text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
2(x − δx, y − δy) − 2
Z Z
=⇒ arg min J(δx, δy) = arg max
δx,δy
δx,δy
D
E1(x − δx, y − δy)E2(x, y)dxdy +
D
Z Z
E1(x − δx, y − δy )E2(x, y)dxdy
Z Z
D
2(x, y)dxdy
E2
Since the first and third terms are constant, and since we are minimizing the negative of a scaled correlation objective, this is
equivalent to m... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
above looks similar to our BCCE constraint from optical flow!
• Gradient-based methods are cheaper to compute but only function well for small deviations δx, δy .
• Correlation methods are advantageous over least-squares methods when we have scaling between the images (e.g. due to
optical setting differences): E1(x, y... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
match.
Are there any issues with this approach? If parts/whole images of objects are obscured, this will greatly affect correlation
computations at these points, even with proper normalization and offsetting.
With these preliminaries set up, we are now ready to move into a case study: a patent for object detection and... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
training/template image to obtain boundary points.
3. Connect neighboring boundary points that have consistent directions.
4. Organize connected boundary points into chains.
5. Remove short or weak chains.
6. Divide chains into segments of low curvature separated by conrner of high curvature.
7. Create evenly-spac... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
and construct
features for feature matching using a Histogram of Oriented Gradients (HoG) [1].
4
• For running this framework at multiple scales/resolutions, we want to use different probes at different scales.
• For multiscale, there is a need for running fast low-pass filtering. Can do so with rapid convolutions, w... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
ly
• At different levels of resolution
• Hexagonally, rather than on a square grid - there is a
4
π
detection is used, and to break ties, we arbitrarily set 3 of the 6 inequalities as ≥, and the other 3 as >.
advantage of work done vs. resolution. Here, hexagonal peak
What is pose?
Pose is short for position and ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
_______________________
6
MIT OpenCourseWare
https://ocw.mit.edu
6.801 / 6.866 Machine Vision
Fall 2020
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
Summary from last week
• Linear systems
f1(t)
x1(t)
f2(t)
x2(t)
a1f1(t)+
a2f2(t)
a1x1(t)+
a2x2(t)
• Translational & rotational mechanical elements & systems
x(t)
M
f(t)
K
fv
M x¨ + fvx˙ + Kx = f
T (t)
T(t)
θ(t)
q(t)
J
D
K
Figures by MIT OpenCourseWare.
• Solving 1st order linear ODEs with constant coefficients
Jω˙ + bω... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/63c439dbeddf8111d5e6c7b95440b5a3_lecture03.pdf |
s)
input, output expressed as functions of new variable s
Benefits:
• Simplifies solution
•
• particularly useful in control
s-domain offers additional insights
2.004 Fall ’07
Lecture 03 – Monday, Sept. 10
Laplace transform: definition
Given a function f (t) in the time domain we define its
Laplace transform F (s) as
... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/63c439dbeddf8111d5e6c7b95440b5a3_lecture03.pdf |
ω)
=
σ − jω
σ2 + ω2 .
Alternatively, we can represent
the complex number s in polar form s = |s| ejφ,
jω
σ2 + ω2
where |s| =
φ ≡ 6 s = atan (ω/σ) the phase of s.
is the magnitude and
1/2
¡
¢
It is straightforward to derive
1
s
=
1
|s|
e−
jφ ⇒
2.004 Fall ’07
=
1
|s|
1
s
¯
¯
¯
¯
¯
¯
¯
¯
and
1
s
= −6 s.
Lecture 03 – Mond... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/63c439dbeddf8111d5e6c7b95440b5a3_lecture03.pdf |
a = 1 sec−
1
Nise Table 2.1
2.004 Fall ’07
Lecture 03 – Monday, Sept. 10
Laplace transforms of commonly used functions
f(t)
d(t)
u(t)
tu(t)
tnu(t)
e-atu(t)
sin wtu(t)
cos wtu(t)
F(s)
1
1
s
1
s2
n!
sn + 1
1
s + a
w
s2 + w2
s
s2 + w2
Figure by MIT OpenCourseWare.
Nise Table 2.1
2.004 Fall ’07
Lecture 03 – Monday, Sept... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/63c439dbeddf8111d5e6c7b95440b5a3_lecture03.pdf |
�
δ(t) = 1;
(unit energy) and
+
∞
Z
−∞
δ(t)f (t) = f (0)
(sifting.)
Properties of the Laplace transform
Let F (s), F1(s), F2(s) denote the Laplace transforms of f (t), f1(t), f2(t),
respectively. We denote L [f (t)] = F (s), etc.
• Linearity
L [K1f1(t) + K2f2(t)] = K1F1(s) + K2F2(s),
where K1, K2 are complex constan... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/63c439dbeddf8111d5e6c7b95440b5a3_lecture03.pdf |
)
=
K1
s + 3
+
K2
s + 5
.
(2)
That would be convenient because we know the inverse Laplace transform of
the 1/(s + a) function (it’s a decaying exponential) and we can also use the
linearity theorem to finally find f (t). All that’d be left to do would be to find
the coefficients K1, K2.
This is done as follows: first multip... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/63c439dbeddf8111d5e6c7b95440b5a3_lecture03.pdf |
ω(t) + bω(t) = Ts(t),
where Ts(t) = T0u(t)
(step function)
and ω(t = 0) = 0
(no spin—down).
Ts(t) ω(t)
Taking the Laplace transform of both sides,
J sΩ(s) + bΩ(s) =
T0
s
⇒ Ω(s) =
T0
b
1
s
(J/b)s + 1
=
T0
b
1
s(τ s + 1)
,
´
where τ ≡ J/b is the time constant (see also Lecture 2).
³
We can now apply the partial fraction ... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/63c439dbeddf8111d5e6c7b95440b5a3_lecture03.pdf |
(t),
where now Ts(t) is an arbitrary function,
but still ω(t = 0) = 0
(no spin—down).
Ts(t) ω(t)
Proceeding as before, we can write
Ω(s) =
Ts(s)
Js + b
⇔
Ω(s)
Ts(s)
=
1
Js + b
.
Generally, we define the ratio
L
output
h
L
input
i
= Transfer Function; in this case, TF(s) =
1
Js + b
.
h
We refer to the (TF)−
i
1 of a sing... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/63c439dbeddf8111d5e6c7b95440b5a3_lecture03.pdf |
= f (t).
x(t)
M
f(t)
K
fv
Figures by MIT OpenCourseWare.
F(s)
1
Ms2 + fvs + K
X(s)
Impedances X(s) =
Forces .
£ P
x(t)
¤
£ P
x(t)
¤
Kx(t)
fv
dx
dt
d2x
dt2M
M
f(t)
KX(s)
fvsX(s)
Ms2X(s)
Figures by MIT OpenCourseWare.
M
f(t)
2.004 Fall ’07
Lecture 03 – Monday, Sept. 10
Summary
• Laplace transform
L [f (t)] ≡ F (s) =
+
... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/63c439dbeddf8111d5e6c7b95440b5a3_lecture03.pdf |
Computational Ocean
Acoustics
• Ray Tracing
• Wavenumber Integration
• Normal Modes
• Parabolic Equation
13.853
COMPUTATIONAL OCEAN ACOUSTICS
1
Lecture 7
Wavenumber Integration
• Range-independent – Integral Transform solution
• Exact depth-dependent solution
– Global Matrix Approach
– Propagator Matrix Approach
– In... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/63eadcae081807f761362a412d313b77_lect_72.pdf |
USTICS
6
Lecture 7
Blows up – k ,z large
r
< 1 inside layer m
13.853
COMPUTATIONAL OCEAN ACOUSTICS
7
Lecture 7
h
zm-1
zm
z
e-g (Z-Z
)
m-1
e-g (Zm-Z)
e-g (Z-Z
)
m-1
e-g (Zm-Z)
gh ~ 1
gh à1
Layer m-1
m
m+1
Layer m-1
m
m+1
Interface
m-1
m
Interface
m-1
m
–
Φm
+Φm
Diagonal
Unstable
13.853
COMPUTATIONAL OCEAN ACOUSTICS
–
... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/63eadcae081807f761362a412d313b77_lect_72.pdf |
Layer 3 Lower halfsp.
2 ‘separate’ systems: No
coupling between upper
and lower matrices =>
No error propagation
from bottom to top =>
Numerically stable
Intfc 1
Intfc 2
Intfc 3
layer 2
x x
x
x
x
0
x
0
00
0 0
0 0
x x
xx
x x
xx
0 0
00
0 0
x x
xx
x x
x x
A
A
+
2
-
2
+
3
-
3
+
4
+
4
A
A
A
B
=
x
x
x
0
0
0
13.853
Bloc... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/63eadcae081807f761362a412d313b77_lect_72.pdf |
7
Evanescent Wave Tunneling
See Fig. 4.14 and 4.15 in Jensen, Kuperman, Porter and Schmidt.
Computational Ocean Acoustics. New York: Springer-Verlag, 2000.
13.853
COMPUTATIONAL OCEAN ACOUSTICS
13
Lecture 7
Evanescent Wave Tunneling
See Fig. 4.14 in Jensen, Kuperman, Porter and
Schmidt. Computational Ocean Acoustics... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/63eadcae081807f761362a412d313b77_lect_72.pdf |
8.022 (E&M) - Lecture 2
Topics:
Energy stored in a system of charges
(cid:132)
(cid:132) Electric field: concept and problems
Gauss’s law and its applications
(cid:132)
Feedback:
Thanks for the feedback!
(cid:132)
Scared by Pset 0? Almost all of the math used in the course is in it…
(cid:132)
(cid:132) Math... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
Qqr
2
r
= −
Coulomb
.
F
= −
1 to r2?
q
(cid:71)
d s
(
W r
1
)
r
→
2
=
(cid:71)
∫
F
I
(cid:71)
•
∫
ds = −
r
1
r
2
ˆ
Qqr
ˆ
i
dr
2
r
=
Qq Qq
−
r
r
1
2
(cid:132)
Does this result depend on the path chosen?
(cid:132)
No! You can decompose any path in segments // to the radial direct... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
Q
( )
0 no other charges: F=0
P
1
=
(cid:71)
(cid:71)
∫
F ds
•
=
∞
(cid:71)
∫
F
I
W
=
(cid:71)
ds
W
+
•
=
1 2
+
W
+ =
1 2
W
1 2 3
+ +
q q
1 2
r
12
+
W
1 3
+
2 3
+
q q
1 2
+
r
12
q q
1 3
+
r
13
q q
2 3
r
23
U
i N
=
j N
=
= ∑ ∑
1
2
i
1
=
j
q q
i
r
ij
j
j
1
=
i
... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
: the denser the lines, the stronger the field.
Faucet
Sink
Demo
+
-
Properties:
(cid:132)
(cid:132)
eld lines never cross (if so, that’s where E=0)
Fi
They are orthogona to equipotential surfaces (wil see this ater).
l
l
l
September 8, 2004
8.022 – Lecture 1
8
4
Electric field of a ring of charge
Pro... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
)
2
2
(cid:198) Integrating on r: 0(cid:198)R:
(cid:71)
r R
=
E
2
z
σ π
(cid:71)
dE
r R
=
=
=
ˆ
z
=
∫
r
=
0
2
(
r
+
z
dr
3
)
2
2
∫
r
=
0
ˆ
z
2
πσ
ˆ
zz
⎛
⎜
|
⎝
1
z
|
−
1
2
+
R
2
z
⎞
⎟
⎠
P
z
R
r
September 8, 2004
8.022 – Lecture 1
11
Special case 1: R(cid:198)infinit... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
⎟
⎠
What happens when h>>R?
(cid:132)
Physicist’s approach:
P
z
R
(cid:132)
The disk will look like a point charge with Q=
2
σπ r2
(cid:198)
E=Q/z
(cid:132)
Mathematician's approach:
(cid:132)
Calculate from the previous result for z>>R (Taylor expansion):
(cid:71)
E
=
2
πσ
ˆ
zz
⎛
⎜
|
⎝
1
z
... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
a river
The water velocity is described by
(cid:71)
v x y z v x v y v z v v v
( ,
)
+
y
z
(cid:71)
ˆ
A An≡
Immerse a squared wire loop of area A in the water (surface S)
,
y
+
≡
ˆ
ˆ
)
(
≡
ˆ
,
,
x
x
z
(cid:132) Define the loop area vector as
Q: how much water will flow through the loop? E.g.:
What ... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
(cid:132)
At any point in space, dA is perpendicular to the surface
It points towards the “outside” of the surface
(cid:132)
(cid:132)
Examples:
.
.
.
ˆn
Intuitively:
(cid:132) “dA i
s oriented in such a way that if we have a hose inside the surface
the flux through the surface will be positive”
September 8, ... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
θ
(cid:71)
i
E d
S
(cid:198)
The total flux through the cylinder is zero!
September 8, 2004
8.022 – Lecture 1
18
9
ΦE through closed empty surface
Q1:
Is this a coincidence due to shape/orientation of the cylinder?
(cid:132) Clue:
(cid:132)
Think about interpretat on of
through the surface…
i
(cid:132) ... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
i
d A
E
=
∫
S
Q
2
R
d A
=
Q
2
R
∫
S
d A
=
Q
2
R
4
R
π
2
=
4
Q
π
September 8, 2004
8.022 – Lecture 1
21
ΦE through a generic surface
What if the surface is not spherical S?
Impossible integral?
Use intuition and interpretation of flux!
(cid:132) Version 1:
(cid:132)
(cid:132)
Consider ... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
(cid:132)
No, it’s useful only when the problem has symmetries
September 8, 2004
8.022 – Lecture 1
23
Applications of Gauss’s law:
Electric field of spherical distribution of charges
Problem: Calculate the electric field (everywhere in space) due to a
spherical distribution of positive charges or radius R.
(NB... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
(cid:71)
S ym m etry: E is constant on S and (cid:38) to dA .
(cid:118)∫ S1
(cid:71)
E dA E 4πr = 4πQ
enclosed
=
i
(cid:71)
1
2
→ E =
Q
r 2
For r>R, sphere looks
like a point charge!
2) Inside the sphere (r<R)
S2
r
+
R
S1
r
Apply Gauss on a sphere S2 of radius r:
(cid:71)
A g ain : Φ = (cid:118)∫ S ... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
8.022 – Lecture 1
r
+
r
R
S1
R
r
26
13
Another application of Gauss’s law:
Electric field of spherical shell
Problem: Calculate the electric field (everywhere in space) due to a
positively charged spherical shell or radius R (surface charge density
σ)
Physicist’s solution:apply Gauss
1) Outside the sp... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
(cid:132)
(cid:132)
Trick #2: apply Gauss’s theorem
(cid:132) Φtot = Φ + Φ + Φbottom
Symmetry: E // z axis
(cid:132)
side
top
(cid:198) Φ =0 and Φtop = Φbottom
side
(cid:118)
∫
(cid:71)
(cid:71)
i
E dA
cylinder
(cid:71)
(cid:71)
i
E dA
=
2
Φ =
(cid:118)
∫
cylinder
∫
top
=
4
Q
π
enclosed
E... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
expansions are more than limits…
) and “massage” the result
, or ln(1+x) or e
x<<1
you
until
x
September 8, 2004
8.022 – Lecture 1
29
Summary and outlook
(cid:132)
What have we learned so far:
(cid:132) Energy of a system of charges
Concept of electric field E
(cid:132)
(cid:132)
To describe the effect ... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/63f25e1a6788d1f29da7870ec50d7b91_lecture2.pdf |
34
1.13. Exactness of the tensor product.
Proposition 1.13.1. (see [BaKi, 2.1.8]) Let C be a multitensor cate
gory. Then the bifunctor ⊗ : C × C → C is exact in both factors (i.e.,
biexact).
Proof. The proposition follows from the fact that by Proposition 1.10.9,
the functors V ⊗ and ⊗V have left and right adjoin... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
the tensor product of bimodules. But the tensor product functor is not
C-bilinear on morphisms (it is only R-bilinear).
Definition 1.13.3. A multiring category over k is a locally finite k-
linear abelian monoidal category C with biexact tensor product. If in
addition End(1) = k, we will call C a ring category.
Thus,... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
2 → X1 ⊗ I2 → 0.
X1 ⊗ X2 → I1 ⊗ I2 → 0.
Arguing similarly, we show that the sequence
0
→
I1 ⊗ I2 → Y1 ⊗ Y2
is exact. This implies the statement.
�
Proposition 1.13.5. If C is a multiring category with right duals, then
the right dualization functor is exact. The same applies to left duals.
Proof. Let 0 X
show ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
the sequence
0 X ⊗ T
→
→
Y ⊗ T
→
Z ⊗ T
is exact, by the exactness of the functor ⊗T . This implies that the
�
sequence Z ∗
0 is exact.
Y ∗ X ∗
→ → →
Proposition 1.13.6. Let P be a projective object in a multiring cate
gory C. If X ∈ C has a right dual, then the object P ⊗ X is projective.
Similarly, if X ∈ C h... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
⊗ F ( )
• →
•
F (• ⊗ •), and F (1) = 1.
(ii) A quasi-tensor functor (F, J) is said to be a tensor functor if J
is a monoidal structure (i.e., satisfies the monoidal structure axiom).
Example 1.14.2. The functors of Examples 1.6.1,1.6.2 and Subsection
1.7 (for the categories Vecω
G) are tensor functors. The identity... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
, which is zero. So K ⊗ J = 0.
Now tensoring the exact sequence 0 K
→ → → →
1
J
and applying Proposition 1.13.1, we get that J = 0, so a = 0.
0 with J,
�
Let {pi}i∈I be the primitive idempotents of the algebra End(1). Let
1i be the image of pi. Then we have 1 = ⊕i∈I 1i.
Corollary 1.15.2. In any multiring catego... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
ned, map Cij
to Cji.
Exercise 1.15.6. Prove Proposition 1.15.5.
Proposition 1.15.5 motivates the terms “multiring category” and
“multitensor category”, as such a category gives us multiple ring cate
gories, respectively tensor categories Cii.
Remark 1.15.7. Thus, a multiring category may be considered as a
2-cat... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
X is simple and X ⊗X ∗ = 0 (because the coevaluation morphism
is nonzero) we obtain that X ⊗ X ∗ = X. So we have a surjective
composition morphism 1 → X ⊗ X ∗ → X. From this and (1.15.1) we
have a nonzero composition morphism 1 � X � 1. Since End(1) = k,
→
�
this morphism is a nonzero scalar, whence X =
1.
∼
Cor... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
A) ⊗ (V �, W �, A�) = (V ⊗ V �, W ⊗ W �, A ⊗ A�),
with obvious associativity isomorphisms, and the unit object (k, k, Id).
Of course, this category has neither right nor left duals.
1.16. Grothendieck rings. Let C be a locally finite abelian category
over k. If X and Y are objects in C such that Y is simple then we ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
][Xk ⊗ Xp : Xl].
k
�
[Xj ⊗ Xp : Xk][Xi ⊗ Xk : Xl].
k
Thus the associativity of the multiplication follows from the isomor
�
phism (Xi ⊗ Xj ) ⊗ Xp
∼= Xi ⊗ (Xj ⊗ Xp).
Thus Gr(C) is an associative ring with the unit 1. It is called the
Grothendieck ring of C.
The following proposition is obvious.
Proposition 1... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
G
more details.
→
→
→
Here are some examples of groupoids.
(1) Any group G is a groupoid G with a single object whose set of
morphisms to itself is G.
40
(2) Let X be a set and let G = X × X. Then the product groupoid
G(X) := (X, G) is a groupoid in which s is the first projection,
t is the second projection, u... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
The unit object is 1 = ⊕x∈X 1x,
where 1x is a 1-dimensional vector space which sits in degree idx in G.
The left and right duals are defined by (V ∗)g = (∗V )g = Vg−1 .
We invite the reader to check that the component subcategories
C(G)xy are the categories of vector spaces graded by Mor(y, x).
We see that C(G) is ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
��nitely many non-zero diagonals, and morphisms are matrices of
linear maps. The tensor product in this category is defined by the
formula
(1.17.2)
(V ⊗ W )il =
�
Vij ⊗ Wjl,
j
and the unit object 1 is defined by the condition 1ij = kδij . The left
and right duality functors coincide and are given by the formula... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
elian category C is said to be finite if
it is equivalent to the category A − mod of finite dimensional modules
over a finite dimensional k-algebra A.
Of course, the algebra A is not canonically attached to the category
C; rather, C determines the Morita equivalence class of A. For this
reason, it is often better to ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
the exactness property of F , and the condition that
P is a generator (i.e., covers any simple object) translates into the
property that F is faithful (does not kill nonzero objects or morphisms).
Moreover, the algebra A = End(P )op can be alternatively defined as
End(F ), the algebra of functorial endomorphisms of ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
-vector spaces, such that F (1) = k, equipped with an isomorphism
J : F ( ) ⊗ F ( )
F (• ⊗ •). If in addition J is a monoidal structure
(i.e. satisfies the monoidal structure axiom), one says that F is a fiber
functor.
• →
C
•
→
G →
Example 1.19.2. The forgetful functors VecG → Vec, Rep(G) Vec
are naturally fiber f... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
it ε : C
k such that
→
(i) Δ is coassociative, i.e.,
(Δ ⊗ Id) Δ = (Id ⊗ Δ) Δ
◦
◦
as maps C
C ⊗3;→
(ii) one has
(ε ⊗ Id) Δ = (Id ⊗ ε) Δ = Id
◦
◦
as maps C
→
C (the “counit axiom”).
Definition 1.20.2. A left comodule over a coalgebra C is a vector
→
C ⊗ M (called the
space M together with a linear map π : M
coacti... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
Exercise 1.20.4. (i) Show that any coalgebra C is a sum of finite
dimensional subcoalgebras.
Hint. Let c ∈ C, and let
(Δ ⊗ Id) ◦ Δ(c) = (Id ⊗ Δ) ◦ Δ(c) =
�
i ⊗ c 2
c 1
i ⊗ c 3
i .
Show that span(c2
i ) is a subcoalgebra of C containing c.
(ii) Show that any C-comodule is a sum of finite dimensional subco
modu... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/63f8d9e6d926a1d4bbb98f9b7790c6f6_MIT18_769S09_lec04.pdf |
1 2 3 4 5 6 7 8 MIT OpenCourseWare
http://ocw.mit.edu
6.851 Advanced Data Structures
Spring 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-851-advanced-data-structures-spring-2012/63fab92cb135438e17b3a11f49e453c8_MIT6_851S12_Lec1.pdf |
Statistical Models
Parameter Estimation
Fitting Probability Distributions
MIT 18.443
Dr. Kempthorne
Spring 2015
MIT 18.443
Parameter EstimationFitting Probability Distributions
1 Statistical Models
Definitions
General Examples
Classic One-Sample Distribution Models
Outline
1 Statistical Models
Definitions... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/6408c60bd43d4da05fc6837a12341c29_MIT18_443S15_LEC2.pdf |
5 Statistical Models
Definitions
General Examples
Classic One-Sample Distribution Models
Statistical Models: General Examples
Example 1. One-Sample Model
X1, X2, . . . , Xn i.i.d. with distribution function F (·).
E.g., Sample n members of a large population at random and
measure attribute X
E.g., n independent... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/6408c60bd43d4da05fc6837a12341c29_MIT18_443S15_LEC2.pdf |
measurement errors
{Ej } are i.i.d. N(0, σ2), with σ2 > 0, unknown.
Semi-Parametric Model: Symmetric measurement-error
distributions with mean µ
{Ej } are i.i.d. with distribution function G (·), where G ∈ G,
the class of symmetric distributions with mean 0.
Non-Parametric Model: X1, . . . , Xn are i.i.d. w... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/6408c60bd43d4da05fc6837a12341c29_MIT18_443S15_LEC2.pdf |
n cases i = 1, 2, . . . , n
1 Response (dependent) variable
yi , i = 1, 2, . . . , n
p Explanatory (independent) variables
xi = (xi,1, xi,2, . . . , xi,p)T , i = 1, 2, . . . , n
Goal of Regression Analysis:
Extract/exploit relationship between yi and xi .
Examples
Prediction
Causal Inference
Approximation
... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/6408c60bd43d4da05fc6837a12341c29_MIT18_443S15_LEC2.pdf |
Fitting Probability Distributions
11 Statistical Models
Definitions
General Examples
Classic One-Sample Distribution Models
Outline
1 Statistical Models
Definitions
General Examples
Classic One-Sample Distribution Models
MIT 18.443
Parameter EstimationFitting Probability Distributions
12 Statistical Models
D... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/6408c60bd43d4da05fc6837a12341c29_MIT18_443S15_LEC2.pdf |
t
e = e
E [X k ] =
−λ λe
x!
|t=0 , k = 0, 1, 2, . . .
tx
e
MIT 18.443
Parameter EstimationFitting Probability Distributions
14 Statistical Models
Definitions
General Examples
Classic One-Sample Distribution Models
Berkson (1966) Data: National Bureau of Standards experiment
measruing 10, 220 times between succ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/6408c60bd43d4da05fc6837a12341c29_MIT18_443S15_LEC2.pdf |
EstimationFitting Probability Distributions
16 Statistical Models
Definitions
General Examples
Classic One-Sample Distribution Models
Classic Probability Models
Normal Distribution
Two parameters:
µ : mean
σ2 : variance
Probability density function:
1 (x − µ)2
σ2
, −∞ < x < ∞.
f (x | µ, σ2) = √
1 −
2πσ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/6408c60bd43d4da05fc6837a12341c29_MIT18_443S15_LEC2.pdf |
stand and apply optimality principles in parameter
estimation.
Important Methodologies
Method-of-Moments
Maximum Likelihood
Bayesian Approach
MIT 18.443
Parameter EstimationFitting Probability Distributions
19 MIT OpenCourseWare
http://ocw.mit.edu
18.443 Statistics for Applications
Spring 2015
For information a... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/6408c60bd43d4da05fc6837a12341c29_MIT18_443S15_LEC2.pdf |
3.052 Nanomechanics of Materials and Biomaterials Thursday 03/08/07
I
Prof. C. Ortiz, MIT-DMSE
LECTURE 9: QUANTITATIVE ASPECTS INTRA- AND
INTERMOLECULAR FORCES
Outline :
LAST LECTURE : INTRODUCTION TO INTRA- and INTERMOLECULAR FORCES.............................. 2
BRIDGING THE GAP BETWEEN LENGTH SCALES .......... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
- and intermolecular forces are electrostatic in origin → key to life on earth (e.g. water, cell
membranes, protein folding, etc.), also materials science (what holds matter together?).
-strength measured relative to the thermal energy (room temperature) : kBT= 4.1 ● 10-21 J : "ruler"
-Classifications; primary or ch... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
.4 0.6 0.8
1
→
-0.04
Interatomic or
Intermolecular Separation
Distance, r(nm)
w(r) or U(r) → f(r)
(one atom, ion, or
molecule)
-1st step is to
assume a
mathematical form of
the potential
rr
repulsive
regime
attractive
regime
kc
0
)
N
n
(
F
,
e
c
r
o
F
Tip-Sample Separation
Distance, D (nm)
W(D) → F(D)
(net ... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
⎡
⎢
⎢
⎣
⎛
⎜
⎝
σ
r
⎞
⎟
⎠
-
σ
r
⎛
⎜
⎝
6
⎞
⎟
⎠
⎤
⎥
⎥
⎦
(4)
F (
LJ
m = 6, n = 12
) =
-6A 12B
+
13
r
r
7
(5)
B
E = "binding energy" or "bond dissociation energy";
or depth of potential well
r = distance at which U(r )
s
F(r )= minimum = F
r = equilibrium bond length
e
= distance at which U(r )= minimum, F(r )= 0
e
r = σ =... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
ipole-dipole
H-H,
Schematic
-
-
-
-
-
-
r
r
u θ r Q
r Q
u θ
-
-
Cu2+ Cu2+
Cu2+ Cu2+
-
-
-
-
Q2
Q2
Q1
Q1
fixed dipole
u r Q
r Q
u
freely rotating dipole
φ
φ
u1 θ1
u1 θ1
r
r
θ2
θ2
u2
u2
r
r
u2
u2
fixed dipole
w(r)= -
charge-induced dipole
freely rotating dipoles
rQ
rQ
α
α
dipole-induced dipole
u θ
u θ r
r
α... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
(r)= -
o
⎞
Q Q
1
2
⎟
4
⎠πε
Qucos
4
πε
o
2
2
2
2
-2
r
(
3
θ
⎞
⎟
⎠
k T
B
2
u u
1
)2
4
πε
o
(
6 4
πε
o
[
⎛
⎜
⎝
⎛
2
Q u
⎜
⎜
)
⎝
⎛
u u 2cos cos
1 2
⎜
⎝
⎛
⎜
⎜
⎝
⎛
⎜
⎜
⎝
⎛
⎜
⎜
⎝
⎛
⎜
⎜
⎝
⎛
⎜
⎜
⎝
)2
4
πε
o
(
2
1+ 3cos
α
)2
(
2
2
α
)2
3h
(
4
πε
o
4
πε
o
4
πε
o
2
να
2
α
⎞
⎟
⎟
⎠
⎞
⎟
⎟
⎠
⎞
⎟
⎟
⎠
Q
u
2
4
u
(
(
)
r
r
r
-6
-4
-6
2
)
2... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
electric polarizability (C2 m2 J-1), r = distance between
interacting atoms or molecules (m), kB= Boltzmann's constant = 1.381●10-23 J K-1, T = absolute temperature (K), h = Planck's constant = 6.626●10-34
J s, ν = electronic absorption (ionization) frequency (s-1), εo=dielectric permittivity of free space = 8.854 ●1... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
�
⎞
⎟
⎠
θ
-2
r (1)
w(r)= -
Qucos
4
πε
o
r= charge-dipole separation distance (nm)
u= electric dipole moment = ql (Cm)
q= charge of dipole (C)
l = separation distance between dipole charges(m)
Q= charge of the ion (C)
θ = dipole angle relative to horizontal
Charge
Charge
Q=ze
Q=ze
dipole moment = ql
dipole momen... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
(4πε )
o
⎞
⎟
⎠
-6
r
=
C
induced
-6
r (3)
Dispersion Energy : Induced dipole-induced dipole :
w(r)
dispersion
=
⎛
⎜
⎝
2
-3h α
ν
2
4(4πε )
o
⎞
⎟
⎠
-6
r
=
C
-6
r (4)
dispersion
Biomolecular Adhesion :
-controlled by bonds between molecular “ligands”
and cell surface “receptors” which exhibit the
“lock-n-key principle... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
,
kBT
0.05-40
0.4
2-10
0.02-16
0.17
0.8-4
2.5
5
13
10-40
25
62
125
380
630
840
1
2
5
4-16
10
25
50
150
250
340
Interaction
dispersion
hydrophobic
H-bonding
ion-ion
covalent
Material
metals
ceramics and glasses
semiconductors
diamond
water
inert gases
solid salt crystals
alkanes... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/64401fe8772bb8ed83da5fdd237df7f5_lec9.pdf |
Coordinate Systems and Separation of Variables
Revisiting the homogeneous wave equation…
2
+∇
ψ
1
2
c
2
∂
ψ
2
t
∂
=
0
where previously in Cartesian coordinates, the Laplacian was given by
2
=∇
2
∂
x
∂
+
2
2
∂
y
∂
+
2
2
∂
z
∂
2
We are now faced with a spherical polar coordinate system, with the
motivation that we might... | https://ocw.mit.edu/courses/2-067-advanced-structural-dynamics-and-acoustics-13-811-spring-2004/6463b606a9e9c7e4d58be5a222f8ad48_lect_8_2.pdf |
of r only…
1
2
r
⎡
⎢
⎣
∂
r
⎛
⎜
⎝
2
r
∂
r
∂
⎞
+⎟
⎠
2
k
⎤
ψ
⎥
⎦
0)(
r
=
Which is known to have the two solutions…
rψ
)(
=
rA
(
)/
⎧
⎨
rB
)/
(
⎩
exp(
exp(
ikr
)
ikr
)
−
(Makes sense as the field
decreases with r as we expect)
As we want to consider the sphere as the only source in the medium (radiation
condition), we ca... | https://ocw.mit.edu/courses/2-067-advanced-structural-dynamics-and-acoustics-13-811-spring-2004/6463b606a9e9c7e4d58be5a222f8ad48_lect_8_2.pdf |
Spherical Harmonics
Re[ 1
1Y
]
Im[ 1
1Y
]
0
1Y
[
]
Various quadrupole orientations in terms of
Spherical Harmonics
Im[ 1
2Y
]
Im[ 2
2Y
]
Re[ 1
2Y
]
Propagating fields from spherical boundaries
rp
,(
,
)
φθ
=
∞
∑
n
=
0
h
h
n
n
(
(
kr
ka
)
)
n
∑
m
−=
Y
n
m
n
,(
)
φθ
∫
ap
,(
′
′
,
φθ
Y
)
m
n
′
′
(
,
φθ
*)
d
Ω′
Where th... | https://ocw.mit.edu/courses/2-067-advanced-structural-dynamics-and-acoustics-13-811-spring-2004/6463b606a9e9c7e4d58be5a222f8ad48_lect_8_2.pdf |
Spring 2009
Spring 2009
Lecture 9: Introduction to
Lecture 9: Introduction to
Program Analysis and
Optimization
Optimization
Outline
Introduction
• Introduction
• Basic Blocks
• Common Subexpression Elimination
• Copy Propagation
C
ti
P
• Dead Code Elimination
ead Code
at o
• Algebraic Simplification
• Sum... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
• Nodes Represent Computation
– Each Node is a Basic Block
– Basic Block is a Sequence of Instructions with
o
p
a
o
a
q
u
u o
• No Branches Out Of Middle of Basic Block
• No Branches Into Middle of Basic Block
• Basic Blocks should be maximal
– Execution of basic block starts with first
instruction
– Includes a... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
1;
return s;
Saman Amarasinghe
17
6.035 ©MIT Fall 2006
s = 0;
s = 0;
a = 4;
i = 0;
k == 0
k == 0
s = 0;
a = 4;
i = 0;
k == 0
k == 0
b = 2;
b = 1;
b = 2;
b = 1;
i < n
s = s + a*b;
i + 1;
i = i + 1;
i
i < n
return s;
s = s + a*b;
i = i + 1;
return s;
Saman Amarasinghe
18
6.035 ©MIT Fall 2006
s = 0;
s = 0;
a = 4;
i = 0;... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
Split point has multiple successors – conditional
branch statements only split points
• Merge point has multiple predecessors
• Each basic block
– Either starts with a merge point or its
predecessor ends with a split point
– Either ends with a split point or its successor
i
starts with a merge point
t
t
i h
t
t... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
Introduced as part of instruction flattening
a o
– Introduced by optimizations/transformations
Typically assigned to only once
– Typically assigned to only once
odu d a
u o
p
p
a
• Program Variables
– Declared in original program
Declared in original program
– May be assigned to multiple times
– May transfer val... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
+v2 v3
v3+v4 v5
v5+v2 v6
v1+v2 t1
v1+v2 t1
v3+v4 t2
v5+v2 t6
Saman Amarasinghe
27
6.035 ©MIT Fall 1998
y
Value Numbering Summary
g
• Forward symbolic execution of basic block
• Each new value assigned to temporary
– a=x+y; becomes a=x+y; t=a;
–
Temporary preserves value for use later in pro... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
y=a+b;
– y=a+b; t=y; x=b; z=t
y a+b; t y; x b; z t
– Why? Because computes with symbolic values
x=b; z=a+x becomes
i
• Finds common subexpressions even if variable
• Finds common subexpressions even if variable
that originally held the value was overwritten
y a+b;
– y=a+b;
– y=a+b; t=y; y=1; z=t
– Why? Because... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
are unnecessary
S
dead code elimination
Saman Amarasinghe
32
6.035 ©MIT Fall 1998
Problems II
• Expressions have to be identical
– a=x+y+z; b=y+z+x; c=x*2+y+2*z–(x+z)
• We use canonicalization
• We use algebraic simplification
Saman Amarasinghe
33
6.035 ©MIT Fall 1998
Copy Propagation
p g
py
• Once ag... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
x+y
a
b = a+z
c = x+y
a = b
• After CSE
a = x+y
t1 = a
b = a+z
t2 = b
c = t1
c
t1
a = b
Saman Amarasinghe
37
6.035 ©MIT Fall 2006
Copy Propagation Example
p g
py
p
Basic Block
After CSE
Aft CSE
a = x+y
t1t1 = a
Basic Block After
CSECSE and C d Copy PProp
a = x+y
t1
t1 = a
tmp to var
tmp to var
... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
�{{t2}t2}
Saman Amarasinghe
40
6.035 ©MIT Fall 2006
Copy Propagation Example
p g
py
p
Basic Block
After CSE
Aft CSE
Basic Block After
CSECSE and C d Copy PProp
a = x+y
t1t1 = a
b = a+z
t2 = b
c = t1
tmp to var
tmp to var
t1 a
t2 b
t2 b
a = x+y
t1
t1 = a
b = a+z
t2 = b
c = a
var to set
va... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
b = a+z
t2 = b
c = a
a = b
var to set
var to set
a {}
bb {{t2}t2}
Saman Amarasinghe
43
6.035 ©MIT Fall 2006
Outline
Introduction
• Introduction
• Basic Blocks
• Common Subexpression Elimination
• Copy Propagation
C
ti
P
• Dead Code Elimination
ead Code
at o
• Algebraic Simplification
• Summary
... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
Needed Set
{ , }
{a, b}
Saman Amarasinghe
48
6.035 ©MIT Fall 2006
Basic Block After
CSE and Copy Prop
a = x+y
t1 = a
t1
b = a+z
t2 = b
c = a
a = b
Needed Set
{ , }
{a, b}
Saman Amarasinghe
49
6.035 ©MIT Fall 2006
Basic Block After
CSE and Copy Prop
a = x+y
t1 = a
t1
b = a+z
c = a
a = b
Need... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
+y
b = a+z
c = a
a = b
Needed Set
{ , ,
{a, b, z}}
Saman Amarasinghe
55
6.035 ©MIT Fall 2006
Outline
Introduction
• Introduction
• Basic Blocks
• Common Subexpression Elimination
• Copy Propagation
C
ti
P
• Dead Code Elimination
ead Code
at o
• Algebraic Simplification
• Summary
Algebraic Simplifi... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
– a * 2
– a * 8
a*a
a + a
a << 3
Saman Amarasinghe
60
6.035 ©MIT Fall 1998
Opportunities for
Algebraic Simplification
Algebraic Simplification
• In the code
•
In the code
– Programmers are lazy to simplify expressions
– Programs are more readable with full expressions
Programs are more readable with ful... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
Saman Amarasinghe
65
6.035 ©MIT Fall 1998
Canonical Format
• Put expression trees into a canonical
p
format
– Sum of multiplicands
– Variables/terms in a canonical order
Example
– Example
(a+3)*(a+8)*4 4*a*a+44*a+96
p
– Section 12.3.1 of whale book talks about this
Saman Amarasinghe
66
6.035 ©MIT Fall 19... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
Si
S b li
Symbolically Simulate Execution of Program
f P
– CSE and Copy Propagation go forward
Dead Code Elimination goes backwards
– Dead Code Elimination goes backwards
l t E
ti
• Transformations stacked
– Group of basic transformations work together
– Often, one transformation creates inefficient code that ... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/646551e794a46c21a5e504a4a15fe52a_MIT6_035S10_lec09.pdf |
6.772/SMA5111 - Compound Semiconductors
Lecture 4 - Carrier flow in heterojunctions - Outline
• A look at current models for m-s junctions (old business)
Thermionic emission vs. drift-diffusion vs. p-n junction
• Conduction normal to heterojunctions
(current across HJs)
Current flow:
1. Drift-diffusion
2. Balli... | https://ocw.mit.edu/courses/6-772-compound-semiconductor-devices-spring-2003/6478642feaf27a4647109cddf23ce37d_lecture4.pdf |
A*T 2 -qf
e
bm / kT (eqvAB / kT
-1)
*
A : Thermionic emission coefficient
bm : Barrier in metal, = f
f
b
+ (kT /q) ln(NC / N D )
Substituting for f
id / A = q
A*T 2
qNC
N D e
-qf
bm and rearranging :
-1)
b / kT (eqvAB / kT
= A*T 2 /qNC
from which we see RTE
C. G. Fonstad, 2/03
Lecture 4 - Slide 3
Drift-di... | https://ocw.mit.edu/courses/6-772-compound-semiconductor-devices-spring-2003/6478642feaf27a4647109cddf23ce37d_lecture4.pdf |
Ê kT ˆ
Ë q ¯
= N D e -qf
b / kT (eqvAB / kT -1).
For this diode R + = De / w p , the diffusion velocity.
pn
C. G. Fonstad, 2/03
Lecture 4 - Slide 5
Comparing the results -
Thermionic emission
RTE
= A*T 2 /qNC , modeling from metal side
Drift-diffusion
= m
RDD
p-n+ diode
E
pk , the drift velocity at x = 0.
... | https://ocw.mit.edu/courses/6-772-compound-semiconductor-devices-spring-2003/6478642feaf27a4647109cddf23ce37d_lecture4.pdf |
+ Egp - EgN|
x
Lecture 4 - Slide 8
-xp
0
xn
Conduction parallel to heterojunctions
Modulation doped structures
N-n heterojunctions and accumulated electrons
Modulation doped structure with surface pinning and HJ
WHITE BOARD
Back to foils
Carrier scattering and mobilities in accumulated two-dimensional
electr... | https://ocw.mit.edu/courses/6-772-compound-semiconductor-devices-spring-2003/6478642feaf27a4647109cddf23ce37d_lecture4.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.02 Multivariable Calculus
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
18.02 Lecture 1. – Thu, Sept 6, 2007
Handouts: syllabus; PS1; flashcards.
Goal of multivariable calculus: tools to handle prob... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/649253ba60d11b0598cc58e9dcf58142_lec_week1.pdf |
displayed example.
Dot product.
Definition: A� B� = a1b1 + a2b2 + a3b3 (a scalar, not a vector).
Theorem: geometrically, A� · B� = |A�||B� | cos θ.
Explained the theorem as follows: first, A� A� = A� 2 cos 0 = |A� 2 is consistent with the definition.
|
Next, consider a triangle with sides A�, B� , C� = A� − B� . Then th... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/649253ba60d11b0598cc58e9dcf58142_lec_week1.pdf |
than 90◦,
zero if perpendicular.
2) detecting orthogonality.
Example: what is the set of points where x + 2y + 3z = 0? (possible answers: empty set, a point,
a line, a plane, a sphere, none of the above, I don’t know).
Answer: plane; can see “by hand”, but more geometrically use dot product: call A� = �1, 2, 3�,
... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/649253ba60d11b0598cc58e9dcf58142_lec_week1.pdf |
� | sin θ (= 1/2 area of parallelogram). Could get sin θ using dot product to compute cos θ and
2
sin2 + cos2 = 1, but it gives an ugly formula. Instead, reduce to complementary angle θ� = π/2 − θ
by considering A�� = A� rotated 90◦ counterclockwise (drew a picture). Then area of parallelogram
= |A�||B� | sin θ = |A... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/649253ba60d11b0598cc58e9dcf58142_lec_week1.pdf |
b1 b3
c1 c3
�
�
+a3
�
�
�
�
�
�
b1 b2
c1 c2
�
�
.
�
�
Geometrically: det( � B, �
A, � C) = ± volume of parallelepiped. Referred to the notes for more about
determinants.
Cross-product. (only for 2 vectors in space); gives a vector, not a scalar (unlike dot-product).
Definition:
A� × B� =
�
�
�
�
�
�
ˆj kˆ
ˆı
a... | https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/649253ba60d11b0598cc58e9dcf58142_lec_week1.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.