text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
)n≥1.
117
18.01 Calculus
Jason Starr
Fall 2005
A sequence (an)n≥1 converges to a limit L if the sequence becomes arbitrarily close to L, and stays
arbitrarily close to L. More precisely, the sequence converges to L if for every positive number �,
there exists an integer N (depending on the sequence and �) such ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
There are 2 remaining cases. If r = −1, then an = (−1)n diverges. If
r = 1, then an = 1 converges to 1.
2. Tests for convergence/divergence. One useful test for convergence is the Squeezing Lemma.
The squeezing lemma. Let (an)n≥1, (bn)n≥1 and (cn)n≥1 be sequences such that for every index
n,
an ≤ bn ≤ cn.
In othe... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
2005
Monotone Convergence Test. A nondecreasing sequence converges if and only if it is bounded
above. In this case, the limit of the sequence is the least upper bound for the sequence. Similarly,
a nonincreasing sequence converges if and only if it is bounded below and the limit is the greatest
lower bound for t... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
conditionally.
k ak
Examples. The harmonic sequence is the sequence an = 1/n. As will be shown soon, the harmonic
series
1/n diverges to ∞. The alternating harmonic sequence is,
�
n
an =
(−1)n
.
n
The alternating harmonic series,
∞
� (−1)n
,
n
n=1
does converge. This will also be shown soon. Since the se... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
the partial sums. But because
we add positive terms with a much higher frequency than negative terms, the sequence of partial
sums is diverging to +∞. Similarly, we could negative terms with a very high frequency and make
the partial sums diverge to −∞. Now it is not so surprising that by adding the terms in a caref... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
For
to 0, then the series
example, it immediately follows that the series ∞
n=1(−1)n diverges (arguing the opposite is a
favorite pasttime of “mathematical cranks”).
�
�
The most basic test for absolute convergence of a sequence follows from the monotone convergence
test. The sequence of partial absolute sums,
... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
essentially the geometric
sequence. Assuming r = 1, the geometric sequence rn+1 converges if and only if |r| < 1. In this
case, the sequence of partial absolute sums,
1
1 − r
1
1 − r
r n+1 .
−
=
Bn = 1 + |r| + r 2
| | + · · ·
+ |r|n =
1
1 − |r|
+
1
1 − |r|
n+1
|r|
,
also converges. Thus, the geometric seri... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
an+1/an| converges to
a real number r > 1 (in which case, the sequence (an)n≥1 does not converge to 0). There is no
information if the sequence converges to 1 or diverges.
∞
n=1
�
�
�
,
�
�
Similarly, if the following limit,
exists, call it r.
Then the sequence (an)n≥1
to the root test : The series
number r ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
g(x)dx,
1
121
�
�
18.01 Calculus
diverges, then the series
sequence (cn) by,
Jason Starr
Fall 2005
�
∞
n=1
an does not converge absolutely. For both directions, define the
The absolute partial sum of the series
n
�
cn =
n+1
f (x)dx, or cn =
�
n
k=1 ck
�
n
is simply,
�
ck =
f (x)dx, or
n
�
�
n+1 ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
1 by,
The series
∞
n=1 1/ns equals 1 + ∞
n=2
�
�
an =
1
.
ns
1/ns, which is the same as,
∞
�
1 +
n=1
1
.
(n + 1)s
Let f (x) be the function f (x) = 1/xs . Then for each integer n, f (x) ≥ 1/(n + 1)s for every x in
[n, n + 1]. The partial sum of (cn) is,
�
cn =
n 1
xs
1
dx =
�
1
�
1 n
�
�
�
1 ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
million prize for an accepted, refereed solution.
Lecture 31. December 8, 2005
Practice Problems. Course Reader: 7B4, 7B6, 7C1, 7C5, 7D1, 7D2.
1. Power series. Given a real number a and a sequence of real numbers (cn)n≥0, there is an
associated expression, called a power series about x = a,
∞
�
cn(x − a)n =... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
es if x ≥ 1.
| |
n=1
3. Consider the power series,
1 + x + x 2/2 + x /3! + · · · =
3
123
� 1∞
n!
n=0
n
x .
18.01 Calculus
Jason Starr
Fall 2005
The ratio of the nth and (n + 1)st terms in the series is,
(x n+1/(n + 1)!)/(x /n!) =
n
x
.
n + 1
For fixed x, as n grows, this sequence of ratios converges to 0... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
ratio or root test
gives an answer.
�
cn(x − a)n is positive,
2. Analytic functions. If the radius of convergence R of a power series
then the power series defines a function on the interval (a − R, a + R),
f (x) =
∞
�
cn(x − a)n .
n=0
A function defined in this manner is called an analytic function. This is th... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
only 1 of many useful properties of analytic functions. Which functions f (x) are analytic
functions? By the last paragraph, if f (x) is analytic, then it is infinitely differentiable. Are there
other restrictions? Can more than 1 power series about x = a give rise to the same analytic
function?
To answer both of the... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
(i.e., the corresponding coefficients of the 2 series are equal).
Moreover, this gives us alot of information about the first question. For an infinitely differentiable
function f (x) defined at a point x = a, there is a very important power series, the Taylor series
expansion of f (x) about x = a,
�∞
n=0
f (n) (a)
n!... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
same. For definiteness, consider the Taylor series expansion of f (x) =
(1 − x)−1 about the point x = 0.
Step 1. Compute all derivatives of f (x). If this sounds like alot of work, it is! In most
examples, this really comes down to finding an inductive formula for the derivatives of f (x). In the
example, the “zeroth... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
+1 = (n + 1)cn.
126
�
18.01 Calculus
Jason Starr
Fall 2005
Thus the result is proved by induction on k.
In fact, more has been accomplished, since now there is an inductive formula for the numbers cn,
cn = ncn−1 = n(n−1)cn−2 = n(n−1)(n−2)cn−3 =
· · · = n(n−1)(n−2)·· · ··3c2 = n(n−1)(n−2)·· · ··
3·2·
1.
This num... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
series is
simply the geometric series. By the previous lecture, the geometric series converges absolutely with
radius R = 1. Moreover, it converges absolutely to (1 −x)−1 . Notice, this gives another explanation
for the radius R = 1. Since (1 − x)−1 has a vertical asymptote at x = 1, the Taylor series cannot
conver... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
+1
]/[(1 − a)−n−1(x − a)n] = (1 − a)−1(x − a).
This is independent of n. Thus, this constant sequence converges to its constant value (1 − a)−1(x −
a). By the ratio test, the sequence is absolutely convergent if and only if this limit has absolute
value less than 1,
(1 − a)−1(x − a) ≤ 1.
|
|
Rearranging, the serie... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
of convergence is
R = ∞. Therefore, for every x, the power series converges absolutely to e ,
x
e x = �∞
n=0 xn/n!.
This equation is sometimes taken as the definition of e . It has certain advantages to our original
definition of ex . Importantly, it is easy for a computer to determine ex to very high precision usin... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
Starr
Fall 2005
Example 3. Consider the function f (x) = sin(x). The derivatives of f (x) are,
f (x)
=
f �(x) =
f ��(x)
=
f (3)(x)
=
f (n+4)(x)
=
sin(x),
cos(x),
− sin(x),
− cos(x),
f (n)(x)
Together, these give all the derivatives of f (x). Write n = 4l, 4l + 1, 4l + 2 or 4l + 3 for some
nonnegative i... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
0,
∞ (−1)m
�
(2m + 1)!
m=0
x
2m+1
.
The ratio of consecutive terms in the series is,
[(−1)m+1 x
2m+3
/(2m + 3)!]/[(−1)m x 2m+1/(2m + 1)!] = −x /(4m 2 + 8m + 3).
2
This sequence converges to 0. Therefore, by the ratio test, the power series converges absolutely to
sin(x) for every choice of x,
sin(x) = �∞
m=0... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
series expansions of sin(x) and cos(x) about a point x = a, we can follow
the procedure above. However, it is much faster to use the angle addition formulas,
sin(x) = sin(a + (x − a)) = cos(a) sin(x − a) + sin(a) cos(x − a),
cos(x) = cos(a + (x − a)) = cos(a) cos(x − a) − sin(a) sin(x − a).
This gives the Taylor se... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
beyond the scope of this class). However, it is quite easy to write down a power series expansion
for f (x). First of all, the Taylor series for e−t2 about t = 0 is obtained by substituting x = −t2 in
the Taylor series for ex about x = 0,
e−t2 =
∞
�
(−1)nt2n/n!.
n=0
Because this series converges absolutely, the i... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
� f (n)(a)
n!
n=0
(x − a)n + RN,a(x).
The precise version of the questions above is, what bounds exist for RN,a(x)?
To understand the answer, consider the simplest case where N = 0. Then the remainder term is
simply,
R0,a(x) = f (x) − f (a).
By the Mean Value Theorem, for every x there exists a real number c (de... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
−1, 1), then M equals e. To make the remainder
term less than 10−10, it suffices to take N = 12.
3. Review problems. Each of the following problems was discussed in lecture. Here are the
problems and answers, without the discussion.
Problem 1. Let a and b be positive real numbers. There are 2 tangent lines to the ell... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
solid is,
Volume =
3π/10.
Problem 4. Using a trigonometric substitution and a trigonometric identity, compute the an
tiderivative,
√
�
2
1 − x
2x
dx.
The antiderivative equals,
� √
1 − x2
x2
√
dx = −
1 − x2/x − sin−1(x
) +
C.
Problem 5. Using integration by parts, compute the following antiderivative,
�
x ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/400735651ecc95647e4c74067ad2fa3f_ocw01f05sums.pdf |
Lecture 16
8.324 Relativistic Quantum Field Theory II
Fall 2010
8.324 Relativistic Quantum Field Theory II
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2010
Lecture 16
Firstly, we summarize the results of the vertex correction from the previous lecture:
Γµ
1 (k1, k2) ≡
q
k2 + l
µ
k1 + l
l
k2
k1
= γµA(... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
in the
λ −→ 0 limit cancel among virtual and real soft photon emissions, and we can safely take the λ −→ 0 limit in the
end.
3.4: VACUUM POLARIZATION
We will now evaluate the one-loop correction to the photon propagator, and consider the physical interpretation,
recalling the general structure we considered in lec... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
)γν (i/q + m)
[
γµ(i/p + i(1 − x)k/ + m)γν (i
= tr
[
= −tr
]
]
p/ − ixk/ + m)
γµpγ/ ν p/
+ m 2tr [γµγν ] + x(1 − x)tr [γµ /
+terms linear in p+terms with an odd number of γmatrices.
kγν /
k]
We note that the trace of a term with an odd number of γ-matrices gives zero, and that
tr [γµγν ] = 4ηµν ,
tr [γµγν γργσ] =... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
pE
2
and p
→ E
ipd , dd
p →
→
2
pE . We recall that
ˆ
ddpE
(2π)d (pE
(p2 )a
E
2 + D)b
=
Γ(b − a − d )Γ(a + d )
2
2
(4π) Γ(b)Γ( d
2 )
d
2
D−(b−a− ,
d
)
2
and so
(1 −
ˆ
2
)
d
p2
E
ddpE
(2π)d (p2 + D)2
E
ˆ
= −D
Hence, the numerator of (8) can be replaced by
1
ddpE
(2π)d (p2 + D)
E
2 .
(A... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
�
iΠµν (k) =
dx x(1 − x)
− γ + log
−8ie2k2P µν
16π2
T (k)
ˆ 1
0
[
2
ϵ
)]
µν
− i(Z3 − 1)k2P ,
T
4πµ2
D
and so
with
Πµν (k) = k2PT
µν Π(k2),
e2
Π(k2) = −
2
π
ˆ 1
dx x(1 − x)
(
1
ϵ
−
log
1
2
D
2e−γ
4πµ
)
− (Z3 − 1).
0
The physical field condition constrains that Π(k2 = 0) = 0, and so Z3 is fi... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
We see that for q2 ≫ m2 , a large spacelike momentum, from (17) we have
Π(q 2) ≈
=
2
e
2
π
α
3π
log
2
log
q2
2
m
q2
,
m2
ˆ
1
0
dx x(1 − x)
e
π . Then, the internal propagator goes as
where α = 4
2
e2
q2 − iϵ
−→
e2
1
1 − Π(q2) q2 − iϵ
≡
e2(q)
q2 − iϵ
,
(18)
the running coupling constant.
with e2(... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
p′
1
µν = ηµν − qµ
q2
and P T
p2)u2(p2) = 0. We now
want to consider corrections to the Coulomb potential. We consider the low energy, non-relativistic limit, where
we derive most of our intuition about electromagnetism from, and where the notion of a potential makes most
sense. The lowest two diagrams drawn ab... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
− x) log
1 +
⃗2q
m2
x(1 − x)
)
,
and the term provides a correction to the Coulomb potential
δV (⃗r) = e1e2
ˆ
2)
⃗q
d3⃗r e−i⃗q.⃗r Π(
.
2
⃗q
From now on, we will for convenience write q ≡ |⃗q|, r ≡ ∥⃗r∥. The angular integral is given by
(24)
(25)
ˆ π
dθ sin θeiqr cos θ
0
(
eiqr − e−iqr
)
,
2π
3
(2π)
1
1 ... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
π4 ir 0
)
(
dx x(1 − x)
ˆ
∞
−∞
(
log 1 +
dz
iz
e
z
)
2
z
a2
.
(26)
log
2
z
1 + a
2
, can be computed using the complex contour shown in figure 1,
I = 2
ˆ ∞
dλ
a ˆ ∞
e−λ
λ
iπ
= 2iπ
dλ
1
e−aλ
λ
after setting λ
→ aλ. So, our result for the correction to the Coulomb potential is given by
ˆ 1
ˆ
... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
4πr
e˜i(r) = ei (1 + Z(mr)) .
1
2
5
(28)
(29)
Re(s)Im(s)ia Lecture 16
8.324 Relativistic Quantum Field Theory II
Fall 2010
Figure 2: Virtual electron-positron pairs form a screening effect.
We observe that ˜ei(r) −→ ∞ as r −→ 0, and ˜ei(r) −→ ei as r −→ 0, with small experimental corrections.
Physically, we c... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/4011862a0e45c9328c9eae315bab4517_MIT8_324F10_Lecture16.pdf |
Turbulent Flow and Transport
6
6.1
Introduction to Turbulent Boundary Layers
The nature of flow in turbulent boundary layers. Inner and outer regions, eddy
diffusivity distributions, intermittency, etc.
6.2
Integral form of the mean flow boundary layer equations.
6.3
6.4
6.5
Reasons for why the turbulent bou... | https://ocw.mit.edu/courses/2-27-turbulent-flow-and-transport-spring-2002/401aed17397ef5afafa60b237670a4ca_Boundary_layers.pdf |
Welcome to ...
2.717J/MAS.857J
Optical Engineering
MIT 2.717J
wk1-b p-1
This class is about
• Statistical Optics
– models of random optical fields, their propagation and statistical
–
properties (i.e. coherence)
imaging methods based on statistical properties of light: coherence
imaging, coherence tomography
... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/4020129e21b47c48dbf70c986d0f9508_wk1_b.pdf |
-6
Syllabus (summary)
• Review of Fourier Optics, probability & statistics 4 weeks
• Light statistics and theory of coherence 2 weeks
• The van Cittert-Zernicke theorem and applications of statistical optics
to imaging 3 weeks
• Basic concepts of inverse problems (ill-posedness, regularization) and
examples (Rado... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/4020129e21b47c48dbf70c986d0f9508_wk1_b.pdf |
plane
MIT 2.717J
wk1-b p-10
′′
′′
x
y
G 1
,
λf 1
λf 1
Fourier plane
g
1
−
f 1 y ′
f
x
′ −
,
f 1
f 2
Image plane
2
The 4F system
1f
2f
2f
1f
θx
)vuG
(
,
1
u
=
x
θ
sin
λ
θ
sin
λ
y
v
=
( , )
yx
g 1
object plane
MIT 2.717J
wk1-b p-11
′′
′′
x
y
G 1
,
... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/4020129e21b47c48dbf70c986d0f9508_wk1_b.pdf |
r
circ
R
(
g
1
)
∗ h −
f 1
f
2
′
−
,
x
f
′
1 y
f
2
Fourier plane: aperture-limited
Image plane: blurred
(i.e. low-pass filtered)
The 4F system with FP aperture
Transfer function:
circular aperture
′′
r
circ
R
Impulse response:
Airy function
r
′R
jinc
λf
... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/4020129e21b47c48dbf70c986d0f9508_wk1_b.pdf |
Transfer
Function
(MTF)
Transfer
Function
(OTF)
MIT 2.717J
wk1-b p-15
Coherent vs incoherent imaging
1f
1f
2f
2f
2a
( )u H
1
~( )
u H
1
−u
c
u
−2u
c
a
u =
c λf 1
u
2u
c
Coherent illumination
Incoherent illumination
MIT 2.717J
wk1-b p-16
Aberrations: geometrical
Paraxial
(Gaussian)
image poi... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/4020129e21b47c48dbf70c986d0f9508_wk1_b.pdf |
9
Fourier Transform
Properties
The Fourier transform is a major cornerstone in the analysis and representa-
tion of signals and linear, time-invariant systems, and its elegance and impor-
tance cannot be overemphasized. Much of its usefulness stems directly from
the properties of the Fourier transform, which we discuss... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/403577752d3007ed4ea30e8c61cf90d7_MITRES_6_007S11_lec09.pdf |
is reflected in an inverse scaling in
frequency. As we discuss and demonstrate in the lecture, we are all likely to
be somewhat familiar with this property from the shift in frequencies that oc-
curs when we slow down or speed up a tape recording. More generally, this is
one aspect of a broader set of issues relating t... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/403577752d3007ed4ea30e8c61cf90d7_MITRES_6_007S11_lec09.pdf |
transform. This is in fact very heavily exploited in discrete-time signal analy-
sis and processing, where explicit computation of the Fourier transform and
its inverse play an important role.
There are many other important properties of the Fourier transform,
such as Parseval's relation, the time-shifting property, an... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/403577752d3007ed4ea30e8c61cf90d7_MITRES_6_007S11_lec09.pdf |
product of their corresponding Fourier transforms. For the analy-
sis of linear, time-invariant systems, this is particularly useful because
through the use of the Fourier transform we can map the sometimes difficult
problem of evaluating a convolution to a simpler algebraic operation, namely
multiplication. Furthermor... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/403577752d3007ed4ea30e8c61cf90d7_MITRES_6_007S11_lec09.pdf |
ideas of Fourier series and the Fourier transform for
the discrete-time case so that when we discuss filtering, modulation, and sam-
pling we can blend ideas and issues for both classes of signals and systems.
Suggested Reading
Section 4.6, Properties of the Continuous-Time Fourier Transform, pages
202-212
Section 4.7,... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/403577752d3007ed4ea30e8c61cf90d7_MITRES_6_007S11_lec09.pdf |
Fourier Transform Properties
9-5
Example 4.7: eat u(t)
a+jcw
a > o
IxMe)
1/a/2
1/a
-
--
1/a
-a
a
TRANSPARENCY
9.3
The Fourier transform
for an exponential
time function
illustrating the
property that the
Fourier transform
magnitude is even and
the phase is odd.
7r/4
-7/4
IT/2
Time and frequency scaling:
x(at)
... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/403577752d3007ed4ea30e8c61cf90d7_MITRES_6_007S11_lec09.pdf |
Fourier transform.
Differentiation:
x(t-t )
.-
e-jwto X(o)
dx(t)
dt
jw X(W)
Integration:
Linearity:
f tf00
x(T)dr
4-+
-
Jco
X(W) + 7r X(O) b(w)
ax (t) + bx 2 (t)
-.
aX1 (w) + bX 2 (w)
TRANSPARENCY
9.9
Transparencies 9.9
and 9.10 illustrate the
convolution property
and its interpretation
for LTI systems. This
transp... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/403577752d3007ed4ea30e8c61cf90d7_MITRES_6_007S11_lec09.pdf |
-- [S(w) * P(co)]
27r
Convolution: s(t) * p(t)
S(co) P(W)
MARKERBOARD
9.1
-t
%~,
%~'
I
Y~wj~ jQ4I
I
_+)+ Gt (t) =At
(7): (V
j. J)j
&o
,e
-
I
=,ji342.
I
Jh~+I
t ~
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Signals and Systems
Professor Alan V. Oppenheim
The following may not correspond to a particular cou... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/403577752d3007ed4ea30e8c61cf90d7_MITRES_6_007S11_lec09.pdf |
6.851: Advanced Data Structures
Spring 2012
Lecture 3 — February 23, 2012
Prof. Erik Demaine
1 Overview
In the last lecture we saw the concepts of persistence and retroactivity as well as several data
structures implementing these ideas.
In this lecture we are looking at data structures to solve the geometric pr... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
first line segment lying above it. Equivalently, if we imagine
shooting a vertical ray from the query point, we want to return the first map segment that it
intersects (see Figure 2).
We can use vertical ray shooting to solve the planar point location problem in the static case by
precomputing, for each edge, what th... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
a balanced
binary search tree. More specifically, we are doing successor queries. In the last lecture we saw how
to make a partially persistent balanced binary search tree with O(log n) time queries, and we know
that a fully retroactive successor query structure can be made with O(log n) queries as well. Note,
howev... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
ray shooting when the rays do not have to be vertical?
Note that the three dimensional version of this problem is motivated by ray tracing.
2.3 Finding Intersections
The line sweep method can also be used to find intersections in a set of line segments, and this
problem gives a good illustration of the line sweep me... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
data in O(log n)
time per update, and when performing a swap can be done in constant time.
3
Orthogonal range searching
In this problem we’re given n points in d dimensions, and the query is determining which points
fall into a given box (a box is defined as the cross product of d intervals; in two dimensions, this... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
Let’s start with the 1-dimensional case, d = 1. To solve it, we can just sort the points and use
binary search. The query is an interval [a, b]; we can find the predecessor of a and the successor
of b in the sorted list and use the results to figure out whether there are any points in the box;
subtract the indices to ... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
know that all the leaves of the right subtree of that node are in the given
interval. The same thing is true for left subtrees of the right branch. If the left tree branches right
or the right tree branches left, we don’t care about the subtree of the other child; those leaves are
outside the interval. The answer is... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
sorted by their y-coordinate. We’ll also store a pointer in each node of the x range tree to
the corresponding y tree (see Figure 7 for an example). For example, αx, βx, and γx point to
Figure 7: Each of the nodes at the x level have a pointer to all of the children of that node sorted
in the y dimension, which is d... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
takes O(n log n) time for d = 1.
Building the range trees in this time bound is nontrivial, but it can be done.
3.2 Layered Range Trees
We can improve on this data structure by a factor of log n using an idea called layered range
trees (See [9], [17], [15]). This is also known as fractional cascading, or cross link... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
subset is arbitrary, as it depends on the x-coordinate of the
points.
7
Figure 9: Here is an example of cascading arrays in the final dimension. The large array contains
all the points, sorted on the last dimensions. The smaller arrays only contain points in a relevant
subtree (the small subtree has a pointer to ... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
turns out that if we have O(n logd−1 n) space and preprocessing time, we can make the structure
dynamic for free using weight balanced trees.
3.4 Weight Balanced Trees
There are different kinds of weight balanced trees; we’ll look at the oldest and simplest version,
BB[α] trees [14]. We’ve seen examples of height-ba... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
as a perfectly balanced tree.
Our data structure only has pointers in one direction - each parent points to its children nodes, but
children don’t point to their parents, or up the tree in general. As a result, we’re free to rebuild
an entire subtree whenever it’s unbalanced. And once we rebuild a subtree, we can ma... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
or more intervals start or end at infinity.
4 Fractional Cascading
Fractional cascading is a technique from Chazelle and Guibas in [6] and [7], and the dynamic version
is discussed by Mehlhorn and N¨aher in [13]. It is essentially the idea from layered range trees put
into a general setting that allows one to elimin... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
≤ 2n = O(n).
i+1
k
k
1
|Li
2
For each i < k, keep two pointers from each element. If the element came from Li, keep a pointer to
the two neighboring elements from Li
i+1, and vice versa. These pointers allow us to take information
of our placement in Li and in O(1) turn it into information about our placement in Li... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
using smaller α, we can do fractional cascading on any graph, rather than just the single path that
1
10
we had here. To do this, we just (modulo some details) cascade α of the set from each vertex along
each of its outgoing edges. When we do this, cycles may cause a vertex to cascade into itself, but
if we choos... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
��cient dynamic orthogonal point location, segment intersection, and range
reporting, SODA 2008:894-903.
[5] B. Chazelle, Reportin and counting segment intersections, Journal of Computer and System
Sciences 32(2):156-182, 1986.
[6] B. Chazelle, L. Guibas, Fractional Cascading: I. A Data Structuring Technique, Algor... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
[14] J. Nievergelt, E. M. Reingold, Binary search trees of bounded balance, Symposium on Theory
of Computing, 1972. Proceedings, 4th annual symposium.
[15] D.E. Willard, New Data Structures for Orthogonal Range Queries, SIAM Journal on Comput
ing, 14(1):232-253. 1985.
[16] D.E. Willard, New Data Structures for Orth... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
MIT 3.071
Amorphous Materials
10: Electrical and Transport Properties
Juejun (JJ) Hu
1
After-class reading list
Fundamentals of Inorganic Glasses
Ch. 14, Ch. 16
Introduction to Glass Science and Technology
Ch. 8
3.024 band gap, band diagram, engineering
conductivity
2
Basics of electrical c... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
moves after M hops in 1-D is:
r
d M
Average diffusion distance:
+
r
2
D
(1-D)
r
6
D
(3-D)
d
D
1
2
d
D
1
6
2
2
(1-D)
(3-D)
Average spacing between
adjacent sites: d
For correlated hops:
D
1
6
2
f d
0
f
1
Ion hopping frequency:
6
A tale of two valleys
Electric field E = 0
... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
Ze d
E
k T
B
Net ion drift velocity:
DEa
+
ZeEd
v
v
d
v
2
0ZeEd
v
2k T
B
exp
D
E
a
Bk T
8
A tale of two valleys
Electric field E > 0
Ion mobility
2
v Zed
0
2
k T
B
exp
D
E
a
k T
B
Electrical conductivity (1-D, random
hop)
2
nv Zed
0
2
k T
B
... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
ionic conductivity
Dispersion of activation
energy in amorphous
solids leads to slight
non-Arrhenius behavior
1/T (× 1,000) (K-1)
Phys. Rev. Lett. 109, 075901 (2012)
10
0lnlnaBETkTDSlope: aBEkD Theoretical ionic conductivity limit in glass
0
T
exp
D
E
a
k T
B
0
2
fnv Zed
0
6... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
V0 / J > 3
P. W. Anderson
Image is in the public domain.
Source: Wikimedia Commons.
Disorder leads to (electron, photon, etc.) wave function localization
15
Anderson localization in disordered systems
Extended states
(Bloch states)
Localized states
16
Density of states (DOS) in crystalline and
amorphous s... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
�334ERgD1344min2249BBERkTgkTD14expVRH T1489BRgkT
DC conductivity in amorphous semiconductors
ln
Extended state conduction
VRH is most
pronounced at
low temperature
Measured DC conductivity
Variable range hopping
1/T
21
1expexT14expVRHT VRH in As-Se-Te... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
�
Summary
Electronic structure of amorphous semiconductors
Anderson localization: extended vs. localized states
Density of states
Mobility edge
E
Conduction band
Band tail and mid-gap states
Extended state conduction
Free vs. drift mobility
Thermally activated process
Localized stat... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
r
1:
P
1
8.3
1
1
re
u
Lect
inciples
of
Applied
Mathematics
Rodolfo
Rosales
Spring
2014
.
t
e
,
s
flow
c
characteristics,
f
o
d
o
th
Me
s,
covered:
nel
ved.
chan
d
invol
ra
n
e
e
G
syllabus,
grading,
books,
notes,
etc.
s
s
u
c
is
D
.
s
s
a... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/4078bbc89141beb5985af84123dc9afd_MIT18_311S14_Lecture1.pdf |
Finite
N
•
and
s
i
ty
analys
li
tabi
s
y:
r
o
e
ic
th
Bas
•
sis
to
the
anal
ity
l
b
sta
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iscr
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•
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s
d
o
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r
t
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ms
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topics,
ma
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bus
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ee
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tivate
the... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/4078bbc89141beb5985af84123dc9afd_MIT18_311S14_Lecture1.pdf |
•
.
s
e
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flow
wav
r :
s
e
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y
waves
[say,
in
lakes].
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nc
e
ffer
ge
r
nve
.
etc].
ab
nebula].
Di
co
ete
Fourier
Transforms
to
Fourier
Series.
es.
.
nce
... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/4078bbc89141beb5985af84123dc9afd_MIT18_311S14_Lecture1.pdf |
Impact Assessment 2
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
LCA: Slide 84
What is Impact Assessment?
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
LCA: Slide 91
Issue 1: Relevance
•Translating from inventory to impact is
– Introduces numerous assumptions
• What are examples of assumptions?
– Controversial
– Necessity depend... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
priorities system
– Human health
– Biological diversity
– Ecosystem production
capacity (crops…)
– Abiotic resources (metals…)
– Cultural & recreational value
(e.g., aesthetics…)
– Resource depletion
• Energy & material
• Water
• Land use
– Human health
• Toxicological
• Non-toxicological
• Work/living en... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
7
Characterization: Eco-Indicator Damage Model
• Fate
– Where does the emission end up
• Water soluble Æ likely in water supply
•
Insoluble Æ soil
– How durable is the emission
• Some substances degrade quickly, reducing the opportunity for impact.
• Exposure
– How many / much are effected?
• How much of a ... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
Europe
• At target level the occurrence of smog periods is
extremely unlikely
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
LCA: Slide 105
Eco-Indicator 95 Weighting factors
Greenhouse
Ozone layer
A... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
)
10
Comparing Impact Assessment
Change in …
Impact on …
Damage to …
y
r
o
t
n
e
v
n
I
Atmospheric
concentration
Land
availability
Ore
availability
Human
health
Ecosystem
Resources
Species
number
Global
Warming
Ozone
Depletion
…
Massachusetts Institute of Technology
Department of Materials Scien... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
•Substances are included if there is an indication
of impact
– Classes 1 -3 carcinogens are included to the
extent that information is available
•Damages are included if possible
•Fossil fuel cannot be subsituted
– Cost of replacement is high
•DALYs are not age weighted
Massachusetts Institute of Technology
De... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
& Engineering
Respiratory (inorg)
Radiation
Acidification
Fossil Use
Respiratory (org)
Ozone Depletion
Land-use
–
ESD.123/3.560: Industrial Ecology Systems Perspectives
Randolph Kirchain
LCA: Slide 117
Issues with Eco-Indicator
• Weaknesses
• Strengths
– Comparatively
comprehensive
– Provides consistent
... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
Filter design
FIR filters
Chebychev design
linear phase filter design
equalizer design
filter magnitude specifications
•
•
•
•
•
1
FIR filters
finite impulse response (FIR) filter:
n−1
y(t) =
hτ u(t
�
τ =0
τ ),
t
Z
∈
−
(sequence) u : Z
(sequence) y : Z
→
→
R is input signal
R is output signal
hi are call... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
frequency response magnitude (i.e., H(ω) ):
|
|
1
10
0
10
−1
10
−2
10
|
)
ω
(
H
|
−3
10
0
0.5
frequency response phase (i.e.,
2
2.5
3
1
1.5
ω
H(ω)):
)
ω
(
H
3
2
1
0
−1
−2
−3
0
0.5
1
1.5
ω
2
2.5
3
Filter design
5
�
�
Chebychev design
minimize max H(ω)
ω∈[0,π] |
Hdes(ω)
|
−
C is (given) ... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
) =
h =
�
�
cos ωk
sin ωk
−
Hdes(ωk)
Hdes(ωk)
1
0
ℜ
ℑ
h0
..
.
hn−1
Filter design
7
Linear phase filters
suppose
n = 2N + 1 is odd
impulse response is symmetric about midpoint:
•
•
ht = hn−1−t,
t = 0, . . . , n
1
−
then
H(ω) = h0 + h1e −iω +
+ hn−1e −i(n−1)ω
· · ·
= e −iN ω (2h0 cos N ... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
ω
0
≤
≤
ωp
minimum stopband attenuation (
20 log10 δ2 in dB):
−
•
•
H(ω)
|
| ≤
δ2, ωs ≤
ω
≤
π
Filter design
11
Linear phase lowpass filter design
sample frequency
can assume wlog H
(0) > 0, so ripple spec is
•
•
�
1/δ1 ≤
H
(ωk)
δ1
≤
�
design for maximum stopband attenuation:
δ2
minimize
subject to 1... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
band attenuation
•
•
•
•
impulse response h:
0.2
0.1
0
−0.1
−0.2
)
t
(
h
0
2
4
6
8
10
t
12
14
16
18
20
Filter design
14
frequency response magnitude (i.e., H(ω) ):
|
|
1
10
0
10
|
)
ω
(
H
|
−1
10
−2
10
−3
10
0
0.5
frequency response phase (i.e.,
2
2.5
3
1
1.5
ω
H(ω)):
3
2
1
0
−1
−2
... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
Gdes(ω) = e−iDω ,
gdes(t) =
1 t = D
0 t = D
�
sample design:
minimize maxt�
subject to
=D |
g˜(D) = 1
g˜(t)
|
an LP
can use
•
•
g˜(t)2 or
=D
t�
=D
t�
|
g˜(t)
|
�
�
Filter design
18
�
extensions:
can impose (convex) constraints
can mix time- and frequency-domain specifications
can equalize multiple sy... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
Chebychev equalizer design:
˜
minimize max
G(ω)
�
�
�
ω
−
equalized system impulse response g˜
e −i10ω
�
�
�
)
t
(
g˜
1
0.8
0.6
0.4
0.2
0
−0.2
0
5
10
15
20
25
30
35
t
Filter design
22
equalized frequency response magnitude
1
10
0
10
|
)
ω
(
Ge
|
G
|
|
�
−1
10
0
0.5
1
2
2.5
3
1.5
ω
equalized fr... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
specifications
transfer function magnitude spec has form
L(ω)
H(ω)
| ≤
≤ |
U (ω), ω
[0, π]
∈
where L, U : R
R+ are given
→
lower bound is not convex in filter coefficients h
arises in many applications, e.g., audio, spectrum shaping
can change variables to solve via convex optimization
•
•
•
Filter design
26
A... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
question: when is r
answer: (spectral factorization theorem) if and only if R(ω)
Rn the autocorrelation coefficients of some h
Rn?
∈
0 for all ω
∈
≥
spectral factorization condition is convex in r
•
many algorithms for spectral factorization, i.e., finding an h s.t.
• R(ω) =
H(ω)
|
2
|
magnitude design via autoc... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
�
≤
1
10
0
10
2
|
)
ω
(
H
|
−1
10
−2
10
−1
10
0
10
ω
optimal fit:
0.5dB
±
Filter design
31
MIT OpenCourseWare
http://ocw.mit.edu
6.079 / 6.975 Introduction to Convex Optimization
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
Today’s topics:
• UC ZK from UC commitments (this is information theoretic and unconditional; no crypto needed)
• MPC, under any number of faults (using the paradigm of [GMW87])
• MPC in the plain model with an honest majority (using elements of [BOGW88] and [RBO89])
1
UC Zero Knowledge from UC Commitments
To imp... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
set v
←
1. Else set v
←
0.
Finally, output (sid, P, V, G, v) to (sid, V ) and to S, and halt.
[the Blum protocol?]
Claim 1 The Blum protocol security realizes F H
wzk
in the Fcomhybrid model.
Proof Sketch: Let A be an adversary that interacts with the protocol. We construct an idealprocess
adversary S that fo... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
0 to
A. If b = 0 (no cheating allowed), then send c = 1 to A.
(c) Obtain A’s openings of the commitments (either a permutation of the graph, or a Hamiltonian
cycle). If c = 0 (permutation) and all openings are consistent with G, then send b� = 1 to Fwzk;
if some openings are bad then send b� = 0. If c = 1 (cycle) a... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
the F R
F R and fools all environments. There are four cases:
zk
wzkhybrid model; we’ll construct an adversary that interacts with
1. If A controls the verifier: this case is simple. All A expects to see is k copies of (x, b) being delivered
zk, sends k copies to A, and outputs whatever
by the copies of Fwzk. S r... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
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