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upper
part
of
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complex
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(j /h
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p
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(A.192)... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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2.062J / 1.138J / 18.376J Wave Propagation
Spring 2017
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u
Lect
inciples
of
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Mathematics
Rodolfo
Rosales
Spring
2014
acteristics
of
u +c *u
=
0
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traffic
flow]
and
u +c *u
x
0
t
t
x
0
Recap
solution
by
char
=
a*u.
t
or
simple
variable
coefficients,
whe... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/2885f5e781a8df81578b6b487caecf6f_MIT18_311S14_Lecture6.pdf |
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... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/2885f5e781a8df81578b6b487caecf6f_MIT18_311S14_Lecture6.pdf |
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18.311 Principles of Applied Mathematics
Spring 2014
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3.044 MATERIALS PROCESSING
LECTURE 6
Ex. 1: glass fiber (ceramic)
Ex. 2: plasma spray (ceramic and metal)
Ex. 3: hot rolling steel slabs (metal)
look at iron-carbon (steel) phase diagram, red hot is about 900 − 1000◦C,
need to heat into gamma field to make it soft and eliminate ceramic carbide phase
Problem Statement: Ho... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/28a25e06805d1037eecbb968e3e7f747_MIT3_044S13_Lec06.pdf |
)
7700
Solution:
t = 22, 000s ≈ 6 hours
T
Ti −
− Tf
Tf
How to decrease time?:
= f (k, c
1. thinner L → constrained by casting
2. higher h (fluid) → molten metal, salt
3. hotter Tf → high energy, doesn’t drastically change time
p, ρ, t, Lx, h)
4. preheat Ti
⇒
vertical integration, combine casting and rolling temperatures... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/28a25e06805d1037eecbb968e3e7f747_MIT3_044S13_Lec06.pdf |
t) = Θ(x, t)Θ(y, t)
F0,x = F0,y
Θ(x, t) = Θ(y, t) =
√
0.1 = 0.32
F0 = 4
t ≈ 3hrs
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3.044 Materials Processing
Spring 2013
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18.156 Differential Analysis II
Lectures 1-2
Instructor: Larry Guth
Trans.: Kevin Sackel
Lecture 1: 4 February 2015
nR , and we will use coordinates x1, . . . , xn when
Throughout these notes, we will be working typically over
necessary. The convention will be that the Laplacian on nR with the standard Euclidean metric ... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
x)+2n(cid:3) =
2n(cid:3) > 0. Hence the previous lemma applies to show u(cid:3) attains its maximum on the boundary, and taking
the limit as (cid:3) → 0 yields the result.
Corollary 1.3. If Δu = Δv with u, v ∈ C 2(Ω) ∩ C 0(Ω) and u|∂Ω = v|∂Ω, thenu = v on all of Ω.
Proof. We have that Δ(u − v) = 0 with u − v = 0 on ∂Ω,... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
a bit mysterious, and doesn’t really tell us what’s going on under the hood. We
present a second proof which takes into account the role of symmetries. We will only do this for the case of
n = 2 and with x = 0, but this can be easily generalized.
Alternate proof of MVP, n = 2. For θ ∈ SO(2) (cid:8) /2π we have a corres... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
∂i(cid:11)u(y)dy,
|B1/2| S1/2(x)
whereby the notation (cid:10)nor, ∂i(cid:11) means the dot product of the vector ∂i with the normal vector at the point
y ∈ S(x, 1/2). These coefficients don’t actually depend on y, and we see immediately that
for some constant Cn depending only on n.
|∂iu(x)| ≤ Cn max |u|
2
Corollary 1.... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
A big first theorem for us to prove is Schauder’s.
Recall the idea of Holder-regularit
¨
y. For 0 < α ≤ 1, we have the semi-norms
[f ]Cα := supx,y
|
f (y)
f (x)
−
|x − y|α
|
,
and corresponding norms (cid:12)f (cid:12)Cα := [f ]Cα + (cid:12)f (cid:12)C0 , where these are defined on some space of functions. Then
Theorem 1... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
If u ∈ C 3 ( nRcpt
), then
(cid:12)∂2u(cid:12)L2 = (cid:12)Δu(cid:12)L2 .
3
Proof. This follows from two applications of integration by parts:
(cid:7) (cid:6)
(cid:12)∂2u(cid:12)2
L2 =
(∂i∂ju)(∂i∂ju)
i,j
Rn
(cid:7) (cid:6)
= −
i,j
(cid:7) (cid:6)
=
=
i,j
(cid:6)
Rn
(cid:7)
(
Rn
i
2
= (cid:12)Δu(cid:12)L2
(∂2
i ∂ju)(∂j... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
cients of Δ − L are aij − δij, which are bounded in absolute value by (cid:3), so continuing the
chain of inequalities, we have
(cid:12)∂2u(cid:12)2 ≤ (cid:12)Lu(cid:12)2
L2
L2 + 2(cid:3)n
(cid:6)
|Lu||∂2
u| + (cid:3) n2(cid:12)∂2u(cid:12)2
2
L2 .
Applying the Cauchy-Schwarz inequality to the middle term, this yields
(... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
|
2
|∂2(ηu)|2
|∂2(ηu
)
|2
B
(cid:6)
1
≤ 4
|
L(ηu) 2
|
B1
(cid:6)
(cid:6)
(cid:6)
≤ 4
η2 Lu
|
|2 + C(n)
|∇u||∇η| + C(n)
|∂2η||u|
B1
The first term is zero, and for the other two, an application of Cauchy-Schwarz separates the u and η terms
in into different integrals so that we get in total
(cid:12)∂2u(cid:12)2
2
L2(B ) ≤... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
the class is Schauder’s inequality.
2.2 Solving Δu = f (a story from physics)
On our way to Korn’s inequality, we want to understand how we can write explicit estimates on the second
partial derivatives of u in terms of Δu. Here, the story begins with some classical physics theory. We will
make the mathematics a bit mo... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
(cid:5)
Br
Fz(x)dz. By symmetry, V (x) is always parallel to x. It suffices to check that
(cid:6)
(cid:6)
V · nor =
F0
· nor|Br|.
∂B(R)
∂B(R)
The RHS is easily seen to be 4π|Br| (by the calculation we stated at the end of the lemma), while the LHS
is
(cid:11)
(cid:6)
(cid:8)(cid:6)
(cid:9)
(cid:6) (cid:10)(cid:6)
(cid:6)... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
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18.156 Differential Analysis II: Partial Differential Equations and Fourier Analysis
Spring 2016
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Resource: Calculus Revisited
Herbert Gross
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
provided by the author as an individual learning resource.
For information about citing these materials or our Terms of Use, visit: http://oc... | https://ocw.mit.edu/courses/res-18-006-calculus-revisited-single-variable-calculus-fall-2010/28c0a2faa9b3f5f4156b58ca0dcb4155_MITRES_18_006_supp_notes.pdf |
Massachusetts Institute of Technology
6.270 Autonomous LEGO Robot Competition
IAP 2005: Attack of the Drones
Workshop 5 — Servos and Advanced Sensors
Monday, January 10, and Tuesday, January 11, 2005
1
Items to Bring
• Handy Board with Expansion Board
2 Reading
Chapter 5 and Appendix 5 of the course notes
3 S... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/28e4da973faa6d894e41ccc9d60be617_5_sss.pdf |
Sensors
• Phototransistor. This sensor alone is unreliable. Although useful for detecting the starting light, it should
be calibrated for each different lighting environment to which the robot is subjected. The phototransistor is
very sensitive to light, and should be shielded carefully. We suggest using the black he... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/28e4da973faa6d894e41ccc9d60be617_5_sss.pdf |
for break
beam purposes (remember that IR is susceptible to red light and color).
1
Updated January 10, 2005
Massachusetts Institute of Technology
6.270 Autonomous LEGO Robot Competition
IAP 2005: Attack of the Drones
5
Activity
This activity will help your robot orient correctly and dependably. Wire up a pho... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/28e4da973faa6d894e41ccc9d60be617_5_sss.pdf |
be? Now try
doing some readings without the LED, and also with varying light conditions. Try shining a flashlight on the table
or shading the table with your hand. What difference does it make?
This is a fairly simple activity, but these are all considerations you need to make when attaching sensors to your
robot, an... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/28e4da973faa6d894e41ccc9d60be617_5_sss.pdf |
1
Multilevel Memories
Joel Emer
Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Based on the material prepared by
Krste Asanovic and Arvind
CPU-Memory Bottleneck
6.823 L7- 2
Joel Emer
CPU
Memory
Performance of high-speed computers is usually
limited by memory bandwidt... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
electrode (VREF)
Ta2O5 dielectric
word
access
FET
Image removed
due to copyright restrictions.
bit
Explicit storage
capacitor (FET
gate, trench,
stack)
poly
word
line
W bottom
electrode
access fet
TiN/Ta2O5/W Capacitor
October 3, 2005
Processor-DRAM Gap (latency)
6.823 L7- 6
Joel Emer
1000
e
c
n
a
... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
in flight!
October 3, 2005
DRAM Architecture
6.823 L7- 8
Joel Emer
N
s
s
e
r
d
d
A
w
o
R
r
e
d
o
c
e
D
N+M M
Col.
1
bit lines
Col.
2M
word lines
Row 1
Row 2N
Memory cell
(one bit)
Column Decoder &
Sense Amplifiers
Data D
• Bits stored in 2-dimensional arrays on chip
• Modern chips have around 4 logi... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
processor are often highly
predictable:
…
loop: ADD r2, r1, r1
SUBI r3, r3, #1
BNEZ r3, loop
…
PC
96
100
104
108
112
What is the pattern
of instruction
memory addresses?
October 3, 2005
Typical Memory Reference Patterns
Address
n loop iterations
linear sequence
Instruction
fetches
Stack
access... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
storage, e.g., memory
– Address usually computed from values in register
– Generally implemented as a cache hierarchy
• hardware decides what is kept in fast memory
• but software may provide “hints”, e.g., don’t cache or
prefetch
October 3, 2005
A Typical Memory Hierarchy c.2003
6.823 L7- 16
Joel Emer
Split ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
HIT
Not in cache
a.k.a. MISS
Return copy
of data from
cache
October 3, 2005
Read block of data from
Main Memory
Wait …
Return data to processor
and update cache
Q: Which line do we replace?
Placement Policy
6.823 L7- 21
Joel Emer
Block Number
0 1 2 3 4 5 6 7 8 9
1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
g
a
T
t
k
c
o
B
l
t
e
s
f
f
O
b
October 3, 2005
HIT
Data
Word
or Byte
Replacement Policy
6.823 L7- 26
Joel Emer
In an associative cache, which block from a set
should be evicted when the set becomes full?
• Random
• Least Recently Used (LRU)
• LRU cache state must be updated on every access
• true imp... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
29
Joel Emer
Average memory access time =
Hit time + Miss rate x Miss penalty
To improve performance:
• reduce the miss rate (e.g., larger cache)
• reduce the miss penalty (e.g., L2 cache)
• reduce the hit time
What is the simplest design strategy?
October 3, 2005
Write Performance
6.823 L7- 30
Joel Emer
T... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
Row
Column
Precharge
Row’
[ Micron, 256Mb DDR2 SDRAM datasheet ]
October 3, 2005
Data
400Mb/s
Data Rate | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
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2.161 Signal Processing: Continuous and Discrete
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Massachusetts Institute of Technology
Department of Mechanical Engineering
2.161 Signal Processing - Continuous and Di... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
set {fn} unless it is known a-priori that f (t)
contains no spectral energy at or above a frequency of π/ΔT radians/s.
• In order to uniquely represent a function f (t) by a set of samples, the sampling
interval ΔT must be sufficiently small to capture more than two samples per
cycle of the highest frequency componen... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
angular frequency
a and a + 2πm/ΔT , for any integer m, will generate the same sample set.
In the figure below, a sinusoid is undersampled and a lower frequency sinusoid, shown as a
dashed line, also satisfies the sample set.
f(t)
o
o
t
o
Δ T
o
0
o
0
o
o
10–2
This phenomenon is known as aliasing. After ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
3) (cid:4) (cid:6)
(cid:3) (cid:3) (cid:4) (cid:6)
The following figure shows the effect of folding in another way. In (a) a function f (t) with
Fourier transform F (j Ω) has two disjoint spectral regions. The sampling interval ΔT is
chosen so that the folding frequency π/ΔT falls between the two regions. The spectru... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
) (cid:13) (cid:17) (cid:13) (cid:16) (cid:8)
(cid:19) (cid:21) (cid:9)
(cid:6) (cid:11) (cid:12) (cid:6) (cid:8) (cid:17) (cid:18) (cid:15) (cid:8) (cid:10) (cid:17) (cid:19) (cid:16) (cid:20) (cid:6) (cid:11)
(cid:10) (cid:21) (cid:13) (cid:17) (cid:13) (cid:16) (cid:8)
(cid:19) (cid:21) (cid:9)
(cid:7) (cid:3) ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
)
(cid:6)
(cid:4)
(cid:7)
1. Selecting a sampling interval ΔT sufficiently small to capture all spectral components,
or
2. Processing the continuous-time function f (t) to “eliminate” all components at or above
the Nyquist rate.
The second method involves the use of a continuous-time processor before sampling f (t).... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
3) (cid:4) (cid:2) (cid:5)
(cid:15) (cid:15)
(cid:15) (cid:7) (cid:4) (cid:8) (cid:2)
(cid:5) (cid:10) (cid:4) (cid:8) (cid:2)
(cid:5)
(cid:10) (cid:11) (cid:17) (cid:20)
(cid:26) (cid:6) (cid:9)
(cid:4) (cid:6)
(cid:27) (cid:3) (cid:15)
(cid:16)
(cid:28)
(cid:9)
(cid:13)
(cid:9)
(cid:13)
In practice it i... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
cid:16) (cid:15) (cid:4) (cid:6)
(cid:7) (cid:3) (cid:8)
(cid:4) (cid:6)
(cid:2)
(cid:5)
(cid:3) (cid:8)
(cid:4) (cid:6)
(cid:2)
(cid:4) (cid:6)
(cid:7) (cid:10) (cid:3) (cid:8)
(cid:4) (cid:6)
(cid:7) (cid:3) (cid:8)
(cid:4) (cid:6)
(cid:7) (cid:9) (cid:9) (cid:4) (cid:8) (cid:2)
(cid:5)
(cid:9) (cid:8) (ci... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
30)
(cid:21)
(cid:30)
(cid:8)
(cid:15)
(cid:18)
(cid:21)
(cid:6)
(cid:12)
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(cid:6)
(cid:4)
(cid:7)
(cid:9) (cid:9)
(cid:4) (cid:2) (cid:5)
(cid:9) (cid:... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
y(t) = F −1 {F (cid:2)(j Ω)H(j Ω)} = F −1 {F (j Ω)} = f (t).
The filter’s impulse response h(t) is
h(t) = F −1 {H(j Ω)} =
sin (πt/ΔT ) ,
πt/ΔT
(cid:18) (cid:4) (cid:2) (cid:5)
(cid:19)
(cid:23) # (cid:4) (cid:6)
(cid:23) $ (cid:4) (cid:6)
(cid:23) " (cid:4) (cid:6)
(cid:23) (cid:4) (cid:6)
(cid:12)
(cid:4) (c... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
nΔT − σ) dσ
sin (π(t − nΔT )/ΔT )
,
π(t − nΔT )/ΔT
10–5
!
(cid:15)
(cid:20)
(cid:16)
(cid:20)
(cid:16)
(cid:12)
(cid:15)
(cid:15)
(cid:15)
(cid:15)
(cid:11)
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(cid:10)
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(cid:15)
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(cid:12)
(cid:11)
(cid:16)
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(cid:20)
(cid:3)
(cid:9)
(cid:9)
(cid:17)
(cid:9)
(cid:17)
(cid... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
0.3
2
The Discrete Fourier Transform (DFT)
We saw in Lecture 8 that the Fourier transform of the sampled waveform f ∗(t) can be written
as a scaled periodic extension of F (j Ω)
F (cid:2)(j Ω) =
1
ΔT
�
∞
� �
F
j Ω −
n=−∞
��
2nπ
T
We now look at a different formulation of F ∗(j Ω). The Fourier transform of... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
of F ∗(j Ω) in a single
period, from Ω = 0 to 2π/ΔT , that is at frequencies
Ωm =
2πm
N ΔT
for m = 0, 1, 2, . . . , N − 1
(cid:9)
(cid:9) (cid:4) (cid:8) (cid:2)
(cid:9) (cid:17)
(cid:13) (cid:15)
(cid:9)
(cid:8)
(cid:21) (cid:13) (cid:17) (cid:15)
(cid:10) (cid:17) (cid:20)
(cid:21) (cid:18) (cid:15)
(cid:21... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
N m=0
Fm e
j 2πmn/N
for n = 0, 1, 2, . . . , N − 1
which is known as the inverse DFT (IDFT). These two equations together form the DFT
pair.
10–7
(cid:7)
(cid:5)
(cid:20)
%
(cid:30)
(cid:12)
(cid:23)
(cid:8)
(cid:10)
(cid:6)
(cid:19)
(cid:17)
(cid:18)
(cid:15)
(cid:8)
(cid:6)
(cid:10)
(cid:11)
(cid:17)
(cid:12)... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
Fourier transform case, we adopt the notations
DFT
{fn} ⇐⇒ {Fm}
{Fm} = DFT {fn}
{fn} =
IDFT {Fm}
to indicate DFT relationships.
10–8 | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.080 / 6.089 Great Ideas in Theoretical Computer Science
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
6.080/6.089 GITCS
21 February 2008
Lecturer: Scott Aaronson
Scribe: Emilie Kim
Lecture 5
1 Administriv... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
care about
that.
3 Oracles
Oracles are a concept that Turing invented in his PhD thesis in 1939. Shortly afterwards, Turing
went on to try his hand at breaking German naval codes during World War II. The first electronic
computers were used primarily for this purpose of cracking German codes, which was extremely
d... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
A : {0, 1}∗ → {0, 1}
where the input is any string of any length and the output is a 0 or 1 answering the problem, then
we can write
M A
for a Turing machine M with access to oracle A. Assume that M is a multi-tape Turing machine
and one of its tapes is a special “oracle tape”. M can then ask a question, some stri... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
be solvable.
If we solve the Diophantine problem, can we also then solve the halting problem?
5-2
�
This was an unanswered question for 70 years until 1970, when it was shown that there was
no algorithm to solve Diophantine equations because if there were, then there would also be an
algorithm for solving the hal... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
comes K¨onig’s Lemma. Let’s say that you have a tree (a computer science tree, with its
root in the sky and grows towards the ground) with two assumptions:
1) Every node has at most a finite number of children, including 0.
2) It is possible to find a path in this tree going down in any finite length you want.
Claim: ... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
number of possible choices. Further,
we’ve assumed that you can tile any finite region of the plane, which means the tree contains
arbitrarily long finite paths. Therefore, K¨onig’s Lemma tells us that the tree has to have an infinite
path, which means that we can tile the infinite plane. Therefore, the tiling problem i... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
Duper Halting Problem, the Super Duper Duper Halting Problem...
Is there any problem that is in an “intermediate” state between computable and the halting
problem? In other words, is there a problem that is 1) not computable, 2) reducible to the halting
problem, and 3) not equivalent to the halting problem?
This wa... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
a more powerful
system, and then there will be statements within that system that can’t be proved, and so on.
5-5
ComputableHalting Problem(Diophantine, Tiling,...)Super Halting ProblemComputableHalting Problem(Diophantine, Tiling,...)Super Halting Problem? Second Incompleteness Theorem: No consistent, computable sys... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
sentence is provable, and
is therefore a provable falsehood! That can’t happen if we’re working within a sound system of
logic. So the sentence has to be true, but that means that it isn’t provable.
G¨odel showed that as long as the system of logic is powerful enough, you can define provability
within the system. Fo... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
is consistent. G¨odel shows that no
sound logical system can prove its own consistency.
The key claim is that Con(S) G(S). In other words, if S could prove its own consistency,
⇒
then it actually could prove the “unprovable” G¨odel sentence, “This sentence is not provable”.
Why? Well, suppose G(S) were false. Then ... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
Ocean Acoustic Theory
• Acoustic Wave Equation
• Integral Transforms
• Helmholtz Equation
• Source in Unbounded and Bounded Media
• Reflection and Transmission
• The Ideal Waveguide
– Image Method
– Wavenumber Integral
– Normal Modes
• Pekeris Waveguide
13.853
COMPUTATIONAL OCEAN ACOUSTICS
1
Lecture 3
U(t... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/2959dc70927677a2f919c6287be36df4_lect_32.pdf |
Harvard-MIT Division of Health Sciences and Technology
HST.951J: Medical Decision Support, Fall 2005
Instructors: Professor Lucila Ohno-Machado and Professor Staal Vinterbo
From Propositions To Fuzzy Logic and Rules
Staal A. Vinterbo
HarvardMIT Division of Health Science and Technology
Decision Systems Group, BWH... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
�, ∨, (, )}.
Definition
An expression in PL is any string consisting of elements from the sets
V and S, i.e., any string of variables and symbols.
An expression is either a well formed formula (wff) or it is not.
The following wff fomation rules allow us to define wff:
Definition
� A variable alone is a wff
� If α... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
pretation: Truth Value of Expressions
Definition
We define a setting s as a function s : V → {0, 1} assigning to each
variable either the value 0 or the value 1, denoting true or false
respectively.
Definition
An interpretation is a function that takes as input a wff and returns 0 or
1 depending on the setting used... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
)
Fuzzy Stuff
HST 951/MIT 6.873
11 / 56
Staal A. Vinterbo (HST/DSG/HMS)
Fuzzy Stuff
HST 951/MIT 6.873
12 / 56
Propositional Logic
Semantics
Propositional Logic
Semantics
Propositional Logic Semantics
Semantics of Operators: Infix notation
Propositional Logic Semantics
Computing the Interpretation I
Usua... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
∧ is called conjunction (“and”)
(a ∧ b) = ∼ (∼ a∨ ∼ b)
def
(a ∧ b) is often called the “conjunction of a and b”.
→
is called implication (“implies”)
�
def
(a → b) = (∼ a ∨ b)
Is (∼ (∼ a∨ ∼ b)) = ∼ (∼ 1∨ ∼ 0)
= ∼ (0 ∨ 1) =∼ 1 = 0
�
Left side is the antecedent, right side is the consequent. We also
let (b ← a) =... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
of variables a and b and the
resulting value for (a
b).
→
� Is (a
→
b) valid? Satisfiable?
Valid: No. Satisfiable: Yes.
Note:Tables can become Large.
Staal A. Vinterbo (HST/DSG/HMS)
Fuzzy Stuff
HST 951/MIT 6.873
17 / 56
Staal A. Vinterbo (HST/DSG/HMS)
Fuzzy Stuff
HST 951/MIT 6.873
18 / 56
Propositional Logic
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
r ))) → q) valid?
(( p
1
4
∧
1
2
( p ↔ ( q
∧
r ))) → q
)
1
6
1
5
1
8
1
7
1
9
0
1
0
3
Answer: Yes. The settings underlined pose a contradiction.
Note:
If we during the process shown are allowed alternatives, we need to
show a contradiction in all the possible alternative settings in order to
declare our... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
/b] is ((c ∧ d ) → ((c ∧ d ) ∨ c)).
Modus Ponens (also called the rule of detachment) is sometimes
written as
α
α
β
β→
If α and α
the truthfunctional meaning of →.
→
β are theorems, then by MP so is β. This simply reflects
Staal A. Vinterbo (HST/DSG/HMS)
Fuzzy Stuff
HST 951/MIT 6.873
23 / 56
Staal A. Vinter... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
Propositional Consequence: Example
By showing how we would do without the rule:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
↔ (β ∧ γ))given
αgiven
(α
((α → (β ∧ γ)) ∧ ((β ∧ γ) → α))US (a→
((((α → (β ∧ γ)) ∧ ((β ∧ γ) → α))) → ((α → (β ∧ γ)))US ((a∧b)→
(α
(β ∧ γ)
((β ∧ γ) →
β
→ (β ∧ γ))MP (3)+(4)
β)US ((a∧b)→a)
MP (1)+... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
.
� Furthermore, f is the characteristic function of the subset S of U
such that S consists exactly of the elements x in U such that
f (x ) = 1.
� Formally, S = f −1(1) = {x ∈ U f (x ) = 1}. We will denote the
|
characteristic function for the set S ⊆ U as χS .
Staal A. Vinterbo (HST/DSG/HMS)
Fuzzy Stuff
HST 95... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
over Sets
Semantics: Truth Sets
Propositions over Sets
Semantics: Truth Sets for “Syntactic Sugar”
The semantics of p over U is based on truth sets. We define truth sets
of wff of PL(U) according to the following rules: For p ∈ F , and wff α
and β
|
� T (p) = {x ∈ U p(x ) = 1},
� T (∼ α) = U − T (α), and
� T (α ∨... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
number” over the natural numbers modeled by the characteristic
functions even and prime with the usual definitions. Let
α = even ∧ prime. Then we have that
T (α) = T (even) ∩ T (prime) = {2},
and I(α, x ) = 1 if and only if x = 2.
Staal A. Vinterbo (HST/DSG/HMS)
Fuzzy Stuff
HST 951/MIT 6.873
35 / 56
Staal A. Vi... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
is a proposition HeightTall over the set of
all people. We now formulate the ifthen rule as propositions over sets:
(HeightTall ∧ HairDark) → LookHandsome
The application becomes:
(HeightTall ∧ HairDark)
(HeightTall ∧ HairDark) → LookHandsome
(LookHandsome)
In other words we set I(β, x ) =
Effect:
We infer th... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
Fuzzy Stuff
HST 951/MIT 6.873
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Fuzzy Sets
Fuzzy Sets
Fuzzy Sets
Crisp Set Operators Definitions
Fuzzy Sets
Fuzzy Set Operations Example
Let A and B be two subsets of some set U. We define union,
intersection, difference, and complementation using in terms of χA and
χB as follows:
Example
Definition
χA... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
interbo (HST/DSG/HMS)
Fuzzy Stuff
HST 951/MIT 6.873
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Staal A. Vinterbo (HST/DSG/HMS)
Fuzzy Stuff
HST 951/MIT 6.873
46 / 56
Fuzzy Relations
Fuzzy Relations
Fuzzy Relations
Fuzzy Composition
Definition
Let X , Y and Z be three sets and let R and R� be two fuzzy relations
from X to Y and Y to Z , respecti... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
47 / 56
Staal A. Vinterbo (HST/DSG/HMS)
Fuzzy Stuff
HST 951/MIT 6.873
48 / 56
4
A Restricted Fuzzy Logic
A Restricted Fuzzy Logic
Fuzzy Logic
Defining the Fuzzy Logic Language
Fuzzy Logic
Semantics
Recall:
For PL(U), the interpretation I(α, x ) is given by
I(α, x ) = χT (α)(x ).
Definition (Fuzzy Proposit... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
(α β)(x ) =
↔
min(max(1 − µT (α)(x ), µT (β)(x )), max(1 − µT (β)(x ), µT (α)(x ))).
Definition (Fuzzy Interpretation)
If we let WFF (U) be the set of wffs of FPL(U) we define the
interpretation I(α, x ) of a wff α with respect to an element x in U to be
� I(α, x ) = µT (α)(x ).
Staal A. Vinterbo (HST/DSG/HMS)
Fuz... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
characteristic functions of subsets of U.
� that a truth set for a given wff is the set for which the interpretation
is a characteristic function.
� that a propositional rule essentially is the application of modus
ponens to an implication called the rule.
Summary
Fuzzy Sets and Logic
We have learned
� that fuz... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
Lecture 10: Supercurrent Equation
Outline
1. Macroscopic Quantum Model
2. Supercurrent Equation and the London
Equations
3. Fluxoid Quantization
4. The Normal State
5. Quantized Vortices
October 13, 2005
Massachusetts Institute of Technology
6.763 2005 Lecture 10
Macroscopic Quantum Model
1. The wavefunctio... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/29bb21252c2fd3293111508de473d2ac_lecture10.pdf |
closed contour
within a superconductor:
The line integral of each of the parts:
Therefore,
flux
integer
n = - n’
Massachusetts Institute of Technology
6.763 2005 Lecture 10
Fluxoid Quantization
The
flux quantum is defined as
And the
Fluxoid Quantization condition becomes
Fluxiod
Experiments testing fluxi... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/29bb21252c2fd3293111508de473d2ac_lecture10.pdf |
Foundations of Applied Superconductivity. Reading, MA:
Addison-Wesley, 1991. ISBN: 0201183234.
Massachusetts Institute of Technology
6.763 2005 Lecture 10
Induced Currents
To have flux quantization, currents must be induced in the cylinder to add to or
oppose the applied magnetic field.
Induced flux= L i
Image rem... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/29bb21252c2fd3293111508de473d2ac_lecture10.pdf |
conductors
Image removed for copyright reasons.
Please see: Figure 6.1, page 259, from Orlando, T., and K. Delin.
Foundations of Applied Superconductivity. Reading, MA:
Addison-Wesley, 1991. ISBN: 0201183234.
Massachusetts Institute of Technology
6.763 2005 Lecture 10
The Vortex State
nV is the areal density of ... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/29bb21252c2fd3293111508de473d2ac_lecture10.pdf |
current density is then
the electrons become normal. This
ξ
In the absence of any current flux, the superelectrons have zero net velocity
but have a speed of the fermi velocity, v . Hence the kinetic energy with
currents is
F
Massachusetts Institute of Technology
6.763 2005 Lecture 10
9
Coherence Length x
The... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/29bb21252c2fd3293111508de473d2ac_lecture10.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 7: Pressure–Velocity Relations for Inviscid,... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/29d24b6ff1988b3c0fc376b437fa7706_lec07_f08.pdf |
α
(cid:3)
, v
x
(
)
Δ =
(cid:3)
α
v
x
6.642, Continuum Electromechanics Lecture 7
Prof. Markus Zahn Page 1 of 5
Θ
(
' 0
)
β
(cid:108)
= Θ
(cid:3)
(
, v 0
x
)
=
(cid:3)
v
β
x
(cid:108) (
Θ
x
)
=
(cid:108)
Θ
α
sinh kx
(cid:108)
− Θ
β
(
sinh k x
− Δ
)
sinh k
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/29d24b6ff1988b3c0fc376b437fa7706_lec07_f08.pdf |
Θ
dx
(cid:3)
p = j
(
ρ ω
- k U
z
) (cid:108)
Θ
⎡
⎢
⎢
⎣
(cid:108)
Θ
α
(cid:108)
Θ
β
⎤
⎥
⎥
⎦
=
1
k
⎡
⎢
⎢
⎢
⎢
⎣
−
coth k
Δ
−
1
sinh k
Δ
1
sinh k
coth k
⎤ ⎡
⎥
Δ ⎢
⎥
⎢
⎥ ⎢
⎥ ⎣
⎦
Δ
(cid:3)
v
(cid:3)
v
α
x
β
x
⎤
⎥
⎥
⎥
⎦
⎡
(cid:3)
p
⎢
⎢
(cid:3)
p
⎣
α
β
⎤
⎥
⎥
⎦
=
(
ρ ω −
j
k U
z
)
k
⎡
⎢
⎢
⎢
⎢
⎣
−
coth k
Δ
−
1
sinh k
Δ
1
inh k
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/29d24b6ff1988b3c0fc376b437fa7706_lec07_f08.pdf |
Courtesy of MIT Press. Used with permission.
II.
Gravity – Capillary Dynamics
6.642, Continuum Electromechanics Lecture 7
Prof. Markus Zahn Page 3 of 5
Equilibrium:
gx P
+
−ρ
a
0
x 0>
0P =
; v 0=
,
0ξ =
−ρ
b
gx P
+
0
x 0
<
Perturba... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/29d24b6ff1988b3c0fc376b437fa7706_lec07_f08.pdf |
⎦
(cid:3)
v
x1
(cid:3)
v=
x4
≡ 0
(rigid boundaries)
Interface:
v
=
x
∂ξ
t
∂
+
v
y
∂ξ
y
∂
+
v
z
∂ξ
z
∂
⇒
(cid:3)
v
x2
(cid:3)
v
=
x3
(cid:3)
= ωj ξ
Force Balance
'
P
3
( )
ξ
'
- P
2
( )
ξ
= -
⎛
γ ⎜
⎜
⎝
2
∂ ξ
2
y
∂
+
2
∂ ξ
2
z
∂
⎞
⎟
⎟
⎠
P
3
( )
ξ
= P
30
(
0
+ ξ
)
'
+ P
3
( )
ξ
= P
30
(
0
)
+
dP
30
dx
x 0
=
ξ +
(
'
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/29d24b6ff1988b3c0fc376b437fa7706_lec07_f08.pdf |
�
(cid:3)
coth kav
⎤
⎦
x2
= -
2
ρ ω
a
k
(cid:3)
coth ka
ξ
6.642, Continuum Electromechanics Lecture 7
Prof. Markus Zahn Page 4 of 5
2
ω
k
⎡
⎢
⎢
⎣
(
ρ
a
coth ka
+ ρ
b
coth kb + g
)
(
ρ − ρ
b
a
)
2
- k
γ
(cid:3)
ξ
= 0
⎤
⎥
⎥
⎦
Dispersion Relation:... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/29d24b6ff1988b3c0fc376b437fa7706_lec07_f08.pdf |
(
ρ − ρ
b
a
)
coth ka ≈
1
ka
coth kb ≈
1
kb
2
ω
2
k
ρ
a
a
⎛
⎜
⎝
+
ρ
b
b
⎞
⎟
⎠
=
g
(
ρ − ρ
a
b
)
2
ω
2
k
=
2
v
P
=
(
g
ρ
b
ρ − ρ
a
ρ
+
a
a
b
b
)
Non-dispersive gravity wave
6.642, Continuum Electromechanics Lecture 7
Prof. Markus Zahn Page 5 of 5 | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/29d24b6ff1988b3c0fc376b437fa7706_lec07_f08.pdf |
Lecture 4
PN Junction and MOS Electrostatics(I)
Semiconductor Electrostatics in Thermal
Equilibrium
Outline
• Nonuniformly doped semiconductor in thermal
equilibrium
• Relationships between potential, φ(x) and
equilibrium carrier concentrations, po(x), no(x)
–Boltzmann relations & “60 mV Rule”
• Quasineutral situa... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/2a02d79958a93f067bd1ae334492fb6a_MIT6_012S09_lec04.pdf |
0
What is n o(x) that satisfies this condition?
no, Nd
partially uncompensated �
donor charge
Nd(x) +
no(x)
net electron charge
-
Let us examine the electrostatics implications of
n o(x) ≠ Nd(x)
x
6.012 Spring 2009
Lecture 4
5
Space charge density
ρρρρ(x) = q Nd(x) − no(x)
[
]
no, Nd
ρ
partially unc... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/2a02d79958a93f067bd1ae334492fb6a_MIT6_012S09_lec04.pdf |
µµµµn •
Dn
dφφφφ
1
=
dx no
dno
dx
•
Using Einstein relation:
q
dφφφφ
•
kT dx
=
d(ln no)
dx
Integrate:
q
kT
(φφφφ− φφφφref )= ln no − ln no,ref = ln
no
no,ref
Then:
q(φφφφ− φφφφref )
no = no,ref exp
kT
6.012 Spring 2009
Lecture 4
9
Any reference is good
In 6.012, φref = 0 ... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/2a02d79958a93f067bd1ae334492fb6a_MIT6_012S09_lec04.pdf |
p o = −(25m )• ln 10)
ni
(
• log
po
ni
Or
p
o
φφφφ≈ −(60 m )• log
ni
EXAMPLE 2:
18
−3
no = 10 cm ⇒ po = 10 cm
⇒ φφφφ= −(60m ) × −8 = 480mV
2
−3
6.012 Spring 2009
Lecture 4
12
Relationship between φφφφ, no and po :
po, equilibrium hole concentration (cm−3)
p-type
intrinsic
n-type
1019
1018
1016
101... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/2a02d79958a93f067bd1ae334492fb6a_MIT6_012S09_lec04.pdf |
7 cm −−−−3 ) −−−− φφφφ(no ==== 1015 cm −−−−3 ) ==== 120 mV
Example 4: Compute potential difference in thermal
equilibrium between region where po = 1020 cm3 and po
= 1016 cm .
3
φφφφ( po ==== 10 20 cm −−−−3) ==== φφφφmax ==== −−−−550mV
φφφφ( po ==== 10 16 cm −−−−3 ) ==== −−−−60 ×××× 6 ==== −−−−360mV
φφφφ( po ==... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/2a02d79958a93f067bd1ae334492fb6a_MIT6_012S09_lec04.pdf |
.012 Spring 2009
Lecture 4
16
MIT OpenCourseWare
http://ocw.mit.edu
6.012 Microelectronic Devices and Circuits
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/2a02d79958a93f067bd1ae334492fb6a_MIT6_012S09_lec04.pdf |
Maximum Likelihood Estimation
Parameter Estimation
Fitting Probability Distributions
Maximum Likelihood
MIT 18.443
Dr. Kempthorne
Spring 2015
MIT 18.443 Parameter EstimationFitting Probability DistributionsMaximum Likelihood
eliho od
1 Maximum Likelihood Estimation
Framework/Definitions
Outline
1 Maximum Likel... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/2a08e2ffa8c87f3187d3969cc24c168b_MIT18_443S15_LEC4.pdf |
X1, X2, . . . , Xn are observations of a time series
{Xt , t = 1, 2, . . .}
Joint density of X = (X1, X2, . . . , Xn) is given by:
f (x1, . . . , xn | θ) =
f (x1 | θ) × f (x2 | θ, x1) × f (x3 | θ, x1, x2) × · · ·
=
=⇒
lik(θ) =
×f (xn | θ, x1, x2, . . . , xn−1)
n
n
f (xi | θ, {xj ; j < i})
i=1
n
n
f (xi | θ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/2a08e2ffa8c87f3187d3969cc24c168b_MIT18_443S15_LEC4.pdf |
i=1
MIT 18.443 Parameter EstimationFitting Probability DistributionsMaximum Likelihood
elihood
5 Maximum Likelihood Estimation
Framework/Definitions
Specifying the MLE
Example 8.5.A: Poisson Distribution
X1, . . . , Xn i.i.d. Poisson(λ)
f (x | λ) = λx ex!
(cid:80)
n
£(λ) =
i=1
MLE λˆMLE
−λ
[xi ln(λ) − λ − ln(x!... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/2a08e2ffa8c87f3187d3969cc24c168b_MIT18_443S15_LEC4.pdf |
(µ, σ2):
ˆ
σ2
θMLE = (ˆµMLE , ˆMLE )
£(θˆMLE ) maximizes £(θ) = £(µ, σ)
ˆ
= 0 and
θMLE solves:
∂£(µ, σ2)
∂µ
∂£(µ, σ2)
∂σ2
= 0
MIT 18.443
Parameter EstimationFitting Probability DistributionsMaximum Likelihood
elihood
7
Maximum Likelihood Estimation
Framework/Definitions
MLEs of Normal Distribution Param... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/2a08e2ffa8c87f3187d3969cc24c168b_MIT18_443S15_LEC4.pdf |
18.443
−n
=
+ n ln(λˆ) +
ln(xi )
n
n
i=1
n
n
+ n ln(α) − n ln(X ) +
ln(xi )
(cid:80)
n
i=1
ln(xi )
i=1
=⇒ 0 =
Γ/
(α)
Γ(α)
Parameter EstimationFitting Probability DistributionsMaximum Likelihood
elihood
10
Maximum Likelihood Estimation
Framework/Definitions
MLEs of Gamma Distribution Parameters
Note: ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/2a08e2ffa8c87f3187d3969cc24c168b_MIT18_443S15_LEC4.pdf |
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