text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
lines block v from u
– u � v if v cut by u auto
– probability 1/(1 + index (u, v)).
– tree size is (by linearity of E)
n +
�
1/index (u, v)
≤
2Hn
�
u
• result: exists size O(n log n) auto
• gives randomized construction
• equally important, gives probabilistic existence proof of a small
BSP
• so might hope to... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2b92be49ad6bbf6b75da5d5c262a27f1_n1.pdf |
Why to Study
Finite Element Analysis!
That is, “Why to take 2.092/3”
Klaus-Jürgen Bathe
Why You Need to Study
Finite Element Analysis!
Klaus-Jürgen Bathe
Analysis is the key to
effective design
effective design
We perform analysis for:
• deformations and internal forces/stresses
• temperatures and heat tr... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
to impact
Comparison of computation with laboratory test results
In engineering practice, analysis
is largely performed with the use
of finite element computer
programs (such as NASTRAN,
ANSYS, ADINA, SIMULIA, etc…)
These analysis programs are
interfaced with computer-aided
,
) p g
design (CAD) programs Ca... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
ible analysis
Schematic solution results
Example problem:
to show what can go wrong
Smallest six frequencies (in Hz) of 16 element mesh
Consistent mass matrix is used
Mode
number
16el. model
Use of 3x3
16el. model
Use of 2x 2
16x64 element model
use of 3x3 Gauss
Gauss integration Gauss integration
inte... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
components can be
tested
Reliable analysis procedures are crucial
Sleipner platform
Recall the catastrophic failure in 1991 of the
Sleipner platform in the North Sea
• Ref. I. Holand, "Lessons to be learned from
• Ref. I. Holand, "Lessons to be learned from
the Sleipner accident"
Proceedings, NAFEMS World Cong... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
the Open Clip Art Library.
molding
(plastic)
reinforcement
(steel)
Bumper cross-section
Bumper reinforcement
upper binder
pad
initial blank
deformed sheet
lower binder
punch
Stamping on a single action press,
“springs” provide constant holding force
Bumper reinforcement
Material data:
steel, 1.8 mm
f... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
Measured
Max 256
*>247.8°F
240.0
220.0
220.0
200.0
180.0
160.0
140.0
120.0
100.0
*<100.0°F
Max
Max
237.4
237.4
Exhaust Manifold Mesh
Detail showing mesh mismatch
Plot of effective stress in the solid
Plot of pressure in the fluid
Fuel pump
Fuel pump
Blood flow through an artery
Fluid mesh ... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
2.093 Finite Element Analysis of Solids and Fluids I(cid:13)
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
ELECTRIC FORCES ON CHARGES
Lorentz Force Law:
+ × μ
)o
H Newtons
a = f/m = qE/m ≈ eV/mL [m s-2]
(
E v
=
q
f
Kinematics*:
t
v
=
a(t)dt
∫
0
=
v
o
+
ˆ
at
z
cathode
-V
-
E⊥
heated
filament
+
z
z = z + z•v t + at /2
o
ˆ
2
o
⇒
f
=
qE ma
=
anode, phosphors
deflection plates
cathode ray tube ... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/2c166c5e5dc71554eab3b62b6c9979aa_MIT6_013S09_lec05.pdf |
Eo Deepest electrons
experience zero force
Attractive pressure
L5-2
ENERGY METHOD
FOR FINDING FORCES
Force, work, and energy:
dw = f ds ⇒ f =
dw
ds
[N]
C = εοA/s
1
w = CV2 =
2
2
2
1 Q s
1 Q
=
2 C 2 A
ε
o
[J]
=
f =
Q ≠ f(s) if C is open circuit
2
1 Q
dw
2 Aε
ds
o
2
( EA)
1(
ε
2
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/2c166c5e5dc71554eab3b62b6c9979aa_MIT6_013S09_lec05.pdf |
εo E2/2 (as before) Q.E.D.
dw
ds
= −
f =
=
=
2
ε
2
A
= −
P A
e
*C = εA/s
L5-4
LATERAL FORCES – ENERGY METHOD
Energy derivative:
= −
f
(externally applied)
dw
dD
2
w
=
2
s
Q
Q
2C 2 WD
ε
o
=
W
E
+Q
C
A’ = Ws
Fringing
field
s
A’
f
D
-Q
=
f
2
Q s
=
2
2 WD
ε
o
2
( EWD)
ε
o
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/2c166c5e5dc71554eab3b62b6c9979aa_MIT6_013S09_lec05.pdf |
A = 2
2
)
Qd(
2C
d
θ
R
2
θ
2
R
2
Therefore : T
C
=
2
Q s
2 2
2 R
ε
θ
o
=
2
E
ε
o
2
A'
[Nm]
≅
pressure
×
gap-area A'
×
θ
R
+
stator
rotor
-
A’ = 2Rs
R
2
-
+
T
Motor power:
Peak power:
Average power: Pavg = P/2 (duty cycle = ½)
P = Tω
[W]
n = 4
θ
Segmentation advantage:
T [Nm]... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/2c166c5e5dc71554eab3b62b6c9979aa_MIT6_013S09_lec05.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.917 Topics in Algebraic Topology: The Sullivan Conjecture
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
The Adem Relations (Continued) (Lecture 5)
We continue to work with complexes over the finite field F2 with ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
get a feel for how everything works, let’s consider the case where V = F2 is a complex concentrated
in degree 0. In this case, we can identify VhΣ2 with the chain complex C (BΣ2), and we can identify V hΣ2
with the cochain complex C ∗(BΣ2). The norm map induces a map
∗
Hn(BΣ2)
→
H−n(BΣ2).
This is just the usual nor... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
(BΣ2) � F2[t, t−1]/F2[t].
∗
Using this isomorphism, H (BΣ2) has a basis consisting of {tn}n<0. In previous lectures, we used a basis
{xi}i≥0 for H (BΣ2) which was dual to the basis {ti}i≥0 for H∗(BΣ2). By comparing degrees, we see that
these bases are related by the following transformation
∗
∗
xi �→ t−i−1
.
It follo... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
classifies a map F2[−n]
induced maps
→
V . We obtain
f : D2(F2)[−2n] � D2(F2[−n]) → D2(V )
f � : DT (F2)[−2n] � D2(F2[−n]) → DT (V ).
For every integer k, we let Sk(v) ∈ Hn+k(DT (V )) denote the image of tk−n ∈ Hk−n(DT (F2)) under the map
f �. If k ≥ n, then
tk−n ∈ Hk−n(D2(F2)) ⊆ Hk−n(DT (F2)).
In this case, we... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
the Steenrod operations on H∗(V ). For
this, we need to introduce a mild finiteness restriction on V :
(∗) The cohomology groups Hn(V ) are finite dimensional for every n ∈ Z, and vanish for n sufficiently
small.
Assuming condition (∗), we have equivalences
V hΣ2 � V ⊗ (F2)hΣ2
V T Σ2 � V ⊗ (F2)T Σ2
VhΣ2 � V ⊗ (F2)hΣ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
FT Σ2 ) � F2[t, t−1], and that the
action of this ring on H∗(DT (V )) satisfies tmSk(v) = Sm+k(v).
→
⇒
2
The coefficient of tk−l in φ(Sk(v)) is given by
Res(tl−k−1φ(Sk(v))) = Res(φ(Sl−1(v))).
3
�
�
�
�
�
We have a commutative diagram
H∗(V ) Sl−1
� �
H∗(DT (V ))
� �
id
H∗(
V )
Sql �
� H∗(D
2(V ))
� H∗(V )[t, t−... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
→
(Σ2 × Σ2) � Σ2 = G.
Applying Proposition 3 in this case, we obtain the following:
Corollary 4. The inclusion j : Σ2 × Σ2 → G induces a restriction map on cohomology H∗(BG)
Σ2) � F2[t, u]. For k ≥ n, this map carries Sk(un) ∈ Hm+k(BG) to
→
H∗(Σ2 ×
�
(n − l, l)u n+ltk−l .
p
We observe that H (BG) � H−∗(D2(C (BΣ2... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
, l) Sq
−q−l
xp−l.
We are now ready to complete the calculation of the last lecture. Recall that we need to show that for
l
p, q > 0, the homology classes
�
(p − 2l, l) Sq
xp−l ∈ Hp+q(BG)
−q−l
l
4
�
�
�
�
�
�
�
�
�
�
�
(q − 2l�, l�) Sq
−p−l�
xq−l� ∈ Hp+q(BG)
l�
have the same image in H (BΣ4). Invoking Co... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
6.890 Algorithmic Lower Bounds and Hardness Proofs
Fall 2014
Erik Demaine
Class 2 Scribe Notes
1 Useful Problems for Hardness Reductions
This lecture mostly focuses on using 3-Partition to solve 2+ problems by reducing to number problems.
The basic idea is to think of your numbers as integers – fixed-point or rati... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
an} and a target sum t want to find a subset S ⊂ A such that ΣS = t. It is easy to think of
an instance of this problem as a partition, although it’s a generalization. Reducing from Subset Sum that
we can reduce from 2-Partition. 2-Partition to Subset Sum is a strict generalization – not given t –
but we are essentia... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
Using a trick, you
can add ”∞” to each ai – any solution before is still a solution (think of ∞ as some arbitrarily large number
· max A), but all the ai’s become roughly equal to each other and arbitrarily close to t/3. Let’s talk
like n
about a couple of related problems to 3-Partition, and why we’re talking about... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
-Dim Matching to 3-Partition
add big numbers!
• add ∞ to each ai
• add 3∞ to each bi
• add 9∞ to each ci
for some ∞ ≈ 10 × max(A ∪ B ∪ C). The new tf becomes t + 13∞. This forces us to pick one from A, one
from B, and one from C. We must show that the infinities can be treated algebraically despite being so
large... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
edge has cardinality 3. The goal is to find n/3 disjoint hyperedges. This is a 3-Dimensional
Matching problem, converted into a graph (forgetting there are numbers) – but we draw a hyperedge exactly
when ai + bj + ck equals the sum (refer to Num 3-Dim Matching). Because tripartite hypergraphs are
3-uniform, X3C is a ... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
hard (and are not Strongly NP-hard under the assumption that P ! = N P ). On the other hand, 3-Partition,
Num 3-Dim Matching, 3DM, and X3C are all Strongly NP-hard.
Generally, it is natural to assume that your inputs are reasonably encoded – binary, ternary, etc. anything
bigger than 1, i.e. don’t use unary. But tod... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
NP-hard: no Pseudopolynomial time algorithm (and therefore no Weakly Polynomial time)
algorithm
Here is Erik’s favorite diagram applied to these concepts. Draw a difficulty axis: but now instead of P ,
we have Weakly Polynomial, Strongly Polynomial, Pseudopolynomial.
if P = N P :
weakly NP-hard
strongly NP-hard
di... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
at least
Strongly NP-hard (comes from Garay and Johnson paper introducing 3-Partition).
2.2 Rectangle Packing
Given n rectangles and target rectangle B, do they fit into B? Rotation and translation allowed but rectangles
must be disjoint from each other (cannot overlap) so that they fit in B.
This problem is strongl... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
://erikdemaine.org/papers/Jigsaw_GC/paper.pdf
4
ai →
B =
ε
ai
t
(n/3)ε
Now time for some puzzles!
2.3 Edge-Matching Puzzle
This puzzle goes back to the 1890s (See slide 3). The goal is to pack triangles with half frogs on them into
a larger triangle such that the cranial and caudal ends of each frog match up... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
+ ai units long) where touching
colors on the inside are color i – so as to force this rectangle to be built (since there are only two “end pieces”
for color i). To prevent rotation, colors on top and bottom are different from colors on sides (colors % and
$ in this instance).
b
a
c
d
ai
%
%
$
%
%
i
i
%
%... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
would not be a valid reduction. In 3-Partition, we are representing the numbers
in unary so everything remains polynomial in size during the reduction.
2.4 Signed Edge-matching
Next puzzle: Signed edge-matching (Like the frog puzzle). Here, lowercase and uppercase letters for colors,
and e.g. b matches with B. We c... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
the
straight boundary edges (See Slide 10). We’ve reduced 3-Partition to Edge-Matching, Edge-Matching to
Signed Edge-Matching, and Signed Edge-Matching to Jigsaw puzzles. Jigsaw puzzles will now be reduced
to... polyomino packing!
2.6 Polyform Packing Puzzles
Polyform Packing are the medium for the Eternity I Puzz... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
can they fit? Squares can only rotate by a multiple of 90◦ (aka they can’t rotate).
Can we reduce from 3-Partition to Square Packing? This is a result from [Leung, Tam, Wong, Young,
Chin 1990]. Take each ai number, add this huge number B, make the height 3B + t (where t is the target for
the ais’ sum). So the little ... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
+ md + 4ms)
3 (mu
md)
⎞
⎟
⎟
⎠
1 (axions). There is a small eigenvalue
associated with an
f 2
F 2
∼
eigenvalue
∼
⎛
⎜
⎝
1
f
aF
f bF ⎟
⎞
⎠
with
60
=
=
⎛
⎜
⎜
⎝
�
So
∈
M
υ
2 )
( f
1. For f
F
61
+ (mu + md)a +
1
√3
(mu −
md)b = 0(8.4)
md
2
mu −
√6
1
√3
2ms) +
√2
3
(mu + md −
(mu −
md)a + (mu + ... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/2c3cf052ae853f5b31fa7246895672d6_chap8.pdf |
⎜
⎜
⎝
f 2
F 2
2
3 ms
0
0
0
4
3√2 ms 0
f
F −
f
4
3√2 ms
F −
0
4
3ms
⎞
⎟
⎟
⎠
(8.6)
(8.7)
(8.8)
(8.9)
(8.10)
has two vanishing eigenvalues. So η gets infected and the GM-O relation is
badly violated. The general case is a little messy but with mu = md �
ms we
easily arrive at
2
minf ected η →
(mu... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/2c3cf052ae853f5b31fa7246895672d6_chap8.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.341: Discrete-Time Signal Processing
OpenCourseWare 2006
Lecture 7
IIR, FIR Filter Structures
Reading: Sections 6.1 - 6.5 in Oppenheim, Schafer & Buck (OSB).
Signal Flow Graphs
A linear time-invariant discrete-time ... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
Figure 6.11, which is the signal glow graph corresponding to the first order system in
OSB Figure 6.10. By convention, the delay element has been represented by a branch gain of
z−1 .
The signal flow graph representation of a LTI system is not unique. In fact, for any given
rational system function, equivalent sets o... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
and
combining the delay elements give the Direct Form II structure shown in OSB Figure 6.15.
Since delay elements correspond to physical memories in actual implementation, direct form
II structures require less state memory than the direct form I implementation. However, the
total memory requirement for both forms ... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
B Figure 6.18 is an example of the resulting cascade structure. This is a sixth-order system
with direct form II realization for each of its second-order subsystems.
2
Parallel Form
Equivalently, expressing the transfer function as a sum using partial fraction expansion gives a
parallel structure:
Np
�
Ns
�
H(z... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
we have shown that a causal generalized linear phase FIR system satisfies
the following symmetry (anti-symmetry) condition, depending on the type of the system:
h[M − n] = h[n]
or h[M − n] =
−h
[n]
n = 0, 1, . . . , M .
Using this special property, can we further simplify the tap-delay line filter structure to redu... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
ter type specific rules.
All-zero (FIR) Lattice Filters the basic two-port section in an FIR lattice filter has the
following non-recursive butterfly signal flow graph structure:
One section of lattice structure for FIR lattice filters
For the overall system, the input is fed into the two input ports of the first stage, ... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
ole filters
We have scaled all signals by AM (z) because assuming the input is at the p + 1 = M -th
stage and the output is at p = 0, setting Ap+1(z) = AM (z), and A0(z) = 1 gives the desired
all-pole filter: AM (z) . The following figure shows the overall structure of an all-pole lattice filter.
Note since B0(z) = z−1... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
fined as “unquantized,” and the 16-bit quantized coefficients have been listed in OSB Tables
6.1 and 6.2. See OSB Section 6.7.2 for detailed analysis of this example.
Filter coefficient quantization causes the poles and zeros of the system to shift, consequently
distorting its frequency response. The next set of figures... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
20.430 / 2.795 / 6.561 / 10.539
Fields Forces and Flows
in Biological Systems
Fall 2015
Instructors: Mark Bathe, Alan Grodzinsky
11
Textbook:
Fields Forces and Flows in Biological Systems
Garland Science, March 2011
Book cover removed due to copyright restrictions.
Source: Grodzinsky, Alan. Fi... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
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Source: Prof. Paolo Provenz... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
FHWR
PDFURPROHFXOH ,J* WUDQVSRUWLQWXPRUW\SHVDQGIRXQGDQ
XQH[SHFWHGFRUUHVSRQGHQFHEHWZHHQWUDQVSRUWUHVLVWDQFH
DQGWKH PHFKDQLFDOVWLIIQHVV
B,S, Mechanical Engineering, University of Wisconsin, 1998
Employment Opportunities
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Provenzano Lab
•
•
Conn... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
Sep 9 Course introduction, overview, and objectives
Sep 5
Sep 10
Sep 14
Sep 12
Sep 16
Sep 21
Sep 17
Sep 19
Sep 23
I.
CHEMICAL SUBSYSTEM
I. CHEMICAL SUBSYSTEM
Diffusion as a random walk; Stokes-Einstein relation for diffusion coefficient;
Examples of diffusion
Constitutive equations for diffusion (Fick’s ... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
1212
“Biologic” TNF-α Blockers: >$20 Billion/year
(Amgen / Pfizer)
(1998 RA)
Remicade
INFLIXIMAB
(Centocor / J&J)
(1998 Crohn's)
(Abbott)
(2002 RA)
© Various sources. All rights reserv... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
w.mit.edu/help/faq-fair-use/.
Monolayer cell culture
Tissue with same cells
Top view
Side view
ew
Day 1 of culture
© source unknown. All rights reserved. This content is
excluded from our Creative Commons license. For more
information, see http://ocw.mit.edu/help/faq-fair-use/.
Day 6 of culture
© source unknown.... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
~5 pg/mg)
(1-10 ng/ml)
© American Chemical Society. All rights reserved. This
content is excluded from our Creative Commons license. For
more information, see http://ocw.mit.edu/help/faq-fair-use/.
Source: Vajdos, Felix F. et al. "Crystal structure of human
.
insulin-like growth factor-1: detergent binding inhibi... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
•
•
•
• •
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
W
L
down
stream
•
Tissue
c2
•
•
•
•
continuous
recirculation ⇒
"real-time" c2(t)
1919
IGFBP-3 Binding Slows entry of IGF-1 into Tissue!
%
2
1.5
1
0.5
0
,
I
O
T
A
R
M
A
E
R
T
S
P
U
/
M
A
E
R
T
S
N
W
O
D
tissue
100nM IGF-1
100nM IGF-1
I... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
potential; conservation of charge; Electro-quasistatics
Oct 10
Oct 13
Laplacian solutions via Separation of Variables; Electric field boundary
conditions; Ohmic transport; Charge Relaxation; Electrical migration vs. chemical
diffusive fluxes
Oct 14 Electrochemical coupling; Electrical double layers; Poisson–Boltz... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
. Garland
Science, 2011. [Preview with Google Books]
2222
[ + Ohmic Constitutive Law (J σE)]
=
© Garland Science. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Source: Grodzinsky, Alan. Field, Forces and Flows ... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
"Transcranial magnetic stimulation and stroke: a
computer-based human model study." Neuroimage 30, no. 3 (2006): 857-870.
2727
Chap 3: Electrochemical Interactions & Transport
Effects of "Ligand" Molecular Charge on:
• Boltzmann Partitioning into charged tissues, gels
• Binding (to ECM / ICM, receptors....... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
issue
Courtesy of Alan Grodzinsky. Used with permission.
3030
3131 2012
µ-fluidic Chip
© Royal Society of Chemistry. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
S... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
vel, M. G. L. et al. "Electrophoresis of individual microtubules in microchannels."
Proceedings of the National Academy of Sciences 104, no. 19 (2007): 7770-7775.
3434
Zeta Potential (particle charge) Instruments
+ (applied electric field) ▬
Measure “ζ” → Infer effective particle charge
© source unknown. All ... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
18.01 Calculus
Jason Starr
Fall 2005
Lecture 3. September 13, 2005
Homework. Problem Set 1 Part I: (i) and (j).
Practice Problems. Course Reader: 1E1, 1E3, 1E5.
1. Another derivative. Use the 3step method to compute the derivative of f (x) = 1/
is,
Upshot: Computing derivatives by the definition is too much work... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
by MathWorld, part of
Wolfram Research.
The Binomial Theorem says that for every positive integer n and every pair of numbers a and b,
(a + b)n equals,
a + na n−1b + · · · +
n
a n−k bk + · · · + nabn−1 + b .
n
� �
n
k
This is proved by mathematical induction. First, the result is very easy when n = 1; it just ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
an−k bk+1
k
an+1 + nanb +
+ anb +
. . . +
. . . +
+
+
n
. . . + ab
. . . + nab
n + bn+1
Summing in columns gives,
an+1 + (n + 1)anb +
�
. . . + ( k + k−1 )an+1−k bk + ( k+1 +
� �
�
�
�
n
n
n
� �
n
k
)an−kbk+1 +
. . . + (1 + n)ab
n
+ bn+1.
18.01 Calculus
Jason Starr
Fall 2005
Using Pascal’s formul... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
+ · · · + hn .
� �
n
k
Thus, f (a + h) − f (a) equals,
(a + h)n − a = na
n−1h + · · ·
n
� �
n
k
+
a n−k hk + · · · + hn .
Thus the difference quotient is,
f (a + h) − f (a)
h
= na +
n−1
� �
n
2
� �
n
k
a n−2h + · · · +
a n−k hk−1 + · · · + hn−1 .
Every summand except the first is divisible by h. The l... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
) = f (a+h)[g(a+h)−g(a)]+f (a+h)g(a)−f (a)g(a) = f (a+h)[g(a+h)−g(a)]+[f (a+h)−
f (a)]g(a).
5. The quotient rule. Let f (x) and g(x) be differentiable functions. If g(a) is nonzero, the
quotient function f (x)/g(x) is defined and differentiable at a, and,
(f (x)/g(x))� = [f �(x)g(x) − f (x)g�(x)]/g(x)2 .
18.01 Calc... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
positive integer, and make the
induction hypothesis that d(xn)/dx equals nxn−1 . The goal is to deduce the formula for n + 1,
d(xn+1)
dx
= (n + 1)x n .
By the Leibniz rule,
d(x
n+1)
dx
=
d(x × xn)
dx
=
d(x)
dx
n
x + x
d(xn)
dx
= (1)x n + x
d(xn)
dx
.
By the induction hypothesis, the second term can be ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.006 Introduction to Algorithms
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Lecture 6
Hashing II: Table Doubling, Karp-Rabin
6.006 Spring 2008
Lecture 6: Hashing II: Table Doubling,
Karp-Rabin
Lecture Overvi... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/2cc0efa30d154a5ec5e161a0adcfc373_lec6.pdf |
:
insert into new table
= Θ(n + m) time = Θ(n) if m = Θ(n)
⇒
2
wkax}r}w-rkeepignoreignore≡+product as sumlots of mixing Lecture 6
Hashing II: Table Doubling, Karp-Rabin
6.006 Spring 2008
How fast to grow?
When n reaches m, say
•
m + = 1?
=
= n inserts cost Θ(1 + 2 + + n) = Θ(n2)
rebuild every step
⇒
⇒
· · · ... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/2cc0efa30d154a5ec5e161a0adcfc373_lec6.pdf |
cost
for both insert and delete - analysis is harder; (see CLRS 17.4).
String Matching
Given two strings s and t, does s occur as a substring of t? (and if so, where and how many
times?)
E.g. s = ‘6.006’ and t = your entire INBOX (‘grep’ on UNIX)
3
Lecture 6
Hashing II: Table Doubling, Karp-Rabin
6.006 Spring... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/2cc0efa30d154a5ec5e161a0adcfc373_lec6.pdf |
Hashing II: Table Doubling, Karp-Rabin
6.006 Spring 2008
Karp-Rabin Application:
for c in s: hs.append(c)
for c in t[:len(s)]:ht.append(c)
if hs() == ht(): ...
This first block of code is O(| s |)
for i in range(len(s), len(t)):
ht.skip(t[i-len(s)])
ht.append(t[i])
if hs() == ht(): ...
The second block of code is O(|... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/2cc0efa30d154a5ec5e161a0adcfc373_lec6.pdf |
Subclassing and Dynamic Dispatch
6.170 Lecture 3
This lecture is about dynamic dispatch: how a call o.m() may actually invoke the code of different
methods, all with the same name m, depending on the runtime type of the receiver object o.
To explain how this happens, we show how one class can be defined as a subclass... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
:
class Trans {
int amount;
1
Date date;
...
}
2
Extending a Class by Inheritance
Suppose we want to implement a new kind of account that allows overdrafts. We might call it
AccountPlus, and code it like this:
class AccountPlus extends Account {
int creditLimit;
AccountPlus (String n, int c) {
super (n);... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
also has a type, given by its declaration at compile-time. At runtime, a
variable can refer to an object whose type is not the variable’s type; it is sufficient that the object
type be the type of a subclass of the variable type. (For now, by the way, we’re using the term
‘type’ to mean classification by class name, to... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
executes exactly as before. What determines
which checkTrans method gets called is the runtime type of acc – that is, the type of the class
that provided the constructor used to create it. In general, how variables are declared has no effect
whatsoever on the behavior of the program, if it executes successfully witho... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
the code says only that the object at runtime will belong to that class or one of its subclasses.
But at runtime, which checkTrans method is selected for the call at Statement 6 will depend on
the runtime type of the object.
This code is said to be polymorphic, meaning ‘many shapes’, since the same piece of code tex... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
some code that calls post on an object of AccountPlus:
Account a = new AccountPlus ("Zork", 100);
a.post (new Trans (-50, new Date ());
System.out.println (a.balance);
Which checkTrans method gets called inside post? If the method from Account is called, it
will return False, ignoring the credit limit, and the pri... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
very convenient to program with, since they can’t grow or shrink. Suppose we
implement Bank with a vector of accounts instead:
// bad code!
1.
class Bank {
4
2.
3.
4.
5.
6.
7.
9.
10.
Vector accounts;
...
void chargeMonthlyFee () {
for (int i = 0; i < accounts.size(); i++) {
Trans fee = new Trans (-1,... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
occur, and calling a non-existent method is one of them.
For this reason, the code above will actually be rejected by the Java compiler.
Instead we have to write this:
void chargeMonthlyFee () {
for (int i = 0; i { accounts.size(); i++) {
Trans fee = new Trans (-1, new Date ());
if (((Account) accounts.elementAt ... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
this code, since the presence of the downcast
ensures that there will be no attempt to call a method that does not exist.
Students are often confused about downcasts, and think that some kind of conversion is taking
place. This is not true. The downcast is simply a test; no change to the object occurs.
5 Downcasts ... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
that Vector is actually positioned quite deep in
the tree: its code is built by inheritance from the classes AbstractCollection and AbstractList
which provide skeletal implementations of collections and lists respectively.
Not every type is a class, though. Java has specification types, called interfaces, that do not... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
that the types that appear in declarations in
the program text tell you something about what will happen when the program runs:
Static typing: If a variable of (declared) type T holds a reference to an object of (runtime)
type T’, then T’ is a subtype of T.
And we can now explain downcasts like this. In the assignm... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
tests, and unlike typecasts, have no
side effects.
8 | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
6.895 Essential Coding Theory
September 22, 2004
Lecturer: Madhu Sudan
Scribe: Swastik Kopparty
Lecture 5
1 Algebraic Codes
In this lecture we will study combinatorial properties of several algebraic codes. In particular, we will
introduce:
Reed-Solomon Codes based on univariate polynomials over finite fields.
... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
is a bijection.
q
a code
map
1) = n
k + 1.
∈
∈
(k
(k
S
−
−
−
−
≈
−
−
n
�
|
|
q
polynomial interpolation theorem:
Theorem 1 Polynomial Interpolation Theorem: Let F be a field. For any �1, �2, . . . �l+1
pairwise distinct, and any y1, y2, . . . , yl+1
p(�i) = yi,
unique polynomial p(x) with degree
F ,
l such ... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
with degree
[n] and any y1, y2, . . . , yl+1
l such that p(�it ) = yi,
unique poly
≤
[l + 1].
F,
≤
t
�
≤
�
�
≤
This motivates the Reed Solomon Code (1960ish) as defined below:
Given n, q, k, �1, . . . �n, with �i distinct elements of Fq .
•
For c = (c0, c1, . . . ck−1), define polynomial pc(x) = c0 + c1x + . . . ... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
= �i, where � is a generator of the multiplicative
q,with q a prime power,
k + 1]q code.
n
−
�
�
�
group F� . However, today we don’t bother with that.
q
Any code that meets the projection bound (with d = n
Separable (MDS) code.
k + 1) is called a Maximum Distance
−
2.1 Linear Codes and their Duals
Recall... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
C
C
C
�
C
C
−
3 Multivariate Polynomials and Reed Muller Codes
The large alphabet size of Reed Solomon codes makes it not as nice as we would have liked. There is a
natural generalization of these codes to multivariate polynomials which mitigates this problem to some
extent.
Without further ado, we define the multiv... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
m ∈
RM(p) = p(x) : x
•
�
l n, because of the following result which will be proved later: Any non-zero polynomial
�
l qm points. We already know this result for m = 1 (and
d = 1
− q
on Fm of degree
indeed used it to prove the distance of the RS code).
l is zero on at most q
�
q
q
= q
•
Let us... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
= m + 1 and d = 1/2 2m . Now the rate of this code is horrendous,
m + 1 bits get encoded as 2m bits. However, we get great relative distance (= 1/2). This code is called
the Hadamard code.
�
−
q
�
1
·
Last lecture we saw the unnerving phenomenon that there cannot exist a binary code with more than
2 codewords wit... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
judicious choice of interleaving strategy, data on a disk can be spread out so that
any local catastrophe (spatially close bits are destroyed) only affects a few bytes of lots of codewords
(all of which can be recovered) as opposed to many bytes of a single codeword.
2 To recap, let x, y, z be codewords in a code wit... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
p(x) = 0]
�
P rx�RSm [p(x) = 0 pl1 (x1, . . . xm−1) = 0] + P rx�R Sm [pl1 (x1, . . . xm−1) = 0]
|
l−l1 + l1 = |S| .
|S|
|S|
l
�
Note that this also proves (by putting q = 2, l = 1, S = F2) the fact that the inner product of a
nonzero vector with a purely random vector gives a purely random bit. This result shows u... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
Lecture 1
Mean Value Theorem
Theorem 1 Suppose Ω ⊂ Rn , u ∈ C2(Ω), Δu = 0 in Ω, and B = B(y, R) ⊂⊂ Ω, then
u(y) =
1
nωnRn−1
1
Rn
ωn
�
B
udx
�
uds =
∂B
�
∂u
∂Br ∂ν
ds =
�
Br
Δudx = 0. Thus
0 =
ds =
�
∂u
∂Br ∂ν
Proof:By Green’s formula, for r ∈ (0, R),
�
∂u
∂Br ∂r
�
∂u
Sn−1 ∂r
�
∂r Sn−1
(r ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
call u superharmonic.
Also we have �u ≤ 0 =
⇒ u(y) ≥
nωnRn−1 ∂B
1
�
Application: Maximum principle and uniqueness.
1
Theorem 2 Ω ⊂ Rn, u ∈ C2(Ω), Δu ≥ 0, If ∃p ∈ Ω s.t.
u(p) = max u,
Ω
then u is constant.
Proof: Let
M = sup u,
Ω
ΩM = {x ∈ Ω|u(x) = M }.
ΩM is not empty because p ∈ M , ΩM is closed by c... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
�
constant C = C(n, Ω, Ω�) s.t.
sup u ≤ C inf u.
Ω�
Ω�
Proof: Let y ∈ Ω�, B(y, 4R) ⊂ Ω. Take x1, x2 ∈ B(y, R), we have
�
�
u(x1) =
1
ωnRn
udx ≤
1
ωnRn
u(x2) =
1
ωn(3R)n
udx ≥
1
ωn(3R)n
udx,
B(y,2R)
B(x1,R)
�
B(x2,3R)
udx,
B(y,2R)
�
= ⇒ u(x1) ≤ 3n u(x2),
= ⇒ sup ≤ 3n
.
B(y,R)
inf
B(y,R)
Choos... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
Theorem 4 u ∈ C∞, Δu = 0, Ω� ⊂ Ω. Then for multiindex α, there exists constant
C = C(n, α, Ω, Ω�) s.t.
|
sup Dα u ≤ C sup u .
|
|
Ω�
Ω
|
Proof: Since ∂ Δ = Δ ∂ , Du is also harmonic. So by mean value theorem and
divergence theorems, we have for B(y, R) ⊂ Ω,
∂xi
∂xi
Du(y) =
1
ωnRn
�
B(y,R)
Dudx =
1
Rn
ωn
... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
n)ωn
1
2π log |x|
, n > 2,
, n = 2.
Note that away from origin, ΔΓ(x) = 0.
3
Theorem 5 Suppose u ∈ C2(Ω), then for y ∈ Ω, we have
�
u(y) =
(x − y) − Γ(x − y)
)dσ +
Γ(x − y)Δudx.
∂u
∂ν
Ω
�
∂Γ
(u
∂Ω ∂ν
Proof: Take ρ small enough s.t. Bρ = Bρ(y) ⊂ Ω. Apply Green’s 2nd formula to u
and v(x) = Γ(x − y), ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
and ϕ is continuous function on ∂B. Then
�
u(x) =
2 �
−|x |
R2
nωnR
ϕ(x)
ϕ(y)
∂B x−y n ds
|
|
, x ∈ B,
, x ∈ ∂B.
belongs to C2(B) ∩ C0(B) and satisfies Δu = 0 in B.
4 | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
(cid:10) 6.642 Continuum Electromechanics
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
(cid:13)
6.642, Continuum Electromechanics, Fall 2004
Prof. Markus Zahn
Lecture 6: Stress Tensors
I. Maxwell Stress Tensor ... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
Generalized Description
0
( )
a z
=
∼
⎡
Re A e−
⎢
⎣
jkz
( )
b z
=
∼
⎡
Re B e−
⎢
⎣
jkz
⎤
⎥
⎦
⎤
⎥
⎦
2 k
π
k
π ∫
2
0
*
(cid:105) (cid:105)
a z b z dz = Re AB = Re A B
(cid:105) (cid:105)
*
( )
( )
⎤
⎥
⎦
1
2
⎡
⎢
⎣
1
2
⎡
⎢
⎣
force on a wavelength
⎤
⎥
⎦
f =
z
w
π
k
μ
0
=
w
π μ
k
0
r *
(cid:105) (cid:105)r
⎡
Re H H
⎢
z
⎣
x... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
5)
= H =
s
r
(cid:4)
= H = -
χ
(cid:105)
r
1
jk
1
jk
z
z
(cid:105)
Κ
s
jk
(cid:105)
K
r
jk
μ
0
r
⎡
(cid:105)
⎢
xH = k -
μ
⎢
⎣
0
s
(cid:4)
χ
sinh kd
r
(cid:4)
χ
⎤
⎥
+ coth kd
⎥
⎦
⎡
= k -
⎢
0
⎢
⎣
μ
(cid:105)
Κ
(cid:105)
Κ
r
jk sinh kd jk
-
s
⎤
coth kd
⎥
⎥
⎦
*
r
⎡
(cid:105) (cid:105)
Re -K H
⎢
r
⎣
x
μ
0
⎤
⎥
⎦
= -Re +
⎡
⎢... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
s
(
j
)
⎤
⎦
s
K = K sin
0
s
ω⎡
⎣
s
⎡
⎣
⎤
⎦
r
K = K sin
0
r
ω
r
⎡
⎣
(
t - k z' -
δ
)
⎤
⎦
; z' = z - Ut
6.642, Continuum Electromechanics Lecture 6
Prof. Markus Zahn Page 5 of 8
r
= K sin
0
(
⎡
⎣
ω
r
+ kU t - k z -
)
(
⎡
= Re -jK e
⎣
r
0
(
)
j +kU t
ω... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
sinh kd
0
r
K K sin k
0
s
0
δ
III. Electrostatic Machine
Courtesy of MIT Press. Used with permission.
6.642, Continuum Electromechanics Lecture 6
Prof. Markus Zahn Page 6 of 8
f = w
z
2
π
k
2 k
π
∫
0
T
zx
x=0
dz =
2 w
π
k
2 k
π
∫
0
ε
E E
0 z x
dz
x=0... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
⎤
+ V coth kd
⎥
⎥
⎦
r
* r
(cid:105) (cid:4)
Re -jk V E = Re -jk
r
x
ε
0
⎤
⎥
⎦
⎡
⎢
⎣
⎡
⎢
⎢
⎣
2
ε
0
(cid:105)
V
*
r
(cid:105)
-V
s
sinh kd
⎛
⎜
⎜
⎝
+ V coth kd
(cid:105)
r
⎞
⎟
⎟
⎠
⎤
⎥
⎥
⎦
f =
z
2
ε
kw
π
k sinh kd
0
⎡
= Re +jk
⎢
⎣
2
ε
0
(cid:105) (cid:105) *
⎤
V V sinh kd
⎥
r
s
⎦
(cid:105) (cid:105)
⎡
Re jV V
⎢
s
⎣
*
r
⎤
⎥... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
sinh kd
ω
s
r= + kU
ω
f = -
z
ε
wk
π
0
sinh kd
r
V V sin k
δ
0
s
0
6.642, Continuum Electromechanics Lecture 6
Prof. Markus Zahn Page 8 of 8 | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
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