text stringlengths 16 3.88k | source stringlengths 60 201 |
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For each eigenvalue
, we can evaluate the eigenvector
consisting of a set of mesh point values
v
i
, i.e.
j
j
j
V
Tj
=
V
j
v
1
j
v
2
j
v -
N
1
STABILITY ANALYSIS
Eigenvalue and Eigenvector of Matrix A
N
1)
The (
j
V
1) matrix
E
formed by the (
N
A
by
diagonalizes the matrix
(
N
1) columns
1E AE
= L
8
Slide 8... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
(cid:0)
-
-
-
(cid:0)
(cid:0)
(cid:0)
-
-
(cid:1)
(cid:1)
(cid:1)
(cid:1)
-
(cid:2)
(cid:3)
(cid:4)
(cid:5)
(cid:4)
(cid:5)
(cid:4)
(cid:5)
(cid:4)
(cid:5)
(cid:4)
(cid:5)
(cid:4)
(cid:5)
(cid:6)
(cid:7)
-
-
-
-
(cid:8)
(cid:8)
(cid:8)
L
-
-
-
-
(cid:9)
(cid:9)
(cid:9)
16.920J... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
-
16.920J/SMA 5212 Numerical Methods for PDEs
l
t
)
E
1
1
E b
Evaluating,
=
u EU E ce
=
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)
(
Complementary
(transient) solution
Particular (steady-state)
solution
(
(cid:7)(cid:1)(cid:7)(cid:1)(cid:7)(cid:8)(cid:7)
l
t
ce
)
=
where
l
t
1
c e
1
l
t
2
c e
2
l
t
j
c e
j
T
l... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
)
(cid:11)
(cid:11)
16.920J/SMA 5212 Numerical Methods for PDEs
STABILITY ANALYSIS
Use of Modal (Scalar) Equation
It may be noted that since the solution
contribution from all the modes of the initial solution,
which have propagated or (and) diffused with the eigenvalue
l
is expressed as a
, and a contribution fr... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
920J/SMA 5212 Numerical Methods for PDEs
EXAMPLE 1
Continuous Time Operator
Proceeding as before, or otherwise (solving the ODEs directly),
we can obtain the solution
=
=
u
1
u
2
l
x
e
c
1 11
x
e
c
1 21
t
1
+
c
l
t
1
+
e
x
2 12
x
2 22
c
l
t
2
l
t
2
e
l
where
and
l
1
2
are eigenvalues of
A
and
l
eigenvectors... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
)
(cid:3)
(cid:4)
(cid:3)
(cid:4)
(cid:5)
(cid:6)
£
-
-
-
-
-
(cid:7)
(cid:8)
(cid:7)
(cid:8)
(cid:9)
(cid:10)
(cid:11)
(cid:12)
(cid:11)
(cid:12)
(cid:13)
(cid:13)
(cid:13)
-
-
(cid:14)
(cid:14)
(cid:14)
(cid:14)
EXAMPLE 1
Discrete Time Operator
As
= L
A E E
1
,
n
u
= L
E ... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
27)(cid:26)
u
1
u
2
(cid:30)(cid:27)(cid:31)
=
[
c
1
]
c
2
x
x
11
21
x
x
12
22
l
t
1
l
t
2
e
e
to the solution where time is discretized
(cid:27)!
u
1
u
2
$(cid:27)%
n
=
[
c
1
'
]
'
c
2
x
x
11
21
x
x
12
22
l
n
1
l
n
2
14
-
(cid:0)
(cid:1)
L
(cid:2)
(cid:3)
(cid:4)
(cid:5)
(cid:8)
(cid:8)
-
(cid:1... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
EXAMPLE 2
Leapfrog Time Discretization
Consider a typical modal equation of the form
du
dt
=
l
+
u ae
m
t
j
l
where
j
is the eigenvalue of the associated matrix
A
.
(For simplicity, we shall henceforth drop the subscript j).
We shall apply the “leapfrog” time discretization scheme given as
du
dt
+
1
n
u
=
2
n
1... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
+
1
n
+
n
2
=
c
EXAMPLE 2
Leapfrog Time Discretization: Time Shift Operator
The complementary solution
nc satisfies the homogenous equation
Slide 25
+
1
n
c
l
h c
2
n
n
1
c
=
0
n
Sc
l
h c
2
n
n
c
S
=
0
16
(cid:0)
-
-
-
(cid:2)
-
-
-
-
-
-
-
16.920J/SMA... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
h e
h
h
1
Slide 27
EXAMPLE 2
Leapfrog Time Discretization: Stability Criterion
For the solution to be stable, the transient
(complementary) solution must not be allowed to grow
indefinitely with time, thus implying that
(
(
s
s
=
=
1
2
l
+
h
+
2
l
h
1
2
l
h
+
1
2
l
h
2
)
)
<
1
<
1
17
-
-
-
-
–
... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
2 Numerical Methods for PDEs
EXAMPLE 2
Leapfrog Time Discretization
In particular, by applying to the 1-D Parabolic PDE
=
u
u
t
2
u
2
x
the central difference scheme for spatial discretization, we obtain
=
A
u
2
x
2
1
1
2
1
0
0
1
2
1
which is the tridiagonal matrix
EXAMPLE 2
Leapfrog Time Discretization
Accordi... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
14)
(cid:15)
(cid:16)
(cid:17)
16.920J/SMA 5212 Numerical Methods for PDEs
One may note that
l
j
is always real and negative, thereby satisfying
the criterion for stability of the space discretization of a parabolic
PDE, keeping time continuous.
EXAMPLE 2
Leapfrog Time Discretization: Absolute Stability Diagram... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
.e. lim
0
but not the accuracy of the scheme.)
affect the stability
=
h
h
(
)
l
l
s
h
l
STABILITY ANALYSIS
Some Important Characteristics Deduced
4. By comparing the power series solution of the principal root to
l
he
one can determine the order of accuracy of the time discretization
scheme. In this example of leap... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
+
l
(1
n
h c
)]
=
0
characteristic polynomial
l
h
Therefore
and
(
n
s
c
l
= +
) 1
h
=
b s
n
The Euler-forward time discretization scheme is stable if
s
+
l
1
h
<
1
l
or bounded by
h
= -
s
1
s.t.
s
<
l
1 in the
h
-plane.
22
Slide 35
Slide 36
D
-
-
-
”
16.920J/... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
+
h
n
du
n
dt
+
1
n
=
u
1
2
n
u
+
ˆ
u
+
1
n
+
h
+
1
n
+
1
ˆ
du
n
dt
as applied to the typical modal equation
du
dt
=
l +
u
m
tae
of the parabolic PDE. Substituting
du
dt
and
ud ˆ
dt
yields
into the predictor-corrector scheme
hn
)
where
t
= D
n t
=
nh
l
ˆ
u
+
1
n
+
ae
m
h n
(
+
1)
)
m
ae
(
h
+
1
n
ˆ
u
=
n
u
+
h
... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
(cid:13)
(cid:14)
(cid:13)
(cid:14)
(cid:13)
(cid:14)
(cid:13)
(cid:14)
-
-
(cid:13)
(cid:14)
(cid:13)
(cid:14)
(cid:15)
(cid:16)
(cid:15)
(cid:16)
-
R
-
-
16.920J/SMA 5212 Numerical Methods for PDEs
(
s
)
= R
(
S
)
=
S S
1
l
h
l
2
2
h
=
0
1
2
s
s
=
0 (trivial root)
1
2
+
h
h
l
l
2
2
= +
1
i.e. the sc... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
MA 5212 Numerical Methods for PDEs
When h increases from zero, s decreases from 1.0. As h continues to increase, s
reaches a minimum of 0.5 with l h = - 1 and then increases. As h increases further, s
returns to 1.0 with l h = - 2. Prior to this point, the scheme is stable. Increasing h and
thus s beyond this point... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
l h)
s (l
16.920J/SMA 5212 Numerical Methods for PDEs
AND l
l h
•
•
•
The above set of ODEs becomes
+
1
n
u
2
n
1
u
h
=
n
+
Au
n
b
Introducing the time shift operator S
n
Su
A
=
+
n
u
S
S S
h
2
2
hAu
n
+
2
hb
n
1
n
I u
= -
n
b
Premultiplying
=
1
I EE
E
operating on
1
n
u
on the LHS and RHS and introducing
... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
s
s
s
l
l
-
-
(cid:15)
(cid:15)
(cid:15)
(cid:15)
-
-
(cid:16)
(cid:17)
-
L
-
(cid:18)
(cid:19)
(cid:20)
(cid:21)
(cid:22)
(cid:22)
-
(cid:23)
(cid:24)
-
L
-
(cid:25)
(cid:26)
(cid:27)
(cid:28)
(cid:29)
(cid:29)
-
-
(cid:30)
L
-
-
16.920J/SMA 5212 Numerical Methods for PDEs
Hen... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
s
s
l
l
s
s
s
s
s
s
l
l
(cid:6)
(cid:7)
(cid:8)
(cid:9)
(cid:10)
(cid:11)
(cid:12)
(cid:12)
s
s
s
l
l
s
s
s
s
s
s
l
l
16.920J/SMA 5212 Numerical Methods for PDEs
IMPLICIT TIME-MARCHING SCHEME
Thus far, we have presented examples of explicit time-marching
methods and these may be ... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
:8)
(cid:9)
-
-
(cid:10)
(cid:11)
-
-
(cid:12)
(cid:13)
16.920J/SMA 5212 Numerical Methods for PDEs
the characteristic polynomial becomes
(
s
)
= R
(
S
)
=
(
1
l
h
)
S
1
=
0
The principal root is therefore
s
=
1
l
h
1
= +
1
l
h
+
l
2
2
h
+
....
which, upon comparison with
l
he
= +
1
l
+
h
first-o... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
16.920J/SMA 5212 Numerical Methods for PDEs
intensive/expensive compared to the multiplication/
addition operations of explicit schemes.
SUMMARY
•
Stability Analysis of Parabolic PDE
Uncoupling the set.
Integrating each equation in the uncoupled set fi
modal equation.
Re-coupling the results to form final soluti... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
Lecture 11
Acoustics of Speech & Hearing
6.551 - HST 714J
Lecture 11: Electro-Mechano-Acoustic Transformers
I. Ideal Transformers & Transducers
1. Ideal Transformers and Transformation of variables
a. One example of a two-port is an Ideal Electrical transformer, where:
P1
P2
U 2
U1
= T =
where T=“the turns ratio... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/6f129e5077cd757d6833fadc2c3d8d65_lec_11_2004.pdf |
THROAT
BONEY
SKULL
INNER EAR
In the lizards the tympanic
membrane is about 20 times
larger in area than that of the
boney footplate of the ossicle (the
wide part that couples to one of
the inner ear windows). This
leads to a transformer ratio of
1/20 that would suggest.
U 2
U1
=
V AFP
V ATM
≈
1
20
.
The Impeda... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/6f129e5077cd757d6833fadc2c3d8d65_lec_11_2004.pdf |
3
Lecture 11
Acoustics of Speech & Hearing
6.551 - HST 714J
when the voltage induced variations in x, C and F are small. Input voltages produce a
force on and velocity of the moving plate (or diaphragm) that when integrated over the
surface of the moving plate produce a volume velocity and a sound pr... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/6f129e5077cd757d6833fadc2c3d8d65_lec_11_2004.pdf |
MTM
TA
(
)
1 jωCE
E( )2
)+ TM
1 jωCE
(
(
jωCM
)
=
P
E
MTM
TA
1 CE
E( )2
1 CE + TM
CM
14- Oct -2004
page 6
Lecture 11
Acoustics of Speech & Hearing
6.551 - HST 714J
3. Reciprocity Revisited
We have already given you a formal definition of reciprocal networks in terms
of constraints placed on two-port network... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/6f129e5077cd757d6833fadc2c3d8d65_lec_11_2004.pdf |
.3)
page 7
Lecture 11
Acoustics of Speech & Hearing
6.551 - HST 714J
Now lets fix I2 with a current source while setting I1 = 0, i.e. opening the circuit at port
1. The relationship between I2 and E1 can be defined from Eqns 11.1 as
I2 I1=0 = E1 Y21 −
⎛
⎜
⎝
Y11Y22
Y12
⎞
⎟ .
⎠ ... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/6f129e5077cd757d6833fadc2c3d8d65_lec_11_2004.pdf |
the transformation
of the force and the velocity produced by the moving
plate of the transducer to volume-velocity and sound
pressure and has units of area.
3. The middle mechanical-impedance describes the force
necessary to move the outer plate with a given velocity.
4. The left hand transformer describes the tr... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/6f129e5077cd757d6833fadc2c3d8d65_lec_11_2004.pdf |
aphragm A, and the electro-static transducer
constant TES.
The Radiation Impedance
Radiation
The Mechanical Impedance with
P
U
=
jωM A RA
jωM A + RA
F
V
=
1
jωCM
+ A2
⎛
jωM A RA
⎜ ⎜
jωM A + RA
⎝
⎞
⎟ ⎟
⎠
Sound Pressure / Force Sound Pressure / Volt
P
F
=
1
A
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1
jωCM
+ A2
⎛
⎜ ⎜
⎝
A2
⎛ ... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/6f129e5077cd757d6833fadc2c3d8d65_lec_11_2004.pdf |
RA
⎝
1
jωCM
1
TES
1
A
P
E
=
+ A2
⎞
⎟
⎟
⎠
or
Sound Pressure / Volt (Approximates)
14- Oct -2004
page 11
Lecture 11
Acoustics of Speech & Hearing
6.551 - HST 714J
P
E smallω
=
1
A TES
(
(
−ω2CM A2 M A
))
P
E large ω
=
1
A TES
.
Electro-Static Speaker Characteristics
ZmRad
ZmComp
Z... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/6f129e5077cd757d6833fadc2c3d8d65_lec_11_2004.pdf |
America 1982
Courtesy of Acoustical Society of America. Used with permission.
In generating an electrical analog of electro-acoustic transduction in the electro-
dynamic speaker, it is still convenient to break the system into electrical; mechanical
and acoustical sections that are connected by ‘ideal’ transducers mo... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/6f129e5077cd757d6833fadc2c3d8d65_lec_11_2004.pdf |
ESD.86. Markov Processes and their
Application to Queueing II
Richard C. Larson
March 7, 2007
Photo: US National Archives
Outline
(cid:139) Little’s Law, one more time
(cid:139) PASTA treat
(cid:139) Markov Birth and Death Queueing Systems
Queueing System
Arriving Customers
SERVICE
FACILITY
Queue of Waiting Custome... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/6f188063c7051ea94c5db10683665f21_lec9.pdf |
)(λ1 /μ2)P0 = (λ0λ1 /[μ1μ2])P0
Pn +1 = (λn /μn +1)Pn = (λ0λ1...λn /[μ1μ2...μn +1])P0
Telescoping!
Source: Larson and Odoni, Urban Operations Research
λnPn = μn +1Pn +1 n = 0,1,2,...
λ0P0 = μ1P1
λ1P1 = μ2P2 ...
λnPn = μn +1Pn +1
P1 = (λ0 /μ1)P0
P2 = (λ1 /μ2)P1 = (λ0 /μ1)(λ1 /μ2)P0 = (λ0λ1 /[μ1μ2])P0
Pn +1 = (λn /μn +... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/6f188063c7051ea94c5db10683665f21_lec9.pdf |
1− ρ
for ρ< 1
P0 = 1− λ/μ for λ/μ< 1.
Pn = (λ/μ)n P0 = (λ/μ)n (1− λ/μ) for n = 1,2,3,...
L = λW = ρ/(1− ρ)
implies W = (1/λ)ρ/(1− ρ) = (1/μ) /(1− ρ)
Lq = λW q etc.
Mean Wait vs. Rho
25
20
15
10
5
0
Note the Elbow!
Series1
0
0.2
0.4
0.6
0.8
1
Rho
More on M/M/1 Queue
Let w(t) = pdf for time in the system
(including q... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/6f188063c7051ea94c5db10683665f21_lec9.pdf |
a single
server queue with service rate μ
or a 2-server queue each with rate
μ/2?
Can someone draw one or both of the
state-rate-transition diagrams?
Then what do you do?
Final Example:
Single Server, Discouraged Arrivals
λ/2
λ/3
λ/4
λ/5
State-Rate-Transition Diagram, Discouraged Arrivals
Pk =
1
k!
(
λ
μ
)k P0
) ... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/6f188063c7051ea94c5db10683665f21_lec9.pdf |
MEASURE AND INTEGRATION: LECTURE 17
p
a
Inclusions between L spaces. Consider Lebesgue measure on the
space (0, ∞) ⊂ R. Recall that x is integrable on (0, 1) ⇐⇒ a > −1,
and it is integrable on (1, ∞) ⇐⇒ a < −1. Now let 1 ≤ p < q ≤ ∞.
Choose b such that 1/q < b < 1/p. Then x−bχ(0,1) is in L but not in
q . On the oth... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/6f21af6c40de1eccd70349bd3a3b0095_18125_lec17.pdf |
then µ(A)1/p → 1 as p → ∞. If µ(A) = ∞, then
µ(A
= ∞. In both cases, we have
)1/p
Since t is arbitrary,
lim inf �f �p ≥ t.
p→∞
lim inf
p→∞
�f �p ≥ �f �∞
.
Date: October 30, 2003.
1
2
MEASURE AND INTEGRATION: LECTURE 17
For the reverse inequality, we need the assumption that f ∈ L for
some (finite) r. For r ... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/6f21af6c40de1eccd70349bd3a3b0095_18125_lec17.pdf |
1, then
�
�f �1 ≤ �f �p ≤ �f �q ≤ �f �∞
Counting measure and lp spaces. Let X be any set, M = P(X),
and µ be the counting measure. Recall that µ(A) is the number of
points in A if A is finite and equals ∞ otherwise. Integration is simply
.
�
f dµ =
�
f (x)
x∈X
X
for any nonnegative function f , and Lp is de... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/6f21af6c40de1eccd70349bd3a3b0095_18125_lec17.pdf |
Local LP spaces. Let G be an open set in Rn . The local Lp space on
G consists of all Lmeasurable functions f defined a.e. on G such that
for every compact set K ⊂ G, the characteristic function f χK has a
finite Lp norm; that is,
�
f (x)|p dx < ∞
|
if 1 ≤ p < ∞;
K
f is essentially bounded on K
if p =
.∞
This s... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/6f21af6c40de1eccd70349bd3a3b0095_18125_lec17.pdf |
1
1
p − q
p − q
1
We NTS that log �f �r ≤ θ log �f �p + (1 − θ) log �f �q . Note that
rθ
1 = +
p
r(1 − θ)
,
q
and so p/rθ and q/r(1 − θ) are conjugate exponents. Thus, by H¨older’s
inequality,
�
�
�
�
f θ f 1−θ
r
�
�
�
�
f rθ f r(1−θ)
��
�
�
�
�
�
f rθ
f r(1−θ)
1/r
1
p/rθ
�f �r
=
=
≤
=
�
r(1−θ)
rθ... | https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/6f21af6c40de1eccd70349bd3a3b0095_18125_lec17.pdf |
Lecture 03
Support vector machines (SVM).
18.465
As in the previous lecture, consider the classification setting. Let X = Rd , Y = {+1, −1},
and
where |ψ| = 1.
H = {ψx + b, ψ ∈ Rd, b ∈ R}
We would like to maximize over the choice of hyperplanes the minimal distance from the
data to the hyperplane:
where
max min d... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/6f32028c31f130e362caa9e6ffdd66d6_lecture03.pdf |
∂φ
∂ψ
= ψ −
αiyixi = 0
∂φ
∂b
= −
�
�
αiyi = 0
1
Lecture 03
Support vector machines (SVM).
18.465
Hence,
and
Substituting these into φ,
ψ =
αiyixi
�
αiyi = 0.
�
2
n
n
1
2
1
2
φ =
=
=
αiyixi −
��
�
i=1
�
αiαj yiyj xixj −
αi yi
�
�
αj yj xj xi + b − 1
�
�
j=1
�
αiαj yiyj xixj... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/6f32028c31f130e362caa9e6ffdd66d6_lecture03.pdf |
Vector Machines find optimal separating hyperplane in a very high-dimensional
∞ φk (xi)φk(xj ) be a scalar product in X . Notice that we don’t
k=1
∞ φk (xi)φk (xj ), a
k=1
need to know mapping x → φ(x). We only need to know K(xi, xj ) =
′
�
symmetric positive definite kernel.
Examples:
�
(1) Polynomial: K(x1, x2)... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/6f32028c31f130e362caa9e6ffdd66d6_lecture03.pdf |
Lecture 04
Generalization error of SVM.
18.465
Assume we have samples z1 = (x1, y1), . . . , zn = (xn, yn) as well as a new sample zn+1. The
classifier trained on the data z1, . . . , zn is fz1,...,zn .
The error of this classifier is
Error(z1, . . . , zn) = Ezn+1 I(fz1,...,zn (xn+1) 6
= yn+1) = Pzn+1 (fz1,...,zn (xn+... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/6f32095a4c52bfbdf7b9b2a159f6502b_lecture04.pdf |
�
Therefore, to obtain a bound on the generalization ability of an algorithm, it’s enough to
obtain a bound on its leave-one-out error. We now prove such a bound for SVMs. Recall
that the solution of SVM is ϕ =
n+1 αi
i=1
0yixi.
Theorem 4.1.
�
L.O.O.E. ≤
min(# support vect., D2/m2)
n + 1
where D is the diameter... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/6f32095a4c52bfbdf7b9b2a159f6502b_lecture04.pdf |
2
i
α0 =
i
�
i∈supp.vect
In the last step we use the fact that
�
α0 = m2 . Indeed, since |ϕ| = ,
�
i∈supp.vect
1
1
m
i
�
= |ϕ|2 = ϕ · ϕ = ϕ ·
1
2m
0 yixi
αi
�
0(yiϕ · xi)
αi
=
=
�
�
�
=
α0
i
�
0(yi(ϕ · xi + b) − 1) +
αi
α0 − b
i
0
��
�
�
0 yi
αi
�
0
� �� �
We now prove Lemma 4.1.
Pro... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/6f32095a4c52bfbdf7b9b2a159f6502b_lecture04.pdf |
β(i)=0 β(i)=1
β(i)=0
′
′
0, . . . , 0, α1, . . ., . . . , α ℓ, 0, . . . , 0 = α
� �� � � �� � � �� � � �� �
�
� �
�
−
��
+
��
′
n
where β ∈ {0, 1} .
Let t > 0 and suppose α′ + tβ satisfies optimization conditions (1). We know that
′ w(α + tβ) ≤ w(α0).
w(α0) − w(α ) ≥ w(α + tβ) − w(α ).
′
′
′
w(α ) =
′
α... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/6f32095a4c52bfbdf7b9b2a159f6502b_lecture04.pdf |
βi(1 − yiϕ · xi) −
��
βi(1 − yi(ϕ · xi + b)) + tb
′
= t
= t
�
�
βiyixi
2
�
βiyi −
�
0
� �� �
2
βiyixi
�
t2
2
��
βiyixi
2
�
′
= t(1 − y1(ϕ · x1 + b)) −
2
t
2
��
Maximizing the above expression over t, we find
t =
1 − y1(ϕ′ · x1 + b)
βiyixi)
(
2
≥ 0.
�
Substituting this t back into the expression,
S... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/6f32095a4c52bfbdf7b9b2a159f6502b_lecture04.pdf |
�� � �
−
��
�
4
Lecture 04
Generalization error of SVM.
18.465
We have
and α0 − γ satisfies constraint (2) and
w(α0) − w(α ) ≥
′
1
2D2
w(α0 − γ) ≤ w(α ).
′
w(α0) − w(α ) ≤ w(α0) − w(α0 − γ) =
′
... similarly to the previous proof
=
1
2
��
= x1 −
2
γiyixi =
�
k
γi
α0 xi
1
�
i=p
1)2
(α0
2 ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/6f32095a4c52bfbdf7b9b2a159f6502b_lecture04.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
3.23 Fall 2007 – Lecture 8
PERIODICPERIODICPER
oDICPERIODICPER
Image removed due to copyright restrictions... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/6f349951d2f95813100f1a5c119f8ca4_clean8.pdf |
ch Theorem
Bloch Theorem
• n, k are the quantum numbers (band index
and crystal momentum), u is periodic
• From two requirements: a translation can’t
change the charge density, and two
translations must be equivalent to one that is
the sum of the two
Bloch Theorem
(cid:71)
(cid:71)
r R
+
)
(
Ψ
(cid:71)
nk
=
exp
(... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/6f349951d2f95813100f1a5c119f8ca4_clean8.pdf |
Microfabrication for MEMS: Part III
Carol Livermore
Massachusetts Institute of Technology
* With thanks to Steve Senturia, from whose lecture notes some of
these materials are adapted.
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
• Etch rate the same in all crystal directions
> Anisotropic
• For wet etches, rate depends on crystal plane
• For dry etches, directionality determined by process
> Selectivity
• Etch rate of substrate vs. etch rate of mask
> Mask adhesion (for wet etching)
• Increased etching along mask/substrate interface
> Temperat... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
is often used as a final release etch.
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lec... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
[DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 8
Making a V-groove
> The previous etch is allowed to go to “termination”,
i.e. the slowing of etch rate when only {111} planes
are exposed
[110]
Top
View
54.70
Cross Section
54.70
Cross Section
Image by MIT OpenCourseWare.
Adapted from Figure 3.21 in: Sent... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
removed due to copyright restrictions.
Figure 3 on p. 143 in: Enoksson, Peter. "New Structure for Corner Compensation in Anisotropic KOH Etching." Journal
of Micromechanics and Microengineering 7, no. 3 (September 1997): 141-144.
A common approach to corner compensation as shown in Enoksson,
J. Micromech. Microeng. 7 ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
[110]
50
misalignment
Image by MIT OpenCourseWare.
Figure 3.25 in: Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 65. ISBN: 9780792372462.
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
material, e.g. oxide or nitride
» Heavily boron doped silicon, p+, as etch
stop for strong bases
(etches several orders of magnitude
more slowly than lightly doped if
concentration > 5 x 1019 cm–3)
• Electrochemical etch stop
Motorola
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabricat... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 17
Shape
> The higher the pressure, the more isotropic the etch
because reactants are... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
romechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 19
Depth depends on features and layout
> Features of different width etch at different rates (recipe
dependent)
Image removed ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
4. Strip resist; pattern
w
resist
ith ne
mask
w
8. Strip oxide mask
as: Carol
Cite
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
als for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
e, course materi
Livermo... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
~ 500 C
Bonding with an intermediate “glue” layer
» Gold (thermocompression), ~ 300 C
» Polymer or epoxy layer
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Te... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
substrate
Thin top wafer
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 28
W... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 30
Wafer bonding and yield
> Yield in MEMS can require a whole-wafer outlook, unlike IC
processing
> A micron-scale defect can create a mm- to cm-scale defect
•... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
, Lecture 3 - 32
Designing process flows for cleanliness
> If you are planning to do a fusion bond, design your process
flow to prevent exposure of bonding surfaces to junk
• Cleanliness is a good idea for anodic bonding, too, but
anodic bonding is less picky
> Some junk washes off easily, but some doesn’t
> Exampl... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 35
Illustrating surface micromachining
Top View
Cross Section
> Example
• Structu... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
.com.
Used with permission.
Figure 10 on p. 242 in: Harsh, K. F., V. M. Bright, and
Y. C. Lee. "Solder Self-assembly for Three-dimensional
Microelectromechanical Systems." Sensors and Actuators
A: Physical 77, no. 3 (Nov. 1999): 237-244.
Courtesy of Elsevier, Inc., http://www.sciencedirect.com.
Used with permission.
Fi... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
2 (phosphosilicate glass)
(cid:137) LPCVD nitride acts as passivation, electrical isolation layer
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downlo... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
Lithography Anchor1 and RIE PSG
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 46
Step 7
Courtesy of MEMSCAP, Inc. Used with permission.
(cid:137) Lithography Anchor 2 and RIE PSG (oxide-2 and oxide-1)
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechan... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 50
Step 11
Courtesy of MEMSCAP, Inc. Used wi... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 53
One release recipe
(cid:137) Acetone soak to remove photoresist (30 min)
(cid:137) ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
6.777J/2.372J Spring 2007, Lecture 3 - 55
(cid:10)
Outline
> Etching
> Wafer bonding
> Surface micromachining
> Process integration
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachuse... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 3 - 58
Chemical Mechanical Polishing (CMP)
> Often used to planarize interlayer dielectric insulators
> Typical surface roughness less than of 1 nm (but waviness can
be much bigger)
> Combination of mechani... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
remember)
Stringer location
Image by MIT OpenCourseWare.
Adapted from Figure 3.35 in: Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 75. ISBN: 9780792372462.
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
structures
> One alternate approach: include etch features on your mask
that will separate the dies most of the way so they snap apart at
the end
> Either way, must think about this when creating your process
flow
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectrom... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
design your processes on the hairy edge of impossibility.
• Including a very difficult process may be unavoidable, but a) don’t
include a lot of them and b) be prepared to put a lot of work into
making that process robust.
• On the design projects, we will know if your process is too
ambitious. In your thesis or in ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6f533cbf3074f26dad3b2e819442dfc5_07lecture03.pdf |
18.997 Topics in Combinatorial Optimization
March 11, 2004
Lecture 10
Lecturer: Michel X. Goemans
Scribe: Nicole Immorlica
Matroid theory was first formalized in 1935 by Whitney [5] who introduced the notion as an
attempt to study the properties of vector spaces in an abstract manner. Since then, matroids have
pr... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
which graphs G and H does M (G) = M (H)? It is easy to see that the matroid representations
of two different graphs might be the same. For example, for the graphs G and H in Figure 1(a) and
Figure 1(b) respectively, M (G) = M (H).
a
c
d
b
f
e
h
g
a
c
d
b
h
e
f
g
(a) G
(b) H
Figure 1: Switching operati... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
uniquely embeddable, but
this is not necessary), then M (G) = M (G ) where the ∗ operation indicates taking the dual of the
corresponding object. It can be shown that planar graphs are unique in this sense.
∗
∗
Theorem 3 (Tutte) The dual matroid of a graphic matroid M (G) corresponding to graph G is
itself a graphi... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
�
Figure 2: Representation of M and M .
∗
∗
Consider the matrix A = [BT |I (n−m)×(n−m)] (Figure 2(b)). Since Z was a basis, B restricted
to the X1 rows and Y1 columns has full rank. Thus the X1 columns in A also have full rank, and so
∗Z = X1 ∪ Y2 is an independent set of vectors. By a similar argument, it is a maxi... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
is U 2
4
itself.)
4
Theorem 5 A matroid is binary if and only if it excludes U 2 as a minor.
4
Tutte further characterized regular matroids, or matroids representable over any field.
Definition 1 The Fano matroid is the matroid with ground set S = {A, B, C, D, E, F, G} whose
bases are all subsets of S of size 3 ex... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
unpublished work of Reid, later published by Bixby [1] and Seymour [3].
Theorem 7 The ternary matroids are the matroids which exclude U5
minors.
∗
2 , U 2 = U5
5
3 , F7, and F7 as
∗
In 2000, Geelen, Gerards and Kapoor characterized matroids representable over GF (4) [2] by
specifying seven excluded minors, a work ... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
) and U(3,5)
U(2,4)
Ternary,
GF(3)
Figure 4: Classes of matroids.
scale them so that they lie in the z = 1 plane). This new representation, say A(cid:2) , B(cid:2) , C(cid:2), and
G(cid:2), preserves the independence relations and thus is also a representation of F7. Now notice that
span(A, G) ∩ span(C, B) = spa... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
=1Mi of matroids M1 = (S1, I1), . . . , Mk = (Sk, Ik) is the
Definition 2 The matroid union ∨i
matroid M = (∪k Si, I) where I = {∪k
i=1
Ii : Ii ∈ Ii}.
i=1
We will show that M is a matroid; this is not completely obvious. Furthermore, one can charac-
terize the size of a maximal independent subset in the union of matro... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
≥ 0
x(U ) ≤ r(U )
∀s ∈ S
∀U ⊆ S
Note that the second inequality implies xs ≤ 1 as the rank of a single vertex is at most one. We
will show that this polytope is integral and that the vertices are the indicator vectors of independent
sets of the matroid. Certainly all independent sets of the matroid satisfy that x... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
optimal, thus proving Theorem 9. Furthermore, this shows that
the dual is integral for an arbitrary integral weight function, and thus the system is TDI. Together
with the fact that the rank function is integral, this proves that the matroid polytope is integral,
thus proving Theorem 10.
Let’s prove that O(cid:2) =... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
J. F. Geelen, A. M. H. Gerards, and A. Kapoor. The excluded minors for gf (4)-representable
matroids. Journal of Combinatorial Theory Series B, 79, 2000.
[3] P. Seymour. Matroid representation over gf (3). J. Coubin. Theory Ser. B, 26:159–173, 1979.
[4] W. T. Tutte. A homotopy theorem for matroids, i, ii. Trans. Ame... | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/6faef8afbcaec34e49dd0dab12611e0f_co_lec10.pdf |
3.37 (Class 5)
Review:
• Metals have high surface energies
o Share electrons from several levels below the atomic surface
o Longer range distance of attraction
• Friction welding has high interfacial shears
o For good friction welder need a good brake so that don’t break the weld
when it stops
o Inertial fricti... | https://ocw.mit.edu/courses/3-37-welding-and-joining-processes-fall-2002/6fd0647ea5db46590857dd4c7bebcdad_33705.pdf |
3 + 2Fe + Heat
o Raises to about 2000 degC
o Start with a small blasting cap to start the chemical reaction
o Molten iron flows down, makes a small casting
o Get about 50% defective welds
o Cheap, equipment on the order of $100
o See rounded, cone-shaped surface
o Story: MIT prank, thermit welding trolley car wh... | https://ocw.mit.edu/courses/3-37-welding-and-joining-processes-fall-2002/6fd0647ea5db46590857dd4c7bebcdad_33705.pdf |
Pretension or posttension of steel reinforcing elements
o Sometimes need to repair, even tear down building
Question: Changing metallurgical properties of the rails?
o High carbon steels
o Used to be mostly hot-rolled rails, these don’t really degrade
o Some heat-treated rails
o Starting to use 1% Cr rails (not h... | https://ocw.mit.edu/courses/3-37-welding-and-joining-processes-fall-2002/6fd0647ea5db46590857dd4c7bebcdad_33705.pdf |
3.15 Magnetic Fundamentals
C.A. Ross, DMSE, MIT
References:
Jiles, Introduction to Magnetism and Magnetic Materials
Magnetic quantities and units
H = magnetic field, A/m –represents energy gradient, or torque on a dipole
2
–number of magnetic field lines per unit area
B = magnetic flux density, T or Wb/m
M = magne... | https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/7009df3d9c177d5865038f9d6a3377f4_ho9_magnetic_fundamentals.pdf |
cancel out, so strong magnetic effects are found in
materials with unpaired electrons.
One electron has a moment of 1 µBB
Stern-Gerlach and Zeeman experiments indicate the quantization of the magnetization in
atoms.
(Bohr magneton) = 9.27 10
Am
-24
2
We expect large magnetic effects in transition metals (unfilled 3d... | https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/7009df3d9c177d5865038f9d6a3377f4_ho9_magnetic_fundamentals.pdf |
Magnetic energy consists of the following terms:
exchange energy (minimise by having all spins parallel)
magnetostatic energy (minimise by having domains pointing in different directions so
there is no external field)
Zeeman energy (potential energy due to an external magnetic field, E = M.H)
magnetocrystalline energ... | https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/7009df3d9c177d5865038f9d6a3377f4_ho9_magnetic_fundamentals.pdf |
self-energy or demagnetizing energy)
The energy in the field surrounding the magnetized object depends on the way the object is
magnetized. For instance, a long thin object has less magnetostatic energy if it is magnetized
along its length, compared to across its length. This can be expressed in the same way as a
unia... | https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/7009df3d9c177d5865038f9d6a3377f4_ho9_magnetic_fundamentals.pdf |
LECTURE 5
Finite fields
5.1. The finite field method
In this lecture we will describe a method based on finite fields for computing the
characteristic polynomial of an arrangement defined over Q. We will then discuss
several interesting examples. The main result (Theorem 5.15) is implicit in the
work of Crapo and Rota ... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/703f0b5aecce90b89f34d10d9f52ac54_lec5.pdf |
0
Proposition 5.13. Let A be an arrangement defined over Z. Then A has good
reduction for all but finitely many primes p.
Proof. Let H1, . . . , Hj be affine hyperplanes, where Hi is given by the equation
if and only if
vi
Zn). By linear algebra, we have H1
x = ai (vi, ai
Hj =
·
≤
⊕ · · · ⊕
�
(36)
rank
�
Moreov... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/703f0b5aecce90b89f34d10d9f52ac54_lec5.pdf |
0 (mod p).
�
This can only happen for finitely many p, viz., for certain B we must have p det(B),
�
so L(A) ∪= L(Ap) for p sufficiently large.
The main result of this section is the following. Like many fundamental results
|
in combinatorics, the proof is easy but the applicability very broad.
Theorem 5.15. Let A be ... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/703f0b5aecce90b89f34d10d9f52ac54_lec5.pdf |
the remainder of this lecture, we will be concerned with applications of
y
�
Theorem 5.15 and further interesting examples of arrangements.
⇔
⇔
LECTURE 5. FINITE FIELDS
63
Example 5.12. Let G be a graph with vertices 1, 2, . . . , n, so
QAG (x) =
(xi
ij⊆E(G)
�
xj ).
−
Then by ... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/703f0b5aecce90b89f34d10d9f52ac54_lec5.pdf |
certain properties that we will not give
here. (References include [4][7][12].) The Coxeter arrangement A(R) consists of
the hyperplanes κ x = 0, where κ
R. There are four infinite (irreducible) classes
of root systems (all in Rn):
≤
·
ei
ei
{
An−1 =
Dn =
{
Bn = D
Cn = D
n
n
n
}
i < j
= Bn
n
→
}
−
→
ej : 1
... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/703f0b5aecce90b89f34d10d9f52ac54_lec5.pdf |
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