text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
0, first k contains no match
– Assume for induction hypothesis holds for k = k0, and consider k = k0 + 1
– If first k0 contains a match, already returned a match by induction
– Else first k0 do not have match, so if first k0 + 1 has match, match contains k0 + 1
– Then algorithm checks directly whether birthday of stude... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
• Model in this class is called the Word-RAM
• Machine word: block of w bits (w is word size of a w-bit Word-RAM)
• Memory: Addressable sequence of machine words
• Processor supports many constant time operations on a O(1) number of words (integers):
– integer arithmetic: (+, -, *, //, %)
– logical operators: (&&,... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
2
’’’
Find a pair of students with the same birthday
Input: tuple of student (name, bday) tuples
Output: tuple of student names or None
’’’
n = len(students)
record = StaticArray(n)
for k in range(n):
(name1, bday1) = students[k]
# Return pair if bday1 in record
for i in range(k):
(name2, bday2) = record.ge... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
Insertion Sort (L03)
Selection Sort (L03)
Merge Sort (L03)
Counting Sort (L05)
Radix Sort (L05)
AVL Sort (L07)
Heap Sort (L08)
Shortest Path Algorithms
Breadth First Search (L09)
DAG Relaxation (L11)
Depth First Search (L10)
Topological Sort (L10)
Bellman-Ford (L12)
Dijkstra (L13)
Johnson (L14)
Floyd-War... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
Assignments
Problems Sets:
• Problem Set 2 (Scheduling) due
• Problem Set 3 (PDDL Modeling) out soon
Readings:
• Hoffman, Porteous, Sebastia, “Ordered Landmarks in Planning,” Journal of
Artificial Intelligence Research, 22, pp. 215-278, 2004. (Voted most influential
paper during ICAPS 2013).
Karpas, et al., “Temporal ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
food and
water
Get the
Rover
ready for
long trips
Alone
on Mars
at Ares 3
Have
MAV
ready
Be
Rescued
Re-Establish
communication
Drive to
Ares 4
Be at
Ares 4
[0, 4 years]
2/24/2016
Cognitive Robotics
5
Outline
• What Landmarks Are
• How Landmarks Are Discovered
• Using Landmarks
– Subgoals
– Heuristic Estimat... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
B
2/24/2016
Cognitive Robotics
11
Temporal Landmarks
• A temporal fact landmark is a formula over
facts that becomes true from time point ts to
te in every valid plan.
– I need to be at Ares 4 within 4 years
• A temporal action landmark is an action which
occurs at time point t in every valid plan
– I have to launc... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
-complete
• Deciding if a given fact is a landmark is PSPACE-
complete
• Deciding if there is a natural / necessary / greedy-
necessary / reasonable ordering between two
landmarks is PSPACE-complete
2/24/2016
Cognitive Robotics
17
Landmark Discovery
• Theory
A is a landmark ⟺ π’A is unsolvable
where π’A is π without ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
t-at-A
Landmark Discovery (1)
Step 1: Find landmarks candidates and orderings
Example
E
A
B
C
D
Landmark: t-at-D
t-at-A
t-at-D
2/24/2016
Cognitive Robotics
23
Landmark Discovery (1)
Step 1: Find landmarks candidates and orderings
Example
E
A
B
C
D
Landmark: t-at-D
t-at-A
drive-t-A-E
drive-t-A-B
t-at-E
t-at-B
t-at-A... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
27
Landmark Discovery (1)
Step 1: Find landmarks candidates and orderings
initialize the LGG to (G, ∅), and set C := G
while C = ∅ do
set C′ := ∅
for all L’ ∈ C, level(L‘) ≠ 0 do
let A be the set of all actions a such that L′ ∈ add(a), and level(a) = level(L′ ) − 1
for all facts L such that ∀a ∈ A : L ∈ pre(a) do
if L... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
t-at-E and its orderings
Landmarks: {A, D}
t-at-E
t-at-E
t-at-D
{A, E, D}
2/24/2016
Cognitive Robotics
32
Landmark Discovery (1)
Disjunctive landmarks also possible, e.g.,
(o-in-p1 ∨ o-in-p2):
•
If B is landmark and all actions that (first) achieve B have A or
C as precondition, then A ˅ C is a landmark.
2/24/2016
C... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
( v ↦ d’ ) → ( v ↦ d )
2/24/2016
Cognitive Robotics
37
Landmark Discovery (2)
p
E
C
o
B
A
D
t
B
t
A
C
D
p
E
Find landmarks through DTGs, if
•
•
•
s0( v ) = d0 ,
v ↦ d landmark (goal), and
every path from d0 to d passes through d’,
then v ↦ d’ landmark, and ( v ↦ d’ ) → ( v ↦ d )
2/24/2016
Cognitive Robotics
38
Landma... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
a,b,e
e
a,c,d,f
f
a,g
g
a
2/24/2016
Cognitive Robotics
42
Landmark Discovery (3)
Facts
Actions
Facts
Actions
Facts
a
1
a
2
a
1
a,b
b
a,c
c
a,d
d
a,b
4
a,c,d
5
a,d
6
a,b,e
e
a,c,d,f
f
a,g
g
a
2/24/2016
Cognitive Robotics
43
Landmark Discovery (3)
Facts
Actions
Facts
Actions
Facts
a
1
a
2
a
1
a,b
b
a,c
c
a,d
d
a,b
4
a,... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
Robotics
46
Outline
• What Landmarks Are
• How Landmarks Are Discovered
• Using Landmarks
– Subgoals
– Heuristic Estimates
– Admissible Heuristic Estimates
– Enriching the Problem
– Beyond Classical Planning
• Summary
2/24/2016
Cognitive Robotics
47
Using Landmarks
• So what can we do once we have these landmarks?
• ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
t-at-C
p-at-C
o-at-C
o-in-p
o-at-E
Partial Plan: drive-t-B
Goal: o-in-t ∨ p-at-C
2/24/2016
Cognitive Robotics
53
Using Landmarks: Subgoals
p
E
C
A
D
B
o
t
Partial Plan: drive-t-B, load-o-t
Goal: t-at-C ∨ p-at-C
o-at-B
t-at-B
o-in-t
t-at-C
p-at-C
o-at-C
o-in-p
o-at-E
2/24/2016
Cognitive Robotics
54
Using Landmarks: Su... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
-p
o-at-E
Partial Plan: drive-t-B, load-o-t, drive-t-C, unload-o-C, fly-p-C,
load-o-p
Goal: o-at-E
2/24/2016
Cognitive Robotics
58
Using Landmarks: Subgoals
B
A
D
C
t
p
E
o-at-B
t-at-B
o-in-t
t-at-C
o
p-at-C
o-at-C
o-in-p
o-at-E
Partial Plan: drive-t-B, load-o-t, drive-t-C, unload-o-C, fly-p-C,
load-o-p, fly-p-E, un... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
Partial Plan: pickup-B, stack-B-C
Goal: clear-A
2/24/2016
Cognitive Robotics
64
Using Landmarks: Subgoals
on-table-A
on-C-A
clear-C
hand-empty
on-table-B
clear-B
clear-A
holding-B
A
C
B
holding-A
on-B-C
on-A-B
Partial Plan: pickup-B, stack-B-C, unstack-B-C, putdown-B,
unstack-C-A, putdown-C
Goal: holding-A
2/24/2016
C... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
is very fast - the base planner needs to
plan to a lesser depth
• Cons:
– Can lead to much longer plans
– Not complete in the presence of dead-ends
2/24/2016
Cognitive Robotics
69
Outline
• What Landmarks Are
• How Landmarks Are Discovered
• Using Landmarks
– Subgoals
– Heuristic Estimates
– Admissible Heuristic Esti... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
/24/2016
Cognitive Robotics
74
LAMA: Required Again Landmarks
• A landmark A is required again by path π in state s if:
– false-goal: A is false in s and is a goal, or
– open-prerequisite: A is false in s and is a greedy-necessary
predecessor of some landmark that is not accepted
• It’s also possible to use (Buffet a... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
P) is specified as before by s and the
various rules
• L(s, P) is the set of landmarks that we know still needs to be
achieved after reaching state s via the paths in P
(Karpas and Domshlak, 2009)
2/24/2016
Cognitive Robotics
78
Outline
• What Landmarks Are
• How Landmarks Are Discovered
• Using Landmarks
– Subgoals... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
, the sum of costs of landmarks that still need to be
achieved is an admissible heuristic, hL
hL(s , π) := cost(L(s , π)) = ∑ cost(A)
A ∈ L(s, π)
2/24/2016
Cognitive Robotics
82
Admissible Heuristic Estimates:
Admissible Cost Sharing
• How can we find such a partitioning?
• Easy answer - uniform cost sharing - each a... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
0)=1.0
min(1.0)=1.0
min(0,0,0,0)=0
2/24/2016
Cognitive Robotics
86
Admissible Heuristic Estimates:
Optimal Cost Sharing
• The good news: the optimal cost partitioning is poly-time to
compute
– The constraints for admissibility are linear, and can be used in a Linear
Program (LP)
– Objective: maximize the sum of land... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
doing this have been proposed
(Wang, Baier and McIlraith, 2009 and Domshlak, Katz and Lefler, 2010)
2/24/2016
Cognitive Robotics
93
Using Landmarks:
Enriching the Problem
Viewing Landmarks as Temporally Extended Goals:
• Landmarks and their orderings can be viewed as temporally
extended goals (Wang, Baier and McIlr... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
Admissible Heuristic Estimates
– Enriching the Problem
– Beyond Classical Planning
• Summary
2/24/2016
Cognitive Robotics
97
Using Landmarks:
Beyond Classical Planning
• Probabilistic landmarks
– a landmark is a fact which must be true in every
successful trajectory (possible execution)
• Temporal Landmarks
2/24/201... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
24/2016
Cognitive Robotics
104
Durative Action: Turn On Flashlight
Duration: 1 seconds
Start:
Condition: have-flashlight
Effect:
Invariant condition:
End:
Condition:
Effect: light
2/24/2016
Cognitive Robotics
105
Causal Landmarks for Flashlight Match Cellar
If we run a casual landmark discovery, we would
get:
• ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
3
…
2/24/2016
Cognitive Robotics
111
•
Temporal Landmarks for Flashlight Match Cellar
•
The goal must hold from some time point g until the end E
– holdsg:E(fuse-fixed)
The only event which can achieve fuse-fixed is END(fix-fuse), which
must occur exactly at g
– occursg(END(fix-fuse))
Every action that ends must star... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
so
– holdssm:em(light), with sm < sf and em = shf - e
•
•
sm
holds(light)
em
ehf
sff
[10,10]
g
[2,2]
shf
sf
[1,1]
sl
to
holds(light)
el
holds(fuse-fixed)
E
START (find-
flashlight)
END (find-
flashlight)
START (turn-
on-flashlight)
END (turn-on-
flashlight)
START
(fix-fuse)
END
(fix-fuse)
2/24/2016
Cognitive Robotics
1... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
Plan “skeleton”
– Use underlying STN as heuristic to estimate makespan
– Enriching the Problem: “Compile” landmarks into the problem
2/24/2016
Cognitive Robotics
117
Some Results
• Compilation approach
– In the compilation, we limit the size of a disjunction to 1, 4, or ∞
• Comparing performance of planners with and ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
1)
floortile (2011)
parcprinter
(2011)
parking (2011)
parking (2014)
pegsol (2011)
satellite (2014)
sokoban (2011)
3
1
0
0
0
0
1
2
1
20 19 19
12 12 12
19 19 10
4
3
4
3
2
2
0
2
5
18
17
3
2
0
TOTAL
102 97 85
83
crew (2011)
elevators (2011)
floortile (2011)
mapanalyser
(2014)
20 20
20 19
5
5
17 17
6
5
0
0
opensta... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
125
2/24/2016
Cognitive Robotics
120
Summary
• Landmarks provide a way to utilize the implicit
structure of a planning problem
• Landmarks work well in
– Classical planning
– Partially observable planning with sensing (Maliah et al, 2014)
– Oversubscription Planning (Mirkis & Domshlak, 2014)
– Temporal planning
• A... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
Lecture 3
8.321 Quantum Theory I, Fall 2017
11
Lecture 3 (Sep. 13, 2017)
3.1 Even More Math
3.1.1 More on Matrix Representations
Last time, we described that in a given basis, there is an exact correspondence between n×n matrices
and operators, where n is the dimension of the Hilbert space. Let {|a(cid:105)} form an or... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/47a22b9e37fbdba8448bb927004fed18_MIT8_321F17_lec3.pdf |
orthonormal bases {|ai(cid:105)} and {|bi(cid:105)}. How are these bases related? We
can define an operator U by the action
This implies that
by definition of the adjoint.
Note that we can write
U |ai(cid:105) = |bi(cid:105) .
(cid:104)bi| = (cid:104)ai|U † ,
U = U 1 = U
|ai(cid:105)(cid:104)ai| =
(cid:88)
i
|bi(cid:105)... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/47a22b9e37fbdba8448bb927004fed18_MIT8_321F17_lec3.pdf |
U
|aj(cid:105)
dj
|ai(cid:105)(cid:104)ai|U |aj(cid:105) .
(3.12)
Thus, we see that
cij =
(cid:88)
i,j
(cid:104)ai|U |aj(cid:105)dj = Uijdj ,
(3.13)
where we have introduced the shorthand notation in which repeated indices are assumed to be
summed over (from now on, we will explicitly state the cases when we are not us... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/47a22b9e37fbdba8448bb927004fed18_MIT8_321F17_lec3.pdf |
)
where the first expression on the left is not summed over i. Thus, we see that Uik
diagonal matrix. This completes the proof.
† Hk(cid:96)U(cid:96)j is a
3.1.4 Simultaneous Diagonalization
Theorem 3. Two (diagonalizable) operators A, B are simultaneously diagonalizable if and only if
[A, B] = 0, where [·, ·] is the co... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/47a22b9e37fbdba8448bb927004fed18_MIT8_321F17_lec3.pdf |
A
=
a
1
a
2
.
.
.
a
2
. . .
, B =
B1
B
2
,
.
.
.
(3.21)
where the Bi are ki × ki blocks, with ki the number
of occurrences of the eigenvalue ai on the
diagonal of A. Diagonalizing the block Bi only mixes eigenkets of A with the same eigenvalue, so
we can diagonalize B whi... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/47a22b9e37fbdba8448bb927004fed18_MIT8_321F17_lec3.pdf |
) ,
where we have defined the measurement operator
Mai :=
(cid:88)
j:a
j
=a
i
|aj(cid:105)(cid:104)aj| .
This operator is the projector onto the subspace with A = ai.
If we simplify to the case where there is only one eigenket with eigenvalue ai. Then
Prob(A = a
2
i) = |(cid:104)ai|ψ(cid:105)|
.
(3.22)
(3.23)
(3.24)
His... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/47a22b9e37fbdba8448bb927004fed18_MIT8_321F17_lec3.pdf |
:88)
j
= (cid:104)ψ|ψ(cid:105)
= 1 ,
(3.26)
because the state |ψ(cid:105) is assumed to be normalized.
2. For any observable A and state |ψ(cid:105), the expectation value of A is
(cid:88)
(cid:104)A(cid:105) :=
ai Prob(A = ai)
ai
(cid:88)
ai
=
(cid:88)
ai
j:a
j =a
i
= (cid:104)ψ|A|ψ(cid:105) ,
| (cid:105)
ψ aj aj ψ
(c... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/47a22b9e37fbdba8448bb927004fed18_MIT8_321F17_lec3.pdf |
the form
−
R
c+ = cos
θ
2
,
c− = eiφ sin
θ
2
,
(3.30)
with 0 ≤ θ < π and 0 ≤ φ < 2π. Specifying these two angles specifies the state exactly, and we note
that specifying these angles is equivalent to specifying a point on the surface of the unit sphere S2.
This parametrization of the Hilbert space is known as the Bloch ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/47a22b9e37fbdba8448bb927004fed18_MIT8_321F17_lec3.pdf |
2
=
(cid:126)
2
(cid:19)
(cid:18)0 1
1 0
(cid:18) −i
(cid:126)
0
0
i
2
(cid:19)
(|+(cid:105)(cid:104)−| − |−(cid:105)(cid:104)+|) =
(cid:126)σy
2
=
We can check explicitly (left as an exercise) that these operators satisfy
(cid:104)
Sa, Sb(cid:105)
Sa Sb(cid:111)
(cid:110)
,
= i(cid:126)(cid:15)abcSc
(cid:126)
2
=
δab1... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/47a22b9e37fbdba8448bb927004fed18_MIT8_321F17_lec3.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.006 Introduction to Algorithms
Fall 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/47a412008dc74679e9072891be09f351_MIT6_006F11_lec07_orig.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.641 Electromagnetic Fields, Forces, and Motion, Spring 2005
Please use the following citation format:
Markus Zahn, 6.641 Electromagnetic Fields, Forces, and Motion, Spring
2005. (Massachusetts Institute of Technology: MIT OpenCourseWare).
http://ocw.mit.edu (accessed MM DD, ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/47fbb5ff17cb74a5d6a5b9c1eeda9c23_lecture3.pdf |
1 of 12
B. Estimate of Error introduced by EQS approximation
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
E =
_
_
i z = E0 i z
V
d
⎪⎧−εE0
σ = ⎨
su
⎪+εE0
⎩
z = d
z = 0
K 2 b π + π b2 dσsu = 0 ⇒ K = −
dt
r
r
b dσsu = −
2 dt
b ε
2
dE
0
dt
(cid:118)∫ i
H ds =
C
∂ ( E) i da ⇒ H... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/47fbb5ff17cb74a5d6a5b9c1eeda9c23_lecture3.pdf |
=
E0
2
εµ d E
2
4E
0 dt
0 (b2 − r2 ) =
1
ω2εµ(b2 − r2 )
4
Eerror (cid:19) 1 ⇒
E0
2ε 2
ω µb
4
(cid:19) 1
fλ = c =
1
εµ
ω
2π
λ = c ⇒ ω =
2ε 2
2 c ω µ b
4
π
λ
⇒
=
2
π
λ2
2
b (cid:19) 1 ⇒ b (cid:19)
λ
π
f=1 MHz in free space ⇒ λ =
×
3 10 8
6
10
= 300 m
If b (cid:19) 100 m
EQS approximation is va... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/47fbb5ff17cb74a5d6a5b9c1eeda9c23_lecture3.pdf |
with permission.
τem =
L
c
= L εµ
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 3
Page 4 of 12
III. Boundary Conditions
1. Gauss’ Continuity Condition
Courtesy of Krieger Publishing. Used with permission.
(cid:118)∫ ε0E i da = ∫σsdS ⇒ ε0 (E2n
- E1n
) dS = σsdS
S
S
... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/47fbb5ff17cb74a5d6a5b9c1eeda9c23_lecture3.pdf |
��
⎣
H a - H b
= K
⎤
⎦
5. Conservation of Charge Boundary Condition
d
(cid:118)∫ J da +
dt ∫ ρdV = 0
i
V
S
n i ⎡ J a - Jb
⎣
⎤ +
⎦
∂
∂t
σ s = 0
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 3
Page 6 of 12
6. Electric Field from a Sheet of Surface Charge
a. Electric ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/47fbb5ff17cb74a5d6a5b9c1eeda9c23_lecture3.pdf |
0 (x2 + y2 )1
2
cos θ =
σ0ydx
2πε0 (x2 + y2 )
E =
y
+∞
∫
x = −∞
dE
y
=
=
+∞
σ0y
2πε0 ∫ x2 + y2
dx
x = −∞
σ0y 1
2πε0 y
tan−1 x
+∞
y −∞
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 3
Page 9 of 12
⎧ σ0
⎪
⎪2ε0
= ⎨
σ0
⎪−
⎪ 2ε
⎩
0
y > 0
y < 0
Checking Boundary cond... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/47fbb5ff17cb74a5d6a5b9c1eeda9c23_lecture3.pdf |
y > a
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 3
Page 10 of 12
7. Magnetic Field from a Sheet of Surface Current
From a line current I
Hφ =
I
π2 r
_
i = − sin
φ
_
φ i + cos
x
_
φ i
y
Thus from 2 symmetrically located line currents
dH =
x
dI
π(
2 x2 + y2
2
)1
(− si... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/47fbb5ff17cb74a5d6a5b9c1eeda9c23_lecture3.pdf |
)
(13.4)
(13.5)
(13.6)
(13.7)
tS
δ 0eij
0
0
ij d V
(13.8)
(13.9)
MIT 2.094
13. Total Lagrangian formulation, cont’d
tK
0 L =
tK =
0 N L
tF =
0
�
0V
�
0V
�
0V
tBT
0
0 L 0C 0 Ld V
tB
T
tB
0 N L 0
tBN Ld V
0
0
tS
����
matrix
tB tSˆ d V
T
0
0 L 0
����
vector
The iteration (full Newton-Raph... | https://ocw.mit.edu/courses/2-094-finite-element-analysis-of-solids-and-fluids-ii-spring-2011/484bed6dca1e54c56a6b993cdf635560_MIT2_094S11_lec13.pdf |
0
−1
0
0 −1
��
tK0 N L
0
−1
0
1
⎤
⎥
⎥
⎦
�
When θ = 0, 0
tKL doesn’t give stiffness corresponding to u2
2, but 0
tKN L does.
56
MIT OpenCourseWare
http://ocw.mit.edu
2.094 Finite Element Analysis of Solids and Fluids II
Spring 2011
For information about citing these materials or our Terms of Use, visit: ht... | https://ocw.mit.edu/courses/2-094-finite-element-analysis-of-solids-and-fluids-ii-spring-2011/484bed6dca1e54c56a6b993cdf635560_MIT2_094S11_lec13.pdf |
15.082J, 6.855J, and ESD.78J
Sept 16, 2010
Lecture 3. Graph Search
Breadth First Search
Depth First Search
Intro to program verification
Topological Sort
Overview
Today: Different ways of searching a graph
a generic approach
breadth first search
depth first search
program verification
data structures to supp... | https://ocw.mit.edu/courses/15-082j-network-optimization-fall-2010/487a2b4c96b8a42c4c3e6372b71b7be6_MIT15_082JF10_lec03.pdf |
0; {that is, it has no predecessor}
LIST = {s}
while LIST ≠ ø do
select a node i in LIST;
if node i is incident to an admissible arc (i,j) then
mark node j;
pred(j) := i;
add node j to the end of LIST;
else delete node i from LIST
The algorithm in the book also keeps track of the
order in which nodes are marked.
6
... | https://ocw.mit.edu/courses/15-082j-network-optimization-fall-2010/487a2b4c96b8a42c4c3e6372b71b7be6_MIT15_082JF10_lec03.pdf |
LIST;
if node i is incident to an admissible arc (i,j) then
mark node j;
pred(j) := i;
add node j to the end of LIST;
else delete node i from LIST
13
Algorithm Invariants
22
2
2
22
44
4
4
4
4
4
3
88
8
8 4
11 1
1
1
5
55
5
5
5
5
77
7
7
8
Select Node 2
LIST
1
3
2
6
66
6
6
6
6
7
99
9
9
9
9
4
85
6
9
7
Invariants: whenever... | https://ocw.mit.edu/courses/15-082j-network-optimization-fall-2010/487a2b4c96b8a42c4c3e6372b71b7be6_MIT15_082JF10_lec03.pdf |
Otherwise, by Invariant 2,
j would have been marked when (i, j) was scanned.
Therefore, no node in T is reachable from s.
17
Cutset Theorem
Corollary of algorithm’s correctness. There
is no directed path from s to t if and only if
the following is true:
there is a partition of the node set N into
subsets S and T ... | https://ocw.mit.edu/courses/15-082j-network-optimization-fall-2010/487a2b4c96b8a42c4c3e6372b71b7be6_MIT15_082JF10_lec03.pdf |
Coolidge believed that the world was flat.
False. But Andrew Jackson did.
Finding all connected components in an
undirected graph
Breadth first search will find a connected
component of an undirected graph in time
proportional to the number of arcs in the
Comp 1
component. (A component of an undirected
graph is a... | https://ocw.mit.edu/courses/15-082j-network-optimization-fall-2010/487a2b4c96b8a42c4c3e6372b71b7be6_MIT15_082JF10_lec03.pdf |
G with
no arcs coming in.
COROLLARY 2.
If G
has no directed
cycle, then one can
relabel the nodes so
that for each arc (i,j),
i < j.
27
Topological ordering
Animation
INITIALIZE as follows:
for all i ∈ N do indegree(i) := 0;
for all (i,j) ∈ A do indegree(j) := indegree(j) + 1;
LIST := ∅;
next := 0;
for all i ∈ ... | https://ocw.mit.edu/courses/15-082j-network-optimization-fall-2010/487a2b4c96b8a42c4c3e6372b71b7be6_MIT15_082JF10_lec03.pdf |
order);
Running time is O(n+m) using simple data
structures and algorithms.
Very important for preprocessing.
32
MIT OpenCourseWare
http://ocw.mit.edu
15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/15-082j-network-optimization-fall-2010/487a2b4c96b8a42c4c3e6372b71b7be6_MIT15_082JF10_lec03.pdf |
Course 18.327 and 1.130
Wavelets and Filter Banks
Modulation and Polyphase
Representations:
Noble Identities;
Block Toeplitz Matrices
and Block z-transforms;
Polyphase Examples
Modulation Matrix
Matrix form of PR conditions:
[F0 (z) F1 (z)] H0(z) H0(-z)
H1(z) H1(-z)
&'(&'(&'(&'(
Modulation matrix, H m(z)
= [ 2z œ... | https://ocw.mit.edu/courses/18-327-wavelets-filter-banks-and-applications-spring-2003/48b610fc0f1cc941fb7d1677d48b7a1f_Handout5.pdf |
-1]
éééé2
H odd(z)
z -1
y[n]
+
Polyphase
Form
8
4
2. Upsampling and filtering
x[n]
åååå2
F(z)
y[n]
F(z) = F even(z2) + z -1 F odd (z2)
x[n]
åååå2
F even(z2)
F odd(z2)
y[n]
+
z -1
9
x[n]
x[n]
åååå2
åååå2
F even(z2)
F odd(z2)
yeven[n]
F even(z)
yodd[n]
F odd(z)
åååå2
åååå2
y[n]
y[n] ... | https://ocw.mit.edu/courses/18-327-wavelets-filter-banks-and-applications-spring-2003/48b610fc0f1cc941fb7d1677d48b7a1f_Handout5.pdf |
] + z -1 h0[3]
h1[0] + z -1 h1[2] h1[1] + z -1 h1[3]
= H0,even (z)
H1,even (z)
H0,odd (z)
H1,odd (z)
This is the polyphase matrix for a 2-channel filter bank.
11
12
6
Similarly, for the synthesis filter bank:
Fb =
4444
4444
f0[0] f1[0]
f0[1] f1[1]
4444
0
0
4444
0
0
3333
f0[2] f1[2]
f0[3] f1[3] ... | https://ocw.mit.edu/courses/18-327-wavelets-filter-banks-and-applications-spring-2003/48b610fc0f1cc941fb7d1677d48b7a1f_Handout5.pdf |
= z-? (simple delay)
???
because Hp
-1(z) must be a polynomial.
• Condition for orthogonality: Fp(z) is the transpose
of Hp(z), i.e.
Hp
T(z-1) Hp(z) = I
i.e. Hp(z) should be paraunitary.
Relationship between Modulation
and Polyphase Matrices
&
&
&
&
h0,even[n] = h0[2n]
)
)
)
)
'
'
'
'
)
)
)
)
h0,odd[n] = h0[2n... | https://ocw.mit.edu/courses/18-327-wavelets-filter-banks-and-applications-spring-2003/48b610fc0f1cc941fb7d1677d48b7a1f_Handout5.pdf |
pp
N
N-point DFT
Matrix
-1
FN = 1 FN
N
Complex conjugate: replace w with w = e -
2pppp
N
17
So, in general
H m(z) F-1
N
= H p(zN) DN(z)
N = # of channels in filterbank
(N = 2 in our example)
18
9
Polyphase Matrix
Example: Daubechies 4-tap filter
h0[0] =
1+µµµµ3
4 µµµµ2
h0[1] =
3 + µµµµ3
4µµµµ2 ... | https://ocw.mit.edu/courses/18-327-wavelets-filter-banks-and-applications-spring-2003/48b610fc0f1cc941fb7d1677d48b7a1f_Handout5.pdf |
µµµ3) (6 + 4µµµµ3) - (6 - 4µµµµ3)
(6 + 4µµµµ3) - (6 - 4µµµµ3) (12 + 6µµµµ3) + (12 - 6µµµµ3)
=
³ µµµµ3/4
µµµµ3/4 ½
22
11
BTB =
1
4 µµµµ2
3 - µµµµ3 3 + µµµµ3
1 - µµµµ3 -(1 + µµµµ3)
1
4 µµµµ2
3 - µµµµ3
3 + µµµµ3 - (1 + µµµµ3)
1 - µµµµ3
1
32
=
=
(12 œ 6µµµµ3) + (12 + 6µµµµ3) (6 - 4µµµµ3) - (6 + 4µµµµ3)... | https://ocw.mit.edu/courses/18-327-wavelets-filter-banks-and-applications-spring-2003/48b610fc0f1cc941fb7d1677d48b7a1f_Handout5.pdf |
Key Concepts for this section
1: Lorentz force law, Field, Maxwell’s equation
2: Ion Transport, Nernst-Planck equation
3: (Quasi)electrostatics, potential function,
4: Laplace’s equation, Uniqueness
5: Debye layer, electroneutrality
Goals of Part II:
(1) Understand when and why electromagnetic (E and B)
interaction ... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/48b63e1bf2248a838f837d85cb37a515_fields_lec5.pdf |
r
ext
(cid:71)
E
∂
t
∂
(cid:71)
B
∇ × =
(cid:71)
J
μμ μμεε
+
r
0
0
0
r
r
(cid:71)
J
=
μ με
+
μr : relative magnetic permeability of the medium
μο : free space permeability (4π×10-7 H/m)
μ of various media
μr for water : very close to 1
μr (Ni)~600, μr (Fe)~5000
Mobility of various ions in water
Species
Mobility
Ui (c... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/48b63e1bf2248a838f837d85cb37a515_fields_lec5.pdf |
-18
In silicon (semiconductor), n×p~1020 (constant)
In aqueous solutions, [H+][OH-] = 10-14 = Kw
(pH= -log10[H+])
Electrolytes (biological systems) are conductors.
E
electrode
V=Vext
Charge Relaxation in electrolyte
+
+
+
+
+
+
+
+
τ ~10-9 sec
+
+
+
ρ=0
+
+
+
+
+
<ρ>=0 within the medium
Fixed charges
+
+
+
τ ~10-9... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/48b63e1bf2248a838f837d85cb37a515_fields_lec5.pdf |
6.02 Fall 2012
Lecture #16
• DTFT vs DTFS
• Modulation/Demodulation
• Frequency Division Multiplexing
(FDM)
6.02 Fall 2012
Lecture 16 Slide #1
Fast Fourier Transform (FFT) to compute
samples of the DTFT
signals of finite dura
for
tion
For an x[n] that is zero outside of the interval [0,L-1], choose
P ≥ ... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/48f208ee650147f6c3d6f7bcc602cc66_MIT6_02F12_lec16.pdf |
2012
Lecture 16 Slide #3
Fast Fourier Transform (FFT) to compute
samples of the DTFT
signals of finite dura
for
tion
For an x[n] that is zero outside of the interval [0,L-1], choose
P ≥ L (with P preferably a power of 2; we’ll assume that it’s at
least a multiple of 2, i.e., even):
X(Ω
P−1
k ) = ∑ x[m]e− jΩ... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/48f208ee650147f6c3d6f7bcc602cc66_MIT6_02F12_lec16.pdf |
[0,P-1] recovers the original x[n] in this
interval
• Evaluating it for n outside this interval results in periodic
replication of the values in [0,P-1], producing a periodic signal x[n]
• So this eqn. is also called a DT Fourier Series (DTFS) for the
periodic signal x[n]. Notation: Ak=X(Ω
k)/P=Xk/P, Fourier co... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/48f208ee650147f6c3d6f7bcc602cc66_MIT6_02F12_lec16.pdf |
demodulation
Signal centered at 0
6.02 Fall 2012
Signal centered at Ω
c
Lecture 16 Slide #7
Modulation by Heterodyning or
Amplitude Modulation (AM)
x[n]
×
t[n]
Re(X )
k
(cid:1)Ω
m
+Ω
m
Im(Xk)
A
B
cos(Ω
cn)
To get this nice picture, the
baseband signal needs to be
band-limited to some range of
freque... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/48f208ee650147f6c3d6f7bcc602cc66_MIT6_02F12_lec16.pdf |
[n]?
6.02 Fall 2012
Lecture 16 Slide #10
Demodulation Frequency Diagra m
Re(T )
k
(cid:1)Ω
c
+Ω
c
R(Ω)=T(Ω)
Im(Tk)
A/2
B/2
Note combining of signals around 0
results in doubling of amplitude
D(Ω)
(cid:1)2Ω
c
Re(Dk)
(cid:3)Ω
m
Ω
m
Im(Dk)
A/2
+2Ω
c
B/2
6.02 Fall 2012
What we want
Lecture 16 Sli... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/48f208ee650147f6c3d6f7bcc602cc66_MIT6_02F12_lec16.pdf |
Ω
c
Im(Tk)
A/2
B/2
–2Ω
c
D(Ω)
+
+2Ω
c
6.02 Fall 2012
Lecture 16 Slide #14
… produces
Note combining of signals around 0
results in cancellation!
6.02 Fall 2012
Lecture 16 Slide #15
Channel Delay
Time delay of D samples
x[n]
t[n]
×
D
d[n]
×
LPF
y[n]
cos(Ω
cn)
cos(Ω
cn)
Cutoff @ ±kin
Gain = 2... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/48f208ee650147f6c3d6f7bcc602cc66_MIT6_02F12_lec16.pdf |
then
w[n]
= I[n]2 + Q[n]2
=| x[n − D] | cos2 θ+ sin2 θ
=| x[n − D] |
OK for recovering x[n] if it
never goes negative, as in
on-off keying
6.02 Fall 2012
jQ
x[n-D]sin(θ)
θ
I
x[n-D]cos(θ)
Constellation diagrams:
x[n] = { 0, 1 }
Q
Q
I
I
transmitter
receiver
Lecture 16 Slide #18
Dealing With Phase
Amb... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/48f208ee650147f6c3d6f7bcc602cc66_MIT6_02F12_lec16.pdf |
of 2 Mbit/s, DQPSK is used. In reaching 5.5 Mbit/s and the full-rate of 11 Mbit/s, QPSK is employed, but has
to be coupled with complementary code keying. The higher-speed wireless LAN standard,
IEEE 802.11g-2003 has eight data rates: 6, 9, 12, 18, 24, 36, 48 and 54 Mbit/s. The 6 and 9 Mbit/s modes
use OFDM modulatio... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/48f208ee650147f6c3d6f7bcc602cc66_MIT6_02F12_lec16.pdf |
R[n]
×
+
-Ω
G -Ω
R -Ω
B
Ω
B
Ω
R
Ω
G
cos(Ω
Rn)
xG[n]
×
cos(Ω
Gn)
Channel “performs addition” by
superposing signals (“voltages”) from
different frequency bands.
6.02 Fall 2012
Lecture 16 Slide #22
AM Radio
AM radio stations are on 520 – 1610 kHz (‘medium wave”)
in the US, with carrier frequencies of dif... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/48f208ee650147f6c3d6f7bcc602cc66_MIT6_02F12_lec16.pdf |
Mechanisms of
Diffusion in
Materials
3.205 L4 11/7/06
1
A final point on interdiffusion…
The composition profiles resulting from inter-
diffusion are generally constrained by phase
equilibria. Consider the an Ir–Re diffusion couple
annealed at 2400°C:
Equilibrium diagram
Diffusion couple
3.205 L4 11/7/06
2... | https://ocw.mit.edu/courses/3-205-thermodynamics-and-kinetics-of-materials-fall-2006/48fa40fa1298230d445991e2325d132d_lecture04_slides.pdf |
-crystalline) polymers.
Interstialcy
Reptation
3.205 L4 11/7/06
6
Atomistic mechanisms, cont’d
Jumps of vacancies and interstitials are thermally
activated, and jumps occur at a jump rate, Γ’,
Figure removed due to copyright restrictions.
See Figure 7.2 in Balluffi, Robert W., Samuel M. Allen, and W. Craig Cart... | https://ocw.mit.edu/courses/3-205-thermodynamics-and-kinetics-of-materials-fall-2006/48fa40fa1298230d445991e2325d132d_lecture04_slides.pdf |
205 L4 11/7/06
10
! R2=nr2"f and D=#r26f Self-diffusion of interstitials
Involves only random walk of the interstitial species,
and thus
Note that interstitial motion is uncorrelated, so f = 1.
3.205 L4 11/7/06
11
! DI="Ir26f=z# " Ir26f=z$r26exp%GImkT()[]&1=z$r26expSImk[]&exp%HImkT()[] Self-diffusion of vacancie... | https://ocw.mit.edu/courses/3-205-thermodynamics-and-kinetics-of-materials-fall-2006/48fa40fa1298230d445991e2325d132d_lecture04_slides.pdf |
6.02 Fall 2012
Lecture #10
• Linear time-invariant (LTI) models
• Convolution
6.02 Fall 2012
Lecture 10, Slide #1
Modeling Channel Behavior
codeword
bits in
1001110101
x[n]
generate
digitized
symbols
modulate
DAC
NOISY & DISTORTING ANALOG CHANNEL
ADC
demodulate
& filter
sample &
threshold
y[n]
1... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/49242956b17f47b9c85bd6784aa5d6b8_MIT6_02F12_lec10.pdf |
2[n] be the response to an arbitrary x2[n].
If, for arbitrary scalar coefficients a and b, we have:
ax1[n]+ bx2[n]
S
ay1[n]+ by2[n]
then system S is said to be linear. If the input is the
weighted sum of several signals, the response is the
superposition (i.e., same weighted sum) of the response
to those signals.... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/49242956b17f47b9c85bd6784aa5d6b8_MIT6_02F12_lec10.pdf |
10, Slide #10
Unit Step Decomposition
“
“Rectangular-wave” digital
signaling waveforms, of the sort
s
we have been considering, are
w
easily decomposed into time-
e
shifted, scaled unit steps --- each
s
transition corresponds to another
t
shifted, scaled unit step.
s
e.g., if x[n] is the transmission of
e
1001... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/49242956b17f47b9c85bd6784aa5d6b8_MIT6_02F12_lec10.pdf |
2]δ[n-2].
In general:
∞
x[n] = ∑ x[k]δ[n − k]
k=−∞
l
For any particular index, only
one term of this sum is non-zero
i
2012
6.02 Fall 2012
6.02 Fall
Lecture 10, Slide #17
Modeling LT
I Systems
If system S is both linear and time-invariant (LTI), then we can
use the unit sample response to predict the resp... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/49242956b17f47b9c85bd6784aa5d6b8_MIT6_02F12_lec10.pdf |
output signal y in terms of the input x and
unit-sample response h. Some constraints are needed to ensure
this infinite sum is well behaved, i.e., doesn’t “blow up” --- we’ll
discuss this later.
∗
We use to denote convolution, and write y=x h. We can thus
write the value of y at time n, which is given by the above... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/49242956b17f47b9c85bd6784aa5d6b8_MIT6_02F12_lec10.pdf |
y[n]
y[n]
y[n]
6.02 Fall 2012
Lecture 10, Slide #22
Spot Quiz
input
response
Unit step response: s[n]
1
0.5
0 1 2 3 4 5 … n
x[n]
1
0.5
0 1 2 3 4 5 6 7 8 9 n
x[n]
S
y[n]
Find y[n]:
1. Write x[n] as a function of
unit steps
2. Write y[n] as a function of
unit step responses
3. Draw y[n]
6.02 F... | https://ocw.mit.edu/courses/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/49242956b17f47b9c85bd6784aa5d6b8_MIT6_02F12_lec10.pdf |
Suspensions and suspension
mechanics
1
Suspensions may be distinguished by
the shape and size of the discrete phase
Debye Length
in 1mM NaCl
Visible Light
micro-CT
resolution
(Optical limit)
gap in PP rheometer
10-10
10-9
10-8
10-7
10-6
10-5
gap in CP rheometer
10-3
10-4
10-2
[m]
Au nanoparticle
Colloidal
silica
... | https://ocw.mit.edu/courses/2-341j-macromolecular-hydrodynamics-spring-2016/4940640c8f594be8046e044b571e02c6_MIT2_341JS16_Lec23-slides.pdf |
PS and PQ?
© source unknown. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/.
<0.75
6
Particle shape is also important
Art glitter, oblate
Glass beads, spherical
1000um
2000um
1000um
500um
prolate, high L/d
Wollastonite... | https://ocw.mit.edu/courses/2-341j-macromolecular-hydrodynamics-spring-2016/4940640c8f594be8046e044b571e02c6_MIT2_341JS16_Lec23-slides.pdf |
All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/.
Zarraga, Hill, and Leighton J. Rheol. 2000 (44)
10
Turn on Brownian motion,
dependence on Pe
© John Wiley and Sons, Inc. All rights reserved. This content is excluded fr... | https://ocw.mit.edu/courses/2-341j-macromolecular-hydrodynamics-spring-2016/4940640c8f594be8046e044b571e02c6_MIT2_341JS16_Lec23-slides.pdf |
etc..
nearest neighbors
15
Radial symmetry is not guaranteed
In general, particle
distribution is a function of
an angle and distance
Investigate hydrodynamics to understand ordering by
hydrodynamic forces
16
θLocal density(,)Bulk densitygrθr Experiments: Parsi & Gadala-Maria
High Pe # spheres
© The Society of Rh... | https://ocw.mit.edu/courses/2-341j-macromolecular-hydrodynamics-spring-2016/4940640c8f594be8046e044b571e02c6_MIT2_341JS16_Lec23-slides.pdf |
drodynamic
Brownian
© AIP Publishing LLC. All rights reserved. This content is excludedfrom our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/.
21
Foss and Brady JFM (2000)
Brownian hard spheres
Viscosity = f(Pe)
Total
Hydrodynamic
Brownian
Replot as
Stress = f(Pe)
Total
Bro... | https://ocw.mit.edu/courses/2-341j-macromolecular-hydrodynamics-spring-2016/4940640c8f594be8046e044b571e02c6_MIT2_341JS16_Lec23-slides.pdf |
A review of microstructure in concentrated suspensions and its implications
for rheology and bulk flow” (2009) 48:909-923
25
11122Nσσ22233Nσσ10.420.5 Experimental agreement?
© AIP Publishing LLC. All rights reserved. This content is excludedfrom our Creative
Commons license. For more information, see https://o... | https://ocw.mit.edu/courses/2-341j-macromolecular-hydrodynamics-spring-2016/4940640c8f594be8046e044b571e02c6_MIT2_341JS16_Lec23-slides.pdf |
transient Couette flows
© AIP Publishing LLC. All rights reserved. This content is excludedfrom our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/.
30
MIT OpenCourseWare
https://ocw.mit.edu
2.341J / 10.531J Macromolecular Hydrodynamics
Spring 2016
For information about citi... | https://ocw.mit.edu/courses/2-341j-macromolecular-hydrodynamics-spring-2016/4940640c8f594be8046e044b571e02c6_MIT2_341JS16_Lec23-slides.pdf |
8.871 Lecture 2
M. Padi
October 20, 2004
1
Electrically and Magnetically Charged Branes
We would like to build a general understanding of branes, the gauge fields to which they are coupled, and
the gauge fields living on their world volume. First, we review electromagnetism in four dimensions. Define
F (2) = dA(1) ... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/4944d4a313d2399badde634815961996_lec2.pdf |
9+1 dimensions, is electrically
charged under B(2), with a charge Q. We therefore write a source equation d � H (3)
. The string is
localized in 8 of the 9 space coordinates and hence we have a delta function in 8 directions. On the other
hand, the NS5-brane is magnetically charged under B(2) with a magnetic charge... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/4944d4a313d2399badde634815961996_lec2.pdf |
an M2-brane which is electrically charged under C (3) and an M5-brane
which is magnetically charged under C (3). Similarly, we can find the branes which are charged under massless
gauge fields in Type IIA string theory. Let us first find the massless fields of this theory. Recall that Type
IIA theory has (1,1) supersymme... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/4944d4a313d2399badde634815961996_lec2.pdf |
dimensions. The
3-form has 8 = 56 components and the 2-form has 2 = 28 components. We can check this decomposition
8
by computing 28 + 56 = 84. Similarly, the graviton decomposes into gab (traceless and symmetric), ga(10)
�
and g(10)(10), where a, b ⊗= 10. These objects have 35, 8 and 1 degrees of freedom respectiv... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/4944d4a313d2399badde634815961996_lec2.pdf |
Note that there are three kinds of forms. The 3-form couples electrically to a D2-brane and
magnetically to a D4-brane. The 2-form, B(2), couples electrically to the F1 string, and magnetically to the
NS5-brane. (One might ask why B(2) couples to the fundamental string and NS5-brane, not the D1-brane
and D5-brane. I... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/4944d4a313d2399badde634815961996_lec2.pdf |
’ll briefly go over the massless multiplet in string theories with 16
supercharges, and thus deduce the types of branes in the Type I and heterotic theories. First, the gravity
·9 − 1 = 35 components. Then there is the 2-form B(2) which
multiplet contains the graviton, g, which has 8
2
has 28 components as usual. Th... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/4944d4a313d2399badde634815961996_lec2.pdf |
and E8 × E8, respectively. As a result we have
two additional 5 branes together with what we listed above: the SO(32) small instanton and the E8 small
instanton
µ
2
2 Central Charges and Branes
The existence of branes (solitonic solutions) and the central charges in the supersymmetry algebra are
intimately ... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/4944d4a313d2399badde634815961996_lec2.pdf |
list of objects in Type IIA theory. We find that the 2-form Bµ� becomes Bµ(10), a 1-form, and Bab
(a, b ⊗= 10) a 2-form in 10 dimensions. These correspond to the F1-string and the D2-brane. The 1-form Pµ
becomes a 1-form in 10 dimensions and a scalar P10. The scalar corresponds to a D0-brane. Similarly, the
5-form de... | https://ocw.mit.edu/courses/8-871-selected-topics-in-theoretical-particle-physics-branes-and-gauge-theory-dynamics-fall-2004/4944d4a313d2399badde634815961996_lec2.pdf |
15.081J/6.251J Introduction to Mathematical
Programming
Lecture 8: Duality Theory I
Slide 1
Slide 2
1 Outline
• Motivation of duality
• General form of the dual
• Weak and strong duality
• Relations between primal and dual
• Economic Interpretation
• Complementary Slackness
2 Motivation
2.1 An idea from L... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/496d9a385fd1d7e1aa82837d91adb23e_MIT6_251JF09_lec08.pdf |
xj
i ∈ M1
i ∈ M2
i ∈ M3
j ∈ N1
j ∈ N2
j ∈ N3
p ′ b
pi ≥ 0
pi ≤ 0
p >
i <0
p ′ Aj ≤ cj
p ′ Aj ≥ cj
p ′ Aj = cj
i ∈ M1
i ∈ M1
i ∈ M3
j ∈ N1
j ∈ N2
j ∈ N3
3.1 Example
x1 + 2x2 + 3x3
min
s.t. −x1 + 3x2
= 5
2x1 − x2 + 3x3 ≥ 6
x3 ≤ 4
x1 ≥ 0
x2 ≤ 0
x3 free,
max
s.t.
5p1 + 6p2 + 4p3
p1 free
p... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/496d9a385fd1d7e1aa82837d91adb23e_MIT6_251JF09_lec08.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.