text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
(log n + k) time, so we avoid the extra log factor from the
previous strategy, allowing us to solve the 2 − d problem in O(log n) time in total. For arbitrary
d, we can use this technique for the last dimension; we can thereby improve the general query to
O(logd−1 n + k) time for d > 1.
3.3 Dynamic Point Sets
We c... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
use the number of nodes in the subtree, the number of leaves in
the subtree, or some other reasonable definition. We also haven’t selected α. If α = 2 , we have a
problem: the tree must be perfectly balanced at all times. Taking a small α, however, (say, α = 1 ),
10
works well. Weight balancing is a stronger property... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
tree can be
rebuilt in θ(k) time, which is easy).
So, for layered range trees, we have O(logd n) amortized update, and we still have a O(logd−1 n)
query.
3.5 Further results
For static orthogonal range searching, we can achieve a O(logd−1 n) query for d > 1 using less
space: O n
[5]. This is optimal in some mode... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
with k separate binary searches, resulting in
a runtime of O(k log n). Fractional cascading allows this problem to be solved in O(k + log n).
To motivate this solution we can look at the layered range trees above and think about how we can
retain information when moving from one list to the next. In particular, if w... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
nd the
x with a binary search. Now to find the location of x in Li
i
two neighboring elements in Li
i using the extra pointers. Then these elements
have exactly one element between them in Li
To find our actual location in Li
i+1, we simply
do a comparison with that intermediate element. This allows us to turn the inf... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
graph has locally bounded in-degree, meaning for each x ∈ R, the number of incoming
edges to a vertex whose range contains x is bounded by a constant.
We can support a search query that finds x in each of k vertices, where the vertices are reachable
within themselves from a single node of the graph so the edges used ... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
Structuring Technique, Algorithmica,
1(2):133-162, 1986.
[7] B. Chazelle, L. Guibas, Fractional Cascading: II. Applications, Algorithmica, 1(2):163-191,
1986.
[8] D. Dobkin, R. Lipton, Multidimensional searching problems, SIAM Journal on Computing,
5(2):181-186, 1976.
[9] Gabow H., Bentley J., Tarjan R., Scaling ... | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/405bf317ea6dffc96e77badf1833f56f_MIT6_851S12_L3.pdf |
72. 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 Orthogonal Queries, Techincal Report, January 1979.
[17] D.E. Willard, Ph.D. dissertation, Harvard University, 1978.... | 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 |
3210 (2009)
5
A tale of two valleys
Electric field E = 0
Assuming completely random hops,
the average total distance an ion
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 site... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
T
B
Hopping frequency ← :
v
1
2
v
0
exp
D
E
a
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
Elec... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
��
DEa
+
ZeEd
D
k TB
E
a
0
T
exp
D
E
a
k TB
9
Temperature dependence of 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 io... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
ine solids
All electronic states are labeled with real Bloch wave vectors k
signaling translational symmetry
All electronic states are extended states
No extended states exist in the band gap
13
Band structures in defect-free crystalline solids
Figure removed due to copyright restrictions. See Figure 12, ... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
opping conduction via localized states
Fixed range hopping: hopping between nearest neighbors
Hopping between dopant atoms at low temperature
Variable range hopping (VRH)
Hopping between localized states near EF
E
Mobility edge
EF
R
g
DOS
y
z
x
19
expR Variable range hopping
Hopping prob... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/406b79662069dca93109298f745ae175_MIT3_071F15_Lecture10.pdf |
J. Appl. Phys. 101, 063520 (2007)
22
Summary
Basics of electrical transport
Conductivity: scalar sum of ionic and electronic contributions
Einstein relation
Ionic conductivity
Occurs through ion hopping between different preferred “sites”
Thermally activated process and non-Arrhenius behavior
23
... | 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 |
solution
of
pde
by
Finite
N
•
and
s
i
ty
analys
li
tabi
s
y:
r
o
e
ic
th
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the
anal
ity
l
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sta
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iscr
D
y
i
•
.
s
d
o
al
meth
r
t
c
e
p
s
d
n
a
FFT
•
.
ms
r
o
f
s
n
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r
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ce
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l
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a
L
d
n
a
r
ie
r
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•
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b
y
Other
topics,
ma
.
bus
a
e... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/4078bbc89141beb5985af84123dc9afd_MIT18_311S14_Lecture1.pdf |
o
i
s
o
l
p
x
e
m,
ves
hock
wa
boo
ic
n
[so
S
•
.
s
e
Traffic
flow
wav
r :
s
e
Oth
•
•
y
waves
[say,
in
lakes].
Solitar
Diffusion
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 |
3/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
LCA: Slide 89
3
Your thoughts:
What do you see as the key issues?
What is most challenging step?
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Rando... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
to
ancient structures)
• What about aesthetics? Comfort?
• Key issue: Double counting
– Boundary between categories is fuzzy
• Oil depletion vs. Emissions from oil use
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Ra... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
of impact
– E.g., Impact of CO2 release = 1
Impact of methane release = 21
• Mid point vs end-point
– Increase in acidification vs. Increase in species
depletion
– Impact indicator vs. damage indicator
– Less uncertainty vs. easier to value
– Eco-indicator 99 is an endpoint / damage-based
method
– Eco-indicator... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
LCA: Slide 104
8
Eco-Indicator 95 Weighting factors
• Distance to target
– The further away current conditions are to an
established target the more se... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
LCA: Slide 106
9
Eco-Indicator 99
• Extension of Eco-Indicator 95
• Focus is on weighting method
– Don’t weigh impact categories
– Weigh only different types of damage
• Limits type of damage categories to 3
– Damage to human health
• Expressed as number of years of life lost and number of years
of life live... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
of Housing, Spatial
Planning and the Environment (VROM). Used with permission.(cid:13)
Source: Eco-indicator 99: Manual for Designers
11
(cid:10)
Weighting via Panel
Hierarchist
Egalitarian
Individualist
Human
Health
Ecosystem
Resources
40%
40%
20%
30%
50%
20%
55%
25%
20%
Massachusetts Institute o... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
ages are assumed to be recoverable
•Fossil fuels cannot be depleted
– Ignored
•DALYs are age weighted
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
LCA: Slide 114
13
Weighting via Panel
•Surveyed... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/40a6b5be9e12c243b8a5596e906617be_lec17.pdf |
– Provides consistent
mechanism for
weighting
– Well documented
– Limited to three
impacts
• Human health
• Biodiversity
• Resource depletion
– Highly European
focused
– Controversial panel
weighting
– Still many inventory
items to model
Massachusetts Institute of Technology
Department of Materials Scienc... | 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 |
2
14
16
18
20
Filter design
4
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... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
where
· · ·
· · ·
cos(n
sin(n
−
1)ωk
1)ωk
−
−
�
�
A(k) =
b(k) =
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(... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
, π]
•
•
Filter design
10
specifications:
maximum passband ripple (
20 log10 δ1 in dB):
±
1/δ1 ≤ |
H(ω)
| ≤
δ1,
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... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
•
•
•
Filter design
13
example
linear phase filter, n = 21
passband [0, 0.12π]; stopband [0.24π, π]
max ripple δ1 = 1.012 (
0.1dB)
±
design for maximum stopband 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 respon... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
or maxω |
H(ω)
|
•
Filter design
16
Chebychev equalizer design:
minimize max
ω∈[0,π]
�
convex; SOCP after sampling frequency
(ω)
−
Gdes(ω)
G
�
�
�
�
�
�
Filter design
17
time-domain equalization: optimize impulse response g˜ of equalized
system
e.g., with Gdes(ω) = e−iDω ,
gdes(t) =
1 t = D
0 t = D
�... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
9
t
Filter design
20
1
10
0
10
|
)
ω
(
G
|
−1
10
0
0.5
1
2
2.5
3
1.5
ω
)
ω
(
G
3
2
1
0
−1
−2
−3
0
0.5
1
1.5
ω
2
2.5
3
design 30th order FIR equalizer with G
(ω)
e−i10ω
≈
Filter design
�
21
�
Chebychev equalizer design:
˜
minimize max
G(ω)
�
�
�
ω
−
equalized system impulse response... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
.6
0.4
0.2
0
−0.2
0
5
10
15
20
25
30
35
t
Filter design
24
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 frequency response phase
G
3
2
1
0
−1
−2
−3
0
)
ω
(
Ge
0.5
1
�
1.5
ω
2
2.5
3
Filter design
25
�
�
Filter magni... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
. , rn−1)
n
t
| ≥
•
•
Rn
∈
Filter design
27
Fourier transform of autocorrelation coefficients is
R(ω) =
−iωτ
e
rτ = r0 +
�
τ
n−1
�
t
=1
2rt cos ωt = H(ω)
|
2
|
always have R(ω)
0 for all ω
≥
can express magnitude specification as
•
•
L(ω)2
R(ω)
≤
≤
U (ω)2 , ω
[0, π]
∈
. . . convex in r
Filter design
28... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/40b33990b6fdba13b9d464c143b07fb4_MIT6_079F09_lec09.pdf |
10 D(ω)
|
|
−
•
D is desired transfer function magnitude
(D(ω) > 0 for all ω)
find minimax logarithmic (dB) fit
•
reformulate as
minimize
subject to D(ω)2/t
t
R(ω)
≤
≤
tD(ω)2 ,
ω
0
≤
≤
π
convex in variables r, t
constraint includes spectral factorization condition
•
•
Filter design
30
example: 1/f (... | 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 |
) If H(G, h) = 1 or b = b� = 1 then 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 a... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
the bit b from Fwzk. If b = 1 (i.e., cheating is allowed), then send the challenge c = 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, t... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
the
same b, output (x, b), else output ⊥.
zk
Claim 2 Parallel composition of k copies of Fwzk realizes Fzk.
Proof: Let A be an adversary in 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 ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
then fail; this can happen only when all bi are 1, which occurs with probability 2−k.)
wzk
• Otherwise give (x, ⊥) to F R
zk.
3. If A controls both parties or neither party, the simulation is trivial. Handling adaptive corruptions is
trivial as well (no party has any secret state).
We analyze S: when the verifier ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
the environment, but are otherwise
passive. In the second, the environment gives inputs directly to the parties, and the adversary merely listens
(i.e., it cannot change the inputs). We will use the first variant to model semihonest adversaries.
Here is the 1outofm oblivious transfer functionality, F m (we name t... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
to see T ’s input (v0, v1), plus the two values (y0, y1) received from R. S can
easily simulate this view, where (v0, v1) are taken as T ’s inputs in the ideal process and (y0, y1) are
random.
• A corrupts R: A expects to see R’s input i, the function f , and the bits (t0, t1). S works as follows:
obtain vi from F ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
k times (due to semihonesty, the receiver will always ask for bits from
the same string). For adaptive adversaries with erasures, the protocol can easily be made to work — R just
erases x0, x1 before sending (y0, y1). Without erasures, we need to do something slightly different.
3.1 Evaluating general functionalitie... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
previous activation (initially, they
are all set to 0).
(d) Pi sets its shares of the adversary/simulator input lines to be 0.
When Pi is instead notified by P1−i, it proceeds as above except with input bits equal to 0.
Now all inputs lines to the circuit have been shared between the two parties.
2. Evaluate the ci... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
keeps its share of each localstate line to be used in
the next activation; outputs to the adversary are ignored.
Claim 4 Let F be any standard ideal functionality. Then the above protocol realizes F in the Fothybrid
model for semihonest, adaptive adversaries.
Proof Sketch: For any A, we construct an S that fools... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
, each message m of Q is followed by a proof of the NP statement:
“there exist input x and random input r that are the legitimate openings of the commitments above,
and such that the message m is the result of running the protocol on x, r, and the messages I received
so far.”
Consider the construction of a UC “GMW ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
the Fzkhybrid model. The protocol uses Com, which is any perfectly
cp
binding, noninteractive commitment scheme.
1. On input (sid, C, V, commit, w) [i.e., to commit to value w], C computes a = Com(w, r) for random r,
adds w to the list W , adds a to the list A, adds r to the list R, and sends (sid, C, V, prove, a... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
all Z. First, S runs A. Then we consider two cases:
• The committer is corrupted: in the commit phase, S obtains from A the message (sid, C, V, prove, a, (w, r))
zk . If Rc holds on a and (w, r), then S sends (sid, C, V, commit, w) to Fcp. In the proof phase, S
zk . If Rp holds on (x, A) and (W, R),
to F Rc
obtain... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
because the
commitments are perfectly binding (and hence cannot correspond to the good witnesses). We can fix this
problem by using “equivocable commitments.”
Research Question 2 Can Fcp be realized unconditionally (i.e., in some hybrid model without computa
tional assumptions)?
Now that we have Fcp, we can constru... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
outgoing message m. In this case, send (sid.0, Q0, Q1, prove, m) to Fcp, where the relation used
by Fcp is:
Rp = {((m, M, r2), (x, r1)) : m = P0(x, r1 ⊕ r2, M )}
4. Receive the ith message m. Q0 receives (sid.1, Q1, Q0, prove, (m, M, s1)) from Fcp. Q0 verifies that
s1 is the value it sent in Step 2, and that M is the ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
it is unconditional, and offers perfect simulation. It works even
for adaptive adversaries. However, it requires S to be able to change the inputs of the semihonest parties
(hence our choice of the specific semihonest model above). �
6
4 Extending to the multiparty case
There are a number of challenges we must c... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
A. Wigderson. How to play any mental game. In STOC 1987,
pages 218–229. ACM, 1987.
[RBO89]
Tal Rabin and Michael BenOr. Verifiable secret sharing and multiparty protocols with honest
majority (extended abstract). STOC 1989, pages 73–85, 1989.
7 | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
6.241 Dynamic Systems and Control
Lecture 8: Solutions of State-space Models
Readings: DDV, Chapters 10, 11, 12 (skip the parts on transform methods)
Emilio Frazzoli
Aeronautics and Astronautics
Massachusetts Institute of Technology
February 28, 2011
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
F... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
case of zero input, i.e., u = 0; in this case, the state-space equations
are written as the difference equations
y [0] = C [0]x0
y [1] = C [1] A[0] x[0]
x[0] = x0
x[1] = A[0] x[0]
x[2] = A[1] A[0] x[0] y [2] = C [2] A[1] A[0] x[0]
. . .
. . .
y [k] = C [k] Φ[k, 0] x[0]
x[k] = Φ[k, 0] x[0]
where we defined the ... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
In other words, x[k] = Γ[k, 0]U[k, 0], where
Γ[k, 0] = Φ[k, 1]B[0] Φ[k, 2]B[1]
�
�
. . . B[k − 1] ,
⎡
⎤
u[0]
u[1]
⎥
U = ⎢
⎥
⎢
⎣
. . .
⎦
u[k − 1]
.
The output is
y [k] = C [k]Γ[k, 0]U[k, 0].
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
Feb 28, 2011
4 / 19
Summary (DT)
In general, state... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
(t) is sufficiently well behaved so
that there exists unique state/output signals x and y . (e.g., A is
piecewise-continuous).
Define a state transition function Φ(t, τ ) such that, for all t, τ ∈ T,
∂
∂t
Φ(t, τ ) = A(t)Φ(t, τ ),
Φ(t, t) = I .
The function Φ can in general be computed numerically, integrating a
differe... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
x(t0) = 0; moreover,
d
dt
x(t) =
Φ(t, τ )B(τ )u(τ ) dτ =
� t
t0
d
dt
� t
∂
∂t
t0
� t
Φ(t, τ )B(τ )u(τ ) dτ + [Φ(t, τ )B(τ )u(τ )]τ =t
= A(t)
Φ(t, τ )B(τ )u(τ ) dτ + B(t)u(t) = A(t)x(t) + B(t)u(t).
t0
Similarly for the output.
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
Feb 28, 2011
8 /... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
check that the matrix exponential satisfies the conditions for the
state transition function.
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
Feb 28, 2011
10 / 19
Similarity Transformations
The choice of a state-space model for a given system is not unique.
For example, let T be an invertible matrix... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
of the state-space model is then
x[k] =
n
�
αi vi λk
i ,
i=1
which is called the modal decomposition of the unforced response.
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
Feb 28, 2011
12 / 19
�
Modal contributions
Since α = V −1x(0), one can also write
x[k] =
n
�
λk
i vi wi
�x0,
which sh... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
44
1.21. Bialgebras. Let C be a finite monoidal category, and (F, J) :
C → Vec be a fiber functor. Consider the algebra H := End(F ). This
algebra has two additional structures: the comultiplication Δ : H →
H ⊗ H and the counit ε : H
k. Namely, the comultiplication is
defined by the formula
→
Δ(a) = α−1 Δ(a)),
F,F
(... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
ε satisfying properties (i),(ii) of Theorem 1.21.1 is called
a bialgebra.
Thus, Theorem 1.21.1 claims that the algebra H = End(F ) has a
natural structure of a bialgebra.
Now let H be any bialgebra (not necessarily finite dimensional).
Then the category Rep(H) of representations (i.e., left modules) of
H and its s... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
equivalence and isomorphism of monoidal functors;
2) finite dimensional bialgebras H over k up to isomorphism.
Proof. Straightforward from the above.
�
Theorem 1.21.3 is called the reconstruction theorem for finite dimen
sional bialgebras (as it reconstructs the bialgebra H from the category
of its modules using a ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
roduct Δ(x) = x ⊗ x, and
counit ε(x) = 1, x ∈ G. Note that the bialgebra k[G] may be defined
for any G (not necessarily finite).
·
Exercise 1.21.6. Let H be a k-algebra, C = H −mod be the category
of H-modules, and F : C → Vec be the forgetful functor (we don’t
assume finite dimensionality). Assume that C is monoida... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
H, the monoidal category of right H-comodules will
be denoted by H − comod, and the subcategory of finite dimensional
comodules by H − comod.
1.22. Hopf algebras. Let us now consider the additional structure
on the bialgebra H = End(F ) from the previous subsection in the case
when the category C has right duals. I... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
) for b = η ⊗ ν, where η, ν ∈ H, which is straightforward.
Now the first equality of the proposition follows from the commuta
coevF (X)
evF (X)
F,F
∼
tivity of the diagram
(1.22.2)
coevF (X)
� F (X) ⊗ F (X)∗ ⊗ F (X)
F (X)
Id
� �
F (X)
η1
� �
F (X)
JX,X∗
F (coevX )
� �
� F (X ⊗ X ∗) ⊗ F (X)
ηX⊗X∗
F (coevX ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
An antipode on a bialgebra H is a linear map
H which satisfies the equalities of Proposition 1.22.1.
S : H
→
Exercise 1.22.3. Show that the antipode axiom is self-dual in the
following sense: if H is a finite dimensional bialgebra with antipode
SH , then the bialgebra H ∗ also admits an antipode SH ∗ = S∗
H .
The ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
it.
�
Proof. Let
(Δ ⊗ Id) Δ(a) = (Id ⊗ Δ) Δ(a) =
◦
◦
�
2
1
3
a
⊗ a
⊗ a
i ,
i
i
(Δ ⊗ Id) Δ(b) = (Id ⊗ Δ) Δ(b) =
◦
◦
i
�
1
2
j ⊗ bj ⊗ b
b
3
j .
j
Then using the definition of the antipode, we have
�
�
S(ab) =
S(a
2
1
i b)ai S(ai ) =
3
S(a
11
3
i bj )ai bj S(bj )S(ai ) = S(b)S(a).
22
3
i
i,j
Thus... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
addition S is invertible, then C also admits left duals, i.e.
is rigid (in other words, C is tensor category). Namely, for any object
X, the left dual ∗X is the usual dual space of X, with action of H given
by
ρ∗X (a) = ρX (S−1(a))∗,
and the usual evaluation and coevaluation morphisms of the category
Vec.
Proof.... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
a Hopf algebra.
Remark 1.22.10. We note that many authors use the term “Hopf
algebra” for any bialgebra with an antipode.
Thus, Corollary 1.22.6 states that if H is a Hopf algebra then Rep(H)
is a tensor category. So, we get the following reconstruction theorem
for finite dimensional Hopf algebras.
Theorem 1.22.11... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
if and only if G is a group, with
S(x) = x−1 , x ∈ G.
Exercises 1.21.5 and 1.22.12 motivate the following definition:
Definition 1.22.13. In any coalgebra C, a nonzero element g ∈ C
such that Δ(g) = g ⊗ g is called a grouplike element.
Exercise 1.22.14. Show that if g is a grouplike of a Hopf algebra H,
then g is i... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
show that m = 0. If not, we can assume
that m = 1 by replacing H with Hm−1.
We have a map S� : H1 → H1 inverse to S. For a ∈ H, let the triple
coproduct of a be
Consider the element
b =
�
1
2
3
ai ⊗ ai ⊗ ai .
i
�
S�(S(ai
1))S(ai
3 .
2)ai
i
On the one hand, collapsing the last two factors using the antip... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
= Δop). Let S be an antipode on H. Show that
S2 = 1.
op
op
(iii) Assume that bialgebras H and H cop have antipodes S and S�.
Show that S� = S−1, so H is a Hopf algebra.
Exercise 1.22.17. Show that if A, B are bialgebras, bialgebras with
antipode, or Hopf algebras, then so is the tensor product A ⊗ B.
Exercise 1... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
end(F ). So the algebra End(F )
(which may be infinite dimensional) carries the inverse limit topology,
in which a basis of neighborhoods of zero is formed by the kernels KX
of the maps End(F ) → End(F (X)), X ∈ C, and Coend(F ) = End(F )∨,
the space of continuous linear functionals on End(F ).
The following theore... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
, and I is naturally a left C-comodule (the comod
ule structure is induced by the coevaluation morphism F (X)∗ ⊗ X →
F (X)∗ ⊗ F (X) ⊗ F (X)∗ ⊗ X).
(iii) Let us regard F as a functor C → C − comod. For M ∈ C −
comod, let θM : M ⊗I M ⊗C ⊗I be the morphism πM ⊗Id−Id⊗πI ,
and let KM be the kernel of θM . Then the functor... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
). Thus Theorem
1.23.1 implies the following “infinite” extensions of the reconstruction
theorems.
Theorem 1.23.2. The assignments (C, F ) �→ H = Coend(F ), H �→
(H − Comod, Forget) are mutually inverse bijections between
1) k-linear abelian monoidal categories C with a fiber functor F , up
to monoidal equivalence a... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
that do not have left duals, i.e., are not
tensor categories (namely, H − comod).
In the next few subsections, we will review some of the most im
portant basic results about Hopf algebras. For a much more detailed
treatment, see the book [Mo].
1.24. More examples of Hopf algebras. Let us give a few more
examples ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
Prim(H).
Exercise 1.24.3. (i) Show that Prim(H) is a Lie algebra under the
commutator.
(ii) Show that if x is a primitive element then ε(x) = 0, and in
presence of an antipode S(x) = −x.
Exercise 1.24.4. (i) Let V be a vector space, and SV be the symmet
ric algebra V . Then SV is a Hopf algebra (namely, it is the... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
1 = qx.
Define the coproduct on H by Δ(g) = g ⊗ g, Δ(x) = x ⊗ g + 1 ⊗ x. It is
easy to show that this extends to a Hopf algebra structure on H. This
Hopf algebra H is called the Taft algebra. For n = 2, one obtains the
Sweedler Hopf algebra of dimension 4. Note that H is not commutative
or cocommutative, and S2 = 1... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
2Z acts on xi by gxig−1 = −xi. De
fine the coproduct on H by making g grouplike, and setting Δ(xi) :=
xi ⊗ g + 1 ⊗ xi (so xi are skew-primitive elements). Then H is a Hopf
algebra of dimension 2n+1 . For n = 1, H is the Sweedler Hopf algebra
from the previous example.
�
54
Exercise 1.24.10. Show that the Hopf alg... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
software
studio
an overview of Rails
Daniel Jackson
1 what is Rails?
an application framework
› full stack: web server, actions, database
a programming environment
› eg, rake (like make), unit testing
an open-source community
› many plugins
2 history of Rails
genesis in Basecamp
› project management tool... | https://ocw.mit.edu/courses/6-170-software-studio-spring-2013/41673afb1ac2f91f3724fd7d28c5da14_MIT6_170S13_10-rails-ovrvw.pdf |
missing specs
› not clear what’s going on
› magic changes over time
an example
› which fields in forms are logged?
› next slide...
11 Pull request by jeyb on GitHub.
12 in summary...
rich environment
many libraries
code generation
helpful community
friendly online guides
invisible magic
quirky conventions
n... | https://ocw.mit.edu/courses/6-170-software-studio-spring-2013/41673afb1ac2f91f3724fd7d28c5da14_MIT6_170S13_10-rails-ovrvw.pdf |
MATH 18.152 COURSE NOTES - CLASS MEETING # 9
18.152 Introduction to PDEs, Fall 2011
Professor: Jared Speck
Class Meeting # 9: Poisson’s Formula, Harnack’s Inequality, and Liouville’s
Theorem
1. Representation Formula for Solutions to Poisson’s Equation
We now derive our main representation formula for solution’s to Poi... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
( )∇ ( ) ( − )
ˆN σ Φ x σ dσ.
Recall also that
(1.0.4)
where
(1.0.5)
and
(1.0.6)
G(x, y) = Φ(x − y) − φ(x, y),
∆yφ(x, y
) = 0,
x ∈ Ω,
(
G x, σ
) =
0 when x Ω and σ ∂Ω.
∈
∈
The expression (1.0.3) is not very useful since don’t know the value of
fix this, we will use Green’s identit
and recalling that ∆yφ x, y
) =
(
y. Ap... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
. We’ll use
⊂
that
P x, σ) = ∇ ˆN G(x, σ from (1.0.2)
a technique
works for special domains.
(
def
Warning 2.0.1. Brace yourself for a bunch of tedious computations that at the end of the day will
lead to a very nice expression.
(
)
The basic idea is to hope that φ x, y from the decomposition G x, y
(
)
φ x, y , where
... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
∣y∣ = R, which implies that
(2.0.9)
Simple algebra then leads to
1
4π∣x − y
∣
=
q
∗ − ∣
4π∣x
y
.
(2.0.10)
∗
∣
x
−
y 2
∣ =
q2 x y 2.
∣ − ∣
When ∣y∣ = R we
,
use
x∗ 2
∣
∣ −
(2.0.10) to compute
2x∗ ⋅ y + R2 = ∣x∗ − y
that
∣2 = q2∣ − y∣2 = q2
x
(∣
x∣2 − 2x ⋅ y + R ,
)
2
the
Euclidean dot product. Then performing simple
alg... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
R2
x∣
∣
2
x.
φ(x, y
) =
1
4π x R
∣
∣
R
2
− ∣
x∣2 x y
∣∣
,
(
φ 0, y
) =
1
4πR
,
where we took a limit as x
→ 0
Next, using (2.0.8), we have
in
(2.0.16)
to derive (2.0.17).
(2.0.18)
G(x, y) = −
(2.0.19)
(
G
0, y
)
= −
1
4π x − y∣
∣
+
1
4π ∣x∣∣
R
2
R
x 2 x − y
∣
∣
∣
,
1
∣
4π y
∣
+
1
4πR
.
For future use, we
also
compute
t... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
x
∣
∣
x 2
R2
)
.
( −
1
=
= −
−
x σ
∣ − ∣3
σ
π
4 x
+
∣
∣
1 x 2
4π R2
R2
−
x 2 x σ
∣
∣
− σ∣3
∣x
Using (2.0.22) and the fact
that
(2.0.23)
∇ ( ) (
ˆN σ G x, σ
ˆ σ
( ) = 1
N
R σ, w
e deduce
def
) = ∇ (
σG x, σ N σ
) ⋅
( ) =
ˆ
R
2
−
4π
∣
∣
x 2
R
1
∣ − ∣
x σ 3
.
4
MATH 18.152 COURSE NOTES - CLASS MEETING # 9
Remark 2.0.2. I... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
�
1
4 R
π
.
R2
x−p
R
−
(x
∣2
p
) − (y − p
)∣
,
≠
x p,
Furthermore, if x
∈ BR(p) and σ ∈ ∂BR(p), then
(2.0.25c)
∇
(
ˆN (σ)G x, σ
) =
∣
− 2
R2 − ∣x p
4πR
1
∣ − ∣
3
x σ
.
We can now easily derive a representation formula for solutions
to the Laplace equation on a ball.
Theorem 2.1 (Poisson’s formula). Let BR(p) ⊂
= (
)
(
... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
unit ball in R
∂BR(
p
n.
)
(
g σ
∣
− σ
dσ,
n
Proof. The identity (2.0.27) follows immediately from Theorem 1.1 and Lemma 2.0.1.
(cid:3)
3. Harnack’s inequality
We will now use some of our tools to prove a famous inequality for Harmonic functions. The
theorem provides some estimates that place limitations on how slow/fa... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
∂BR 0
Applying the first inequality to (3.0.30), and using the non-negativity of
(i.e. σ
we have that ∣x R x σ
∣
≤
e
g, w
− ∣ ≤
deduce
R
),
∈
∣ +
∣
x R.
that
(3.0.31)
)
∂BR(0
Now recall that by the mean value property, we have that
u(x) ≤
R
R2
+ ∣x
− ∣
x
∣
1
∣2 4πR
∫
( )
g σ dσ.
(3.0.32)
u( ) =
0
1
4πR2
Thus, combining ... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
), if x Rn and R is sufficiently large,
we have that
u M . Observe that v
0
( ) ≥
+ ∣
≥
∈
∣
def
=
(3.0.34)
Rn−2(R − ∣
∣)
x
−
( + ∣
∣)
R x n 1
v(0) ≤
v
)
(x
≤
−
( + ∣
n 2 R
R
∣)
∣
−
(
x
R
∣)
x
n−1
)
v(0 .
Allo
wing R
(and therefore u is to
→ ∞
o).
in (3.0.34), we conclude that v x
( ) = ( )
0
v
. Th
us, v is
a constant-va... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
18.413: ErrorCorrecting Codes Lab
February 24, 2004
Lecturer: Daniel A. Spielman
Lecture 6
6.1
Introduction
Begin by describing LDPC codes, and how they are described by many local constraints. Point out
that random graphs locally look like trees (from the birthday paradox), and so we will learn to do
belief p... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
= a1] =
|{a2 : (a1, a2) ∈ C}|
|C|
Ppost [X1 = a1|Y1Y2 = b1b2] = cb1,b2Pprior [X1 = a1] Pext [X1 = a1 Y1Y2 = b1b2] ,
|
so it suffices to prove
Lemma 6.2.2.
Pext [X1 = a1|Y1Y2 = b1b2] = cb1,b2Pext [X1 = a1 Y1 = b1] Pext [X1 = a1 Y2 = b2] .
|
|
61
Lecture 6: February 24, 2004
62
Proof. We begin by examining the ri... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
we examine the lefthandside:
Pext [X1 = a1|Y1Y2 = b1b2] = cb1,b2P [Y1Y2 = b1b2 X1 = a1]
= cb1,b2
�
|
P [Y1Y2 = b1b2|X1X2 = a1a2] P [X2 = a2 X1 = a1]
|
= cb1,b2
a2:(a1,a2)∈C
�
a2:(a1,a2)∈C
= cb1,b2P [Y1 = b1|X1 = a1]
|
P [Y1 = b1|X1 = a1] P [Y2 = b2 X2 = a2] P [X2 = a2 X1 = a1]
|
�
a2:(a1,a2)∈C
P [Y2 = b2|X... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
uniformly subject to this condition.
The variables (X1, X2, X3) then satisfy what the book calls the “Markov” property. That is, for
all a1, a2, a3,
P [X1X3 = a1a3|X2 = a2] = P [X1 = a1 X2 = a2] P [X3 = a3 X2 = a2] .
|
|
In this case, we can say that all the information that X3 contains about X1 is transmitted throu... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
2 = b2 X2 = a2]
|
�
a3:(a2,a3)∈C23
P [Y3 = b3|X3 = a3] P [X3 = a3 X2 = a2]
|
P [X2 = a2|X1 = a1] P [Y2 = b2 X2 = a2] Pext [X2 = a2 Y3 = b3]
|
|
6.5 Trees
A hypergraph is given by a collection of vertices x1, . . . , xn and a collection of edges e1, . . . , em,
where each ei ⊆ {x1, . . . , xn}. A path in a hypergra... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
uniformly subject to constraints, each of which involves
only the variables in an edge, and the corresponding hypergraph is a tree, then Lemma 6.4.1 can
be extended to an algorithm for belief computation in the tree. | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
6.763 Applied Superconductivity
Lecture 1
Terry P. Orlando
Dept. of Electrical Engineering
MIT
September 8, 2005
Outline
• What is a Superconductor?
• Discovery of Superconductivity
• Meissner Effect
• Type I Superconductors
• Type II Superconductors
• Theory of Superconductivity
• Tunneling and the Josephs... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
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