text stringlengths 16 3.88k | source stringlengths 60 201 |
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Combustion Stoichiometry
Air: Oxygen 21%, Nitrogen (nitrogen + argon) 79%
Fuel: Hydrocarbons (CaHb), oxygenates (CaHbOc)
Examples:
Gasoline
Diesel fuel
Natural gas (mostly methane)
Coal
Methanol
Ethanol
CnH1.87n
CnH1.75n
CH3.8
CnH0.8n
CH3OH
C2H5OH
LHV
44 MJ/kg
43 MJ/kg
45 MJ/kg
30 MJ/kg
20 MJ/kg
2... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/68a85760cbd84316b8a086354841a873_MIT2_61S17_lec4.pdf |
5
Fuel H to C ratio
3.5
4
2/24/2015
1
Lean and rich combustion
Fuel-lean combustion
– major products: CO2, H2O, O2, N2
– minor products: HC, CO, H2, NO
Fuel-rich combustion
– major products: CO2, H2O, CO, H2, N2
– minor products: HC, O2, NO
Equivalence ratio: Normalized A/F or F/A ratio... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/68a85760cbd84316b8a086354841a873_MIT2_61S17_lec4.pdf |
.mit.edu/help/faq-fair-use.
Figure 4-22
Exhaust gas composition from several diesel engines in mole fractions on a dry
basis as a function of fuel/air equivalence ratio.31
© McGraw-Hill Education. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/68a85760cbd84316b8a086354841a873_MIT2_61S17_lec4.pdf |
molar
concentration
Value corresponds to
equilibrium composition
of water-shift reaction at
~ 1740oK
H2O + CO H2 + CO2
2/24/2015
4
Equilibrium combustion products:
Dissociation effects
P=30 atmospheres
© McGraw-Hill Education. All rights reserved. This content is excluded from... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/68a85760cbd84316b8a086354841a873_MIT2_61S17_lec4.pdf |
2/3/2014
Engine general working principle
Pressure
force
Piston
connected
to load
• Pressure force pushes a load
– Expansion process; the higher the
expansion, the more work is produced
• Pressure created by combustion
• End pressure limited by ability to exhaust
– Need compression process to generate high ... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/68b07c151fd2842460241680498d0968_MIT2_61S17_lec1.pdf |
― by application: Car, Truck, Marine, Rail, Stationary generation, …
― by basic engine design: reciprocating, rotary, in-line block, V-block, radial, oppose piston, pre-/open chamber
― by working cycle: 2-stroke, 4-stroke, naturally aspirated , turbo-charged, super-charged, turbo-compound
― by fuel: gasoline, diesel... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/68b07c151fd2842460241680498d0968_MIT2_61S17_lec1.pdf |
the Petroleum
Industry
Spark plug dominated the market of
ignition devices
ICE dominated the market of
automotive power plant
Tetra-ethyl lead as anti-knock agent Thomas Midgley, under the direction of Carles Kettering at GM
Spark plug was invented by Edmond Berger in 1839. Albert
Champion was the most successf... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/68b07c151fd2842460241680498d0968_MIT2_61S17_lec1.pdf |
pressure is
unambiguous since it is uniform in the cylinder (except in knocking), whereas temperature is not.
― Empirically for most efficient operation, peak pressure for SI engine is at 14-17o CA-ATC; for diesel is at 7-10o
CA-ATC.
― The very rapid pressure rise in the beginning of diesel combustion is the cause ... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/68b07c151fd2842460241680498d0968_MIT2_61S17_lec1.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.685 Electric Machines
Single Phase Induction Motors, Modeling of Inductances and Resistances
(cid:13)c 2011 James L. Kirtley Jr.
1
Introduction
This short note is a description of the analysis of single phase induction mot... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
=
n odd
X
µ0
4
nπ
Naia
2pg
(cid:18)
kwna sin npθ +
Nbib
2pg
π
kwnb sin n cos npθ
2
(cid:19)
where Na is the number of turns in the main winding, Nb is the number of turns in the auxiliary
winding, p is the number of pole pairs, g is the effective air-gap and kwna and kwnb are the winding
factors for the nth harmonic fiel... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
0
= 2α
nLan
Note the sign of the effective turns ratio, which does not affect the self inductances but does
affect the direction of rotation of the various magnetic field components. Harmonics number 3 and
7 (and one of the zigzag components) rotate in the reverse direction than would be indicated by
the sequence order of ... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
��eld is caused by currents in the stator as well as the rotor but that the higher order
space harmonic voltage components are produced only by rotor currents. To refer the rotor current
back to the stator, note that, if a current IF were in the stator it would make a magnetic field:
Br
= −
µ0 π
4 NaI
F kwna
2pg
then th... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
magnetizing inductance, slot leakage
and rotor resistance for the higher order harmonic terms will be of the form:
Lagn =
X2,n
=
R2,n =
wanRℓ
a k2
n2p2g
4 µ0N 2
π
8ℓN 2
wna
a k2
NR
a k2
NR
8ℓN 2
wna Rslot
ωL
xlot
+
ω
4 µ0N 2
π
2
a kwnaRℓ
g
(cid:18)
1
(NR + np)2
+
1
(NR − np)2
(cid:19)
Note that rotor resistance must be... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
rotor to the other is, in electrical degrees, θsk, the skew factor
for the nth harmonic can be shown to be:
ksn =
sin( nθsk )
2
( nθsk )2
The magnetizing and leakage inductance components are then:
Xφn =
X1n =
2
ωLagnksn
ωLagn(1 − ksn)
2
6 Operation: Fundamental only
Admitting that space harmonics may be important here... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
ΛR = L φI a + jαLφI b + LAI R
F −
I
I
R
Voltage equations are, in the stator coordinate system:
jXφ
2
αXφ
2
jXφ
2
αXφ
2
V a = (jXa + Ra)I a +
I
F +
I
R
0 =
V b = (jXb + Rb)I b −
αXφ
jXφ
2
2
αXφ
jXφ
2
2
I a −
I a +
0 =
I
b +
I +
b
I
I F +
jXA R2
+
s
2
R
I
F
2
jXA
2
(cid:18)
(cid:18)
(cid:19)
R2
2(2 − s)
+
I R
(cid:19)
T... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
the production of
voltage in the stator. Each rotating component (forward and backward at each harmonic order)
will have its own voltage balance equation. Considering only one of the harmonics, which we will
refer to as n, the voltage equations become:
V A = (jXa + Ra)I a +
I
F +
I +
R
F n +
jXφ
2
αXφ
2
jXφ
2
αXφ
2
I
j... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
2 R2
2
2
Inserting the definition for harmonic order slip, we find torque due to the nth harmonic is:
Tmn =
np
ω
2 R2
2sn+
IF n|
|
(cid:18)
− |IRn|
2 R2
2sn− (cid:19)
Adding harmonic terms is straightforward and though, if there are a lot of space harmonics
considered, it yields a relatively large coupling matrix, the re... | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
jXφn
2
jXφn
2
8
MIT OpenCourseWare
http://ocw.mit.edu
6.685 Electric Machines
Fall 2013
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-685-electric-machines-fall-2013/68c5951b80b2480b6d02bfd08d416ad6_MIT6_685F13_chapter12.pdf |
L. McCray 9/12/05
On Knowing and Doing
Or; Rationality and Its Practical Limits in Organizational Choice
Or; Knowledge and Intuition in Decision Making
Or; The Powers and Limits of Policy Analysis
Overview
Rationality and Administrative Choice – A Review [Simon, Updated]
– Area 1 -- Formalistic Analysis and the Federa... | https://ocw.mit.edu/courses/ids-900-integrating-doctoral-seminar-on-emerging-technologies-fall-2005/68d0a27d821a7d40ee33a3091a452a23_lec1.pdf |
1947 – basis of his Nobel Prize
– Updated it in 1997, with significant new commentary
• Simon the Mensch
– An amiable fellow, but . . . does the tidiness of a man’s office reflect his soul?
– A blemished record on emerging technologies
___________________________
On Knowing and Doing, 9/12/05
5
Rationality and Adminis... | https://ocw.mit.edu/courses/ids-900-integrating-doctoral-seminar-on-emerging-technologies-fall-2005/68d0a27d821a7d40ee33a3091a452a23_lec1.pdf |
Leeches will heal many ills; leeches will heal no ills”
– “You’ve just got bronchitis – go home and rest”
– “That altitude alarm is wrong, I’m sure we’re at 1000 feet”
– “We can make Enron’s stock price rise indefinitely”
– “It’s a coincidence that Brazil and Africa look like adjacent puzzle pieces”
• The key problem –... | https://ocw.mit.edu/courses/ids-900-integrating-doctoral-seminar-on-emerging-technologies-fall-2005/68d0a27d821a7d40ee33a3091a452a23_lec1.pdf |
licitation of outside comments on every proposed policy
– Formal agency responses to comments about data, analysis, assumptions
– Often, an “advanced notice” to elicit general ideas
– Effective enforcement by judges in subsequent appeals hearings
• Upshot – A Solidly Entrenched Reform, Lots of Work for Contractors
____... | https://ocw.mit.edu/courses/ids-900-integrating-doctoral-seminar-on-emerging-technologies-fall-2005/68d0a27d821a7d40ee33a3091a452a23_lec1.pdf |
little problems – quality and credibility
• Require Public Analyses by the Decision Makers [e.g., NPRM, EIS]
– Helpful, but there is a little problem of trust
• Employ Outside Contract Analysts
– Helpful, but there is a little problem of trust
• Engage Outside “Blue Ribbon” Expert Panels
– Helpful, but there is a littl... | https://ocw.mit.edu/courses/ids-900-integrating-doctoral-seminar-on-emerging-technologies-fall-2005/68d0a27d821a7d40ee33a3091a452a23_lec1.pdf |
in Federal Regulation
• But the Exceptional Cases Are Surely Interesting
– NAAQS
– Airline Safety?
___________________________
On Knowing and Doing, 9/12/05
19
Area 3 -- Adaptation as a Rational Strategy
C. Why Doesn’t it Happen More Often?
• Darned If I Know
• Some Simple Theories
– Bureaucracies Just Always Resist C... | https://ocw.mit.edu/courses/ids-900-integrating-doctoral-seminar-on-emerging-technologies-fall-2005/68d0a27d821a7d40ee33a3091a452a23_lec1.pdf |
Functions of two variables
Examples: Functions of several variables
f (x, y) = x2 + y2
f (x, y) = xy2ex+y
f (x, y, z) = xy log z
Ideal gas law: P = kT /V .
f (1, 2) = 5 etc.
⇒
Dependent and independent variables
In z = f (x, y) we say x, y are independent variables and z is a dependent variable. This
indicates tha... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/68e54de412d9ab63d40838c414467472_MIT18_02SC_notes_13.pdf |
of its features.
1. First we draw the axes. The z-axis points up, the y-axis is to the right and the x-axis
comes out of the page, so it is drawn at the angle shown. This gives a perspective with the
eye somewhere in the first octant.
2. The yz-traces are those curves found by setting x = a constant. We start with t... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/68e54de412d9ab63d40838c414467472_MIT18_02SC_notes_13.pdf |
The geometry of linear equations
The fundamental problem of linear algebra is to solve n linear equations in n
unknowns; for example:
y = 0
2x
x + 2y = 3.
−
−
In this first lecture on linear algebra we view this problem in three ways.
The system above is two dimensional (n = 2). By adding a third variable z
we ... | https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/6913aa2f37dada8fdda334168b30339b_MIT18_06SCF11_Ses1.1sum.pdf |
of vector
1
2
2
1
added to y copies of vector
equals the vector
. As we see
�
�
from Figure 2, x = 1 and y = 2, agreeing with the row picture in Figure 2.
0
3
−
−
�
�
�
�
Figure 2: A linear combination of the column vectors equals the vector b.
In three dimensions, the column picture requires us to find a linea... | https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/6913aa2f37dada8fdda334168b30339b_MIT18_06SCF11_Ses1.1sum.pdf |
columns of A.
+ 2
�
� �
=
�
�
�
�
�
�
.
12
7
5
3
2
1
You may also calculate the product Ax by taking the dot product of each
row of A with the vector x:
2 5
3
1
1
2
=
�
�
� �
�
Linear Independence
2 1 + 5 2
1 1 + 3 2
·
·
·
·
=
�
�
12
7
.
�
In the column and matrix pictures, the right hand side of th... | https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/6913aa2f37dada8fdda334168b30339b_MIT18_06SCF11_Ses1.1sum.pdf |
PDSS Critical Parameter Development & Mgt. Process Quick Guide
12 Steps for a Critical Parameter Development Project (incl. 6 Check Points)
Step 1: Create a CP Project Charter
-
-
establish goal, objectives, team members, roles & responsibilities, time line & scope
define clear, specific & measurable CP project resul... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
Step Prevention Process
Select the appropriate groups of flows that matter the most; again – which are NUD?
Align critical noise parameters with the appropriate sub-groups
Step 7: Prove measurement systems are capable
- MSA & Gage R&R Studies for Critical Ys, sub-ys & controlling Xs (for both leading & lagging indi... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
signal/noise resolution
o Contingency & Corrective Action plans
� Alternative action plan
� Process specific LSS-based corrective action process plan
• Kaizen event or 6σ Project?
Conduct CP summary
reviews & make
Cpk>>>Cp adjustments
as needed during steady
state mfg.
Step 12: Evaluate Quality and Implement ... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
CP Project
Charter
Step 2: Create
a cross-
functional team
of experts to
help ID a
thorough set of
CPs
Step 3:
Generate /
Assess
Requirement
Clarity,
Classification &
Flow-down
Documentation
Step 4:
Generate I-O
C Diagrams, P-
Diagrams,
Noise
Diagrams,
Boundary
Diagrams &
Math Models
Step 5:
St... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
, Quality Planning, SPC Studies, Capability
Studies, CP Allocation for Production & Assembly Processes,
CP Documentation & CP Scorecards, CP Deployment in
Production & Supply Chain Environments Module
© 2010 PDSS Inc.
page 4 of 11
PDSS Critical Parameter Development & Mgt. Process Quick Guide
Recommendation for ... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
C a pa b ility
C p
1 .4 9
C P U
C P L
C p k
C p m
P p
P P U
P P L
P pk
1 .5 1
1 .4 7
1 .4 7
*
8
9
1 0
1 1
1 2
O v er a ll C a pa b ility
O b s e rv ed P er fo r m a n ce
E xp . " W ith in " P e r fo rm a nc e
E xp . "O v er a ll" P e r fo rm an c e
1 .4 6
1 .4 7
1 .4 4
1 .4 4
P P M < L S L
... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
6
P o te nt ial (W ith in) C a pa b ility
C p
C PU
C PL
C p k
C p m
P p
P PU
P PL
P pk
1 .4 9
1 .5 1
1 .4 7
1 .4 7
8
*
9
1 0
1 1
1 2
O v er a ll C a pa b ility
O b s e rv ed P er fo r m a n ce
E xp . " W ith in " Pe r fo rm a nc e
E xp . "O v er a ll" Pe r fo rm an c e
1 .4 6
1 .4 7
1 .4 4
1... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
Parameter Development & Mgt. Process Quick Guide
Problem Prevention Steps:
The 6 Steps for Problem Prevention:
• Design a Diagram of detailed functions in their serial & parallel flows
• These are value-added functions that possess inherent robustness
• Rank & prioritize the value of each function relative to one an... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
: Cp & Cpk
DOE, S/n: COV, std. deviation
If these items are problematic and not in control then these are NUD parameters that are very good
candidates for Critical Parameter status… until proven under control and able to be re-classified as
Easy, Common & Old.
Measurability:
G ag e R &R (ANO VA) f or Thic kn ess
... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
a
r
e
v
A
8 .2
8 .1
8 .0
7 .9
Pa rt
1
2
3
4
5
6
7
8
9
1 0
O p e ra tor
Fre d
J oe
M a ry
Stability:
I and MR Chart for C2
5
0
-5
l
e
u
a
V
l
i
a
u
d
v
d
n
I
i
Subgroup
0
50
100
e
g
n
a
R
g
n
v
o
M
i
10
5
0
Tunability:
ANOVA: Y plus Noise versus A, B, C
Factor Type Levels Values
A fixed ... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
- Data Means for Y plus Noise
A
B
C
30
25
20
15
10
-1
1
-1
1
-1
1
i
e
s
o
N
s
u
p
Y
l
A
1
- 1
B
1
-1
30
20
10
30
20
10
C
Effect
Sum of Squares
epsilon2
%epsilon2
A
B
C
AC
AB
Total
69.82
69.82/2233.39=0.031
3.1%
401.70
401.70/2233.39=0.18
1620.35
1620.35/2233.39=0.72
18%
72%... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
1
1
1
1
1
1
1
8
15
SS
69.82
401.70
1620.35
27.82
110.85
0.00
0.00
2.85
2233.39
F
MS
P
69.82 196.14 0.000
401.70 1128.38 0.000
1620.35 4551.60 0.000
78.14 0.000
110.85 311.38 0.000
0.00 0.969
0.00 0.967
27.82
0.00
0.00
0.36
Interaction Plot (data means) for Y plus Noise
- 1
1
1
-
1
... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
]= 14%
• Resistor D
[10,764/105,227] x [100]= 10%
• Capacitor C
[6,281/105,227] x [100]= 6%
• Resistor B
[3,335/105,227] x [100]= 3%
• Resistor A
[1,580/105,227] x [100]= 1.5%
• Transistor C
[716/105,227]
• Transistor F
x [100]= 0.68%
Source DF Seq SS Adj SS Adj MS
F
P
Res A
Res B
Cap C
Res D
1
1
... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
ro c e s s C a p a b ilit y A n a ly s is fo r C 2
US L
W ith in
O v e ra ll
C p
C P U
C P L
C p k
C p m
P p
P PU
P PL
P pk
O v era ll C apa bilit y
1. 49
1. 51
1. 47
1. 47
*
1. 46
1. 47
1. 44
1. 44
8
9
1 0
1 1
1 2
O bs e rv ed P erf or m a nc e
E xp . "W it hin " Pe rf orm an c e
Ex p. "O... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
s
e
R
C
.
p
a
C
B
r
o
t
s
i
s
e
R
A
r
o
t
s
i
s
e
R
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page 8 of 11
PDSS Critical Parameter Development & Mgt. Process Quick Guide
Example of getting Functions right and driving your CP development methodology:
If I have 5 Functions, then I have at le... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
each y…
X variables come in 4 varieties:
X mean shifter = controllable parameter that has a strong ability to move the mean of y
X std. dev. shifter = controllable parameter that has a strong ability to move the value of σσσσ
X COV shifter = controllable parameter that has a strong ability to move both the mean of y... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
σσσ2 from Δ
Δ Xb, …); σσσσ2 of y3 = f(σσσσ2 from Δ
Δ Xb, …); σσσσ2 of y4 = f(σσσσ2 from Δ
Δ Xb, …); σσσσ2 of y5 = f(σσσσ2 from Δ
Δ Xa, σσσσ2 from Δ
Δ Xa, σσσσ2 from Δ
Δ Xa, σσσσ2 from Δ
Δ Xa, σσσσ2 from Δ
Δ Xa, σσσσ2 from Δ
Δ Xb,…)
Δ Xb,…)
Δ Xb,…)
Δ Xb,…)
Δ Xb,…)
We will know the model is complete by developing bo... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
Δ
PDSS Critical Parameter Development & Mgt. Process Quick Guide
What about Noise and its affect of the Functions (ys) and the relationship of (X * Noise)
interactions.
The Noise Diagrams:
For the Function, what noises cause it to vary?
Unit-Unit
Noises
External
Noises
Functi... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/69261b72bed60ec44f5e50c12fa7fade_MITESD_33SUM10_lec07b.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.014 Calculus with Theory
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-014-calculus-with-theory-fall-2010/693969587a934022831b3f1ae5cf5514_MIT18_014F10_ChEnotes.pdf |
Chapter 2 Learning to Program in Python
6.01— Spring 2011— April 25, 2011
21
Chapter 2
Learning to Program in Python
Depending on your previous programming background, we recommend different paths through
the available readings:
•
If you have never programmed before: you should start with a general introduction ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
Learning to Program in Python
6.01— Spring 2011— April 25, 2011
22
>>> 5 + 5
10
>>> x = 6
>>> x
6
>>> x + x
12
>>> y = ’hi’
>>> y + y
’hihi’
>>>
So, you can use Python as a fancy calculator. And as you define your own procedures in Python,
you can use the shell to test them or use them to compute useful results.
2.... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
= a - 10
else:
s = s + 10
a = a + 10
Chapter 2 Learning to Program in Python
6.01— Spring 2011— April 25, 2011
23
There is no way to put more than one statement on a single line.3 If you have a statement that is
too long for a line, you can signal it with a backslash:
aReallyLongVariableNameThatMakesMyLinesLong =... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
you don’t try to use a variable
in a way that is inconsistent with its declaration.
In Python, however, things are a lot more flexible. There are no variable declarations, and the
same variable can be used at different points in your program to hold data objects of different
types. So, the following is fine, in Pytho... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
we might package a set of
utility functions together into a single file, called utility.py. This file is called a module in
Python.
Now, if we want to use those procedures in another file, or from the the Python shell, we will
need to say
import utility
so that all those procedures become available to us and to the ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
Chapter 2 Learning to Program in Python
6.01— Spring 2011— April 25, 2011
25
Ultimately, for debugging big programs, it is most useful to use a software development environ
ment with a serious debugger. But these tools can sometimes have a steep learning curve, so in
this class we will learn to debug systematicall... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
is important here, too. All of the statements of the procedure have to be indented
one level below the def. It is crucial to remember the return statement at the end, if you want
your procedure to return a value. So, if you defined f as above, then played with it in the shell,4
you might get something like this:
>>>... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
. So,
then, when we tried to apply g to the result of g(4), it ended up trying to evaluate g(None),
which made it try to evaluate None + 1, which made it complain that it did not know how to
add something of type NoneType and something of type int.
Whenever you ask Python to do something it cannot do, it will compl... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
is added to 1, and the
result, 5, is printed by the Python read-eval-print loop.
The book Think Python, which we recommend reading, was translated from a version for Java, and
it has a lot of print statements in it, to illustrate programming concepts. But for just about every
thing we do, it will be returned values... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
>>> 7 > 8
False
>>> -6 <= 9
True
We can also test whether data items are equal to one another. Generally we use == to test for
equality. It returns True if the two objects have equal values. Sometimes, however, we will be
interested in knowing whether the two items are the exact same object (in the sense discussed in... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
conditional statement, it starts by evaluating <boolean-
Expr>, getting either True or False as a result.6 If the result is True, then it will eval
uate <statementT1>,...,<statementTk>; if it is False, then it will evaluate <state
mentF1>,...,<statementFn>. Crucially, it always evaluates only one set of the statement... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
values 0, 0.0, [],
’’, and None as if they were False and everything else as True.
7 or, more esoterically, another object that can be iterated over.
Chapter 2 Learning to Program in Python
6.01— Spring 2011— April 25, 2011
30
result = 0
for x in [1, 3, 4]:
result = result + x * x
At the end of this execution... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
usage is range(n), which returns a list of integers going from 0 up to, but not including, its
argument. So range(3) returns [0, 1, 2].
Exercise 2.2.
Write a procedure that takes n as an argument and returns the sum of the
squares of the integers from 1 to n-1. It should use for and range.
Chapter 2 Learning to P... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
while p*2 < n:
p = p*2
return p
Lists
Python has a built-in list data structure that is easy to use and incredibly convenient. So, for
instance, you can say
>>> y = [1, 2, 3]
>>> y[0]
1
>>> y[2]
3
>>> y[-1]
Chapter 2 Learning to Program in Python
6.01— Spring 2011— April 25, 2011
32
3
>>> y[-2]
2
>>> len(y)
3
>... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
[1:]
[1, 2, 3, 4, 5, 6, 7, 8, 9]
>>> b[3:]
[3, 4, 5, 6, 7, 8, 9]
>>> b[:7]
[0, 1, 2, 3, 4, 5, 6]
>>> b[:-1]
[0, 1, 2, 3, 4, 5, 6, 7, 8]
>>> b[:-2]
[0, 1, 2, 3, 4, 5, 6, 7]
Iteration over lists
What if you had a list of integers, and you wanted to add them up and return the sum? Here are
a number of different ways of ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
style you might have learned to write in a Java class (actually, you
would have used for, but Python does not have a for that works like the one in C and Java).
def addList1(l):
sum = 0
listLength = len(l)
i = 0
while (i < listLength):
sum = sum + l[i]
i = i + 1
return sum
It increments the index i from 0 thr... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
Python
6.01— Spring 2011— April 25, 2011
34
In section ??, we will see another way to do addList, which many people find more beautiful
than the methods shown here.
List Comprehensions
Python has a very nice built-in facility for doing many iterative operations, called list comprehen
sions. The basic template is ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
1000003]
>>> [a[0] for a in [[’Hal’, ’Abelson’],[’Jacob’,’White’],
[’Leslie’,’Kaelbling’]]]
[’Hal’, ’Jacob’, ’Leslie’]
>>> [a[0]+’!’ for a in [[’Hal’, ’Abelson’],[’Jacob’,’White’],
[’Leslie’,’Kaelbling’]]]
[’Hal!’, ’Jacob!’, ’Leslie!’]
Imagine that you have a list of numbers and you want to construct a list contai... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
’],[’Jacob’,’White’],
[’HalAbelson’, ’JacobWhite’, ’LeslieKaelbling’]
[’Leslie’,’Kaelbling’]]]
Another built-in function that is useful with list comprehensions is zip. Here are some examples
of how it works:
> zip([1, 2, 3],[4, 5, 6])
[(1, 4), (2, 5), (3, 6)]
> zip([1,2], [3, 4], [5, 6])
[(1, 3, 5), (2, 4, 6)]
Here ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
plus1(x) + 2
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
TypeError: unsupported operand type(s) for +: ’NoneType’ and ’int’
Chapter 2 Learning to Program in Python
6.01— Spring 2011— April 25, 2011
36
• Weird results from math can result from integer division
>>> 3/ 9
0
• “Unsubscriptabl... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
last):
File "<stdin>", line 1, in <module>
File "<stdin>", line 2, in fizz
File "<stdin>", line 2, in fizz
...
File "<stdin>", line 2, in fizz
RuntimeError: maximum recursion depth exceeded
• “Key Error” means that you are trying to look up an element in a dictionary that is not present.
>>> d = {’a’:7, ’b’:8}
>>> d[’... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
, if you have a bug
in the calculation (or you want to change the program), you will have to change it multiple
times. It is also inefficient.
• Avoid repetition of a pattern of computation.
You should use a function instead, again to avoid having to change or debug the same basic
code multiple times.
• Avoid nume... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
computation of the magnitude of a vector is a useful and understandable
operation in its own right, and should probably be put in its own procedure. That leads to this
procedure:
def mag(v):
return math.sqrt(sum([vi**2 for vi in v]))
def normalize3(v):
return [vi/mag(v) for vi in v]
This is especially nice, because ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
, which we are just going to sum up; that’s somewhat less efficient than adding the values
up in a loop. However, as we said at the outset, for almost every program, clarity matters more than efficiency. And
once you have something that’s clear and correct, you can selectively make the parts that are executed frequentl... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
4), (2, 5), (3, 6)]
2.5.3 Bank transfer
What if we have two values, representing bank accounts, and want to transfer an amount of
money amt between them? Assume that a bank account is represented as a list of values, such as
[’Alyssa’, 8300343.03, 0.05], meaning that ’Alyssa’ has a bank balance of $8,300,343.03,
an... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
- fee
Here, we’ve used a destructuring assignment statement to give names to the components of the
account. Unfortunately, when we want to change an element of the list representing the account,
we still have to index it explicitly. Given that we have to use explicit indices, this approach in
which we name them migh... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
:], b)
10 We’ll see other approaches to this when we start to look at object-oriented programming. But it’s important to apply
basic principles of naming and clarity no matter whether you’re using assembly language or Java.
11 To get versions of basic Python operations in the form of procedures, you need to do import... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
this time in the imperative style:
def isSubset(a, b):
for x in a:
if not x in b:
return False
return True
It works by going through the elements in a, in order. If any of those elements is not in b, then we
can immediately quit and return False. Otherwise, if we have gotten through all of the elements
in a, an... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
procedure works by going through both lists and making a list of the
items that are in common (basically computing the intersection). Then, it
checks to see whether the intersection is of the same length as a. There are
several problems here:
• Using the idea of computing the intersection and then seeing whether
i... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
True
for itera in range(len(a)):
tempokay = False
for iterb in range(len(b)):
if a[itera] == b[iterb]:
tempokay = True
if tempokay == False:
okay = False
return okay
continued
Chapter 2 Learning to Program in Python
6.01— Spring 2011— April 25, 2011
44
Bad Example 3.
Previous bad example, being made over... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
foundAllAsSoFar = foundAllAsSoFar and foundThisA
return okay
Chapter 2 Learning to Program in Python
6.01— Spring 2011— April 25, 2011
45
Exercise 2.4.
Now, see if you can, first, explain, and then improve this example!
def issubset(a,b):
i = 0
j = 0
while i < len(a):
c = False
while j < len(b):
if a[i] ==... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/697e081c3d8090b020b1a8c9b493f880_MIT6_01SCS11_chap02.pdf |
Lecture 6
PN Junction and MOS Electrostatics(III)
MetalOxideSemiconductor Structure
Outline
1.
2.
3.
Introduction to MOS structure
Electrostatics of MOS in thermal
equilibrium
Electrostatics of MOS with applied bias
Reading Assignment:
Howe and Sodini, Chapter 3, Sections 3.73.8
6.012 Spring 2009
Lecture 6
1... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6986289796927e9b4947924e823ec5a2_MIT6_012S09_lec06.pdf |
2
Remember: n opo=ni
Fewer holes near Si / SiO2 interface
⇒ ionized acceptors exposed (volume charge)
6.012 Spring 2009
Lecture 6
4
Space Charge Density
•
In semiconductor: spacecharge region close Si /SiO2
interface
– can use depletion approximation
In metal: sheet of charge at metal /SiO2 interface
•
• Ove... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6986289796927e9b4947924e823ec5a2_MIT6_012S09_lec06.pdf |
s
+
Almost done ….
6.012 Spring 2009
Lecture 6
9
Still do not know xdo ⇒⇒⇒⇒ need one more equation
Potential difference across structure has to add up to φB:
φφφφB = VB,o + Vox,o =
2
qNaxdo
2εεεεs
+
qNa xdotox
εεεεox
Solve quadratic equation:
xdo =
εεεεs
εεεεox
tox 1 +
2εεεε
2 φφφφB ... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6986289796927e9b4947924e823ec5a2_MIT6_012S09_lec06.pdf |
and diffusion
• Electrostatics qualitatively identical to thermal
equilibrium (but amount of charge redistribution is
different)
2
• np = ni
6.012 Spring 2009
Lecture 6
13
Apply VGB>0: potential difference across structure
increases ⇒ need larger charge dipole ⇒ SCR expands into
semiconductor substrate:
Sim... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/6986289796927e9b4947924e823ec5a2_MIT6_012S09_lec06.pdf |
6
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-2009/6986289796927e9b4947924e823ec5a2_MIT6_012S09_lec06.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 7
Hashing III: Open Addressing
6.006 Spring 2008
Lecture 7: Hashing III: Open Addressing
Lecture Overview
• Open Addressing... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/699bc3c0c21033be722093c8c937698e_lec7.pdf |
, m-1)> h: U x {φ,1, . . . , m-1} {φ,1, . . . , m-1} permutationall possible keyswhich probeslot to probecollisionφ1234567m-1collisioninsert586 , . . .133 , . . .204 , . . .496 , . . .481 , . . .probe h(496, φ) = 4probe h(496, 1) = 1probe h(496, 2) = 5 Lecture 7
Hashing III: Open Addressing
6.006 Spring 2008
Delete(k... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/699bc3c0c21033be722093c8c937698e_lec7.pdf |
1);;;..; Lecture 7
Hashing III: Open Addressing
6.006 Spring 2008
Analysis
Open addressing for n items in table of size m has expected cost of ≤
where α = n/m(< 1) assuming uniform hashing
Example: α = 90% =
⇒
10 expected probes
1
1 − α
per operation,
Proof:
Always make a first probe.
With probability n/m, fir... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/699bc3c0c21033be722093c8c937698e_lec7.pdf |
without assuming simple uniform hashing!
CLRS 11.3.3
Perfect Hashing
Guarantee O(1) worst-case search
•
idea: if m = n2 then E[� collisions] ≈ 1
2
=
⇒
get φ after O(1) tries . . . but O(n2) space
• use this structure for storing chains
Figure 5: Two-level Hash Table
• can prove O(n) expected total space!
• if... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/699bc3c0c21033be722093c8c937698e_lec7.pdf |
£
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31... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/699d8f5156bc430d254e93f44743b7c2_projects.pdf |
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⁄(cid:150)K (cid:128... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/699d8f5156bc430d254e93f44743b7c2_projects.pdf |
:1)
(cid:23)(cid:41)(cid:26)(cid:1)(cid:48)(cid:46)(cid:55)(cid:1)(cid:48)(cid:35)(cid:27)(cid:1)(cid:41)(cid:27)(cid:54)(cid:48)(cid:1)(cid:27)(cid:26)(cid:34)(cid:27)(cid:60)(cid:1)(cid:46)(cid:27)(cid:44)(cid:27)(cid:23)(cid:49)(cid:41)(cid:34)(cid:1)(cid:48)(cid:35)(cid:36)(cid:47)(cid:1)(cid:44)(cid:46)(cid:42)(ci... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/69b5b28067ecf1769a6143453d77eba1_MIT6_0002F16_lec3.pdf |
Preface
In the beginning group theory dealt with finite groups, for example permu-
tation groups of the type which appear in Galois theory of field extensions.
However, the step from permutation groups to transformation groups is
a simple and natural one. Consider for example the set of isometrics of the
plane R2 . Cle... | https://ocw.mit.edu/courses/18-755-introduction-to-lie-groups-fall-2004/69cdb4486d2c77120f239db7d107930f_preface.pdf |
-lt-lg
). Then t-lg
z
. Thus JVC(2)is described by three parameters.
Let O(n) denote the orthogonal group in R n that is g9 0(n) if and only
if ( .tI. g. v) = (vU,) for all vectors 'u and v, ( ) denoting the scalar product.
Consider the case ',, = 3. Each E 0(3) has a real eigenvalue A and IXl = 1.
Thus k maps a lin... | https://ocw.mit.edu/courses/18-755-introduction-to-lie-groups-fall-2004/69cdb4486d2c77120f239db7d107930f_preface.pdf |
pt be a group of diffeomorphisms of R2 depending on 1-parameter
t, so
= I, ps+t =
ostp.It is said to leave the equation
dy _ Y(x,y)
X(x, y)
dx
stable if each cptpermutes the solutions. Consider the vector field
{,= d()
(it J.1
t=o
)
}
ay
Theorem 1. If Pt leaves the equation stable then the function
(X - Y)-l
is ... | https://ocw.mit.edu/courses/18-755-introduction-to-lie-groups-fall-2004/69cdb4486d2c77120f239db7d107930f_preface.pdf |
dU and we see easily that we can take
U = arctan (
-
so solution is y = :rtan (+2
+- C) .
(ii) Another example (Lie 1874).
Transformation in (x, t, u) leaves equation stable if it permutes the solutions.
Lie determined this group. It is a product of a 6-dim group and an infinite-
dimensional group. Most subgroups of... | https://ocw.mit.edu/courses/18-755-introduction-to-lie-groups-fall-2004/69cdb4486d2c77120f239db7d107930f_preface.pdf |
T i,i=1 \04, t=okO
and proved the following fundamental result.
Theorem. The bracket
seatiSf7es
[Tl,
e] = T
T
- T o T.
Where the ce are constants satzsfying
&
fP=q
m
,
k1
1k'
I
+CP C=
mqk
P
q )= 0
ky(lk}
This leads to the concept of a Lie algebra.
q
3
Definition.
is a bilinear map (X, Y)
A Lie algebra is ... | https://ocw.mit.edu/courses/18-755-introduction-to-lie-groups-fall-2004/69cdb4486d2c77120f239db7d107930f_preface.pdf |
own
but got tied up with differential geometry, particularly with the advent of
fibre bundles.
In the mid-twenties applications to physics came about and this has
continued to our days. A journal on mathematical physics will have Lie
groups on most pages. A very important application came in the mid-
fifties to the the... | https://ocw.mit.edu/courses/18-755-introduction-to-lie-groups-fall-2004/69cdb4486d2c77120f239db7d107930f_preface.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.265/15.070J
Lecture 9
Fall 2013
10/2/2013
Conditional expectations, filtration and martingales
Content.
1. Conditional expectations
2. Martingales, sub-martingales and super-martingales
1 Conditional Expectations
1.1 Definition
Recall how we define conditional expectatio... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/69dc2223b9957bbcc754de94dfc64cfd_MIT15_070JF13_Lec9.pdf |
every A ∈ G, we have E[X1{A}] = E[Y 1{A}].
For simplicity, from now on we write Z ∈ F to indicate that Z is measurable
with respect to F. Also let F(Z) denote the smallest σ-field such with respect
to which Z is measurable.
Theorem 1. The conditional expectation E[X|G] exists and is unique.
Uniqueness means that if... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/69dc2223b9957bbcc754de94dfc64cfd_MIT15_070JF13_Lec9.pdf |
variable Y : Ω → R defined as
for ω ∈ A and
Y (ω) = E[X|A] =
E[X1{A}]
P(A)
c1
Y (ω) = E[X|Ac] =
E[X1{Ac}]
P(Ac)
c2
for ω ∈ Ac . We claim that Y = E[X|G]. First Y ∈ G. Indeed, assume
for simplicity c1 < c2. Then {ω : Y (ω) ≤ x} = Ø when x < c1, = A
2
for c1 ≤ x < c2, = Ω when x ≥ c2. Thus Y ∈ G. Then we need ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/69dc2223b9957bbcc754de94dfc64cfd_MIT15_070JF13_Lec9.pdf |
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