
205
63
53
48
48
94.60
97.49
74.07
65.27
80.91
51.72
14.81
66.67
93.20
74.34
79.20
77.56
88.89
45.28
35.42
62.50
93.46
92.82
75.68
71.24
81.65
71.43
66.67
76.92
92.59
75.72
79.54
72.60
81.16
60.00
54.84
69.77
1418
73.20
75.49
Type
Sentential head ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
59
115
81
79
58
45
28
27
11
96.36
96.63
78.12
40.00
94.99
74.93
90.54
83.78
90.98
66.31
74.44
60.38
86.96
88.89
51.90
25.86
66.67
75.00
3.70
9.09
96.85
94.51
60.98
33.33
94.99
78.57
93.49
80.37
84.67
74.70
72.43
68.57
90.91
85.71
49.40
48.39
63.83
52.50
12.50
100.00
2242... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/2fcb90c84d2859c5292fdebfad49b464_lec4.pdf |
6.045: Automata, Computability, and
Complexity
Or, Great Ideas in Theoretical
Computer Science
Spring, 2010
Class 10
Nancy Lynch
Today
• Final topic in computability theory: Self-Reference
and the Recursion Theorem
• Consider adding to TMs (or programs) a new,
powerful capability to “know” and use their own
desc... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
Self-referencing machines/programs
• One more example:
• P3: On input w:
– Obtain < P3 >
– Run P3 on w
– If P3 on w outputs a number n then output n+1.
• A valid self-referencing program.
• What does P3 compute?
• Seems contradictory: if P3 on w outputs n then P3
on w outputs n+1.
• But according to the usual semantics... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
possibly partial) 2-argument function t: Σ* × Σ* → Σ*. Then
there is another TM R that computes the function r: Σ* →
Σ*, where for any w, r(w) = t(<R>, w).
<M>
w
• Example: P2, revisited
– Computes length of input.
– What are T and R?
– Here is a version of P2 with an extra
input <M>:
– T2: On inputs <M> and w:
• If... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
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