text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
Q0
:
:
independent of i
independent of i, j
1
1
2
2
g3(Q0)2 − g3(Mun.)
MU
Q0
1
1
gi(Q0)2 −
1
2β◦
i
ln
gi(Q0)2 − gj (Q0)2
βi◦
βj◦
−
N.B.: of course
Master formula:
2 g =
1
�2
g
5
3
β0 =
11
− 3
e2 +
4
3
T 1 +
2
1
3
T0
�
Minimal Standard Model (MSM):
real
weyl
1
2
× 1
× 2
β(3) =
β(2) =
4
... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
2
4
2
3 ×
4 ×
2
+ 3 ×
12 × 2
+ 3 ×
1 ×
+ 3 ×
2
1
1
Gaugions
1
2 Higgsions
Sf erminos
Extra Higgs
1×
2
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
(3.36)
3.1. SU(5) UNIFICATION
=
25
6
Higgsions:
Δβ(1) =
3 4
5{3 ×
2
(2
×
×
1
( )2)
2
1
( )2 + 3
6
×
1
3 ×
1
3 ×
3[6
×
1 ... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
β(3)
β(3)
Δβ(3)
β(3)
Δβ(2)
β(2)
doublets.
Also change in MU :
15
(3.37)
(3.38)
(3.39)
(3.40)
(3.41)
(3.42)
(3.43)
(3.44)
(3.45)
(3.46)
(3.47)
6 effective
⇒
16
CHAPTER 3. GRAND UNIFIED THEORY
ln
MU
1
βj
Q ∝ βi −
1
βi −
1016.5
)SU SY �
(
1012 11 +2
9·
βj
βj
�
(
1
βi −
1017
102 →
(3.48... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
3
CdR
1
3
1
2
− −ν
e
1
2
�
T µν: antisymmetric
T αβ
T αi
T ij
¯: 3, 1,
: 3, 2,
: 1, 1,
2
3
1−
6
1
•
Symmetry Breaking
Adjoint (traceless)
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
→
SU (5)
2
2
2
3
−
3
−
SU (2)
×
×
SU (3)
SU (3)
⎞
M
⎟
⎟
⎟
⎟
⎟
⎟
⎠
U (1)
�
SU (2)
�
Still need SU (2)
U (1)
×
⇒
U (1).
(3.52)
(3.5... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
5 =
0 1
1 0
�
�
ΨL =
η
η
−
�
CΨ� = γ2Ψ
�
�
C�2
=
�
∗
iσ2η∗
iσ2η∗
�
¯
(eΨ ) =
�
iσ2η�
iσ2η�
�
η�a �abηb
∝
0 1
1 0
� �
η
η
−
�
�
�
where, a and b are Dirac indices.
Therefore,
η(i)�
a �abη(j)
b
Note symmetric in i
⇔
j, due to Fermi statistics.
(3.58)
(3.59)
(3.60)
(3.61)
(3.62)
... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
uRu� d2
L
2F4
ReL
T 31F1)
(�12345T 12T 45ϕ3)(ϕ∗
3
¯
u¯R3e¯Ru¯R2dR1
19
(3.70)
(3.71)
(3.72)
(3.73)
(3.74)
(3.75)
(3.76)
(3.77)
Single appearance of �αβγ.
Phenomenology
M large.
⇒
•
Implementing SB; Hierarchy problem ϕ+ϕ, ϕ+Aϕ, ϕ+A2ϕ, trA2ϕ+ϕ, trA2 ,
trA3 , trA4 , tr(A2)2 . Need big vev for A, small for ... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
g =
g
2
2
1
2
1
2
2
g
g
�2
g
�2
−
=
3
2
g
5
3
(1 +
5
)g2
=
3
8
Expct.
0.22.
≈
Also, of course,
sin 2 θw
=
2
g
gSU (2) = 1
gSU (3)
(3.78)
(3.79)
(3.80)
(3.81)
(3.82)
(3.83)
(3.84)
3.3. SO(10) UNIFICATION
3.3 SO(10) Unification
SU (6)?
SU (5) in SO(10): 5 complex components
5
6
×
2
, F ab , T µν
Z... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
jkδlnXm
δjkδlmXn −
−
δjlδknXm}
+δjlδkmXn −
−
−
δnlT km
= δnkT lm
δmkT ln + δmlT kn
Γ matrices “=” √rotation
(spinor rep.)
1[Γk, Γl] satisfy the SO(10) commutators.
4
Γk, Γl}
{
= 2δkl
[Γk, Γl] = 2(ΓkΓl −
δkl)
1
16
[[Γk, Γl], [Γm, Γn]] =
=
1
4
1
4
[ΓkΓl, ΓmΓn]
(ΓkΓlΓmΓn −
ΓmΓnΓkΓl)
Claim:
−
So
Now use
2... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
:
ΓmΓl =
1
2
([Γm, Γl] + 2δml) etc.
(3.98)
1
2
δnkΓmΓl + δmkΓnΓl −
(
−
δnlΓkTm + δmlΓkΓn) =
δnk[Γm, Γl] + δmk[Γn, Γl]
(
−
−
δnl[Γk, Γm] + δml[Γk, Γn])
(3.99)
1
4
Compare to
QED.
δnkΓlm
δnlΓkm
−
−
δmkΓln + δmlΓkn
Construction of Γ matrices:
U −
1(R)T µU (R) = RµT ν
ν
Γ1 = σ1 ⊗
Γ2 = σ2 ⊗
Γ3 = σ3 ⊗
Γ4 = σ... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
. .
23
(3.107)
(3.108)
1
1
1
1
⊗
⊗
⊗
⊗
1
1
1
σ3 ⊗
1
⊗
⊗
σ3 ⊗
So we diagonalize:
SO(2)
SO(2)
SO(2)
⊗
⊗
⊗
SO(2)
⊗
SO(2)
⊂
SO(10)
(3.109)
This gives us a 25 = 32 – dimensional representation of SO(10) by
R(e
iθabTab)
iθab(
= e
1
− 4 [Γa,Γb])
(3.110)
It is not quite irreducible.
Note
K =
iΓ1Γ2
−
Γ10... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
+σ3 ⊗
the 16 has even number of signs, (16) has odd number of signs.
Back to SU (5), υ�Jυ invariant, with
σ3 ⊗
σ3 ⊗
σ3 ⊗
σ3
(3.114)
(3.115)
24
CHAPTER 3. GRAND UNIFIED THEORY
J
=
υ
Δυ�Jυ
→
=
=
⇒spinor
Similarly we identify
0
1
1 0
−
⎛
.
.
.
−
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
υ + �Gυ
�υ�(GT J + JG)υ
GJ + JG)υ
�υ�(
−
... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
1
6
Analysis of 16 standard model Q
5 + signs
(R12 + R34 + R56)
−
1
4
−
#s:
R78 −
R910 ∝
U (1)Y
1 state
+ + + ++ >
U (1) singlet
|
×
SU (3)
SU (2)
×
1 + signs
(3.123)
(3.124)
(3.125)
(3.126)
5 state, 2 types
>
+
| − −
−−
3.3. SO(10) UNIFICATION
25
+
− −−
− − −−
>
>
| −
+
|
Y˜ =
1
6
1)
(
−
+
−
1
4
2... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
let
2
−3 ≡
+
−
+ +
+ +
+
>
>
u
C
R
=
−
−
−
|
|
|
�
|
|
1
Y = (
6
1)
−
−
1
4
(2)
(3.127)
(3.128)
(3.129)
(3.130)
(3.131)
26
CHAPTER 3. GRAND UNIFIED THEORY
+ + +
−
+ +
+
−
− −
+
−
−
+ +
−
|
|
|
|
>
−
+ >
+ >
+ >
SU (3) 3
SU (2) 2
u
d
�
�
L
Y =
1
6
(1) =
1
6 ≡
Comments:
(3.132)
1. The const... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
the form
SU (2)
×
×
µij N¯i Lµjϕ∗
R
µ
(3.135)
where, i, j are formly indices and µ = SU (2) index.
By 2nd order perturbation theory we induce Majorana Masses for the νL.
<Φ>
<Φ>
M
Figure 3.3: Majorana Masses
m
2µ
∼ M
(3.136)
3.3. SO(10) UNIFICATION
Breaking Scheme: Higgs ϕ in 16
< ϕN > = 0
SO(10)
SU (... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
1
+ γ = 1
β
−
αB + βL + γY = 5B
There are 45adjoint and 10vectro as before.
Fermion masses: bilinears in ϕ
5L
4Y
−
−
Γ1 =
Γ2 =
. . .
σ1 ⊗
σ2 ⊗
1
1
⊗ · · · ⊗
⊗ · · · ⊗
1
1
ϕ∗ transforms as e[Γµ,Γν ]∗
(Cϕ∗)� = Ce[Γµ,Γν ]∗
ϕ∗
Cϕ∗
= e[Γµ,Γν ]∗
ΓµC
CΓ∗ =
µ
±
C = Γ1Γ3Γ5Γ7Γ9
With + sign:
(3.146)
(3.147... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
�
−
tensor + 2104
�
��
�
−
tensor
��
�
(3.155)
Totally
(3.156)
(3.157)
3.3. SO(10) UNIFICATION
29
Final comments on SO(10) vs. SU (5): 3 RH neutrinos vs. SU (5)
quantization.
�(B
Q
∝
−
L)
U (1). Charge
×
(3.158)
p : 1 + �, e :
N.B.: mechanics of charge quantization. n
�, n : �, v :
1
−
−
−
�.
pev¯.... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/1bb0ca3337af13703289df734a424e7c_chap3.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.005 Elements of Software Construction
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Today
conclusion
¾ take-away messages
¾ what to do next
project 3 awards
6.005 quiz game
HKN evaluations
HKN evaluations... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/1bb173ac0fc980981d33851b7de8af6b_MIT6_005f08_lec23.pdf |
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 3-1
Lecture 3 - Carrier Statistics in Equilibrium
(cont.)
February 9, 2007
Contents:
1. Equilibrium electron concentration
2. Equilibrium hole concentration
3. np product in equilibrium
4. Location of Fermi level
Reading assignment:
del Ala... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
location of EF from additional arguments (such as
charge neutrality)
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MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microe... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
de
h2
⎞
3/2 ∞
E − Ec
(cid:2)
⎠
Ec 1 + exp E−
EF
kT
dE
Refer energy scale to Ec and normalize by kT . That is, define:
η =
E − Ec
kT
ηc =
EF − Ec
kT
Then:
⎛
no = 4π
⎝
2m ∗
dekT
h2
⎞
3/2 ∞
(cid:2)
⎠
0 1 + eη−ηc
dη
√
η
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic ... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 3-7
Fermi integral of order 1/2:
)
x
(
2
/
1
1E+02
1E+01
1E+00
1E-01
1E-02
1E-03
1E-04
1E-05
e x
x3/2
non-degenerate
degenerate
-10
-8
-6
-4
-2
2
4
6
8
10
0
x
Key result again:
no = NcF1/2(ηc... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 3-9
� Degenerate regime
More complicated behavior of F1/2(x) for high values of x (see Ad
vanced Topic AT2.3).
Degenerate semicond... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
ex
ponential with kT as characteristics energy.
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - S... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
T , or po (cid:10) Nv:
po (cid:7) Nv exp
Ev − EF
kT
Fermi level well above valence band edge.
� Degenerate regime:
If ηv (cid:11) 1, or Ev − EF (cid:11) kT , or po (cid:11) Nv, more complicated
dependence of po on EF .
Fermi level inside valence band.
Cite as: Jesús del Alamo, course materials for 6.720J Integrate... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
a given semiconductor, nopo depends only on T and is indepen
dent of precise location of EF .
But only if semiconductor is non-degenerate.
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Techn... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
is close to the middle of the bandgap
In Si at 300 K, Ei is 1 meV above midgap
Ec
Ei
Ev
[Consistent with use of Maxwell-Boltzmann statistics in ni expres
sion]
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massach... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
po = ni = NcNv exp −
√
Eg
2kT
• In non-degenerate semiconductor nopo is a constant that only
depends on T :
2
nopo = ni
• In intrinsic semiconductor, EF is close to middle of Eg.
• In extrinsic semiconductor, EF location depends on doping level:
– n-type non-degenerate semiconductor:
no (cid:7) ND,
EF − Ec (... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/1bccffbfa7b4487be99b8af8b46186ef_lecture3.pdf |
3.032 Mechanical Behavior of Materials
Fall 2007
shear bands (red) forming in
polycrystalline elemental metal
with many line and point defects
Images removed due to copyright restrictions.
Please see:
http://www-geol.unine.ch/03_france/granites/Granites-Thumbnails/3.jpg
shear bands in granite
(complex crystal) form... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/1bd633873707143d7607ffb3b6d08fb6_lec20.pdf |
Lecture 20 (10.26.07)
3.032 Mechanical Behavior of Materials
Fall 2007
fibrils of polymer
hydrocarbon chains
aligned within fibril
crazing in amorphous polymer
rupture of fibrils under
tensile load
Image sources: http://www.kern-gmbh.de/index_glossar.html?http://www.kern-gmbh.de/kunststoff/service/glossar/crazin... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/1bd633873707143d7607ffb3b6d08fb6_lec20.pdf |
20.110/5.60 Fall 2005
Lecture #10
page
1
Chemical Equilibrium
Ideal Gases
Question: What is the composition of a reacting mixture of ideal
gases?
e.g.
½ N2(g, T, p) + 3/2 H2(g, T, p) = NH3(g, T, p)
What are
p p
,
H
2
N
2
, and
p at equilibrium?
NH3
Let’s look at a more general case
νA A(g, T, p) + νB B(g... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/1bdff1fc0dbacbe7a759e6ca10d918d3_l10.pdf |
20.110/5.60 Fall 2005
Lecture #10
page
2
where ε is an arbitrary small number that allows to let the reaction
proceed just a bit.
We know that
µ
i
(
)
T p
,
g,
=
o
µ
i
(
T RT p
i
ln
+
)
⎡
⎢
⎣
p
i
1 bar
implied
⎤
⎥
⎦
where
o
(
i Tµ
)
is the standard chemical potential of species “i” at 1 bar
and in a pure (not... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/1bdff1fc0dbacbe7a759e6ca10d918d3_l10.pdf |
= ∆
− ∆ o
H T S
o
rxn
rxn
or
∆
o
o
= ∆
G G
form
(
products
)
− ∆
o
G
form
(
reactants
)
If
ε∆
G
<
( ) 0
then the reaction will proceed spontaneously to form
more products
ε∆
G
>
( ) 0
then the backward reaction is spontaneous
ε∆
G
=
( ) 0
No spontaneous changes ⇒
Equilibrium
20.110J / 2.772J / 5.601JThermodyn... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/1bdff1fc0dbacbe7a759e6ca10d918d3_l10.pdf |
,XK pT .
(
)
Recall that all pi values are divided by 1 bar, so Kp and KX are
both unitless.
________________________________________________
Example: H2(g) + CO2(g) = H2O(g) + CO(g)
T = 298 K
p =1 bar
H2(g) CO2(g) H2O(g) CO(g)
a
b
a-x
b-x
0
x
0
x
Initial #
of moles
# moles
at Eq.
Total # moles at E... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/1bdff1fc0dbacbe7a759e6ca10d918d3_l10.pdf |
and b = 2 mol
We need to solve
(
1
−
2
x
)(
2
x
=
9.7 10
x
−
6
−
x
)
A) Using approximation method:
K << 1, so we expect x << 1 also.
Assume
1
−
x
≈
1, 2
−
x
≈
2
⇒
x ≈
0.0044 mol (indeed
1
(
<<
2
x
)(
2
−
x
≈
2
x
2
−
x
)
=
9.7 10
x
−
6
1)
B)
Exactly:
2
x
−
3
x x
2
+
2
=
K
p
=
9.7 10
x
−
6
2
x
(
x
=
1 9.7 10
x
... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/1bdff1fc0dbacbe7a759e6ca10d918d3_l10.pdf |
20.110/5.60 Fall 2005
Lecture #10
page
5
Effect of total pressure: example
N2O4(g) = 2 NO2(g)
Initial mol #
n
# at Eq.
n-x
Xi’s at Eq.
−
n x
+
n x
0
2x
2x
n x+
Total # moles at Eq. =
n – x + 2x = n + x
K
p
=
2
p
NO
2
p
NO
2
4
=
2
2
p X
NO
2
pX
NO
2
4
=
p
2
2
x
⎞
⎛
⎟
⎜
+⎝
n x
⎠
−
n x
⎞
⎛
⎟
⎜
+⎝
⎠
n x
=
p
2
... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/1bdff1fc0dbacbe7a759e6ca10d918d3_l10.pdf |
��
⎟
⎠
∴
If p increases, α decreases
Le Chatelier’s Principle, for pressure:
An increase in pressure shifts the equilibrium so as to decrease the
total # of moles, reducing the volume.
In the example above, increasing p shifts the equilibrium toward the
reactants.
---------------
20.110J / 2.772J / 5.601JThermo... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/1bdff1fc0dbacbe7a759e6ca10d918d3_l10.pdf |
=
1
−
⎛
⎜
⎜
⎝
2
pK
p
1 3
⎞
⎟
⎟
⎠
In this case, if p↑ then x↑ as expected from Le Chatelier’s principle.
20.110J / 2.772J / 5.601JThermodynamics of Biomolecular SystemsInstructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/1bdff1fc0dbacbe7a759e6ca10d918d3_l10.pdf |
2.160 Table of Contents
1. Introduction
Physical modeling vs. Black-box modeling
System Identification in a Nutshell
Applications
Part 1 ESTIMATION
2. Parameter Estimation for Deterministic Systems
2.1 Least Squares Estimation
2.2 The Recursive Least-Squares Algorithm
2.3 Physical meanings and properties of ma... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/1bf206dfb15f096397afc979ba28bddb_updtd_table_cont.pdf |
Model Structure
6.2.2 Linear Regressions
6.2.3 ARMAX Model Structure
6.2.4 Pseudo-linear Regressions
6.2.5 Output Error Model Structure
6.3 State Space Model
6.4 Consistent and Unbiased Estimation: Preview of Part 3, System ID
6.5 Times-Series Data Compression
6.6 Continuous-Time Laguerre Series Expansion
6.7 ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/1bf206dfb15f096397afc979ba28bddb_updtd_table_cont.pdf |
otic Variance
14 Experiment Design
14.1 Review of System ID Theories for Experiment Design
Key Requirements for System ID
14.2 Design Space of System ID Experiments
14.3 Input Design for Open-Loop Experiments
14.4 Practical Requirements for Input Design
14.5 System ID Using Random Signals
14.6 Pseudo-Random Bin... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/1bf206dfb15f096397afc979ba28bddb_updtd_table_cont.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
Prof. Markus Zahn
Lecture 4: Continuum Electromechanics (Melcher) – Sections 2.18... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/1bf674b3da895a32ac2d670975ead983_lec04_f08.pdf |
-
ρ
ε
⇒ Φ
=
∫
V
4
π
ρ
ε
dV
r - r '
∇
2f = 0
⇒
f = 0
⇒
C = A
C. Vector Poisson’s Equation Solutions
2
∇
A = - J
μ ⇒
( )
A r =
μ
π ∫
4
V
)
(
J r ' dV
r - r '
6.642, Continuum Electromechanics Lecture 4
Prof. Markus Zahn Page 1 of 6
_
Courtesy ... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/1bf674b3da895a32ac2d670975ead983_lec04_f08.pdf |
∂
r θ
∂ ⇒
A
∂
r
∂
-
A
∂
r
∂
dr +
A
∂
θ
∂
dθ = dA = 0
⇒
3. Axisymmetric Cylindrical
A =
Λ
(
r, z
r
)
−
i
θ
1
B = × A = -
r
∇
∂Λ
z
∂
1
−
i +
r
r
∂Λ
r
∂
−
i
z
dz
dr
=
B
B
z
r
=
1
r
1
r
-
∂Λ
r
∂ ⇒
∂Λ
z
∂
∂Λ
r
∂
dr +
∂Λ
z
∂
dz = d = 0
Λ
Λ
(
r, z = constant
)
4. Axisymmetric Spherical
A =
Λ
r, θ
)
(
r sin θ
−
i
φ
B ... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/1bf674b3da895a32ac2d670975ead983_lec04_f08.pdf |
Free Regions
∇
i
E = 0
⇒
E = × A
∇
II. Vector Potential Transfer Relations with J =0 (Section 2.19)
∇
2 A = 0
[Vector Laplace’s Equation]
A. Cartesian Coordinates
2
∇
A =
2
∇
⎡
⎢
⎣
−
−
A i + A i + A i
y
∇
∇
2
2
x
−
z
x
y
⎤
⎥
⎦
z
A = i R e A x e
(cid:105) (
⎡
⎣
)
-jky
⎤
⎦
−
z
(cid:105) (
)
A x =
α
(cid:105)
(cid... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/1bf674b3da895a32ac2d670975ead983_lec04_f08.pdf |
Δ
α
β
(cid:105)
A
(cid:105)
A
⎤
⎥
⎥
⎦
B =
x
A
∂
z
y
∂
⇒
B = -jk A
⇒
(cid:105)
(cid:105)
x
(cid:105)
B
α
x
(cid:105)
B
β
x
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
= -jk
α
β
(cid:105)
A
(cid:105)
A
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
6.642, Continuum Electromechanics Lecture 4
Prof. Markus Zahn Page 4 of 6
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/1bf674b3da895a32ac2d670975ead983_lec04_f08.pdf |
�
⎤
⎥
⎥
⎦
m
β
⎞
⎟α
⎠
β
(cid:105)
+ A
-
α
β
⎛
⎜
⎝
⎛
⎜
⎝
⎡
⎢
⎢
⎣
m
⎞
⎟
⎠
r
α
m
⎞
⎟
⎠
-
α
r
⎛
⎜
⎝
m
⎞
⎟
⎠
⎤
⎥
⎥
⎦
⎤
⎥
⎥
⎦
6.642, Continuum Electromechanics Lecture 4
Prof. Markus Zahn Page 5 of 6
H = -
θ
1 A
∂
r
μ ∂
H =
r
1 A
∂
r θ
μ ∂
⇒
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/1bf674b3da895a32ac2d670975ead983_lec04_f08.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.641 Electromagnetic Fields, Forces, and Motion
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 2: Differential Form of Maxwell’s Equati... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/1c014f0d8c71c1be1eb4fd3198b1d543_MIT6_641s09_lec02.pdf |
i da = ∫ ∇ i (ε0E dV =
)
∫ ρ dV
S
V
V
∇ i ε0E = ρ
)
(
µ H i da = ∇ i µ H dV = 0
( 0 )
∫
V
0
S
(cid:118)∫
∇ i (
)
µ0H = 0
II. Stokes’ Theorem
1. Curl Operation
) i
A ds = ∫ Curl A
(
i
S
(cid:118)∫
C
da
(cid:118)∫
)n
Curl A = lim C
(
da
n →0
A ds
i
da
n
6.641, Electromagnetic Fields, Forces, and... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/1c014f0d8c71c1be1eb4fd3198b1d543_MIT6_641s09_lec02.pdf |
x (
∆y
y
(
) - A x, y
)⎤
⎤
⎦
⎥
⎥
⎦
∆x
= da z ⎢
⎡ ∂Ay
⎣ ∂x
-
∂Ax ⎤
⎥
∂y ⎦
)z
(
Curl A
= (cid:118)∫
i
A ds
=
da
z
A
∂
y -
x
∂
A
∂
x
y
∂
By symmetry
(
Curl A
) = (cid:118)∫ A ds
i
y
da
y
∂A
= x
∂z
-
∂A
z
∂x
Curl A = (cid:118)∫ A ds
=
i
(
)x
dax
∂Az -
∂y
∂Ay
∂z
Curl A = i x ⎢
− ⎡ ∂A
⎣ ∂... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/1c014f0d8c71c1be1eb4fd3198b1d543_MIT6_641s09_lec02.pdf |
∇ × A
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 2
Page 6 of 10
2. Stokes’ Integral Theorem
lim ∑ (cid:118)∫ A ds
i = (cid:118)∫ A ds
i
N→∞
i
C
N
i=1 dCi
N→∞
= ∑ (∇ × A) i da
i
i=1
= ∫ (∇ × A) i da
S
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Mark... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/1c014f0d8c71c1be1eb4fd3198b1d543_MIT6_641s09_lec02.pdf |
��
0 = ∇ i
⎡
⎢J + ε0
⎢
⎣
⎤
∂E
⎥
∂t ⎥
⎦
0 = ∇ i J +
∂ρ
∂t
3. Magnetic Field
∇ i ⎨∇ × E = - µ0
⎧
⎪
⎪
⎩
⎫
∂H⎪
⎬
∂t ⎪
⎭
0 = -
∂
t ⎣
∂
⎡∇ µ0
⎦ ⇒ ∇ i (µ0H) = 0
⎤
i H
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 2
Page 8 of 10
4. Vector Identity
b
(
∫ i
E dl = Φ a
) − Φ ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/1c014f0d8c71c1be1eb4fd3198b1d543_MIT6_641s09_lec02.pdf |
+ (y − y ')2
⎡(x − x ')2
1
+ (z − z ')2
⎤
⎦
2
∇ µ0
i ( H) = 0 ⇒ µ H = ∇ × A
0
∇ 2 A = − µ 0 J, ∇ i A = 0
A x, y, z
(
) = ∫∫∫
x ',y ',z ' 4π ⎡
(
)
µ0 J x ', y ', z ' dx dy dz
+ (y − y ')2
'
'
⎣(x − x ')2
+ (z − z ')2 ⎤
⎦
'
1
2
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lectu... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/1c014f0d8c71c1be1eb4fd3198b1d543_MIT6_641s09_lec02.pdf |
18.175: Lecture 5
More integration and expectation
Scott Sheffield
MIT
18.175 Lecture 5
1 Outline
Integration
Expectation
18.175 Lecture 5
2 Outline
Integration
Expectation
18.175 Lecture 5
3 Recall Lebesgue integration
� Lebesgue: If you can measure, you can integrate.
� In more words: if (Ω, F) is a m... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/1c016821763783c34c48816b1ac66969_MIT18_175S14_Lecture5.pdf |
then gdµ =
fdµ.
<
|
|f |dµ.
When (Ω, F, µ) = (Rd , Rd , λ), write
fdµ| ≤
<
(cid:73)
�
(cid:73)
�
<
fdµ + b gdµ.
<
f (x)dx = 1E fdλ.
<
E
18.175 Lecture 5
5 Outline
Integration
Expectation
18.175 Lecture 5
6 Outline
Integration
Expectation
18.175 Lecture 5
7 Expectation
(cid:73)
�
�
(cid:73)
<
Given... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/1c016821763783c34c48816b1ac66969_MIT18_175S14_Lecture5.pdf |
proof: Rescale so that lf lplg lq = 1. Use
some basic calculus to check that for any positive x and y we
have xy ≤ x p/p + y q/p. Write x = |f |, y = |g | and integrate
to get
Cauchy-Schwarz inequality: Special case p = q = 2. Gives
<
|fg |dµ ≤ lf l2lg l2. Says that dot product of two vectors is
at most product ... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/1c016821763783c34c48816b1ac66969_MIT18_175S14_Lecture5.pdf |
= lim infn→∞ fn(x). Then truncate, used
bounded convergence, take limits.
18.175 Lecture 5
11
More integral properties
(cid:73)
�
Monotone convergence: If fn ≥ 0 and fn ↑ f then
(cid:90)
(cid:90)
fndµ ↑
fdµ.
(cid:73)
�
�
(cid:73)
�
(cid:73)
Main idea of proof: one direction obvious, Fatou gives other.
Dom... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/1c016821763783c34c48816b1ac66969_MIT18_175S14_Lecture5.pdf |
18.445 Introduction to Stochastic Processes
Lecture 10: Hitting times
Hao Wu
MIT
16 March 2015
Hao Wu (MIT)
18.445
16 March 2015
1 / 8
Recall
Consider a network (G = (V , E), {c(e) : e ∈ E}). The effective
resistance is defined by
R(a ↔ z) = (W (a) − W (z))/||I||.
Consider a random walk on the network, the Green’s func... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/1c4e0f0bf28f0ec9f0fd6bc8f7b896f1_MIT18_445S15_lecture10.pdf |
x, y ) ∈ Ω × Ω, there
is a bijection ϕ : Ω → Ω such that
ϕ(x) = y ; P(ϕ(z), ϕ(w)) = P(z, w), ∀z, w.
Example : simple random walk on N-cycle, on hypercube.
Lemma
For a transitive Markov chain on finite state space Ω, the uniform
measure is stationary.
Hao Wu (MIT)
18.445
16 March 2015
5 / 8
Commute time
Definition
Suppos... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/1c4e0f0bf28f0ec9f0fd6bc8f7b896f1_MIT18_445S15_lecture10.pdf |
τb] = Eb[τa].
Hao Wu (MIT)
18.445
16 March 2015
7 / 8
Summary
For random walk on network
t(cid:12) ≤ thit ≤ 2 maxw Eπ[τw ].
Ea[τba] = cGR(a ↔ b).
For random walk on transitive network
t(cid:12) ≤ thit ≤ 2t(cid:12).
Ea[τb] = Eb[τa].
2Ea[τb] = cGR(a ↔ b).
Hao Wu (MIT)
18.445
16 March 2015
8 / 8
MIT OpenCourseWare
http:... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/1c4e0f0bf28f0ec9f0fd6bc8f7b896f1_MIT18_445S15_lecture10.pdf |
Utility Theory
Week 4 Framing
Required Reading:
de Neufville, Richard, Applied Systems Analysis: Engineering Planning and Technology
Management, McGraw-Hill, New York, 1990. Chapters 18, 19, 20, 21.
McManus, H. L., and Ross, A. M., SSPARC Book Material for Lecture 4.
Gumbert, C. C., Violet, M. D., Hastings, D. E.... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/1c5ef3f7a6f50076b2608123521822a9_04000lec4outlnv3.pdf |
to the Space Based Radar (SBR), Master of
Science Thesis in Aeronautics and Astronautics, Massachusetts Institute of Technology.
Seshasai, Satwiksai, “Knowledge Based Approach to Facilitate Engineering Design,”
Masters Thesis in Electrical Engineering, Massachusetts Institute of Technology, May
2002.
Scott, M. J.... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/1c5ef3f7a6f50076b2608123521822a9_04000lec4outlnv3.pdf |
1.4 Backward Kolmogorov equation
When mutations are less likely, genetic drift dominates and the steady state distributions are
peaked at x = 0 and 1. In the limit of µ1 = 0 (or µ2 = 0), Eq. (1.63) no longer corresponds
to a well-defined probability distribution, as the 1/x (or 1/(1 − x)) divergence close to x = 0
(or x... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/1c5ef58df27b3a6ae906ba0d5aaba9ed_MIT8_592JS11_lec5.pdf |
is called Markovian, after the Russian mathematician Andrey Andreyevich
Markov (1856-1922). We can use this probability to construct evolution equations for the
probability by focusing on the change of position for the last step (as we did before in deriving
Eq. (1.35)), or the first step. From the latter perspective, w... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/1c5ef58df27b3a6ae906ba0d5aaba9ed_MIT8_592JS11_lec5.pdf |
.
(1.66)
Using the normalization condition for R(δy, y) and the definitions of drift and diffusion
coefficients from Eqs. (1.36) and (1.37), we obtain
∂p(x, t|y)
∂t
= v(y)
∂p
∂y
+ D(y)
∂2p
∂y2 ,
(1.67)
which is known as the backward Kolmogorov equation. If the drift velocity and the diffusion
coefficient are independent of p... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/1c5ef58df27b3a6ae906ba0d5aaba9ed_MIT8_592JS11_lec5.pdf |
(1.59). However, as we
noted already, in the context of absorbing states the function p∗ is not normalizable and
thus cannot be regarded as a probability. Nonetheless, we can express the results in terms
of this function. For example, the probability of fixation, i.e. Π1(y) is obtained with the
boundary conditions Π1(0)... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/1c5ef58df27b3a6ae906ba0d5aaba9ed_MIT8_592JS11_lec5.pdf |
xed, with a probability that decays with population
size as Π1 = e−(2N −1)|s|. The probability of loss of the mutation is simply Π0 = 1 − Π1.
1.4.2 Mean times to fixation/loss
When there is an absorbing state in the dynamics, we can ask how long it takes for the
process to terminate at such a state. In the context of ra... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/1c5ef58df27b3a6ae906ba0d5aaba9ed_MIT8_592JS11_lec5.pdf |
and Ohta (1968)3, we first examine the numerator of the above ex-
pression, defined as
(Writing limT →∞
equation by parts to get
T
0 rather than simply
R
0
Z
∞
0
Ta(y) = lim
T →∞
T
dt t
∂p(xa, t|y)
∂t
.
(1.76)
is for later convenience.) We can integrate this
Ta(y) = lim
T →∞
= lim
T →∞
R
T p(xa, T |y) −
dt p(xa, t|y)
T
(... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/1c5ef58df27b3a6ae906ba0d5aaba9ed_MIT8_592JS11_lec5.pdf |
80) reduces to
y(1 − y)
4N
∂2T0
∂y2 = −(1 − y) ⇒
∂2T0
∂y2 = −
4N
y
.
(1.81)
3M. Kimura and T. Ohta, Genetics 61, 763 (1969).
18
After two integrations we obtains
T0(y) = −4Ny (ln y − 1) + c1y + c2 = −4Ny ln y ,
(1.82)
where the constants of integration are set by the boundary conditions T0(0) = T0(1) = 0,
which follow... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/1c5ef58df27b3a6ae906ba0d5aaba9ed_MIT8_592JS11_lec5.pdf |
y)
dt
.
(1.86)
(Note that the above PDF is properly normalized as S(∞) = 0, while S(0) = 1.) The mean
survival time is thus given by
hτ (y)i× = −
1−
∞
dt t
0
Z
0+
Z
dx
dp(x, t|y)
dt
=
1−
∞
dx
dt p(x, t|y) ,
(1.87)
0+
Z
0
Z
where we have performed integration by parts and noted that the boundary terms are zero.
Applying... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/1c5ef58df27b3a6ae906ba0d5aaba9ed_MIT8_592JS11_lec5.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
ESD.70J / 1.145J Engineering Economy Module
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
ESD.70J Engineering Economy
Fall 2009
Session Two
Michel-Alexandre Cardin
Prof. Richard de Neufville
ESD.70J Engineering Eco... | https://ocw.mit.edu/courses/esd-70j-engineering-economy-module-fall-2009/1c610901c86e62fbe5cc1e555297b7ed_MITESD_70Jf09_lec02.pdf |
tab “RAND”
2. Type “=Entries!C9*((1-
Entries!C25)+2*Entries!C25*RAND())” in cell C3
3. Type “=Entries!C10*((1-
Entries!C25)+2*Entries!C25*RAND())” in cell D3
4. Type “=Entries!C11*((1-
Entries!C25)+2*Entries!C25*RAND())” in cell E3
5. Press “F9” several times to see what happens
ESD.70J Engineering Economy Module - Ses... | https://ocw.mit.edu/courses/esd-70j-engineering-economy-module-fall-2009/1c610901c86e62fbe5cc1e555297b7ed_MITESD_70Jf09_lec02.pdf |
to the random demand generator,
specifically, Plan A!E5 = Rand!C3; Plan A!G5 = Rand!D3; Plan A!I5 =
Rand!E5
In “Simulation” sheet, type “=‘Plan A’!C16” in cell B8 (“=‘Plan A’!C16”
is the output of result for NPVA)
Create the Data Table. Select “A8:B2008”, click “Table” under “Data”
menu, in “column input cell” put “A... | https://ocw.mit.edu/courses/esd-70j-engineering-economy-module-fall-2009/1c610901c86e62fbe5cc1e555297b7ed_MITESD_70Jf09_lec02.pdf |
Cell D3 type “=MIN(B$9:B$2008)”
ESD.70J Engineering Economy Module - Session 2
15
Give it a try!
Check with your neighbors…
Check the solution sheet…
Ask me questions…
ESD.70J Engineering Economy Module - Session 2
16
Deterministic vs. dynamic results
• From the base case spreadsheet, we learn
NPVA is $162.1 million... | https://ocw.mit.edu/courses/esd-70j-engineering-economy-module-fall-2009/1c610901c86e62fbe5cc1e555297b7ed_MITESD_70Jf09_lec02.pdf |
values, and the range =Simulation!$I$7:$I$27 for Y
values. Click “OK”
9. Right-click the curve and change “Weight” to 3
10. Hit “command =” or “F9” and watch the target curve
move !
ESD.70J Engineering Economy Module - Session 2
20
Explanation
• We set up 20 data buckets and count how
many data points fall into eac... | https://ocw.mit.edu/courses/esd-70j-engineering-economy-module-fall-2009/1c610901c86e62fbe5cc1e555297b7ed_MITESD_70Jf09_lec02.pdf |
This is not a particularly realistic model,
though it is simple and sufficient for today’s
purposes
• Next session explores alternative probability
distributions from which to sample and
stochastic models. STAY TUNED!
ESD.70J Engineering Economy Module - Session 2
26 | https://ocw.mit.edu/courses/esd-70j-engineering-economy-module-fall-2009/1c610901c86e62fbe5cc1e555297b7ed_MITESD_70Jf09_lec02.pdf |
Turbulent Flow and Transport
4
Free Shear Flows I: Jets, Wakes, etc.−Solutions Based on
Simple Mean−Flow Closure Schemes
4.1 Mean−flow closure schemes for free shear flows.
4.2
4.3
4.4
for
4.5
Spreading of a velocity discontinuity with downstream distance in steady flow.
The nature of the laminar flow soluti... | https://ocw.mit.edu/courses/2-27-turbulent-flow-and-transport-spring-2002/1c80d3bfd54cb73fffc3a301657e8cfb_Free_shear_flows.pdf |
Turbulent Jets:103−113, 120−125.
Rajaratnam. "Turbulent Jets." Elsevier, 1976.
Rodi. In "Studies in Convection" B. E. Launder. ed. Academic
Press, 1975: 79 ff.
Townsend. Chap.6 in The Structure of Turbulent Shear Flow. 2nd ed.
Cambridge, 1976.
Handouts:
Selected experimental data & summaries. | https://ocw.mit.edu/courses/2-27-turbulent-flow-and-transport-spring-2002/1c80d3bfd54cb73fffc3a301657e8cfb_Free_shear_flows.pdf |
Introduction to Engineering
Introduction to Engineering
Systems, ESD.00
Lecture 6
Lecturers:
Professor Joseph Sussman
Dr Afreen Siddiqi
Dr. Afreen Siddiqi
TA: Regina Clewlow
Uncertainty Lecture 2 Outline
Uncertainty Lecture 2-- Outline
Global Climate Change
High-impact, Low-probability events
Decision-making Und... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/1cb961bffd1d5e437be957c66ad547fd_MITESD_00S11_lec06.pdf |
tsunamis
5th biggest earthquake in recorded history,
biggest ever in Japan
biggest ever in Japan
Huge loss of life, injuries, property damage
Japan likely the most prepared nation in the
th
k di
world for earthquake disastters
ld f
Uncertainty: High-impact, Low
Probability Events
Probability Events
Very high-i... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/1cb961bffd1d5e437be957c66ad547fd_MITESD_00S11_lec06.pdf |
]] deal for the comppanyy that sold
What might you do instead of buying an annuity?
What might you do instead of buying an annuity?
Uncertainty: Compound probabilities
Uncertainty: Compound probabilities
Assume independence
________A_________x_______B________
Electrical example: P(A) is probability that A is cond... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/1cb961bffd1d5e437be957c66ad547fd_MITESD_00S11_lec06.pdf |
: Bayes Theorem
Uncertainty: Bayes Theorem
’
The MIT Snow Day example
Uncertainty: Bayes Theorem
Uncertainty: Bayes Theorem
’
The birthday example: How many birthdays until a
mat h?tch?
More on Decision-making Under
Uncertainty:
Uncertainty:
Decision-making under uncertainty
Decision trees:
Example: Football ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/1cb961bffd1d5e437be957c66ad547fd_MITESD_00S11_lec06.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
Mar 11, 2008
Lecturer: Scott Aaronson
Scribe: Yinmeng Zhang
Lecture 10
1 Administriv... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/1cce8117881c1d1e5a21935bdf76d377_lec10.pdf |
we will see some more reductions.
10-1
PNPNP-completeNP-hard 3 SAT reduces to 3SAT
Though SAT is NP-complete, specific instances of it can be easy to solve.
Some useful terminology: a clause is a single disjunction; a literal is a variable or the negation
of a variable. A Boolean formula in conjunctive normal form (... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/1cce8117881c1d1e5a21935bdf76d377_lec10.pdf |
∨ed together, we first need to specify an order in
which to take the ∧s or ∨s. For example, if we saw (x1 ∨ x2 ∨ x3) in our formula, we should parse
it as either ((x1 ∨ x2) ∨ x3) which becomes OR(x1, OR(x2, x3)), or (x1 ∨ (x2 ∨ x3)) which becomes
OR(OR(x1, x2), x3). It doesn’t matter which one we pick, because ∧ and ... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/1cce8117881c1d1e5a21935bdf76d377_lec10.pdf |
of these variables such that for every gate, the output has the right
relationship to the input/inputs, and the final output is set to true.
So let’s look at the NOT gate. Call the input x and the output y. We want that if x is true
then y is false, and if x is false then y is true – but we’ve seen how to write if-th... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/1cce8117881c1d1e5a21935bdf76d377_lec10.pdf |
Once we’ve got these gadgets, we have to somehow stick them
together.
In this case, all we have to do is AND together the Boolean formulas for each of the gates, and
the variable for the final output wire. The Boolean formula we get from doing this is true if and
only if the circuit is satisfiable. Woo.
4
3COLOR is... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/1cce8117881c1d1e5a21935bdf76d377_lec10.pdf |
colors, and because they’re both connected to the Neither vertex in the palette, one of them will
have to be true, and one of them will be false.
Here is a gadget for the AND gate, courtesy of Alex.
10-3
palette¬xxTFN It’s obvious it works, right? The main things to note are as follows. The inputs and output
are c... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/1cce8117881c1d1e5a21935bdf76d377_lec10.pdf |
are 3-colorable iff there’s a short proof of the Riemann
Hypothesis in ZF set theory. Chew on that.
Looking at the somewhat horrifying AND gadget, you might wonder if there’s a way to draw the
graph so that no lines cross. This turns out to be possible, and implies that 3PLANAR-COLOR
(given a planer graph, is it 3-c... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/1cce8117881c1d1e5a21935bdf76d377_lec10.pdf |
otherwise! This, of course, is what
motivates the question of whether P=NP — what makes it one of the central questions in all of
math and science.
6 Tricky Questions
Cook defined a language A to NP-complete when it itself was in NP, and every NP problem could
be solved in polynomial time if given oracle access to ... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/1cce8117881c1d1e5a21935bdf76d377_lec10.pdf |
14. Instability of Superposed Fluids
Figure 14.1: Wind over water: A layer of fluid of density ρ+ moving with relative velocity V over a layer
of fluid of density ρ− .
Define interface: h(x, y, z) = z
The unit normal is given by
−
η(x, y) = 0 so that ∇h = (
ηx,
−
ηy, 1).
nˆ =
∇h
∇h
|
|
=
−
ηy, 1)
1/2
ηx,
(
−
... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/1d1f0dbdfd935d542ea0242a4649fada_MIT18_357F10_Lecture14.pdf |
∂φ±
∂z
zˆ
from which
∂η
∂t
=
1
V +
2
∂φ±
∂x
)
(
−
(
∓
ηx) +
∂φ±
(
∂y
−
ηy) +
∂φ±
∂z
(14.1)
(14.2)
(14.3)
(14.4)
Linearize: assume perturbation fields η, φ± and their derivatives are small and therefore can neglect
their products.
Thus ηˆ
ηy, 1) and ∂η =
ηx,
(
1 V ηx +
2
∂φ±
∂z
∂t
≈
−
−
⇒
±
∂... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/1d1f0dbdfd935d542ea0242a4649fada_MIT18_357F10_Lecture14.pdf |
V
(
∓
∂φ±
∂x
)
+ p± + ρ±gη = G(t)
(14.7)
so
p−
−
p+ = (ρ+
−
ρ−)gη + (ρ+
∂φ±
∂t
−
ρ−
∂φ−
∂t
) +
V
2
(ρ−
∂φ−
∂x
+ ρ+
∂φ+
∂x
) =
−
σ(ηxx + ηyy)
(14.8)
is the linearized normal stress BC. Seek normal mode (wave) solutions of the form
η = η0e
iαx+iβy+ωt
φ± = φ0±e
∓kz iαx+iβy+ωt
e
where
Appl... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/1d1f0dbdfd935d542ea0242a4649fada_MIT18_357F10_Lecture14.pdf |
�−(ω +
1
2
iαV ) + gk(ρ−
]
ρ+) +
−
1
2
iαV ρ+(ω
[
−
1
2
iαV ) + ρ−(ω +
1
2
iαV )
]
−
so
ω2 + iαV
ρ−
ρ+
−
ρ− + ρ+
(
ω
)
−
1
α2V + k2C0
2
4
2 = 0
where C 2
k.
0
Dispersion relation: we now have the relation between ω and k
σ
ρ−+ρ+
ρ−−ρ+
ρ−+ρ+
+
≡
g
k
(
)
ω =
1
2
i
ρ−
ρ+
−
ρ− + ρ+
)
(
k ... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/1d1f0dbdfd935d542ea0242a4649fada_MIT18_357F10_Lecture14.pdf |
W. M. Bush
14.1. Rayleigh-Taylor Instability
Chapter 14. Instability of Superposed Fluids
14.1 Rayleigh-Taylor Instability
We consider an initially static system in which heavy fluid overlies light fluid: ρ+ > ρ−, V = 0. Via
(14.15), the system is unstable if
C 2
0 =
ρ−... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/1d1f0dbdfd935d542ea0242a4649fada_MIT18_357F10_Lecture14.pdf |
The base state and the per-
turbed state of the Rayleigh-Taylor system,
heavy fluid over light.
1. The system is stabilized to small λ disturbances by
σ
2. The system is always unstable for suff. large λ
3. In a finite container with width smaller than 2πλc,
the system may be stabilized by σ.
4. System may be stabilize... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/1d1f0dbdfd935d542ea0242a4649fada_MIT18_357F10_Lecture14.pdf |
Equivalently,
2
ρ−ρ+V > (ρ−
ρ+) g
−
λ
2π
+ σ
2π
λ
(14.18)
Note:
1. System stabilized to short λ disturbances by
surface tension and to long λ by gravity.
2. For any given λ (or k), one can find a critical
V that destabilizes the system.
Marginal Stability Curve:
Figure 14.4: Kelvin-Helmholtz instability: a ... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/1d1f0dbdfd935d542ea0242a4649fada_MIT18_357F10_Lecture14.pdf |
·103
70
70
J
⇒
·
≈
650cm/s is the mini-
3.8 cm , so λc = 1.6cm. They thus correspond to capillary
−1
MIT OCW: 18.357 Interfacial Phenomena
58
Prof. John W. M. Bush
MIT OpenCourseWare
http://ocw.mit.edu
357 Interfacial Phenomena
Fall 2010
For information about citing these materials or our Term... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/1d1f0dbdfd935d542ea0242a4649fada_MIT18_357F10_Lecture14.pdf |
6.897: Selected Topics in Cryptography
Lecturer: Ran Canetti
Lectures 3 and 4:
ZK as function evaluation and
sequential composition of ZK
• Review the definition of ZK and PoK
• Give SFE-style definition of ZK and show
equivalence to the standard one.
• The Blum protocol for Hamiltonicity:
– Commitment schemes
– ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/1d52d1e1de07e4fd8cf3ea9e3f689583_lecture3_4.pdf |
.)
Review: Zero-Knowledge
[Goldwasser-Micali-Rackoff 85]
• Zero-Knowledge:
For any verifier V* there exists a machine S
such that for all x,w,z, S(x,R(x,w),z) ≈ V*
P(x,w)(z).
“Whatever V* can gather from interacting with P, it could have
computed by itself given R(x,w).”
(Distribution ensembles D,D’ are computat... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/1d52d1e1de07e4fd8cf3ea9e3f689583_lecture3_4.pdf |
2-party function:
F
R ((x,w), - , - ) = ( - , (x,R(x,w)) , (x,R(x,w)))
zk
Theorem: A two-party protocol securely
R (with respect to
realizes Fzk
non-adaptive adversaries) if and only if
it is a ZK PoK for R.
Proof: Homework.
Example:
Blum’s protocol for Graph Hamiltonicity
[Blum 8?]
Commitment schemes
Intuitive i... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/1d52d1e1de07e4fd8cf3ea9e3f689583_lecture3_4.pdf |
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