text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
157),
and
if
we
apply
the
inverse
Fourier
transform
to
the
result-
ing
equation,
we
obtain
0
0
U
N
^
2
w(x
,
z
,
t
)
ik
( (k)
s
i
n
(
k
z
+
k
U
t
+
k
x
)dk
N/U
2
;
8
U
;
0
s
Z
{
1
:
2
N
(7.158)
+
ik
( (k)
e
x
p
(
k
z )
sin(+k
U
t
+
k
x
)dk
^
2
0
N/U
s
;
;
U
Z
9
}
In
terms
of
the
fxed
reference
frame,
the
vertical
velo... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
Next,
we
consider
an
example
of
a
localized
topography,
the
\witch
0
;
of
Agnesi",
for
which
( (x) (fxed reference
fram
e)
is
given by the equation
A
0
( (x)
,
(7.160)
1 +
(
x/b)
2
and
its
Fourier
transform
is
given
by
the
equation
^
( (k)
b
exp(
A
kb
):
(7.161)
0
;j
j
For
this
particular
example,
the
vertical
veloci... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
1 (x,
z ) A
b
exp(
b
k
)
sin
(
k
z
+
kx
)dk
0
8
;
j
j
;
U
0
s
N/U
2
N
2
Z
{
1
:
2
2
N
+
exp(
b
k
k
z )
sin(+kx
)dk
N/U
s
;
j
j ;
;
U
9
Z
}
41
(7.164)
By
inspecting
equation
(7.164),
if
the
buoyancy
frequency
N
is
zero,
the
stream
function
1
has
a
simple
expression
given
by
the
equation
1 (x,
z )
,
(7.165)
1 +
[
x/(b
+... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
the
stream
with
horizontal
constant
speed
U
passing
over
a
localized
topography
z
( (x).
We
assume
the
buoyancy
frequency
N
constant
along
the
atmosphere.
For
a
reference
frame
fxed
on
the
ground,
we
have
the
governing
equation
(7.137)
for
the
fow
vertical
velocity
w .
For
this
governing
equation
we
h
a
ve
the
boundary... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
42
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
s
i
x
a
z
-10
0
10
20
x axis
Figure
8:
Stream
lines
for
the
case
of
zero
value
for
the
buoyancy
frequency
N
.
A
1
0
and
b
.
4
43
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
s
i
x
a
z
-10
0
10
20
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
Fourier
transform
pair
given
by
equations
(7.141)
and
(7.142).
For
w^ (k
, z , t
)
(Fourier
transform
of
w(x
, z , t
)),
we
consider
the
time
dependence
0
w^(k
, z , t
) w^
(k
, z
)
ex
p
(
i! t):
(7.167)
0
;
0
The
governing
equation
for
w^
(k
, z
)
is
given
by
equation
(7.146)
and
it
has
also
to
satisfy
the
boundary
co... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
)
,
(7.170)
(
!
;
;
or
we
can
obtain
a
dispersion
relation,
which
follows
kN
!
:
(7.171)
2
2
p
k
+
m
This
dispersion
relation
will
be
necessary
to
obtain
the
group
velocity
o
f
t
h
e
w
ave
distur-
bances
generated
by
the
topography,
which
will
b
e
necessary
to
discuss
how
to
deform
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
(U t
+
x
))dk
(7.173)
;
2
sinh(mh)
;1
Z
This
inverse
Fourier
has
closed
form
solution,
which
is
basically
the
sum
of
the
residue
of
part
of
the
poles
of
the
integrand
in
equation
(7.173).
To
obtain
these
p
o
l
e
s
,
w
e
use
1
2
z
sinh
z
z
1
+
(7.174)
2 2
l
l=1
Y
Therefore,
we
can
write
sinh(m(h
z ))
;
1
m
(h;z )
2
2
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
1,
there
is
no
real
N
l�
U
h
2
poles,
which
implies
no
waves
associated
with
the
localized
topography.
In
this
case
we
have
just
a
local
evanescent
wavefeld
close
to
the
localized
topography.
To
evaluate
the
integral
in
equation
(7.173),
we
consider
a
closed
contour,
which
is
the
original
integration
contour
along
the
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
les.
This
decision
is
associated
if
we
have
waves
downstream
or
upstream
of
the
localized
topography,
and
to
carry
it
out,
we
need
to
compute
the
group
velocity,
which
is
given
by
2
2
m
N
! m
C
g
,
(7.177)
2
2
3/2
(m
+
k
)
k
k
;
!
j
j
where
!
/k
is
the
phase
velocity
of
the
wave
following
the
topography,
which
has
valu... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
.
Therefore,
the
only
contribution
comes
from
the
poles
inside
the
closed
contour
illustrated
in
the
fgure
10.
Now,
the
expression
for
the
vertical
velocity
w(x
, z , t
)
can
be
written
as
follows:
0
Case
(U t
+
x
)
>
0.
We
assume
that
the
frst
L
poles
are real, and the
poles
for
0
•
l >
L
are
pure
imaginary.
We
have
t... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
U
(
(k)
e
;
2
2
;
2
sinh(m(k , N
/U
)h)
j
k= (N/U
)
;(j�
/h
)
p
j=1
X
L
�
�
i
sinh(m(k , N
/U
)(h
z ))
^
(ik(U t
+x
))
0
+
Res
ikU
(
(k)
;
e
2
2
;
2
sinh(m(k , N
/U
)h)
j
k=; (N/U
)
;(j�
/h
)
p
j=1
X
1
�
�
sinh(m(k , N
/U
)(h
z ))
(ik(U t
+x
))
0
^
+
iRes
ikU
(
(k)
e
;
2
2
;
j
sinh(m(k , N
/U
)h)
k=i
(j�
/h
)
;(N/U
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
of
l
is
one,
the
critical
speed
for
a
given
value
of
the
buoyancy
frequency
N
is
U
.
For
current
values
U >
,
there
is
no
wave
�
�
N
h
N h
disturbance
downstream
of
the
localized
topography.
To
illustrate
the
fow
for
this
problem,
we
change
from
the
moving
reference
frame
to
the
fxed
reference
frame
(x
+
U t
x),
and
we... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
Case
x >
0.
We
assume
that
the
frst
L
poles
are
real,
and
the
poles
for
l >
L
are
•
pure
imaginary.
We
have
that
49
L
i
sinh(m(k , N
/U
)(h
z ))
^
1 (x,
z )
Res
(
(k)
exp(ikx)
;
2
2
;
2
sinh(m(k , N
/U
)h)
j
k= (N/U )
;(j�
/h)
p
j=1
X
L
�
�
i
sinh(m(k , N
/U
)(h
z ))
^
+
Res
(
(k)
exp(ikx)
;
2
2
;
2
sinh(m(k , N
/U
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
184)
The
equations
(7.183)
and
(7.184)
can
b
e
written
in
a
simple
way
in
terms
of
the
quantities
e
, 1
and
a
defned
in
the
appendix
A.
j
j
j
Case
x >
0,
where
the
frst
L
poles
are
assumed
real
numbers,
and
the
other
poles
•
are
in
the
upper
part
of
the
complex
k
plane.
L
1
1 (x,
z )
/
e
sin(a
x)
/
1
exp(
a
x),
(7.185... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
1 (x,
z )
/
1
exp(a
x),
(7.187)
j
j
j
;
j=L+1
X
We
chose
for
( (x)
the
same
topography
we
considered
in
the
previous
section.
The
stream
lines
for
this
fow
are
illustrated
in
fgure
11,
12
and
13.
For
stream
speeds
that
approach
the
critical
values
from
below,
the
group
velocity
l�
N
h
C
for
the
l-th
lee
wave
approache... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
/U
)
(j /h
)
(A.188)
j
2
2
±
±
;
p
For
N/U
/h
we
have,
j
ia
i
(j /h
)
(N
/U
)
(A.189)
j
2
2
±
±
;
p
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
a
fnite
atmosphere
with
U
/N
2
and
A
1/2.
We
expect
to
see
a
superposition
of
four
lee
waves
for
0
this
value
of
N
/U
.
53
30
25
20
s
i
x
a
z
15
10
5
-10
-5
0
5
10
15
20
25
x axis
30
35
40
45
50
Figure
13:
Stream
function... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
Next,
we
give
the
expression
for
the
residues
in
equations
(7.183)
and
(7.184).
Case
x
0.
We
consider
in
this
case
the
real
p
o
l
e
s
for
which
j
1, : : :
, L
.
We
•
2
frst
give
t
h
e
residue
at
the
real
poles.
55
Res
(
(k)
exp(ikx)dk
(
(
a
)e
,
(A.190)
;
k=±a
j
j
j
^
^
sinh(m(k , N
/U
)(h
z ))
�
�
sinh(m(k , N
/U
)h)
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
cos(j )
(j /h
)
(N
/U
)
2
2
;
p
Case
x 0.
We
consider
for
this
case
the
poles
in
the
lower
part
of
the
complex
•
k
plane. The residue at these
poles
follows:
Res
(
(k)
exp(ikx)dk
( (
ia
)(
i1
)
;
k=;ia
j
j
j
^
^
sinh(m(k , N
/U
)(h
z ))
�
�
sinh(m(k , N
/U
)h)
j
;
;
(A.194)
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
r
1:
P
1
8.3
1
6
re
u
Lect
inciples
of
Applied
Mathematics
Rodolfo
Rosales
Spring
2014
acteristics
of
u +c *u
=
0
[linearized
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 |
ariabl
tic
v
e
characteris
liminate
th
E
•
e
the
solution
is
defined.
Show
wher
le:
re
a
ll
l
l
x
x
e
e
a
a
ro
p
P
V
<
∞
-‐
n
o
em
,
t
>
0.
∞
x
<
:
u +c *u
=
a*u.
I
1
mp
E
x
t
0
1)
,
(x
u
y,
+
y*
-‐∞
r
o
f
x)
g(
∞
x
<
... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/2885f5e781a8df81578b6b487caecf6f_MIT18_311S14_Lecture6.pdf |
y
>
0.
MIT OpenCourseWare
http://ocw.mit.edu
18.311 Principles of Applied Mathematics
Spring 2014
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/2885f5e781a8df81578b6b487caecf6f_MIT18_311S14_Lecture6.pdf |
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 |
125m, α =
k
ρcp
=
(cid:2)
35
(cid:3)
kg 0
m3
.8
(cid:4)
kJ
kg K
(cid:5)
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. prehe... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/28a25e06805d1037eecbb968e3e7f747_MIT3_044S13_Lec06.pdf |
when Θ = 0.1 @ x = 0 and y = 0
By Symmetry:
Θ(x, t) = f (F0,x)
Θ(y, t) = f (F0,y)
Θ(x, y, 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
MIT OpenCourseWare
http://ocw.mit.edu
3.044 Materials Processing
Spring 2013
For information about citing these materials or our Terms of Use, visit... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/28a25e06805d1037eecbb968e3e7f747_MIT3_044S13_Lec06.pdf |
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 |
Δu(cid:3))(x) = (Δu)(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 ... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
constant is u(x).
(cid:5)
− u is constant, and
Sr
This proof is somehow 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.
A... | 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, which is only found in the domain of integration, not the
integrand. One sees that
(cid:6)
1
∂iu(x) =
(cid:10)nor, ∂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 de... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
they are worth attention.
One might wonder whether the Maximum Principle, Mean Value Property, or Regularity results that
harmonic functions enjoy are shared by variable coefficient operators. We will be discussing in what ways
this is the case. A big first theorem for us to prove is Schauder’s.
Recall the idea of Holder-... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
related problem. Suppose u : B1 → R is C 2 with |Δu(x)| ≤ 10−6 and
|u(x)| ≤ 1 on B1. Then can we say |∇u(0)| ≤10 6?
Lecture 2: 6 February 2015
2.1 Almost Harmonic Functions
Proposition 2.1. 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 integrat... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
∂2u(cid:12)L2 ≤ 2
(cid:12)Lu(cid:12)L2 .
|∂2u|2
|Δu|2
|Lu + (Δ− L)u|2
(cid:6)
(cid:6)
|Lu|2 + 2
|Lu||(Δ − L)u| +
|(Δ − L)u|2
Now, the coefficients 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(c... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
| + |u|(cid:12)L (B1)
2
for some C(n) depending only on n.
4
Proof. The idea is to apply Proposition 2.2 to a localized version ofu. Namely, take η a smooth bump
function supported on B1 with η = 1 on B1. Then
(cid:12)∂2u(cid:12)2
L2(B1) =
=
≤
(cid:6)
B1/2
(cid:6)
B1/2
(cid:6)
2
|
∂ u
|
2
|∂2(ηu)|2
|∂2(ηu
)
|2
B
(cid:... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
does hold in the Lp-norm, and this is the Calderon-Zygmund inequality from the 1950s.
3. The inequality does not hold int the C 1-norm.
4. The inequality does hold in the C α-norm. This is Korn’s inequality from the early 1900s. This will
be proved in Lecture 5.
Our goal over the next week will be to work especially to... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
3R with smooth boundary, and z ∈ Ω, then
(cid:6)
∂Ω
Fz(x)
· nor = +4π.
(If z ∈/ Ω, the same proof will show that this integral is just 0.)
Proof. Set U = Ω \ Br(z) where r is small enough so that the entire ball lies inside Ω. Then the divergence
theorem shows that
(cid:6)
(cid:6)
(cid:6)
(cid:6)
0 =
divFz =
Fz
· nor =... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
, we have that for a domain Ω as in Newton’s Theorem,
(cid:6)
(cid:6)
Δu =
∇u · nor.
Ω
∂Ω
We can take this instead to be our definition of the Laplacian in the sense of distributions. Doing this, note
that from the Lemma, we had
(cid:6)
∇Γ · nor =
Ω
(cid:12)
4π,
0,
∈ Ω
0
0 ∈/ Ω
so that in the distributional sense,
ΔΓ = ... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/28b906907cba5b677e70eafe5e7da274_MIT18_156S16_Lec1-2.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
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 |
the mount on the servo.
4 Advanced Sensors
4.1 Reflectance 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... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/28e4da973faa6d894e41ccc9d60be617_5_sss.pdf |
IR LED/Phototransistor. Useful for both breakbeam and reflectance sensing—probably better 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
Ac... | https://ocw.mit.edu/courses/6-270-autonomous-robot-design-competition-january-iap-2005/28e4da973faa6d894e41ccc9d60be617_5_sss.pdf |
best readings. Does the angle at which
the LED hits the table surface affect the reading? How close to the table does your sensor need to 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 do... | 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 |
s
October 3, 2005
One Transistor Dynamic RAM
6.823 L7- 5
Joel Emer
1-T DRAM Cell
TiN top 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 ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
put (T) = Number in Flight (N) / Latency (L)
CPU
Misses in
flight table
Memory
Example:
--- Assume infinite bandwidth memory
--- 100 cycles / memory reference
--- 1 + 0.2 memory references / instruction
⇒ Table size = 1.2 * 100 = 120 entries
120 independent memory operations in flight!
October 3, 2005
DRAM A... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
charges bit lines to known value, required before next row access
Each step has a latency of around 20ns in modern DRAMs
Various DRAM standards (DDR, RDRAM) have different ways of encoding the
signals for transmission to the DRAM, but all share the same core
architecture
October 3, 2005
Multilevel Memory
Strateg... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
latency: Register << SRAM << DRAM why?
why?
• bandwidth:
on-chip >> off-chip
On a data access:
hit (data ∈ fast memory) ⇒ low latency access
miss (data ∉ fast memory) ⇒ long latency access (DRAM)
Fast mem. effective only if bandwidth requirement at B << A
October 3, 2005
Management of Memory Hierarchy
6.823 L... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
533MHz, 32-bit bus, 2.
• 1GB/s1GHz, 2x32-bit bus, 16GB/s
• Up to 8GB DRAM, 400MHz, 128-bit bus,
6.4GB/s
• North Bridge Chip
• PCI-X Expansion, 133MHz, 64-bit bus, 1
GB/s
October 3, 2005
18
Five-minute break to stretch your legs
Inside a Cache
Address
Address
Processor
CACHE
Data
Data
Main
Memory
copy o... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
7
Fully
(2-way) Set
Associative Associative
Direct
Mapped
block 12
can be placed
anywhere
anywhere in
set 0
(12 mod 4)
only into
block 4
(12 mod 8)
October 3, 2005
Direct-Mapped Cache
Tag
Index
Block
Offset
t
V
Tag
k
Data Block
b
t
=
HIT
October 3, 2005
2k
lines
Data Word or Byte
Direct ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
• FIFO with exception for most recently used block
This is a second-order effect. Why?
October 3, 2005
Block Size and Spatial Locality
6.823 L7- 27
Joel Emer
Block is unit of transfer between the cache and memory
Tag
Word0 Word1 Word2
Word3
Split CPU
address
block address
4 word block,
b=2
offset
b
32-b b... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
005
6.823 L7- 31
Joel Emer
Write Policy
• Cache hit:
– write through: write both cache & memory
• generally higher traffic but simplifies cache coherence
– write back: write cache only
(memory is written only when the entry is evicted)
• a dirty bit per block can further reduce the traffic
• Cache miss:
– n... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/29244173f6b6a78eed99d3b2b50196b6_l07caches.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
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 |
intervals ΔT , can not be unambiguously
reconstructed from its sample 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 cap... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
nΔT + φ
where m is an integer, giving the following important result:
Given a sampling interval of ΔT , sinusoidal components with an 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
... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
6)
(cid:3) (cid:3) (cid:4) (cid:6)
(cid:3) (cid:3) (cid:4) (cid:6)
(cid:3) (cid: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
ch... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
, there is nothing that can be done to eliminate the
effects of aliased frequency components. The only way to guarantee that the sample set
unambiguously represents the generating function is to ensure that the sampling theorem
criteria have been met, either by
10–3
(cid:6)
(cid:5)
(cid:6)
(cid:7)
(cid:7)
(cid:6)
(... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
(cid:10) (cid:6) (cid:8) (cid:8) (cid:15) (cid:25) (cid:12) (cid:11) (cid:16) (cid:17) (cid:20)
(cid:14) (cid:15) (cid:3) (cid:8) (cid:6)
(cid:5) (cid:9) (cid:11) (cid:9) (cid:12) (cid:13) (cid:9) (cid:9) (cid:14) (cid:2)
(cid:10) (cid:4) (cid:8) (cid:2)
(cid:29)
(cid:3) (cid:4) (cid:2) (cid:5)
(cid:15) (cid:15) ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
) (cid:6)
(cid:3) (cid:8)
(cid:4) (cid:6)
(cid:11) (cid:3) (cid:8)
(cid:4) (cid:6)
(cid:2)
If it is assumed that the sampling theorem was obeyed during sampling, the repetitions in
F (cid:2)(j Ω) will not overlap, and in fact f (t) will be entirely specified by a single period of F (cid:2)(j Ω).
Therefore to recon... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
(cid:17) (cid:6) (cid:11)
(cid:24) (cid:23)
(cid:17) (cid:19)
(cid:21) (cid:13) (cid:8)
(cid:30) (cid:19)
(cid:21) (cid:13)
(cid:10) (cid:4) (cid:8) (cid:2)
(cid:12)
(cid:7) (cid:3) (cid:8)
(cid:4) (cid:6)
(cid:3) (cid:8)
(cid:4) (cid:6)
(cid:2)
(cid:3) (cid:4) (cid:2) (cid:5)
(cid:15) (cid:7) (cid:4) (cid:8) (ci... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
3) " (cid:4) (cid:6)
(cid:23) (cid:4) (cid:6)
(cid:12)
(cid:4) (cid:6)
" (cid:4) (cid:6)
$ (cid:4) (cid:6)
# (cid:4) (cid:6)
(cid:2)
and note that the impulse response h(t) = 0 at times t = ±nΔT for n = 1, 2, 3, . . . (the
sampling times). The output of the reconstruction filter is the convolution of the input
... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
(cid:16)
(cid:17)
(cid:20)
(cid:3)
(cid:9)
(cid:9)
(cid:17)
(cid:9)
(cid:17)
(cid:15)
(cid:15)
or in the case of a finite data record of length N
f (t) =
N −1 �
n=0
fn
sin (π(t − nΔT )/ΔT )
π(t − nΔT )/ΔT
.
This is known as the cardinal (or Whittaker) reconstruction function. It is a superposition
of shifted sin... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
now look at a different formulation of F ∗(j Ω). The Fourier transform of the sampled
function f (cid:2)(t)
� ∞
F (cid:2)(j Ω) =
f (cid:2)(t) e−j Ωtdt =
� ∞ ∞ �
f (t)δ(t − nΔT ) e−j Ωt dt
−∞
�
∞
=
−∞ n=−∞
f (nΔT ) e−j ΩnΔT
n=−∞
10–6
by reversing the order of integration and summation, and using the siftin... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
(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) (cid:25)
(cid:15) (cid:7) (cid:15)
(cid:15) (cid:4) (cid:8) (cid:2)
(cid:5)
(cid:7) (cid:3) (cid:8) (cid:4) (cid:6)
(cid:3) (cid:8)
(cid:6)
(cid:3) (cid:8)
(cid:4) (cid:6) ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
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)
(cid:12)
(cid:2)
(cid:4)
(cid:10)
•
The DFT operations are a transform pair between two sequenc... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/2934569026a399e9062597924c730a78_lecture_10.pdf |
m}
{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
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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 |
of integers.
But who cares? That’s why Turing brought up the halting problem; we actually 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
comp... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
hard relative to which other problems. It can tell us things like
“this problem can’t be solvable, because if it were, it would allow us to solve this other unsolvable
problem”.
Given:
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 wr... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
one by one in order, and then pause when we find the solution. However, if
there is no solution, it would just go on forever. So our question to the oracle would be, “Does this
program halt or not?”, and if so, the Diophantine equation would be solvable.
If we solve the Diophantine problem, can we also then solve the... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
, if you could solve the halting
problem, then can you decide whether a set of tiles will tile the plane or not? Can you tile a 100x100
grid? Can you tile a 1000x1000 grid? These questions can be answered by a Turing machine. But
suppose every finite region can be filled, why does it follow that the whole infinite plan... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
subtrees must be infinite, and we can keep
going on forever. The end result is an infinite path.
So how does K¨onig’s Lemma apply to the tiling problem? Imagine that the tree is the tree of
possible choices to make in the process of tiling the plane. Assuming you’re only ever placing a tile
adjacent to existing tiles... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
same as the original halting problem. We can
follow each machine step by step and ask the oracle, but the oracle won’t have the answer for the
Super Halting Problem.
If there were the Super Turing Machine that could solve the Super Halting Problem, then you
could feed that Super Turing Machine itself as input and c... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
1930, five years before Turing invented Turing machines, G¨odel showed that this was impossible.
G¨odel’s theorems were a direct inspiration to Turing.
G¨odel’s Incompleteness Theorem says two things about the limits of any system of logic.
First Incompleteness Theorem: Given any system of logic that is consistent (c... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
He started out with the paradox of the liar: “This
sentence is not true.” It can’t be either true or false! So if we’re trying to find an unprovable
statement, this seems like a promising place to start. The trouble is that, if we try to express this
sentence in purely mathematical language, we run into severe proble... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
we
can define sentences that talk about their own provability. The end result is that “This sentence is
not provable” gets “compiled” into a sentence purely about integers.
What about the Second Incompleteness Theorem? Given any reasonable logical system S, let
G(S) = “This sentence is not provable in S”
Con(S) = “... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/2941b1c3c81ee47665e6819c772f82a9_lec5.pdf |
provable in
S. The system will never be able to prove its own consistency.
5-6 | 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 |
Propositional Logic Syntax
Components
Formation Rules
The PL language consists of
� an infinite set of variables V = {a, b, . . .}, and
� a set of symbols S = {∼, ∨, (, )}.
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 exp... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
6
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Propositional Logic
Semantics
Propositional Logic
Semantics
Propositional Logic Semantics
Setting: variable value assigments
Propositional Logic Semantics
Interpretation: Truth Value of Expressions
Definition
We define a setting s as a ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
∼ {0, 1} → {0, 1}
∨ : {0, 1} × {0, 1} → {0, 1}
∼ (0) = 1
∼ (1) = 0
∨
0 1
0 0 1
1 1 1
Table: Truth table for disjunction ∨
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Propositional Logic
Semantics
Propositio... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
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Propositional Logic
Semantics
Propositional Logic
Semantics
Propositional Logic Semantics
Example: Computing the Interpretation I
Example
I(∼ (∼ a∨ ∼ b)) = ∼ I(∼ a∨ ∼ b)
= ∼ (∼ I(a)∨ ∼ I(b))
= ∼ (∼ s(a)∨ ∼ s(b))
If we let s(a) = 1 and s(b) = 0, then
Propositional Logic
Syntactic “Sugar... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
only if Is (α) = 1 for every setting s. A wff α is
satisifiable if there exists a setting s such that Is (α) = 1, and
unsatisfiable if no such setting s exists.
Example
The wff (α∨ ∼ α) is valid, while (α∧ ∼ α) is unsatisifiable.
Propositional Logic
Testing for validity: Truth Table Method
The truth table for (a
→
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
( q
∧
r ))) → q
)
1
6
1
5
1
8
1
7
1
9
0
1
0
3
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Fuzzy Stuff
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Propositional Logic
Validity and Satisfiability
Propositional Logic
The PL Logic System
Propositional Logic
Examp... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
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Propositional Logic
The PL Logic System
Propositional Logic
The PL Logic System
Propositional Logic
The PL Logic System: Uniform Substitution
Propositional Logic
The PL Logic System: Modus Ponens
� The result of uniformly replacin... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
’ ⇒’. As in:
US:
MP:
� α ⇒� α[β1/a1, β2/a2, . . . , βn/an]
� α, (α → β) ⇒� β
Clear:
we can manipulate wff by using the rules defining operators and
semantics.
Definition
The wff β is a propositional consequence of wff α if and only if
α ↔ β ∧ γ for some wff γ.
We formulate this as a derived transformation rule:... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
not study, Alf
has a good time. If Alf does not get good grades, Alf does not
have a good time.
What can we say about Alf?
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Propositional Logic
Propositional Consequenc... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
.873
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Propositions over Sets
Propositions over Sets
Propositions over Sets
Propositions: Defined
Propositions over Sets
Syntax
Now, a proposition over a set is a proposition that describes a property
of the elements of that set. Such propositions are modeled by
characteristic functions.
Example
Let N ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
� T (β).
Analogous to the PL case:
� T (α ∧ β) = T (α) ∩ T (β),
� T (α → β) = (U − T (α)) ∪ T (β), and
� T (α ↔ β) = ((U − T (α)) ∪ T (β)) ∩ ((U − T (β)) ∪ T (α)).
Example
For the natural numbers and the proposition even the truth set is
T (even) = {2, 4, 6, . . .}.
Staal A. Vinterbo (HST/DSG/HMS)
Fuzzy Stuff
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
MS)
Fuzzy Stuff
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Propositions over Sets
Propositional Rules
Propositions over Sets
PL(U) ⊇ PL
Propositional Rules
The implication view
We state that PL is “contained in” PL(U). Indeed, PL is contained in
PL({0... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
Tall ∧ HairDark)
(HeightTall ∧ HairDark) → LookHandsome
(LookHandsome)
In other words we set I(β, x ) =
Effect:
We infer the unknown proposition LookHandsome.
Definition
Given a rule (α
the computation of I(β, x ) as I(α, x ).
→
β). The application of this rule to a data point x is
�
I(α, x ) = 1,
1
0 otherwi... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
51/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∩B (x ) = min(χA(x... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
(χR (x , y ), χR� (y , z))}
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Fuzzy Stuff
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Fuzzy Stuff
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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 relati... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
7, .8)}
1, .5, .7} = .
= max{.
7
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Fuzzy Stuff
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4
A Restricted Fuzzy Logic
A Restricted Fuzzy Logic
Fuzzy Logic
Defining the Fuzzy Logic Language
Fuzzy Logic
Semantics
Recall:
Fo... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
actic Sugar”
Fuzzy Logic
Examples
Example
Analogous to the PL(U) case we can show that
� µT (α∧β)(x ) = min(µT (α)(x ), µT (β)(x )),
� µT (α→β) = max(1 − µT (α)(x ), µT (β)(x )), and
� µT (α β)(x ) =
↔
min(max(1 − µT (α)(x ), µT (β)(x )), max(1 − µT (β)(x ), µT (α)(x ))).
Definition (Fuzzy Interpretation)
If w... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-fall-2005/29aab5bf96b7690620369891254d3f44_proplogic_sets.pdf |
a setting such that its interpretation
is 1.
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Fuzzy Stuff
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Summary
Summary
Summary
Propositions over sets
We have learned
� about the propositional language PL(U), over propositio... | 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 |
One can show that this is true as
long as
or
Massachusetts Institute of Technology
6.763 2005 Lecture 10
3
Flux Quantization
3. Take the line integral of the supercurrent equation around a closed contour
within a superconductor:
The line integral of each of the parts:
Therefore,
flux
integer
n = - n’
Mass... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/29bb21252c2fd3293111508de473d2ac_lecture10.pdf |
is quantized in the bulk limit.
Massachusetts Institute of Technology
6.763 2005 Lecture 10
5
Flux Quantization Experiments
Flux trapped in
hollow cylinder
Deaver and
Fairbank, 1961,
measure
and show that q*=2e;
Cooper Pairs.
Image removed for copyright reasons.
Please see: Figure 5.3, page 249, from Orland... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/29bb21252c2fd3293111508de473d2ac_lecture10.pdf |
.
The MQM wavefunction for the superconductor is the spatial average of this phase
coherent wavefunction and is preserved with an applied field. The coherence persists
over the macroscopic scale of the superconductor.
In the normal state, the applied field causes dissipation; this energy loss causes the
phase of t... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/29bb21252c2fd3293111508de473d2ac_lecture10.pdf |
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