text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
randomized test PZ∣X , put g
Moreover,
⊃
R(
P, Q
direction.
) ⊃ R (
det P, Q . By Theorem 10.1,
))
(x) = PZ=0∣X=x. Then g is a
measurable function.
P [Z = 0] = ∑ g(x
)P
x
(x) = EP
[g
(X
)] =
∫
0
Q
[ = ] = ∑
Z
0
x
( ) ( ) =
g x Q x
[ (
g
X
EQ
)] = ∫
0
1
1
P
[g(X) ≥ t]dt
Q
[g(X) ≥ t]dt
114
where we applied the formula E... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
� {±∞}
. The likelihood
R
dQ ≤ τ }. Formally, we assume that dP = p(x)dµ and
F (x) ≜
⎧⎪⎪⎪⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪⎪⎪⎪⎩
⎨
log p(x)
q(x)
+∞
,
−∞
,
/
n a,
>
, p(x) > 0, x
( )
q
x) >
p(
0, q(x
( ) >
(
) =
0, q x
p x
(x) =
0, q(x) =
p
0
) = 0
0
0
Notes:
•
Q x
( ) exceeds a certain
LRT is a deterministic test. The intuition is that upon o... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
[ (
) { ≥ }]
exp τ EQ g X 1 F τ
exp{−τ } ⋅ EP [g(X)1{F ≤ τ }]
(10.3)
(10.4)
115
Below, these and similar inequalities are only checked for the cases of F not taking extended
values, but from this remark it should be clear how to treat the general case.
• Another useful observation:
Q[F = +∞] = P [F = −∞] = 0 .
(10.5)
... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
{
F
[exp
= −∞]
,
dµ p(x) exp{−F (x)}g(F (x)) + g(−∞)Q F
[ = −∞]
(10.8)
(10.9)
(10.10)
where we used (10.5) to justify
restriction to finite values of F .
(1) To show F
is
a s.s, w
e need to show PX∣F = Q
PX∣
F (x∣f ) =
( )
( ∣ )
PX x PF X f x
∣
PF (f )
=
X∣F . For the discrete case we have:
ef Q(x)1{ P (x)
Q(x)
PF (f )
... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
( ∥
d β α D Q
∥ )
Q
P )
where d
(⋅∥⋅)
is the binary divergence.
Proof. Use
data processing with PZ∣X .
Lemma 10.1 (Deterministic tests). ∀E, ∀γ > 0 ∶ P [E] − γQ[E] ≤ P [ log dP
dQ >
log
γ
]
Proof. (Discrete version)
P [E] − γQ[E] = ∑
x E
∈
p(x) − γq(x
) ≤
∑
∈E
x
( (x) − γq(x
p
))1{p(x)>γq(x)}
=
[
P log
dP
dQ
> log γ, X... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
value for the
Lemma 10.2 (Randomized tests). P
[Z = 0] − γQ[Z = 0] ≤ P [ log dP
]
d > log γ .
Q
Proof. Almost identical to the proof of the previous Lemma 10.1:
P [Z = 0] − γQ[Z = 0] = ∑ PZ∣X (0
∣x
x
)(p(x) − γq(x)) ≤ ∑ PZ∣X (0∣x)(p(x) − γq
x
(x))1{p(x)> (x)}
γq
=
[
P log
≤ P [ log
dP
dQ
dP
dQ
> log γ, Z = 0] − Q[ log
... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
of the LLR, whereas
Note: To apply the
to apply the weak converse Theorem 10.4 we need only to know the expectation of the LLR, i.e.,
divergence.
Simil
(
R
P, Q is contained in the intersection of a collection
say c , at which poin
∗
indexed by γ.
of half-spaces
>
]
[
10.5 Achievability bounds on
R
(
P, Q
)
R(
)
Since ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
is attained by the
following test:
⎧⎪⎪⎪
P
log d
dQ > τ
1
⎪⎪
λ log dP
⎨
dQ = τ
⎪⎪⎪⎪
⎪
log dP
dQ < τ
0
⎩
PZ∣X (0∣x
(10.13)
) =
where τ ∈ R and λ ∈ [0, 1] are the unique solutions to α = P [log dP
dQ > τ ] + λP [log dP
dQ
=
]
τ .
Proof of Theorem 10.6. Let t = exp(τ ). Given any test PZ∣X , let g(x) = PZ∣X 0 x
to show tha... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶
]
{
+EQ[g(X)1
]
dP t
dQ > }
{
( EP [(1 − g(X))1
dP
] + λP [
d
Q
„„„„„„„„„„„„„„„„„„„„„„„„„
·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
t}
{
a consequence of the Neyman-Pearson lemma, all the points on the boundary of
Remark 10.2.
the region
R(
As
)
P, Q are attainable. Therefore
− }
R(P, Q) = {(α, β) ∶ βα ≤ β ≤ 1 − β1 α .
Since α ↦ βα is convex on [0, 1], hence continuous, the region
Consequently, the infimum in the definition of βα is in fact a minimum... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
��ned with an ≤ instead of <.
{log dP
dQ <τ }
{log dP
τ
dQ > τ ] for any τ , then we have λ ∈ (0, 1), and (10.13) is equivalent
¯
with probability λ.
probability λ or 1
with
119
t1P[logdPdQ>t]ατt1P[logdPdQ>t]ατ Corollary 10.1. ∀τ ∈ R, there exists (α, β) ∈ R(P, Q) s.t.
α = P [
log
dP
dQ
> ]
τ
β
≤ exp(−τ )P [
log
dP
dQ
... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
to 0?
exponents of the convergence rates of π1 0
∣
⎧⎪⎪
π1∣0 → 0
⎨
⎪⎪
⎩π0∣1 → 0
120
MIT OpenCourseWare
https://ocw.mit.edu
6.441 Information Theory
Spring 2016
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.013/ESD.013J Electromagnetics and Applications, Fall 2005
Please use the following citation format:
Markus Zahn, 6.013/ESD.013J Electromagnetics and Applications, Fall
2005. (Massachusetts Institute of Technology: MIT OpenCourseWare).
http://ocw.mit.edu (accessed MM DD, YYYY... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/27035dfb2ffa2460b38ef54ac5238d69_lec11.pdf |
(z)
= Zn(z)
= −Γ(z)
=
1
Zn(z)
= Yn(z) =
Y (z)
Y0
=
ˆi(z)
Y0vˆ(z)
D. If line is matched, ZL = Z0, ΓL = 0, Zn(z) = 1
II. Load Impedance Reflected Back to the Source
From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
Zn(z = 0) =
�
Zn z = −
=
�
λ
4
ZL =... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/27035dfb2ffa2460b38ef54ac5238d69_lec11.pdf |
A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
To match Z1 to RL ⇒ Z1 = RL/Z2
Z2
=
Z2
2
RL
, Z2
=
√
Z1RL.
3
V0coswtYRs IV. Smith Chart
Zn(z) = r + jx
(z) = 1+Γ(z)
Zn
1−Γ2
Γ(z) = Γr + jΓi
1+Γr +jΓi
1−Γ(z) ⇒ r + jx = 1−Γr −jΓi
r −Γ2
x =
i
(1−Γr)2+Γ2
i
2Γi
(1−Γr)2+Γ2
i
r =
�
�
�... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/27035dfb2ffa2460b38ef54ac5238d69_lec11.pdf |
50 + Z
�
�
Im [Z(z = −l)]
50 + Re [Z(z = −l)]
φ = tan−1
6
V. Standing Wave Parameters
From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
vˆ(z) = Vˆ+e−jkz [1 + Γ(z)]
ˆi(z) = Y0Vˆ+e−jkz [1 − Γ(z)]
⇒ |
⇒ |
vˆ(z) = Vˆ+||1 + Γ(z)|
ˆi(z)
|
| = Y0|Vˆ+||1 − Γ(z)| ... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/27035dfb2ffa2460b38ef54ac5238d69_lec11.pdf |
Y0Vˆ+
= 1 − |ΓL|
= |1 − ΓL|
E. If ZL = RL (real), then ΓL real. If ZL > Z0, VSWR = ZL . If ZL < Z0, VSWR = Z0
ZL
Z0
7
From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
Load Impedance: ZL = Z0
1 + ΓL ejφ
|
|
ejφ
1
ΓL
− |
|
�
VSWR + 1 + (VSWR − 1)e
jφ �
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/27035dfb2ffa2460b38ef54ac5238d69_lec11.pdf |
Introduction to C++
Massachusetts Institute of Technology
January 19, 2011
6.096
Lecture 7 Notes: Object-Oriented Programming
(OOP) and Inheritance
We’ve already seen how to define composite datatypes using classes. Now we’ll take a step
back and consider the programming philosophy underlying classes, known as ob... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
pedals, the wheels, etc. OOP allows programmers to pack away details
into neat, self-contained boxes (objects) so that they can think of the objects more abstractly
and focus on the interactions between them.
There are lots of definitions for OOP, but 3 primary features of it are:
• Encapsulation: grouping related d... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
you can also think of the interface of a class as the set
of buttons each instance of that class makes available. Interfaces abstract away the details
of how all the operations are actually performed, allowing the programmer to focus on how
objects will use each other’s interfaces – how they interact.
This is why C... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
Car and Truck
classes share this code.
The Vehicle class will be much the same as what we’ve seen before:
1
2
3
4
class Vehicle {
protected :
string license ;
int year ;
2
5
6 public :
7
8
9
10
11
12
13 };
Vehicle ( const string & myLicense , const int myYear )
: license ( myLicense ) , year ( myYea... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
1 class Car : public Vehicle { // Makes Car inherit from Vehicle
2
3
4 public :
5
Car ( const string & myLicense , const int myYear , const string
& myStyle )
: Vehicle ( myLicense , myYear ) , style ( myStyle ) {}
const string & getStyle () { return style ;}
6
7
8 };
Now class Car has all the data members and ... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
B:
1. Every A object has a B object. For instance, every Vehicle has a string object (called
license).
2. Every instance of A is a B instance. For instance, every Car is a Vehicle, as well.
Inheritance allows us to define “is-a” relationships, but it should not be used to implement
It would be a design error to mak... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
{ return style ;}
6
7
8
9
10 };
3.2.1 Programming by Difference
In defining derived classes, we only need to specify what’s different about them from their
base classes. This powerful technique is called programming by difference.
Inheritance allows only overriding methods and adding new members and methods. We
c... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
If we have a function that expects a Vehicle object, we can safely pass it a Car
types.
object, because every Car is also a Vehicle. Likewise for references and pointers: anywhere
you can use a Vehicle *, you can use a Car *.
5
4.1
virtual Functions
There is still a problem. Take the following example:
1 Car c... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
references:
1 Car c ( " VANITY " , 2003) ;
2 Vehicle & v = c ;
3 cout << v . getDesc () ;
This will only call the Car version of getDesc if getDesc is declared as virtual.
Once a method is declared virtual in some class C, it is virtual in every derived class of C,
even if not explicitly declared as such. However... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
actually implement it, and therefore cannot be
instantiated.
5 Multiple Inheritance
Unlike many object-oriented languages, C++ allows a class to have multiple base classes:
1 class Car : public Vehicle , public InsuredItem {
2
3 };
...
This specifies that Car should have all the members of both the Vehicle and the ... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/270def7b1f68535b7c3846c606b220eb_MIT6_096IAP11_lec07.pdf |
LECTURE 17
LECTURE OUTLINE
Review of cutting plane method
Simplicial decomposition
•
•
Duality between cutting plane and simplicial
•
decomposition
All figures are courtesy of Athena Scientific, and are used with permission.1 CUTTING PLANE METHOD
Start with any x0
X. For k
0, set
≥
⌘
•
xk+1 ⌘
where
arg min Fk(x),
X
x
⌦... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/2710d17aff6b2989d7c1b2eac2af2c84_MIT6_253S12_lec17.pdf |
X is much
simpler than minimizing f over X.
−
3◆
SIMPLICIAL DECOMPOSITION METHOD
f (x0)
x0
f (x1)
x1
X
˜x2
f (x2)
x2
f (x3)
x3
˜x4
x4 = x
˜x1
˜x3
Level sets of f
•
(initially x0
Given current iterate xk, and finite set Xk ⌦
x0
{
Let x˜k+1 be extreme point of X that solves
X, X0 =
⌘
).
}
•
X
minimize
∇
subject to x
f (x... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/2710d17aff6b2989d7c1b2eac2af2c84_MIT6_253S12_lec17.pdf |
many ex-
treme points), so case (a) must eventually occur.
The method will find a minimizer of f over X
•
in a finite number of iterations.
5 COMMENTS ON SIMPLICIAL DECOMP.
Important specialized applications
•
Variant to enhance e⌅ciency. Discard some of
•
the extreme points that seem unlikely to “partici-
pate” in the o... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/2710d17aff6b2989d7c1b2eac2af2c84_MIT6_253S12_lec17.pdf |
x
n
⌘ �
⌅
F (x) = max
j=1,...,⌫
y
j� x
−
f (yj)
⇤
[this follows using x�jyj = f (xj) +f (yj), which is
◆f (xj) – the Conjugate Subgra-
implied by yj
⌘
dient Theorem]
⌅
7
◆
INNER LINEARIZATION OF FNS
f (x)
F (y)
f (y)
x0
x1
0
x2
x
Slope = y0
O
F (x)
Slope = y1
f
Outer Linearization of f
Slope = y2
f
y0
y1
f
0
y2
y
Inn... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/2710d17aff6b2989d7c1b2eac2af2c84_MIT6_253S12_lec17.pdf |
f is differentiable
Ck+1(x)
Ck(x)
c(x)
Slope:
f (xk)
−⇥
Const.
f (x)
−
xk
xk+1
˜xk+1
x
Given Ck: inner linearization of c, obtain
xk ⌘
arg min f
⌦�
x
n
⇤
Obtain x˜k+1 such that
(x) +C k(x)
⌅
•
�
•
•
•
f (xk)
◆c(˜xk+1),
−∇
⌘
x˜k+1}
and form Xk+1 = Xk ∪ {
9 NONDIFFERENTIABLE CASE
Given Ck: inner linearization of c, obtain... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/2710d17aff6b2989d7c1b2eac2af2c84_MIT6_253S12_lec17.pdf |
2(x),
x
⌦�
n
min f (⌃) +f
⌅
⌦�
1
n
2 ( ⌃
−
)
Primal and dual approximations
•
•
2,k(x)
1(x) +F
min f
n
x
⌦�
F2,k and F
•
tions of f and f
2
2
f
min 1 (⌃) +F 2,k(
⌅
⌦�
n
⌃)
−
2,k are inner and outer approxima-
x˜i+1 and gi are solutions of the primal or the
•
dual approximating problem (and corresponding
subgradient... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/2710d17aff6b2989d7c1b2eac2af2c84_MIT6_253S12_lec17.pdf |
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© sources unknown. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/hel... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/27220e77673d0f1dff71a461b1574259_MIT2_062J_S17_Chap4.pdf |
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MIT OpenCourseWare
https://ocw.mit.edu
2.062J / 1.138J / 18.376J Wave Propagation
Spring 2017
For information about citing these materials or our Terms of Use, visit: https://... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/27220e77673d0f1dff71a461b1574259_MIT2_062J_S17_Chap4.pdf |
Spring 2010
Semantic Analysis
Semantic Analysis
Saman Amarasinghe
Massachusetts Institute of Technology
Massachusetts Institute of Technology
Symbol Table Summary
• Program Symbol Table (Class Descriptors)
• Class Descriptors
D
Cl
– Field Symbol Table (Field Descriptors)
i t
Pointer to Field Symbol Table for... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
In a stack
push a; push b; mul; push c; add; pop x
• In single use temporary
registers
i t
t1 = mul a, b
x add t1, c
x = add t1 c
=
• Trees
– Intermediate values are
implicit in the edges
St x
St x
add
mul
ld c
ld a
ld b
Handling Control Flow
Handling Control-Flow
(cid:129) Control-Flow Graph
• Contr... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
– take small stepsp
– don’t try to do too many at once
– don t try to do anything too early
don’t try to do anything too early
– try not to loose any information!
Saman Amarasinghe
9
6.035 ©MIT Fall 2006
Outline
Outline
• Practical Issues in Intermediate
t
di
I
t
I
l
ti
P
i
Representation
Representation... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
iqueness checks
– Uniqueness checks
– Type checks
Saman Amarasinghe
15
6.035 ©MIT Fall 2006
Flow of control checks
Flow of control checks
11
• Flow-control of the program is context
• Flow control of the program is context
sensitive
• Examples:
Declaration of a variable should be visible at use
– Declarat... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
I
l
ti
P
i
Representation
Representation
• What is semantic analysis?
• Type systems
• What to check?
Type Systems
Type Systems
17
• A type system is used to for the type
• A type system is used to for the type
checking
• A type system incorporates
syntactic constructs of the language
– syntactic const... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
• If T is a type expression an array(T, I) is
• If T is a type expression an array(T I) is
also a type expression
– I is a integer constant denoting the number of
elements of type T
– Example:
int foo[128];
];
[
array(integer, 128)
Saman Amarasinghe
26
6.035 ©MIT Fall 2006
Type Expressions: Method Calls
T... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
Saman Amarasinghe
29
6.035 ©MIT Fall 2006
A simple typed language
A simple typed language
• A language that has a sequence of declarations
• A language that has a sequence of declarations
followed by a single expression
P D; E
P D; E
D D; D | id : T
T char
T
|
cha
E literal | num | id |... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
2006
Parser actions
Parser actions
24
E E1 + E2
E E1 + E2
integer and
{ if E1 type == integer and
{ if E1.type
E2 .type == integer then
E.type = integer
yp
g
else
E.type = type_error
yp
yp
}
Saman Amarasinghe
33
6.035 ©MIT Fall 2006
Parser actions
Parser actions
E E1 [E2 ]
E E [E ]
integ... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
of one type to another
type
• Example
int A;
float B;
B = B + A
• Two types of coercion
– widening conversions
– narrowing conversions
id
i
i
Saman Amarasinghe
37
6.035 ©MIT Fall 2006
Narrowing conversions
Narrowing conversions
• Conversions that may loose information
• Conversions that may loose infor... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
what is its type?
Saman Amarasinghe
42
6.035 ©MIT Fall 2006
Overloading
Overloading
28
Some operators may have more than one
• Some operators may have more than one
•
type.
• Example
l
E
int A, B, C;
float X, Y, Z;
A = A + B
X = X + Y
X
X
+
Y
• Complicates the type system
– Example
A = A + X
• Wha... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
Local Symbol Table
Local Symbol Table
• When building the local symbol table, have
• When building the local symbol table have
a list of local descriptors
• What to check for?
duplicate variable names
– duplicate variable names
– shadowed variable names
• When to check?
k?
h
h
– when insert descriptor into loc... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
else can/should be checked?
Saman Amarasinghe
51
6.035 ©MIT Fall 2006
Add Operations
Add Operations
• What does compiler have?
• What does compiler have?
– two expressions
• What can go wrong?
?
Wh t
– expressions have wrong type
– must both be integers (for example)
• So compiler checks type of expressions... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
Expression
• What does it do?
– Look up variable name.
pa
• If in local symbol table, reference local descriptor
• If in parameter symbol table, error
sy
bo tab e e o
• If in field symbol table, reference field descriptor
• If not found, semantic error
a ete
,
–
Check that type of array index expression is int... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
• Do semantic checks when build IR
• Many correspond to making sure entities are
• Many correspond to making sure entities are
there to build correct IR
• Others correspond to simple sanity checks
• Each language has a list that must be checked
• Can flag many potential errors at compile time
Saman Amarasinghe
... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-spring-2010/275980aad79ddb35c3872f07019d3278_MIT6_035S10_lec06.pdf |
18.354J Nonlinear Dynamics II: Continuum Systems
Lecture 1
Spring 2015
1 Math basics
1.1 Derivatives and differential equations
In this course, we will mostly deal with ordinary differential equations (ODEs) and partial
differential equations (PDEs) real-valued scalar or vector fields. Usually, non-bold symbols
will be... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/27c776ae2d0b07942bb58afcdf7f2029_MIT18_354JS15_Ch1.pdf |
the product: For
instance, in the case of a 3D vector field v(t, x), we can obtain a scalar field called divergence
another (pseudo-)vector field called curl
∇ · v ≡ ∂ivi,
and the gradient matrix
∇ ∧ v ≡ ((cid:15)ijk∂jvk)
∇v ≡ (∂ivj).
The (scalar) Laplacian operator (cid:52) in Cartesian coordinates is defined by
(cid:52) ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/27c776ae2d0b07942bb58afcdf7f2029_MIT18_354JS15_Ch1.pdf |
by
z¯ = x − iy
(11)
and corresponds to a reflection at the real axis or, equivalently, at the line (cid:61)(z) = 0.
Addition of complex numbers is linear
z = z1 + z2 = (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2) = x + iy
(12)
corresponding to the addition of the two 2D vectors (x1, y1) and (x2, y2).
complex multipl... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/27c776ae2d0b07942bb58afcdf7f2029_MIT18_354JS15_Ch1.pdf |
exp to the trigonometric sin-and cos-functions.
When dealing with axisymmetric problems it is often advantageous to use the polar
representation of a complex number
√
r = |z| = zz¯ ∈ R+
z = reiφ ,
0 , ,
φ = arctan 2(y, x) ∈ [0, 2π)
(18)
From the properties of the exp-function, it follows that the multiplication of comp... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/27c776ae2d0b07942bb58afcdf7f2029_MIT18_354JS15_Ch1.pdf |
Introduction to Simulation - Lecture 6
Krylov-Subspace Matrix Solution Methods
Jacob White
Thanks to Deepak Ramaswamy, Michal Rewienski,
and Karen Veroy
Outline
• General Subspace Minimization Algorithm
– Review orthogonalization and projection formulas
• Generalized Conjugate Residual Algorithm
– Krylov-subspace
– ... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
k
r
⇒
− ∑ (cid:71)
Mwα
i
i
i
0
=
i sα
'
Residual Minimizing idea: pick
b Mx
= −
=
b
k
k
1
−
to minimize
k
r
≡
(
k
r
T
) (
k
r
)
=
2
2
b
−
⎛
⎜
⎝
SMA-HPC ©2003 MIT
k
1
−
∑
i
=
0
α
i
(cid:71)
Mw
i
T
⎞ ⎛
⎟ ⎜
⎠ ⎝
b
−
k
1
−
∑
i
=
0
α
i
(cid:71)
Mw
i
⎞
⎟
⎠
Arbitrary Subspace
methods
Residual Minimization
Computational Ap... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
M b
,
and a set of search directions
(cid:71)
p
1) Generate
by orthogonalizing
's
j
Algorithm Steps
(cid:71)
w
0
{
,...,
(cid:71)
w −
k
1
}
M '
w s
j
(
Mw
(
Mp
i
j
j
1
−
− ∑
i
=
0
T
) (
) (
T
Mp
i
Mp
i
)
)
p
i
j
For
=
k
0 to
−
1
p
=
w
j
j
r
k
2) compute the minimizing solution
x
T
T
) (
(
) (
(
)
)
T
T
) (
) (... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
j
j
j
)
)
pMp
j
)
M
j
Mp
j
Mp
(
+
(
r
−
Normalize
Update Solution
Update Residual
j
p
j
SMA-HPC ©2003 MIT
Arbitrary Subspace
methods
Subspace Selection
Criteria
,...,
All that matters is the
'
k
1
w
0
Criteria for selecting
w −
{
}
span w
,...,
w −
0
k
= − ∑ (cid:71)
b
Mw
i
i
0
=
}
(cid:19)
w
for
N
k
(cid:71)
(ci... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
)
{
span w
0
x
)
0
(
f x
(cid:71)
w
k
,...,
,...,
∇
}
− =
1
x
k
1
−
(
f x
}
)
{
span r
0
=
0
{
span r
}
k
1
r −
,...,
k
1
−
,...,
r
}
If:
i
Mrα
i
k
1
−
− ∑
i
,...,
0
=
r
k
1
−
}
=
0
,
}
{
0
1 0
k
−
span r Mr
M r
(cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)
Krylov Subspace
,...... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
+ =
r
k
k
x
r
+
−
p
k
1
+
=
r
k
1
+
pα
k
k
Mpα
k
(
Mr
(
Mp
0
k
j
k
− ∑
j
=
Vector inner products, O(n)
Matrix-vector product, O(n) if sparse
Vector Adds, O(n)
k
1
+
T
) (
) (
T
Mp
Mp
j
j
)
)
p
j
O(k) inner products,
total cost O(nk)
If M is sparse, as k (# of iters) approaches n,
3
O n
O n
( )
)
(2 )
total cost
=
Bet... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
⎡
⎤
⎢
⎥
1 2
−⎢
⎥
⎢
⎥−
1
⎥
⎢
⎣
⎦
(cid:8)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:10)
M
1 2
(cid:37)
−
Nodal
Equation
Form
Krylov Methods
Nodal Formulation
“No-leak Example”
Circuit and Matrix
1
2
3
4
1m −
m
2
1
−
⎤
⎡
⎥
⎢
1 2
−⎢
⎥
⎥−
⎢
1
⎥
⎢
⎣
⎦
(cid:8)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
“leaky” Example
Circuit and Matrix
1
2
3
4
1m −
m
m
SMA-HPC ©2003 MIT
1
−
2.01
2.01
⎤
⎡
⎥
⎢
1
−⎢
⎥
⎥−
⎢
1
⎥
⎢
1 2.01
⎣
⎦
(cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)
M
(cid:37)
−
Nodal
Equation
Form
GCR Performance(Random Rhs)
Insulating
Leaky
0
10
-1
10
-2
10
-3
10
R
E
S
I... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
Note: for any
{
1
r
span
r
,
0
k
−
i
α
i
∑
0
=
0α ≠
0
0
r
−
α=
0
SMA-HPC ©2003 MIT
kth order polynomial
1 0
i
+
M r
=
(
)
(
I M M r
ξ
k
−
)
0
Mr
0
}
= span
{
0
r Mr
,
0
}
Krylov Methods
Convergence Analysis
Basic Properties
0
0
,
,
x
...
}
2)
span
k
M r
in GCR, then
{
0
r Mr
,
=
th
is the k
k
If
for all
0
j
... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
k+1
k+1
℘(cid:4)
th
order
Therefore
Any polynomial which satisfies
the zero constraint can be used
to get an upper bound on
kr +
21
2
SMA-HPC ©2003 MIT
Eigenvalues and
Vectors Review
Basic Definitions
Eigenvalues and eigenvectors of a matrix M satisfy
eigenvalue
(cid:71)
(cid:71)
u
M λ=
u
i
i
i
eigenvector
O... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
Simplifying
Assumption
Almost all NxN matrices have N linearly
independent Eigenvectors
↑
(cid:71)
u
2
↓
↑
(cid:71)
u
3
↓
↑
(cid:71)
u
1
↓
↑
(cid:71)
u
⎡
⎢
⎢
⎢
⎣
M
N
↓
⎤
⎥
⎥
⎥
⎦
(cid:34)
(cid:34)
(cid:34)
⎡
↑
(cid:71)
⎢
u
= ⎢
λ λ
2 2
⎢
↓
⎣
↑
(cid:71)
u
1 1
↓
↑
(cid:71)
u
λ
3 3
↓
(cid:34)
(cid:34)
(cid:34)
λ
↑
(cid:71... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
M’s eigenvalues
SMA-HPC ©2003 MIT
Eigenvalues and
Vectors Review
Heat Flow Example
Incoming Heat
Unit Length Rod
1T
NT
(0)T
+
-
sv T=
(0)
SMA-HPC ©2003 MIT
(1)T
+
-
sv T=
(1)
Eigenvalues and
Vectors Review
Heat Flow Example
Continued
2
⎡
⎢
1
−⎢
⎢
0
⎢
⎣
0
1
0
−
(cid:37)
2
(cid:37) (cid:37)
0
−
1
0
⎤
⎥
0
⎥
⎥−
1
⎥
2
... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
+
p
(cid:8)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)
Diagonal
a
λ
1
+…
+
a
1
(
f M U U a I
=
0
)
(
+
a
λ
1
+
+…
a
p
λ
p
)
SMA-HPC ©2003 MIT
Useful
Eigenproperties
Spectral
Decomposition
Decompose arbitrary x in eigencomponents
+
(cid:71)
u
+
α α
1 1
(cid:71)
u
N N
(cid:71)
u
2 2... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
2) If M has only q distinct eigenvalues, the GCR
Algorithm converges in at most q steps
(
(
...
=
Proof: Let
( )
x
(cid:4)
℘
q
λ
2
λ
1
)(
−
−
−
x
x
x
)
λ
q
)
SMA-HPC ©2003 MIT
Summary
• Arbitrary Subspace Algorithm
– Orthogonalization of Search Directions
• Generalized Conjugate Residual Algorithm
– Krylov-sub... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/27d0ff96c912a862ba47b64f1ce0cc7c_lec6.pdf |
4
Convolution
In Lecture 3 we introduced and defined a variety of system properties to
which we will make frequent reference throughout the course. Of particular
importance are the properties of linearity and time invariance, both because
systems with these properties represent a very broad and useful class and be-
cau... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/27da9a018e92fc06d2cf1fbce5cd3e71_MITRES_6_007S11_lec04.pdf |
of both continuous-
time and discrete-time signals as a linear combination of delayed impulses
and the consequences for representing linear, time-invariant systems. The re-
sulting representation is referred to as convolution. Later in this series of lec-
tures we develop in detail the decomposition of signals as linea... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/27da9a018e92fc06d2cf1fbce5cd3e71_MITRES_6_007S11_lec04.pdf |
al and is similar in its properties to the convolution sum for discrete-time
signals and systems. A number of the important properties of convolution that
have interpretations and consequences for linear, time-invariant systems are
developed in Lecture 5. In the current lecture, we focus on some examples of
the evaluat... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/27da9a018e92fc06d2cf1fbce5cd3e71_MITRES_6_007S11_lec04.pdf |
2
0-0
--
*-
x[-I]8[n+1]
n
-1 0
I 2
X[-] x [-2]8[n +2]
0--0-0
-1 0 1 2
n
x[o]8[n]+x(I] 8[n -1]
+ x [-I]8[n+ ]+.--
+X kr
=2 x[k]8[n-k]
k= -c
Signals and Systems
TRANSPARENCY
4.2
The convolution sum
for linear, time-
invariant discrete-time
systems expressing
the system output as a
weighted sum of
delayed unit impuls... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/27da9a018e92fc06d2cf1fbce5cd3e71_MITRES_6_007S11_lec04.pdf |
0
LTI: y[n]
=E x[k] h[n - k]I
Convolution Sum
Convolution
4-5
TRANSPARENCY
4.4
Approximation of a
continuous-time signal
as a linear
combination of
weighted, delayed,
rectangular pulses.
[The amplitude of the
fourth graph has been
corrected to read
x(O).]
TRANSPARENCY
4.5
As the rectangular
pulses in Trans-
parency 4.... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/27da9a018e92fc06d2cf1fbce5cd3e71_MITRES_6_007S11_lec04.pdf |
) =
0
+o
+O k=- o
x(kA)
hk(t) A
+00
=f xT) hT(t) dr
If Time-Invariant:
LTI:
hkj t) = ho(t - kA)
h,(t) = he (t - r)
+01
f x(r) h(t-7) dr-
v(t)
1
-0
Convolution Integral
x(t)
0
t
ti
x (0) h Mt
x(kA)
oA
kA
t
x (0)
x(A)
AA
y(t)
y(t)
oA
0
x(t)
0
t
t
Convolution
4-7
TRANSPARENCY
4.8
Comparison of the
convolution su... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/27da9a018e92fc06d2cf1fbce5cd3e71_MITRES_6_007S11_lec04.pdf |
an input that is a
unit step and a system
impulse response that
is a decaying
exponential for t > 0.
MARKERBOARD
4.2
y(t)f
x(r)h(t-r)dr
x(t) u (t)
h (t )=e~43
u
t)
x (t)
t
h (t)
x (r)
r
O
0
h (t-r)
t
T
Convolution
MARKERBOARD
4.3
v4egva.z:
t).
Te
Lk (t - -C jT
Ct
k Lt-T)aT
t (0
1 U -e
3
o-
eoverkp ~3et ee
h*... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/27da9a018e92fc06d2cf1fbce5cd3e71_MITRES_6_007S11_lec04.pdf |
LECTURE 2
Hilbert Symbols
Let K be a local field over Qp (though any local field suffices) with char(K) (cid:54)= 2.
Note that this includes fields over Q2, since it is the characteristic of the field, and
not the residue field, with which we are concerned. Recall from the previous lecture
the duality
(2.1) Gal2(K) := Galab(K... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/27f59437d11cbc1a6039b250e4c325ca_MIT18_786S16_lec2.pdf |
osition 2.3. The Hilbert symbol satisfies the following properties:
(1) Bimultiplicativity. For all a, b, c ∈ K ×,
(a, bc) = (a, b) · (a, c).
(2) Non-degeneracy. For all a ∈ K ×, if (a, b) = 1 for all b ∈ K ×, then
a ∈ (K ×)2.
Note that (a, b) = (b, a) trivially. Bimultiplicativity says that we can solve
ax2 +by2 = 1 if... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/27f59437d11cbc1a6039b250e4c325ca_MIT18_786S16_lec2.pdf |
(x)y, where y ∈ O×
K. Then the
following are equivalent:
(1) x is a square;
(2) v(x) is even and y is a square;
(3) y mod p is a square in K ×.
Note that we may reduce to x ∈ O×
K. We offer two proofs:
Proof (via Hensel’s Lemma). All explanations aside from that from the
final condition are clear. So suppose x mod p is a... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/27f59437d11cbc1a6039b250e4c325ca_MIT18_786S16_lec2.pdf |
a + b)π + abπ2, and since abπ2 ∈ p2, we are left with
1 + (a + b)π in the associated graded term, hence multiplication simply corresponds
to addition in k. Similarly, for each n ≥ 1, we have (1 + pn)/(1 + pn+1) (cid:39) k by a
similar argument, since n + 1 ≤ 2n. Now, σ acts on the filtration as
K. Clearly O×
O×
K ⊇ 1 + ... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/27f59437d11cbc1a6039b250e4c325ca_MIT18_786S16_lec2.pdf |
quotients are all isomorphic to finite
fields. As a general principle, we can understand many things about A via its
associated graded Gr∗A.
Definition 2.7. Let A be an abelian group. A filtration on A is a descending
sequence of subgroups
A =: F0A ⊇ F1A ⊇ F2A ⊇ · · · ,
and it is said to be complete if A ∼−→ lim
←−n
are t... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/27f59437d11cbc1a6039b250e4c325ca_MIT18_786S16_lec2.pdf |
= f (y0) + f ((cid:15)1) ≡ x mod F2B,
where we have defined y1 := y0 +(cid:15)1. This is an equation of the same form as before,
and we may iterate to find a “compatible” system of yn such that f (yn) = x mod
Fn+1B for each n ≥ 0, where by “compatible” we mean that for each n we have
yn ≡ yn+1 mod Fn+1A. But then there i... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/27f59437d11cbc1a6039b250e4c325ca_MIT18_786S16_lec2.pdf |
here we will view it asymmetrically. Suppose
a) is a degree 2 extension of K (note that if a is
a is not a square, so that K(
a nonzero square, then we need only understand K(
a) to be the corresponding
étale extension of K, isomorphic to K × K).
√
√
Claim 2.11. We have (a, b) = 1 if and only if b is a norm for the ext... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/27f59437d11cbc1a6039b250e4c325ca_MIT18_786S16_lec2.pdf |
.
Theorem 2.12. If L/K is a quadratic extension of local fields, then the norm
N : L× → K × is a homomorphism, and N(L×) ⊆ K × is a subgroup of index 2.
Example 2.13. Consider C/R.
MIT OpenCourseWare
https://ocw.mit.edu
18.786 Number Theory II: Class Field Theory
Spring 2016
For information about citing these mater... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/27f59437d11cbc1a6039b250e4c325ca_MIT18_786S16_lec2.pdf |
Sufficiency
Sufficiency
MIT 18.443
Dr. Kempthorne
Spring 2015
MIT 18.443
Sufficiency
1 Sufficiency
Definition
Example
Theorems
Outline
1 Sufficiency
Definition
Example
Theorems
MIT 18.443
Sufficiency
2 Sufficiency
Definition
Example
Theorems
Sufficient Statistics
Definition: Sufficiency
X1, X2, . . . , Xn iid with ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/27f9a8f9aaacfe7feb9475af1ec13a79_MIT18_443S15_LEC7.pdf |
finition
Example
Theorems
Sufficiency: Example
Example 8.8.1A Bernoulli Trials Let X = (X1, . . . , Xn) be the
outcome of n i.i.d Bernoulli(θ) random variables
The pmf function of X is:
p(X | θ) = P(X1 = x1 | θ) × · · · × P(Xn = xn | θ)
= θx1 (1 − θ)1−x1 × θx2 (1 − θ)1−x2 × · · · θxn (1 − θ)1−xn
= θ
xi (1 − θ)(n−... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/27f9a8f9aaacfe7feb9475af1ec13a79_MIT18_443S15_LEC7.pdf |
we should only
need the information of T (X) = t, since the value of X given
t reflects only the order information in X which is
independent of θ.
MIT 18.443
Sufficiency
6 Sufficiency
Definition
Example
Theorems
Outline
1 Sufficiency
Definition
Example
Theorems
MIT 18.443
Sufficiency
7 Sufficiency
Definition
Examp... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/27f9a8f9aaacfe7feb9475af1ec13a79_MIT18_443S15_LEC7.pdf |
ary A
If T is sufficient for θ, then the maximum likelihood estimate is a
function of T .
Rao-Blackwell Theorem
Let θˆ be an estimator of θ with E [θˆ2] < ∞ for all θ.
Suppose that T is sufficient for θ
Define θ˜ = E [θˆ | T ].
Then for all θ,
E [(θ˜ − θ)2] ≤ E [(θˆ − θ)2].
The inequality is strict unless θ˜ ≡ θ. ˆ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-spring-2015/27f9a8f9aaacfe7feb9475af1ec13a79_MIT18_443S15_LEC7.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.265/15.070J
Lecture 5
Fall 2013
9/16/2013
Extension of LD to Rd and dependent process. G¨artner-Ellis Theorem
Content.
1. Large Deviations in may dimensions
2. G¨artner-Ellis Theorem
3. Large Deviations for Markov chains
1
Large Deviations in Rd
Most of the developmen... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
M (θ) < ∞ for all θ is needed. Known counterex
amples are somewhat involved and can be found in a paper by Dinwoodie [2]
which builds on an earlier work of Slaby [5]. The difficulty arises that there is
no longer the notion of monotonicity of I(x) as a function of the vector x. This
is not the tightest condition and... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
g(θ1, θ2) = 0, we have that
(θ1
2 + θ1θ2 + θ2
2).
x1 − θ1 − θ2 = 0,
1
2
x2 − θ2 − θ1 = 0,
1
2
2
from which we have
θ1 = x1 − x2,
θ2 = x2 − x1
4
3
2
3
4
3
2
3
Then
So we need to find
2
I(x1, x2) = (x1 + x2 − x1x2).
2
2
3
2
(x1 + x 2 − x1x2)
2
inf
x1,x2 3
s.t. 2x1 + x2 ≥ 5
(x ∈ F )
This becomes a non-li... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
1)
which gives x1 =
10
11
, x2 =
35
11
and I(x1, x2) = 5.37. Thus
lim sup
n
1
n
log P(
Sn
n
∈ F ) ≤ −5.37
Applying the lower bound part of the Cram´er’s Theorem we obtain
lim inf
n
1
n
log P(
Sn
n
∈ F )
≥ lim inf
n
1
n
log P(
Sn
n
∈ F o)
=≥
= − 5.37 (by continuity of I).
3
(3)
(4)
−
inf
2... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
the i.i.d. case
n
φn(θ) =
=
1
n
1
n
log E[exp(n(θ, n−1Sn))]
log M n(θ)
= log M (θ)
= log E[exp((θ, X1))].
Loosely speaking G¨artner-Ellis Theorem says that when convergence
φn(θ) → φ(θ)
(5)
takes place for some limiting function φ, then under certain additional technical
assumptions, the large deviation... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
supθ(θa − M (θ))
when a > µ = E[X]. But now we are dealing with the multidimensional case
where such an identity does not make sense.
3
Large Deviations for finite state Markov chains
Let Xn be a finite state Markov chain with states Σ = {1, 2, . . . , N }. The
transition matrix of this Markov chain is P = (Pi,j , ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
have strictly positive compo
nents.
This theorem can be found in many books on linear algebra, for example [4].
The following corollary for the Perron-Frobenious Theorem shows that the
essentially the rate of growth of the sequence of matrices Bn is ρn. Specifically,
Corollary 1. For every vector φ = (φj , 1 ≤ j ≤ ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
Let ρ(Pθ) denote its
Perron-Frobenious eigenvalue.
Theorem 4. The sequence 1 Sn =
tions bounds with rate function I(x) =
n
p1
n p
f (Xn) satisfies the large devia
1≤i≤k
θ∈Rd ((θ, x)−log ρ(Pθ)). Specifically,
6
for every state i0 ∈ Σ, closed set F ⊂ Rd and every open set U ⊂ Rd, the fol
lowing holds:
l... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
Perron-Frobenious theory in fact can
be used to show that such a differentiability indeed takes place. Details can be
found in the book by Lancaster [3], Theorem 7.7.1.
References
[1] A. Dembo and O. Zeitouni, Large deviations techniques and applications,
Springer, 1998.
[2] IH Dinwoodie, A note on the upper boun... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
3.032 Mechanical Behavior of Materials
Fall 2007
Using U(r):
Measure parameters for U(r) in physical model to predict stresses that are high enough for
elastic instabilities to occur (e.g., nucleation of defects in crystals).
Images removed due to copyright restrictions.
Please see: Fig. 1c in Gouldstone, Andrew, et a... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/280e4e7a60634a54a4a94650acce94f7_lec13.pdf |
Properties of
Polymer Nanofibers." Macromolecules 40 (2007): 8483-8489.
Electrospun PEO could be used for filters,
composites, fuel cells, drug delivery, cell
scaffolds, etc. [Rutledge Group, MIT]
Internal energy inside polymer
nanofiber increases for fibers of R < 5
nm.
Curgul, Rutledge and Van Vliet, Macromolecu... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/280e4e7a60634a54a4a94650acce94f7_lec13.pdf |
18. Div grad curl and all that
Theorem 18.1. Let A ⊂ Rn be open and let f : A −→ R be a differ
entiable function.
If �r : I −→ A is a flow line for �f : A −→ Rn, then the function
f ◦ �r : I −→ R is increasing.
Proof. By the chain rule,
d(f �r)
◦
dt
(t) = �f (�r(t)) · �r�(t)
= �r�(t) �r�(t) ≥ 0.
·
�
Corollary... | https://ocw.mit.edu/courses/18-022-calculus-of-several-variables-fall-2010/2811ddbe75dcb771e8ff4b7d4e62dcac_MIT18_022F10_l_18.pdf |
jˆ+
ˆı +
on vector fields:
Definition 18.5. Let A ⊂ R3 be an open subset and let F� : A −→ R3
be a vector field.
The divergence of F� is the scalar function,
which is defined by the rule
div F� : A −→ R,
div F� (x, y, z) = � · F� (x, y, z) =
∂f
∂x
+
∂f
∂y
+
∂f
.
∂z
1
The curl of F� is the vector field
c... | https://ocw.mit.edu/courses/18-022-calculus-of-several-variables-fall-2010/2811ddbe75dcb771e8ff4b7d4e62dcac_MIT18_022F10_l_18.pdf |
F� = �0, and it is called incompressible if
the divergence is zero, div F� = 0.
: −→
�
3R
Proposition 18.7. Let f be a scalar field and F� a vector field.
(1) If f is C2, then curl(grad f ) = �0. Every conservative vector field
is rotation free.
(2) If F� is C2, then div(curl F� ) = 0. The curl of a vector field is
... | https://ocw.mit.edu/courses/18-022-calculus-of-several-variables-fall-2010/2811ddbe75dcb771e8ff4b7d4e62dcac_MIT18_022F10_l_18.pdf |
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