text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
1, t2
0
0
n'(t)
τ
gl
0
0
>>τ
t1 t2,
t
t
t
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - S... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
• In Si around 300K,
– τ ∼ N −1 for low N (trap-assisted recombination),
– τ ∼ N −2 for high N (Auger recombination).
• Order of magnitude of key parameters for Si at 300K:
– τ ∼ 1 ns − 1 ms, depending on doping
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. ... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.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 8: Electrohydrodynamic and Ferrohydrodynamic... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
coth ka
−
1
sinh ka
−
coth kb
−
1
sinh kb
1
sinh ka
⎤
⎥
⎥
⎥
coth ka
⎥
⎦
⎡
(cid:108)
Ψ⎢
⎢
(cid:108)
Ψ
⎣
c
d
⎤
⎥
⎥
⎦
1
sinh kb
coth kb
⎤
⎥
⎥
⎥
⎥
⎦
⎡
(cid:108)
Ψ⎢
⎢
(cid:108)
Ψ
⎣
e
f
⎤
⎥
⎥
⎦
6.642, Continuum Electromechanics Lecture 8
Prof. Markus Zahn Page 1 of 13
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
)
ξ
⎤
⎦
'
P
d
( )
ξ =
P
od
( )
ξ +
(
'
P 0
d
)
=
'
P
e
( )
ξ =
P
oe
( )
ξ +
(
'
P 0
e
)
=
dP
od
dx
dP
oe
dx
x 0
=
x 0
=
ξ +
(
'
P 0
d
)
g
= −ρ ξ +
a
(
0'
P
d
)
ξ +
(
'
P 0
e
)
= −ρ ξ +
b
g
(
'
P 0
e
)
6.642, Continuum Electromechanics Lecture 8
Prof. Markus Zahn Page 2 of 13
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
(cid:108)
Ψ
μ
a
d
= −
μ
b
k c
oth kb
(cid:108)
Ψ
e
6.642, Continuum Electromechanics Lecture 8
Prof. Markus Zahn Page 3 of 13
(cid:108)
Ψ =
d
−μ
b
μ
a
coth kb
coth ka
(cid:108)
Ψ
e
(cid:108)
Ψ −
e
⎡
μ
b
⎢
μ⎣
a
coth kb
coth ka
−
⎤
1
⎥
⎦
=
(
H H
−
b
a
(cid:3)
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
ξ
2
ω
k
(
ρ
a
coth ka
+ ρ
b
coth kb
)
= ρ − ρ
b
(
a
)
g
+ γ
k
2
−
B H H coth ka coth kb
−
(
0
a
− μ
a
)
k
)
b
coth ka
a
+ μ
b
(
μ
b
coth kb
H
a
=
B
0
μ
a
, H
b
= ⇒ −
H H
b
a
B
0
μ
b
=
B
0
⎛
⎜
⎜
⎝
1
μ
a
−
1
μ
b
⎞
⎟
⎟
⎠
(
=
μ − μ
b
a
μ μ
a b
μ
)
B
0
μ − μ
(
2
B
0
b
tanh ka
)2
a
+ μ
μ
b
k
tanh kb
⎤
⎦
a
2
ω
k
(
ρ
a
coth ka... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
f
0=
df
dk
=
0 2 k
= γ
−
c
2
(
2
B
0
)
μ − μ
b
a
(
μ μ μ + μ
b
a b
a
k
c
=
2
2
(
2
B
0
)
μ − μ
b
(
γμ μ μ + μ
b
a b
a
a
)
g
(
ρ − ρ
a
b
)
+ γ
⎡
⎢
⎢
⎣
2
2
(
2
B
0
)
μ − μ
b
(
γμ μ μ + μ
b
a b
a
a
)
2
⎤
⎥
⎥
⎦
)
−
⎡
⎢
⎢
⎣
2
(
2
B
0
)
μ − μ
b
a
(
μ μ μ + μ
b
a b
a
)
2
⎤
⎥
⎥
⎦
1
2
γ
=
0
2
(
2
B
0
)
μ − μ
b
a
(
μ μ μ + μ
b
a... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
Polarization Forces
Courtesy of MIT Press. Used with permission.
μ → ε
a
a
μ → ε
b
b
B
0
D→
0
6.642, Continuum Electromechanics Lecture 8
Prof. Markus Zahn Page 6 of 13
2
ω
k
(
ρ
a
coth ka
+ ρ
b
coth kb
)
=
g
(
ρ − ρ
b
a
)
2... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
e
(cid:108)
k= Φ
a
xa
(cid:108)
k= − Φ
b
xb
1
nh k
si
coth k
⎤
⎡
⎥Δ ⎢
⎥
⎢
⎥ ⎢
⎥ ⎣
⎦
Δ
(cid:3)
v
(cid:3)
v
α
x
β
x
⎤
⎥
⎥
⎥
⎦
⎡
(cid:3)
p
⎢
⎢
(cid:3)
p
⎣
α
β
⎤
⎥
⎥
⎦
=
j
ωρ
k
⎡
⎢
⎢
⎢
⎢
⎣
−
coth k
Δ
−
1
sinh k
Δ
(cid:3)
v
α =
x
(cid:3)
v
β
x
= ω(cid:3)ξ
j
(cid:3)
P
a
=
j
ωρ
a
k
(cid:3)
v
xa
= −
2
ω ρ
a
k
ξ(cid:3)
6.64... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
)
)
g
(
ρ − ρ ξ =
b
a
(cid:3)
)
−
(
ε
a
−
ε
b
)
E
0
dE
0
dx
(cid:3)
ξ − γ
k
2
(cid:3)
ξ −
jk E
y 0
(
ε
a
−
−
ε
b
k
)(
(
ε
a
) (
ε
a
−
ε
b
)
(cid:3)
ξ
jk E
y 0
)
ε
b
+
2
ω
k
(
ρ + ρ
b
a
)
=
g
(
ρ − ρ
a
b
)
2
+ γ
k
+
(
ε
a
−
ε
b
)
E
0
+
2 2
k E
y 0
k
(
dE
0
x
d
(cid:78)
E
0
R
ε
(
ε
a
2
)
−
ε
b
ε
b
)
a
+
Uniform tangentia... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/5938069f486691fdd2376f96b52951ee_lec08_f08.pdf |
Overview
This will be a mostly self-contained research-oriented course designed for undergraduate students
(but also extremely welcoming to graduate students) with an interest in doing research in theoretical
aspects of algorithms that aim to extract information from data. These often lie in overlaps of
two or more ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/594e3ae91cc8e865f25d07dcbd2dd460_MIT18_S096F15_Ses1.pdf |
on time available.
Open Problems
A couple of open problems will be presented at the end of most lectures. They won’t necessarily
be the most important problems in the field (although some will be rather important), I have tried
to select a mix of important, approachable, and fun problems. In fact, I take the opportu... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/594e3ae91cc8e865f25d07dcbd2dd460_MIT18_S096F15_Ses1.pdf |
n, try it!
√
0.4.2 Matrix AM-GM inequality
We move now to an interesting generalization of arithmetic-geometric means inequality, which has
applications on understanding the difference in performance of with- versus without-replacement
sampling in certain randomized algorithms (see [RR12]).
Open Problem 0.2 For an... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/594e3ae91cc8e865f25d07dcbd2dd460_MIT18_S096F15_Ses1.pdf |
W14] A. Israel, F. Krahmer, and R. Ward. An arithmetic-geometric mean inequality for prod
ucts of three matrices. Available online at arXiv:1411.0333 [math.SP], 2014.
[Nik13] A. Nikolov. The komlos conjecture
holds for vector colorings. Available online at
arXiv:1301.4039 [math.CO], 2013.
[RR12] B. Recht and C. Re... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/594e3ae91cc8e865f25d07dcbd2dd460_MIT18_S096F15_Ses1.pdf |
+
k
2
y
+
2
k
z
2
= ω με (cid:22)
k
2
o
Wave Vector k:
Perpendicular to uniform plane wave phase front,
Therefore perpendicular to⎯E and⎯H
L9-2
UPW AT PLANAR BOUNDARY
Case I: TE Wave
x
iE
kz
iH
kx
θi
ik
k
=i
k
o
ok = ω με
rE
θr
rk
ε,μ
kz
εt,μt
θi
y
z
θt
tEtk
“Transverse Electric”
⊥(ci... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/599acc32617af6d8eac2964c8f42cbb5_MIT6_013S09_lec09.pdf |
)(cid:10) (cid:8)(cid:11)(cid:9) (cid:11)(cid:10) (cid:8)(cid:11)(cid:9) (cid:11)(cid:10)
k
r
z
k sin
t
θ =
r
k
i
z
=
=
k
t
z
=
zk
θr = θi Angle of incidence equals angle of reflection
Snell’s Law:
sin
sin
θ
t
θ
i
=
k
o
k
t
=
ω με
ω μ ε
t
t
=
v
t
v
i
=
n
i
n
t
"Snell's Law"
θi
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/599acc32617af6d8eac2964c8f42cbb5_MIT6_013S09_lec09.pdf |
θ =
c
t
)
i
e.g., [
2
ε = ε ⇒
i
]
o
[n
i
=
2]
[
⇒ θ =
c
45 ]
°
L9-5
NON-UNIFORM PLANE WAVES (NUPW)
Normal refraction: θi < θc
θi
Phase fronts
Glass
Air
θt
λglass
z
Lines of
constant phase
>λo
λo
Beyond the critical angle, evanescence:
x
θi
θi > θc
Lines of constant phase
λglass
kz ’... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/599acc32617af6d8eac2964c8f42cbb5_MIT6_013S09_lec09.pdf |
22)
ˆ
z ˆ
j x k
k z
− α
−
′′
jk
=
0
−
E,H e
α
(
j k
′
−
′′
jk
)
i
r
L9-7
EVANESCENT WAVES -- SUMMARY
Names:
“non-uniform plane wave”
“evanescent wave” (
0
=
in direction of decay)
“surface wave”
“inhomogeneous plane wave”
)t(S
(
′
i
j k jk r
−
Lossless medium:
−
E,H e
α
′
′′
0
k
• =
j
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/599acc32617af6d8eac2964c8f42cbb5_MIT6_013S09_lec09.pdf |
6.867 Machine learning, lecture 4 (Jaakkola)
1
The Support Vector Machine and regularization
We proposed a simple relaxed optimization problem for finding the maximum margin sep
arator when some of the examples may be misclassified:
minimize
�θ�2 + C
1
2
n
�
ξt
t=1
subject to yt(θT xt + θ0) ≥ 1 − ξt and ξt ≥ 0... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
and θ0.
�
Cite as: Tommi Jaakkola, course materials for 6.867 Machine Learning, Fall 2006. MIT OpenCourseWare
(http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].(cid:13)(cid:10)
−3−2−10123−1−0.500.511.522.53−3−2−10123−1−0.500.511.522.53 6.867 Machine learning, lecture 4 (Jaakk... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
cases. One simple model of noisy
labels in linear classification is a logistic regression model. In this model we assign a
Cite as: Tommi Jaakkola, course materials for 6.867 Machine Learning, Fall 2006. MIT OpenCourseWare
(http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].(cid... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
with the linear prediction, may seem a little arbitrary (but perhaps not more so than the
hinge loss used with the SVM classifier). We will derive the form of the logistic function
later on in the course based on certain assumptions about class-conditional distributions
P (x|y = 1) and P (x|y = −1).
In order to bett... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
on [DD Month YYYY].(cid:13)(cid:10)
6.867 Machine learning, lecture 4 (Jaakkola)
4
L(θ, θ0) is known as the (conditional) likelihood function and is interpreted as a function
of the parameters for a fixed data (labels and examples). By maximizing this conditional
likelihood with respect to θ and θ0 we obtain maximu... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
t=1
n
� �
�
log 1 + exp −yt(θT xt + θ0)
��
(10)
(11)
(12)
(13)
t=1
We can interpret this similarly to the sum of the hinge losses in the SVM approach. As
before, we have a base loss function, here log[1 + exp(−z)] (Figure 1b), similar to the hinge
loss (Figure 1a), and this loss depends only on the value of ... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
nding the minimizing θˆ and θˆ
0 including simple gradi
ent descent. In a simple (stochastic) gradient descent, we would modify the parameters in
response to each term in the sum (based on each training example). To specify the updates
we need the following derivatives
log �
1 + exp �
−yt(θT xt + θ0) ��
d
dθ0
... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
equals zero. Setting the gradient to zero is also a necessary
condition of optimality:
d
dθ0
d
dθ
(−l(θ, θ0) = −
yt[1 − P (yt|xt, θ, θ0)] = 0
(−l(θ, θ0)) = −
ytxt[1 − P (yt|xt, θ, θ0)] = 0
n
�
t=1
n
�
t=1
(19)
(20)
Cite as: Tommi Jaakkola, course materials for 6.867 Machine Learning, Fall 2006. MIT Open... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
/1 labels: ˜yt = (1 + yt)/2 so that
y˜t ∈ {0, 1}. Then the above optimality conditions can be rewritten in terms of prediction
errors [˜yt − P (y = 1|xt, θ, θ0)] rather than mistake probabilities as
n
�
[˜yt − P (y = 1|xt, θ, θ0)] = 0
t=1
n
�
xt[˜yt − P (y = 1|xt, θ, θ0)] = 0
t=1
and
�
θ0
n
�
[˜yt − P (y = 1... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
+ exp −yt(θT xt + θ0)
is strictly decreasing). Thus, as a result, the maximum like
lihood parameter values would become unbounded, and infinite scaling of any parameters
corresponding to a perfect linear classifier would attain the highest likelihood (likelihood
of exactly one or the loss exactly zero). The resulting... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
t=1
log �
1 + exp �
−yt(θT xt + θ0) ��
(26)
since it seems more natural to vary the strength of regularization with λ while keeping the
objective the same.
Cite as: Tommi Jaakkola, course materials for 6.867 Machine Learning, Fall 2006. MIT OpenCourseWare
(http://ocw.mit.edu/), Massachusetts Institute of Technol... | https://ocw.mit.edu/courses/6-867-machine-learning-fall-2006/59a63d2efbe8aa01041937ff539a449a_lec4.pdf |
6.896 Quantum Complexity Theory
September 16, 2008
Lecturer: Scott Aaronson
Lecture 4
1 Review of the last lecture
1.1 BQP
BQP is a class of languages L ⊆ (0, 1)∗, decidable with bounded error probability ( say 1/3 ) by a
uniform family of polynomial-size quantum circuit over some universal family of gate. In to... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
In terms of computational complexity, the schrodinger picture ( αx|x�) and Heisenberg’s density
matrix (ρ) both lead to an exponential-space simulation since we need to calculate whole evolution
of state vectors. On the other hand, the Feynman’s path integral, summing up all the histories,
leads to a polynomial-spac... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
of 1/3 and 2/3. This class is
physical not realistic for we cannot know whether the probability is 1/2 or 1/2 − 1/2|x| without
running algorithm exponential time. However, in terms of complexity theory, we can prove that
BQP ⊆ P P .
P P is the decision version of #P , which means we cannot count the number of accep... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
first, we still don’t know where N P sits in the diagram and how N P relates to BQP . We
conjecture that N P �⊆ BQP , which means that quantum computer cannot solve N P complete
problems in polynomial time. However, we have no idea as for whether BQP �⊆ N P or not.
Another interesting question is that if P = N P , th... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
(unitary
operation) and get the answer. Then we apply CNOT-gate to the answer and keep it in some
safe location that won’t be touched again. Then we run the entire subroutine backwards to erase
everything but the answer.
Figure 4: The idea of uncomputing.
4-4
The uncomputing step will partly erase the unnecessar... | https://ocw.mit.edu/courses/6-845-quantum-complexity-theory-fall-2010/59b32d603954c2f22a331ffb689706f3_MIT6_845F10_lec04.pdf |
(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
Last
modi(cid:0)ed(cid:1)
September
(cid:2)(cid:3)(cid:4)
(cid:2)(cid:5)(cid:5)(cid:6)
Many(cid:0)body
phenomena
in
condensed
matter
and
atomic
physics
(cid:0)
Lectures
(cid:1)(cid:2)
(cid:3)(cid:4)
Bose
condensation(cid:4)
Symmetry(cid:5)
breaking
and
quasiparticles(cid:4)
In... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
BEC
2
h
(cid:10)
(cid:9)(cid:1)
2(cid:1)3
T
(cid:7)
(cid:5) n
(cid:2) (cid:5)
(cid:7)
(cid:7) (cid:12)
(cid:8)(cid:12)(cid:11)(cid:13)(cid:9)(cid:8)(cid:8)(cid:8)
(cid:3)(cid:9)(cid:4)
BEC
m
(cid:6)
(cid:3)(cid:12)(cid:7)(cid:9)(cid:4)
2(cid:1)3
at
density
n(cid:1)
there
is
no
condensate(cid:14)
n
(cid:7)
(cid:15)(cid:... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
(cid:1)
serves
well
to
illustrate
the
new
features
of
Bose
condensation
of
interacting
particles(cid:14)
spontaneous
symmetry
breaking(cid:1)
the
o(cid:6)(cid:0)diagonal
long(cid:0)range
order(cid:1)
and
collective
excitations(cid:5)
(cid:0)(cid:1)(cid:0)
Spontaneous
symmetry
breaking
Weakly
interacting
Bose
gas
with
a... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
length(cid:1)
to
be
discussed
b
e
R
(cid:0)
(cid:11)
l
o
w(cid:5)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
a
coherent
state(cid:1)
a
BEC
(cid:7)
N
BEC
(cid:1)
which
is
equivalent
to
replacing
the
operator
a
(cid:17)
by
a
c(cid:0)numb
e
r
0
0
p
j
i
j
i
p
N
(cid:5)
This
can
be
achieved
if
the
BEC
state
is
understood
as
a
coherent
state(cid:1)
which
requires
considering
the
problem
(cid:3)(cid:13)(cid:4)
in
the
(cid:20)big(cid... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
(cid:3)(cid:22)(cid:4)
(cid:0)
j
j
0
0
j
i
(cid:0)
j
i
j
i
m�0
m(cid:21)
(cid:0)
(cid:1)
p
p
+
V
(cid:3)
2
(cid:0)
(cid:0)
(cid:1)(cid:1)
X
1
2
p
V (cid:10)
(cid:2)
m
which
have
the
desired
property
(cid:10)
(cid:17)
(cid:10)
(cid:7)
(cid:10)
(cid:10)
(cid:5)
These
states
do
not
correspond
to
any
speci(cid:16)c
numb
e... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
operator
e
(cid:1)
applied
to
(cid:10)
(cid:1)
produces
a
state
of
the
same
energy(cid:1)
with
a
phase
j
i
2
V
(cid:3)
(cid:3)
0
2
of
(cid:10)
shifted
by
(cid:5)(cid:5)
Since
the
overlap
of
coherent
states
obeys
(cid:10)
(cid:10)
(cid:7)
e
(cid:1)
(cid:0)
j
(cid:0)
j
(cid:1)
^
i(cid:4)N
any
two
di(cid:6)erent
states
(c... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
expectation
value(cid:1)
obtain
(cid:17)
(cid:17)
H
H (cid:2)
H (cid:0)
U
(cid:3)(cid:10)(cid:4) (cid:7)
(cid:10)
(cid:3)N
(cid:10)
(cid:7)
(cid:10)
(cid:3)
(cid:10)
(cid:3)(cid:24)(cid:4)
(cid:17)
(cid:11)
4
2
h
jH
(cid:0)
j
i
j
j
(cid:0)
j
j
(cid:9)
(cid:25)
the
so(cid:0)called
Mexican
hat
potential(cid:5)
The
energy... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
relation
between
the
density
and
chemical
potential(cid:1)
(cid:3)
(cid:7)
(cid:11)n(cid:5)
1
From
the
symmetry
point
of
view(cid:1)
the
systuation
is
quite
interesting(cid:5)
The
microscopic
hamiltonian
(cid:3)(cid:13)(cid:4)
has
global
U
(cid:3)(cid:11)(cid:4)
symmetry(cid:1)
since
it
is
invariant
under
adding
a
cons... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
states
with
(cid:26)uctuating
particle
number(cid:5)
One
can
instead
start
with
the
density
matrix
of
the
Bose
gas
(cid:3)(cid:13)(cid:4)
ground
state
(cid:27)
in
the
coordinate
representation(cid:1)
(cid:5)
j
i
R(cid:3)x(cid:2)
x
(cid:4) (cid:7)
(cid:27)
(cid:10)
(cid:17) (cid:3)x
(cid:4)
(cid:10)(cid:3)x(cid:4)
(cid:... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
:15)
(cid:1)
k
k
n
(cid:7)
n
(cid:3)(cid:9)(cid:1) (cid:4)
(cid:0) (cid:3)
(cid:4)
(cid:8)
f
(cid:3)
(cid:4)
(cid:3)(cid:11)(cid:11)(cid:4)
k
k
k
0
3
In
a
B
o
s
e
where
f
(cid:3)
(cid:4)
i
s
a
smooth
function(cid:5)
Accordingly(cid:1)
t
h
e
density
matrix
(cid:3)(cid:11)(cid:15)(cid:4)
has
two
terms(cid:1)
k
R(cid:3)x(... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
vanish
at
several
interatomic
distances(cid:4)(cid:5)
The
(cid:16)nite
limit
n
(cid:7)
lim
(cid:27)
(cid:10)
(cid:17)
(cid:3)x (cid:4)
(cid:10)(cid:3)x(cid:4)
(cid:17)
(cid:27)
suggests
that
the
quantities
(cid:10)(cid:3)x(cid:4)(cid:1)
(cid:17)
0
x
x
(cid:5)
(cid:5)
(cid:1)
0
+
+
i(cid:4)
+
i(cid:4)
h
j
j
i
j
(cid:0)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
:0)nite(cid:1)
but
large
system(cid:1)
with
(cid:0)xed
particle
number(cid:1)
the
true
ground
state
(cid:2)TGS(cid:3)
of
a
quantum(cid:4)
mechanical
hamiltonian
is
nondegenerate(cid:5)
This
TGS
is
isotropic
in
(cid:1)
due
to
boundary
e(cid:6)ects
that
split
(cid:0)
the
circular
manifold(cid:5)
The
statement
about
the
a... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
cid:8)
(cid:11)n(cid:3)a
(cid:8)
a
(cid:4)
(cid:3)a
(cid:8)
a
(cid:4)
(cid:3)(cid:11)(cid:13)(cid:4)
k
k
k
k
k
k
k
k
k
(cid:9)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(
(cid:6)
)
k
k
X
(cid:0)
(cid:0)
(cid:1)
where
the
sum
is
taken
over
pairs
(cid:3)
(cid:2)
(cid:4)
Here
we
used
the
value
(cid:3)
(cid:7)
(cid:11)n
obtained
a... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
1)
q(cid:17)
the
hamiltonian
is
represented
as
a
sum
of
k
k
independent
harmonic
oscillators(cid:5)
Indeed(cid:1)
since
a
a
(cid:8)
a
a
(cid:7)
p(cid:17)
p(cid:17)
(cid:8)
(cid:17)q
q(cid:17)
(cid:1)
we
can
k
k
k
k
k
k
k
k
+
+
+
+
rewrite
the
hamiltonian
as
follows(cid:14)
(cid:0)
(cid:0)
(cid:11)
(0)
(0)
2
+
+
(cid... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
8) (cid:9)
(cid:11)n
k
which
acts
on
the
operators
a
(cid:1)
a
as
k
k
+
a
(cid:7)
cosh
(cid:13)
b
sinh
(cid:13)
b
(cid:2)
a
(cid:7)
cosh
(cid:13)
b
sinh
(cid:13)
b
(cid:3)(cid:11)(cid:23)(cid:4)
k
k
k
k
k
k
k
k
k
k
+
+
+
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:3)see
Lecture
(cid:9)(cid:4)(cid:5)
The
transformatio... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
b
(cid:1)
b
(cid:1)
having
energy
k
k
+
(0)
2
2
E
(cid:7)
(cid:12)
(cid:8)
(cid:11)n
(cid:3)(cid:11)n(cid:4)
(cid:3)(cid:11)(cid:28)(cid:4)
k
k
r
(cid:0)
(cid:0)
(cid:1)
The
new
ground
state
is
annihilated
by
all
the
b
(cid:5)
Since
for
the
ground
state
of
the
ideal
k
Bose
gas
a
(cid:27)
(cid:7)
(cid:15)(cid:1)
and
the... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
(cid:4)
(cid:4)
(cid:3)see
Lecture
(cid:9)(cid:4)(cid:1)
one
can
write
the
new
ground
state
as
(cid:27)
(cid:7)
U
(cid:27... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
cid:1)
where
(cid:10)
(cid:7)
(cid:3)(cid:7)(cid:11)(cid:5)
The
linearized
equation
has
solution
0
0
of
the
form
q
(cid:15) (cid:3)
(cid:2) t
(cid:4) (cid:7)
ae
(cid:8)
be
(cid:3)(cid:9)(cid:13)(cid:4)
r
(cid:0)
(cid:0)
kr
kr
i
i(cid:8)t
i
+i(cid:8)t
(cid:10)
with
h(cid:16)
(cid:10)
(cid:7)
(cid:12)
(cid:8)
(cid:11)n
(... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
ations
are
predominantly
of
the
(cid:16)eld
(cid:10)(cid:1)
not
in
the
modulus(cid:1)
just
as
one
expects
in
the
phase
(cid:0)
from
Goldstone
theorem
(cid:3)and
the
above
Mexican
hat
picture(cid:4)(cid:5)
At
large
(cid:1)
however(cid:1)
the
k
normal
modes
have
a
(cid:8)
b
or
a
b
nearly
equal
in
magnitude(cid:1)
which
m... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
3)(cid:9)(cid:22)(cid:4)
0
(cid:9)
(0)
2
(cid:0)
k
�0
2
X
k
B
C
(cid:12)
(cid:8)
(cid:11)n
(cid:3)(cid:11)n(cid:4)
B
C
(cid:0)
r
(cid:6)
A
(cid:0)
(cid:1)
3(cid:1)2
Estimating
the
sum
as
O(cid:3)(cid:11)
(cid:4)(cid:1)
we
(cid:16)nd
that
the
condensate
depletion
is
a
small
e(cid:6)ect(cid:5)
In
contrast(cid:1)
in
super... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/59d41060e232d13d3da729e329a15c73_lec45.pdf |
Routing
Second main application of Chernoff: analysis of load balancing.
• Already saw balls in bins example
• synchronous message passing
• bidirectional links, one message per step
• queues on links
• permutation routing
• oblivious algorithms only consider self packet.
• Theorem Any deterministic oblivious perm... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
permu
tation
• What if don’t wait for next phase?
– FIFO queuing
– total time is length plus delay
– Expected delay ≤ E[ T (el)] = n/2.
– Chernoff bound? no. dependence of T (ei).
�
• High prob. bound:
– consider paths sharing i’s fixed route (e0, . . . , ek )
– Suppose S packets intersect route (use at least on... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
2 key roles for chernoff
• sampling
• load balancing
• “high probability” results at log n means.
3
The Probabilistic Method—Value of Random Answers
Idea: to show an object with certain properties exists
• generate a random object
• prove it has properties with nonzero probability
• often, “certain propert... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
cs(cs/n)ds
=
≤
≤
[(s/n)d−c−1 e
[(1/3)d−c−1 e
[(c/3)d(3e)c+1]
c d−c]
s
c d−c]
s
s
c+1
c+1
– Take c = 2, d = 18, get [(2/3)18(3e)3]<2−s
– sum over s, get < 1
Existence proof
• No known construction this good.
• N P -hard to verify
• but some constructions almost this good
• recent progress via zig-zag produc... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
• need 2n boundaries, so aim for prob. bound 1/2n2 .
• solve, δ =
�
(4 ln 2n2)/ ˆw.
√
•
So absolute error
8 ˆw ln n
– Good (o(1)-error) if ˆw � 8 ln n
– Bad (O(ln n) error) is ˆw = 2
– General rule: randomized rounding good if target logarithmic, not
if constant
MAX SAT
Define.
•
•
literals
clauses
• NP-com... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/59d8268c8f271c4bc7c5302f797fd6bd_n6.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.265/15.070J
Lecture 1
Fall 2013
9/4/2013
Metric spaces and topology
Content. Metric spaces and topology. Polish Space. Arzel´a-Ascoli Theo
rem. Convergence of mappings. Skorohod metric and Skorohod space.
1
Metric spaces. Open, closed and compact sets
When we discuss p... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
d
1
Problem 1. Show that Lp is not a metric when 0 < p < 1.
Another important example is S = C[0, T ] – the space of continuous func
tions x : [0, T ] → Rd and ρ(x, y) = ρT = sup0≤t≤T Ix(t) − y(t)I, where I · I
can be taken as any of Lp or L∞. We will usually concentrate on the case d = 1,
in which case ρ(x, y) = ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
necessarily connected by an edge, let ρ(u, v) be the length
of a shortest path connecting u with v. Then it is easy to see that ρ is a metric
on the finite set V .
Definition 2. A sequence xn ∈ S is said to converge to a limit x ∈ S (we write
xn → x) if limn ρ(xn, x) = 0. A sequence xn ∈ S is Cauchy if for every E > ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
. Given a set S, consider the metric ρ defined by ρ(x, x) = 0, ρ(x, y) =
1 for x = y. Show that (S, ρ) is a metric space. Suppose S is uncountable. Show
that S is not separable.
Given x ∈ S and r > 0 define a ball with radius r to be B(x, r) = {y ∈ S :
ρ(x, y) ≤ r}. A set A ⊂ S is defined to be open if for every x ∈ A... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
⊂ S is defined to be compact
if every sequence xn ∈ K contains a converging subsequence xnk → x and
x ∈ K. It can be shown that K ⊂ Rd is compact if and only if K is closed and
bounded (namely supx∈K IxI < ∞ (this applies to any Lp metric). Prove that
every compact set is closed.
¯
¯
Proposition 1. Given a metric s... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
implies ρ2(f (x), f (y)) < E.
Problem 5. Show that f is a continuous mapping if and only if for every open
set U ⊂ S2, f −1(U ) is an open set in S1.
Proposition 2. Suppose K ⊂ S1 is compact. If f : K → Rd is continuous then
it is also uniformly continuous. Also there exists x0 ∈ K satisfying If (x0)I =
supx∈K If ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
belong to Bo(xi, δ(xi)). Then f (y), f (z) ∈ Bo(f (xi), E). By triangle
inequality we have If (y) − f (z)I ≤ If (y) − f (xi)I + If (z) − f (xi)I < 2E.
We conclude that for every two points y, z such that ρ1(y, z) < δ/2 we have
If (y) − f (z)I < 2E. The uniform continuity is established. Notice, that in this
proof t... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
bounded closed
sets. What about C[0, T ]? We will need a characterization of compact sets in this
space later when we analyze tightness properties and construction of a Brownian
motion.
Given x ∈ C[0, T ] and δ > 0, define wx(δ) = sups,t:|s−t|<δ |x(t) − x(s)|. The
quantity wx(δ) is called modulus of continuity. Sin... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
�A
1
n
(3)
(4)
Suppose A is compact but (4) does not hold. Then we can find a subsequence
xni ∈ A, i ≥ 1 such that wxni
(1/ni) ≥ c for some c > 0. Since A is compact
then there is further subsequence of xni which converges to some x ∈ A. To
ease the notation we denote this subsequence again by xni . Thus Ixni −x... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
f uniformly on compact sets if and only if for every T > 0
lim sup ρ2(fn(t), f (t)) = 0.
n 0≤t≤T
Point-wise convergence does not imply uniform convergence even on com
pact sets. Consider xn = nx for x ∈ [0, 1/n], = n(2/n − x) for x ∈ [1/n, 2/n]
and = 0 for x ∈ [2/n, 1]. Then xn converges to zero function point-wis... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
∞) are Polish.
6
Problem 7. Use Proposition 3 (or anything else useful) to prove that C[0, T ] is
complete.
That C[0, T ] has a dense countable subset can be shown via approximations
by polynomials with rational coefficients (we skip the details).
3
Skorohod space and Skorohod metric
The space C[0, ∞) equipped ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
es). We can try to use the uniform metric again. Let us consider
the following two processes x, y ∈ D[0, T ]. Fix τ, ∈ [0, T ) and δ > 0 such that
τ + δ < T and define x(z) = 1{z ≥ τ }, y(z) = 1{z ≥ τ + δ}. We see that x
and y coincide everywhere except for a small interval [τ, τ + δ). It makes sense
to assume that ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
inf Iλ − II ∨ Ix − yλ)I ,
(cid:16)
λ∈Λ
for all x, y ∈ D[0, T ], where I ∈ Λ is the identity transformation, and I · I is
the uniform metric on D[0, T ].
Thus, per this definition, the distance between x and y is less than E if there
exists λ ∈ Λ such that sup0≤t≤T |λ(t)−t| < E and sup0≤t≤T |x(t)−y(λ(t))| <
E.
Pro... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
ingsley, Convergence of probability measures, Wiley-Interscience
publication, 1999.
8
MIT OpenCourseWare
http://ocw.mit.edu
15.070J / 6.265J Advanced Stochastic Processes
Fall 2013
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/59e4f0d302a20e75921eaf4e290739dd_MIT15_070JF13_Lec1.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS
by A. Megretski
Lecture 2: Differential Equations As System Models1
Ordinary differential requations (ODE) are the most frequently used tool for modeling
continuous-time ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
= a(x(t), t)
x(t2) − x(t1) =
t2
�
t1
a(x(t), t)dt � t1, t2 ⊂ T.
1Version of September 10, 2003
(2.1)
(2.2)
2
The variable t is usually referred to as the “time”.
Note the use of an integral form in the formal definition (2.2): it assumes that the
function t ∈� a(x(t), t) is integrable on T , but does not r... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
(t) = min{t − c, 0},
where c is an arbitrary real constant. These solutions are not differentiable at the critical
“stopping moment” t = c.
2.1.2 Standard ODE system models
Ordinary differential equations can be used in many ways for modeling of dynamical
systems. The notion of a standard ODE system model describes ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
well-posedness introduces some typical constraints aimed at
insuring their applicability.
Definition A standard ODE model ODE(f, g) is called well posed if for every signal
v(t) ⊂ V and for every solution x1 : [0, t1] ∈� X of (2.4) with x1(0) ⊂ X0 there exists a
solution x : R+ ∈� X of (2.4) such that x(t) = x1(t) f... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
show that no
solution of the ODE
x˙ (t) = 0.5 − sgn(x(t))
satisfying x(0) = 0 exists on a time interval [0, tf ] for tf > 0. Indeed, let x = x(t) be such
solution. As an integral of a bounded function, x = x(t) witll be a continuous function
of time. A continuous function over a compact interval always achieves a ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
∈� Rn of the standard ODE
(same as (2.1)), subject to a given initial condition
x˙ (t) = a(x(t), t)
x(t0) = x0.
(2.7)
(2.8)
Here a : Z ∈� Rn is a given continuous function, defined on Z � Rn × R. It turns out
that a solution x = x(t) of (2.7) with initial condition (2.8) exists, at least on a sufficiently
short ti... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
to Theorem 2.1, for any initial
condition x(0) = x0 there exists a solution of the Riccati equation, defined on some time
interval [0, tf ] of positive length. This does not mean, however, that the correspond
ing autonomous system model (producing output w(t) = x(t)) is well-posed, since such
solutions are not neces... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
Theorem 2.2 Let X be an open subset of Rn . Let a : X × R ∈� Rn be a continuous
function. Then all maximal solutions of (2.7) are defined on open intervals and, whenever
(t0, t1) ∈� X has a finite interval end t = t0 ⊂ R or t¯ = t1 ⊂ R (as
such solution x :
opposed to t0 = −→ or t1 = →), there exists no sequence tk ⊂ ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
continuous dependence on time
The ODE describing systems dynamics are frequently discontinuous with respect to the
time variable. Indeed, the standard ODE system model includes
x˙ (t) = f (x(t), v(t), t),
where v = v(t) is an input, and the ODE becomes discontinuous with respect to t when
ever v is a rectangular i... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
0
|a(x1(t), t) − a(x2(t), t)|dt < π
whenever x1, x2 : [t0, t0 +r] ∈� Rn are continuous functions satisfying |xk (t)−x0| ∀ r
and |x1(t) − x2(t)| < � for all t ⊂ [t0, t0 + r].
Then, for some tf ⊂ (t0, t0 + r) there exists a solution x :
(2.8).
[t0, tf ] ∈� Rn of (2.7) satisfying
Example 2.4 Theorem 2.3 can be used ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
⎨
= 0
for all t ∞= 0. Hence x(t) = ct for some constant c, and x(0) = 0.
7
2.2.4 Differential inclusions
Let X be a subset of Rn, and let � : X � 2Rn
X to a subset of Rn . Such a function defines a differential inclusion
be a function which maps every point of
x˙ (t) ⊂ �(x(t)).
(2.9)
By a solution of... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
x0| ∀ r}
(b) for every ¯
x ⊂ Br (x0) the set �(¯
x) is convex;
(c) for every sequence of ¯
sequence uk ⊂ �(¯
¯
the subsequence ¯
xk ⊂ Br (x0) converging to a limit x ⊂ Br (x0) and for every
xk ) there exists a subsequence k = k(q) � → as q � → such that
uk(q) has a limit in �(¯
x).
¯
Then the supremum
is finite, and... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
be compatible with the “dry friction” interpretation of the sign nonlinearity.
In particular, with the initial condition x(0) = 0, the equation has solutions for every
value of c ⊂ R. If c ⊂ [−1, 1], the unique maximal solution is x(t) ≤ 0, which corresponds
to the friction force “adapting” itself to equalize the ex... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/5a1c1ee61722a192c4479093ffb612d0_lec2_6243_2003.pdf |
6.825 Techniques in Artificial Intelligence
Logic
Lecture 3 • 1
Today we're going to start talking about logic. Now, my guess is that almost
everybody's been exposed to basic propositional logic in the context of
machine architecture or something like that. But, it turns out that that
exposure to logic was just a... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
dealing with sets of states.
• The sentence “It’s raining” stands for all the states
of the world in which it is raining.
Lecture 3 • 3
What if I say "It's raining."? One way to think about what it means -- what
that assertion means, that it's raining -- is to say that it stands for all those
states of the world ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
and a semantics, and a way of manipulating
expressions in the language. We’ll talk about each of these.
6
What is a logic?
• A formal language
• Syntax – what expressions are legal
Lecture 3 • 7
The syntax is a description of what you're allowed to write down, what the
expressions are that are legal in a langua... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
expressions
(which will tell us something new)
• Why proofs?
Two kinds
of inferences an agent
might want to make:
• Multiple percepts => conclusions about the
world
Lecture 3 • 11
In the context of an agent trying to reason about its world, think about a
situation where we have a bunch of percepts. You know, ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
anyway, they talk about logic
in the abstract and then they talk about propositional logic. So, we're just
going to dive right into propositional logic, learn something about how that
works, and then try to generalize later on.
We’ll start by talking about the syntax of propositional logic. Syntax is what
you're a... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
ences (wffs: well formed formulas)
Lecture 3 • 16
So let's define the syntax of propositional logic. We’ll call the legal things to
write down "sentences". So if something is a sentence, it is a syntactically
OK thing in our language. Sometimes sentences are called "WFFs" (which
stands for “well-formed formulas” i... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
ψ are sentences, then so are
(φ),
¬
φ, φ Æψ, φ Çψ, φ
→
ψ, φ
↔
ψ
Lecture 3 • 19
Now, here’s the recursive part. If \phi and \psi are sentences, then so are –
Wait! What, exactly, are \phi and \psi? They’re called metavariables, and
they range over expressions. This rule says that if \phi and \psi are things
that ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
� B)
(C Ç D)
→
A
→
B Ç C
↔
D
(A
→
(B Ç C))
D
↔
• Precedence rules enable “shorthand” form of sentences,
but formally only the fully parenthesized form is legal.
• Syntactically ambiguous forms allowed in shorthand only
when semantically equivalent: A Æ B Æ C is equivalent to
(A Æ B) Æ C and A Æ (B Æ C)
L... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
Lecture 3 • 22
22
Semantics
Lecture 3 • 23
So let's talk about semantics. The semantics of a sentence is its meaning.
What does it say about the world?
We could just write symbols on the board and play with them all day long,
and it could be fun, it could be like doing puzzles. But ultimately the reason
that we... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
i ]
[
Lecture 3 • 25
How could we decide whether A wedge B wedge C is true or not? Well, it
has to do with what A and B and C stand for in the world. What A and B and
C stand for in the world will be given by an object called an “interpretation”.
An interpretation -- and I'm going to depart a little bit from Russe... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
• 26
Similarly, we’ll use a turnstile with a slash through it to say that a sentence is
not true in an interpretation. And since the meaning of every sentence is a
truth value and there are only two truth values, then if a sentence \Phi is not
true (does not have the truth value T) in an interpretation, then it has... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
The sentence consisting of a symbol "false" has truth value "F" in all
interpretations.
All right, now we can do the connectives. We’ll leave out the parentheses.
The truth value of a sentence with top-level parentheses is the same as the
truth value of the sentence with the parentheses removed.
29
Semantics
• M... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/5a2344cf5fdd746421d4ab2b8bc60ee8_Lecture3Final.pdf |
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