text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
f1), x2⌥
f1(x1)
dom(
x1⌥
= inf
x1⌥
n
dom(f1), x2
dom(f2)
⌘
⌘
dom(f2)
f1(x1) + f2(x2) + ⌥⇧(x2
x1)
−
⌥⇧x1
⇤
+ inf
x2⌥
n
2(x2) + ⌥⇧x2
f
−
⇤
⌅
Dual problem: max
⌅
{−
min⌅{−
q(⌃)
or
}
•
−
f (⌃)
1
⇤
−
f
2 (
⌃)
}
−
⌅
=
1 (⌃) +f
f
2 (
minimize
n,
subject to ⌃
⌘ �
⌃)
−
where f
1 and f
2 are the conjugates.
6◆
◆
•
FENCHE... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/19c72f4d87303765c3925112f5926282_MIT6_253S12_lec11.pdf |
1 = x2,
f1(x1) +f 2(x2)
x1
⌘
dom(f1), x2
dom(f2)
⌘
and the fact
ri
dom(f1)
⇤
dom(f2)
= ri
dom(f1)
dom(f2)
⇤
�
�
to satisfy the relative interior condition.
�
⇥
⇥
⇥
For part (b), apply the optimality conditions
(primal and dual feasibility, and Lagrangian opti-
mality).
7✓
GEOMETRIC INTERPRETATION
f
1 ()
q()
f
2 (
)
... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/19c72f4d87303765c3925112f5926282_MIT6_253S12_lec11.pdf |
optimization.
9◆
CONIC DUALITY
Consider minimizing f (x) over x
n
C, where f :
] is a closed proper convex function
•
�
and C is a closed convex cone in
,
−⇣
⇣
→
n.
⌘
(
�
We apply Fenchel duality with the definitions
f1(x) = f (x),
f2(x) =
0
⇣
�
if x
if x
C,
⌘
/ C.
⌘
The conjugates are
f ⌥(⇤) = sup
1
n
x
⌥
⇤
where C ⇤... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/19c72f4d87303765c3925112f5926282_MIT6_253S12_lec11.pdf |
has an optimal solu-
tion if the optimal value of the dual conic problem
is finite, and
ri
dom(f )
�
⌫
⇥
ri(Cˆ) = Ø.
11✓
✓
LINEAR CONIC PROGRAMMING
Let f be linear over its domain, i.e.,
•
f (x) =
c�x if x
if x
⇣
X,
⌘
/ X,
⌘
�
where c is a vector, and X = b + S is an a⌅ne set.
•
•
•
Primal problem is
minimize
c�x
subje... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/19c72f4d87303765c3925112f5926282_MIT6_253S12_lec11.pdf |
⇤ if and only if
p is lower semicontinuous at 0.
⇣
Duality Theorem: Assume that X, f , and gj
•
are closed convex, and the feasible set is nonempty
and compact. Then f ⇤ = q⇤ and the set of optimal
primal solutions is nonempty and compact.
Proof: Use partial minimization theory w/ the
function
F x, u
(
) =
f (x)
X, g(x... | https://ocw.mit.edu/courses/6-253-convex-analysis-and-optimization-spring-2012/19c72f4d87303765c3925112f5926282_MIT6_253S12_lec11.pdf |
Lecture 8
Symmetries, conserved quantities, and the labeling of states
Angular Momentum
Today’s Program:
1. Symmetries and conserved quantities – labeling of states
2. Ehrenfest Theorem – the greatest theorem of all times (in Prof. Anikeeva’s opinion)
3. Angular momentum in QM
4. Finding the eigenfunctions of Lˆ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
When one speaks of a symmetry it is critical to state symmetric with
respect to which operation.
How do symmetries manifest themselves in equations?�Let us suppose that your system is
symmetric with respect to translations in x that would imply that any physical property could not
have an x dependence. In particula... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
�
0
d
dt
ˆ
A 0
States are labeled by specific values of their properties, which do not change with time –
these properties are called constants of motion. We learned that in QM physical properties are
represented by operators and that the values of properties obtained in measurements are
eigenvalues of ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
�
r, t
form:
V
t . Let’s substitute it into the Schrodinger’s equation above:
r
r
t i
r t
t
r
2
2m
1
r
2
2m
V
r
r
1
t
i
t
t
r and the right side of the
Note that the l... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
�
2m
2 V
r
r E r uE
x , E
Then the solutions to time-dependent Schrodinger’s equation will have a form:
E
r, t
uE
r E
t uE
t
E
i
r e
In general, since the Hamiltonian may have many eigenvalues and corresponding eigenfunctions,
the solut... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
�
2
2
2m x2
E uE
x
e
x
x
e
i
i
2 mE
2 x
2mE
2
x
Using the energy as a “label” doesn’t completely and uniquely specify a state.
What about momentum? – If momentum is a constant of motion then we can use it as an
additional label to uniquely specify the eigenstate... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
ˆ x x
What are the eigenfunctions and eigenvalues of the parity operator:
ˆ u x ˆ u x
ˆ u x
ˆ ˆ
u x
u x
u x
u x
ˆ
ˆ
2 1
u x
The eigenfunctions of the parity operator all are either odd or even.
f (−x)= f (x) even
f (−x)−= f (x) odd�
Does Ha... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
�� ˆ
, Hˆ
x Hˆ x
Hˆ ˆ
x
ˆ
2
2
2
2m x
m2 x
1
2
2
2
2
x
2
2m x
2
2
2m x
1
2
m2
x
2
2
x
2
2
2
2m x
1
2
m2
2
x x 0
This means that one can always find a set o... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
Conserved Quantities Example II: Angular momentum and spherical symmetry
Consider a system with a spherical symmetry, such as Hydrogen atom:
Hydrogen atom consists of a proton with a positive charge q = e =
1.6×10-19 C an electron charge – q = – e. Consequently they are bound
by a Coulombic potential:
V r
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
we can define a corresponding observable (Hermitian operator):
ˆ
rˆ pˆ
L
iˆ
x
x
iˆ
y
y
iˆ
z
z
i
x
i
y
i
z
ˆ
iˆ Lˆ iˆ Lˆ iˆ Lˆ
L
x
x
y
y
z
z
iˆ iy iz
x
iˆ iz ix
y
iˆ ix iy
z
z
y
x
z
y
x
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
��
py isin
cos
tan
Lˆ
y zˆpˆ x xˆpˆ z icos
sin
tan
Lˆ
z xˆpˆ y yˆpˆ x i
Lˆ2
2
2
2
1
2
tan sin2 2
1
We call this operator the orbital angular momentum since it has a classical equivalent.... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
quantity also obeys the same commutation rules and is therefore also called an spin angular
momentum. The spin angular momentum has no classical equivalent.
One can show that:
0
Lˆ2, Lˆ
z
Therefore we now that one can fin d a set of eigenfunctions common to Lˆ2 and Lˆ
z :
m
m
Lˆ
zYl , mYl ,
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
m
eigenfunctions Yl , are called spherical harmonics:
m
7
Below are the plots of first several spherical harmonics:
Weisstein, Eric W. "Spherical Harmonic." From MathWorld - A Wolfram Web Resource. Used with permission.
8
MIT OpenCourseWare
http://ocw.mit.edu
3.024 Electronic, Optic... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/19da8ffb4044af736d595f30eebe310d_MIT3_024S13_2012lec8.pdf |
Representation Invariants
6.170, Lecture 8
Fall 2005
8.1
Context
What you’ll learn: How to find representation invariants and avoid representation exposure.
Why you should learn this: An understanding of the theory of abstract types helps you avoid whole
classes of nasty, subtle bugs – or at minimum alerts you to thei... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
the values that the type is designed to support. These
are a figment of our imagination. They’re platonic entities that don’t exist as described, but they
are the way we want to view the elements of the abstract type, as clients of the type. For example,
an abstract type for unbounded integers might have the mathemat... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
(index>=0) {
s.deleteCharAt(index);
}
}
public boolean member(char ch) {
return s.indexOf(String.valueOf(ch))!=-1;
}
}
This works just fine. What are the rep values and the abstract values of this type? The abstract
values are easy, and are given purely by the specification – they can be described by the possible
val... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
spaces for the two implementations graphi
cally, drawing arcs from each rep value to the abstract value it represents:
First implementation
Second implementation
representation
R
abstraction
A
representation
R
abstraction
A
s = ""
cs = {}
s = ""
cs = {}
s = "abc"
s = "cba"
cs = {a,b,c}
s = "abc"
s = "... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
to write special cases in the code. Or sometimes we will want
to impose certain properties on the rep to make the code of the operations more efficient or
easier to write. In the first implementation, the implementer decided that the string s should
not contain duplicates. This made it possible to terminate the remove ... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
(r) = r.s != null &&
r.s contains no duplicates
Like specifications, rep invariants can be expressed in various levels of (in)formality. For example,
r.s contains no duplicates could be written as something like i!=j => r.s[i]!=r.s[j], but
it is probably clearer as is.
For our second implementation of CharSet, the ... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
= node && RI(tree,node.left))) &&
((node.right!=null) =>
(node.right.parent = node && RI(tree,node.right)))
This makes sure that the parent, left and right fields are consistent throughout the tree. Written
this way, it is easy to imagine converting the RI to test code. You could also write the RI in many
other way... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
false, then we know that something has gone wrong – the object could not possibly represent
an abstract value. If the rep invariant is true, we know that the object does indeed represent an
abstract value (though not necessarily the right one; this is not a panacea for detecting all bugs).
Here is what we do:
• Cre... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
type, only that it produces a
well-formed instance. When we study abstraction functions in the next lecture, we’ll look at that
issue.)
Our method says that we can consider the operations one by one, and then appeal to induction
to show that every instance will be well-formed. A crucial aspect of this method is loc... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
Date getStop() { return stop; }
public long getDuration() { return duration; }
}
The rep invariant will capture the need for consistency between the duration, start, and stop fields.
But unfortunately this implementation leaks like a sieve because the Java Date class is mutable,
and it is easy for a client to end up ... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
.add(new HighSociety("Lord Vandersnoot"));
posh.add(new HighSociety("Lady Bassington-Bassington"));
//...
Object[] whosWho = posh.toArray();
whosWho[0] = new Ruffian(); // ouch! who left the door open...
A more subtle variant of this problem arises with iterators. Many Java classes have a method that
returns an itera... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
methods to understand what’s going on before you can confidently add a new
method.
• It helps catch errors. By implementing the invariant as a runtime assertion, you can find bugs
that are hard to track down by other means.
8 | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/1a1f80fa85176ed3cae09ccd74f902d3_lec8.pdf |
White blood cell (e.g., neutrophil) scavenging:
rolling, adhesion, and extravasation
© John Wiley & Sons, Inc. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Source: Man, Shumei, Eroboghene E. Ubogu, and Richard M. Ran... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/1a2b50cdc52990a6dac50a4591a53a61_MIT20_430JF15_Lecture19.pdf |
Inc., http://www.sciencedirect.com. Used with permission.
Source: Hochmuth, Robert M. "Micropipette aspiration of living cells." Journal of Biomechanics 33, no. 1 (2000): 15-22.
5
Complex fluids: Silly Putty
Text about Silly Putty removed due to copyright restrictions.
© source unknown. All rights reserved. This ... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/1a2b50cdc52990a6dac50a4591a53a61_MIT20_430JF15_Lecture19.pdf |
-use/.
Drag on rodlike bacteria versus spherical
cells (21:11-25:00)
Falling rods at low Re—difference in drag in
two directions
Does the rod fall vertically, or not?
Why or why not?
© Education Development Center, Inc. All rights reserved. This content is excluded from our
Creative Commons license. For more in... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/1a2b50cdc52990a6dac50a4591a53a61_MIT20_430JF15_Lecture19.pdf |
6.890: Algorithmic Lower Bounds: Fun With Hardness Proofs
Fall 2014
Prof. Erik Demaine
Lecture 5 Scribe Notes
1 Overview
We’ll start this week with a few more examples of classic Nintendo games, most of which are hard
via reductions from 3SAT or a variant. Then, we’ll look at a few more puzzles, including one who... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
can only push a block once. The lock mechanism from
Push-1 can be modified for a Push-Once environment, so Push-Once is also hard.)
Figure 1: The modification of the Push-1 lock to the Push-Once lock prevents Link from pushing
a block more than once (assuming Link does not have a raft) because blocks can’t be pushed... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
do because she can’t shoot down. The cross over gadget allows her to either come from the left
or right. Depending on which tunnel she chose she has to either roll left or roll right to avoid the
monsters that are moving in opposite directions.
Remark: We actually claim NP-completeness for Metroid because the states... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
and only if they’re all satisfied.
(We imagine that trainers have a larger field of vision than they do in the real game, though the
reduction still requires only a bounded constant range.) With this setup, Pokemon is NP-Hard.
4
Figure 5: Variable, clause, and crossover gadget. The purpose of the clause gadget is... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
I to Generation II. In Generations III, IV, and V, Gastly
can be replaced by Duskull, which is allowed to learn the move Memento, which serves the same
purpose as Self Destruct, via breeding.”
5
2.6 Conway’s Phutball (Philosopher’s Football)
Phutball is a 2-player game played with black and white stones on a sq... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
ball can comprise a large number of individual jumps. However...
2.7 Checkers
Checkers mate-in-1 also seems to involve a large number of jumps, but deciding whether a checkers
position is a mate-in-1 is in P: the problem reduces to the question of finding an Eulerian path on
6
a graph, which is very easy to deci... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
opposite va. This is accomplished through a string of intermediate sums:
C = carry(yi + yi) E {0, 1}
(mod 4)
bi = 2ai
vi = 2bi + C
= 4ai + C = C
di = 2ci + C
ei = di + 1 + C
= 2ci + 1 + 2C
vi = di + ei
= 4ci + 1 + 3C
= 3C + 1 = 1 − C
(mod 4)
di 0 1
ei
vi 0 ei
yi 0 ci
0 di yi 0 ci
zi 0 di
yi 0 bi
yi... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
construction can be simplified considerably by instead reducing from 1-in-3SAT, which is still
NP-complete. We don’t need negations anymore, and we can then check a clause by seeing whether
its three variables sum to 1 (mod 4):
8
gi = 2fi
hi = 2gi
= 4fi
ti = hi + 1
= 4fi + 1
va + vb + vc = ti = 1 (mod 4)
It... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
. If the creases locally intersect, then
the pattern is illegal because a piece of paper can’t both be on top and under itself.
We build our reduction out of pleats (two parallel segments). Two parallel creases can’t both be
“mountains” or both be “valleys,” because then the paper would locally self-intersect, so th... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
.
The reduction is from Not-All-Equal-SAT: a triangle of folds represents a clause, and it is foldable
if and only if the three incoming folds are not all of the same type. We can represent negations
using the splitter gadget and the crossover gadget prevents folds from crossing themselves. For
more, see [DO07].
2... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
crossover gadget, which is necessary for ensuring
the graph’s planarity.
2.10.1 What Can Go Wrong
• In order for a region to be fillable, it needs to have an even number of terminals. But Erik’s
original construction left an odd, unpaired terminal.
12
Figure 15: Empty space parity.
• The original gadget was de... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
Zig-Zag Numberlink; another version, “Classic,” was considered in 2010 and is also hard [KT10].
References
[ADGV12] Greg Aloupis, Erik D Demaine, Alan Guo, and G Viglietta. Classic Nintendo games
are (NP-) hard. arXiv preprint arXiv:1203.1895, 2012.
[ADGV14] Greg Aloupis, Erik D Demaine, Alan Guo, and Giovanni Vigl... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
S Fraenkel, MR Garey, David S Johnson, T Schaefer, and Yaacov Yesha. The
complexity of checkers on an n× n board. In Foundations of Computer Science, 1978.,
19th Annual Symposium on, pages 55–64. IEEE, 1978.
[KT10] 古妻浩一 and 武永康彦 . ナンバーリンクの NP 完全性と問題の列挙 . 電子情報通信学
会技術研究報告. COMP, コンピュテーション, 109(465):1–7, 2010.
[Lyn75]... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/1a2ff45b31e8e695010a55eb322a46f9_MIT6_890F14_Lec5.pdf |
ESD.86
Random Incidence
A Major Source of Selection Bias
Richard C. Larson
February 21, 2007
Examples
(cid:139) Waiting for a bus at 77 Mass. Avenue.
– “Clumping”
(cid:139) Interview passengers disembarking
from an airplane.
Doctoral Exam Question
(cid:139) You arrive at a bus stop where busses arrive
according to ... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/1a54fef489a72a6f19f64f10cabbff4a_lec5.pdf |
have a renewal process.
The Gap We Fall Into by Random Incidence
fW(w)dw=P{length of gap is between w and w+dw}
fW(w)dw is proportional to two things:
(1) the relative frequency of gaps [w, w+dw]
(2) the length of the gap w (!!).
Thus, normalizing so we have a proper pdf,
We can write
fW(w)dw =wfY(w)dw/E[Y], or
fW(w) ... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/1a54fef489a72a6f19f64f10cabbff4a_lec5.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Lecture 13
Fall 2018
MOMENT GENERATING FUNCTIONS
Contents
1. Moment generating functions
2. Sum of a random number of random variables
3. Transforms associated with joint distributions
Moment generating functions, and their close relatives (probability gener... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
tools for proving limit theorems, such as laws of
large numbers and the central limit theorem.
1
1 MOMENT GENERATING FUNCTIONS
1.1 Definition
Definition 1. The moment generating function associated with a random
variable X is a function MX : R
[0,
] defined by
∞
→
MX (s) = E[e
sX ].
The domain DX of MX is defi... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
then
3s
MX (s) = e + e + e ,
2s
s
1
2
1
3
1
6
(1)
which is finite for every s
fX (x) = 1/(ˇ(1 + x2)), for all x, it is easily seen that MX (s) =
s
R. On the other hand, for the Cauchy distribution,
, for all
= 0.
∞
∈
In general, DX is an interval (possibly infinite or semi-infinite) that contains
zero.
... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
the other extreme, if we are told that MX (s) =
for every s
is certainly not enough information to determine the distribution of X.
∞
= 0, this
On this subject, there is the following fundamental result. It is intimately
related to the inversion properties of Laplace transforms. Its proof requires so-
phisticated... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
order of integration and differentiation, we obtain
dMX (s)
|
ds
dmMX (s)
|
dsm
s=0
s=0
d
= E[e
ds
dm
dsm
=
|
sX ]
|
sX ]
E[e
s=0
|
= E[XesX ]
s=0
|
sX ]
= E[X m e
s=0
= E[X],
= E[X m]
s=0
3
6
Thus, knowledge of the transform MX allows for an easy calculation ... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
discrete random variables, the following probability generating function
is sometimes useful. It is defined by
gX (s) = E[s X ],
with s usually restricted to positive values. It is of course closely related to the
moment generating function in that, for s > 0, we have gX (s) = MX (log s).
One difference is that whe... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
= Ge(p)
d
1
MX (s) =
sm
e
m=1
X
p(1
p)m−1
−
(
,
∞
In this case, we also find gX (s) = ps/(1
gX (s) =
, otherwise.
∞
se p
1−(1−p)es
s
, e < 1/(1
otherwise.
−
p);
(1
−
p)s), s < 1/(1
p) and
−
−
d
Example : X = N (0, 1). Then,
MX (s) =
exp(sx) exp(
1
1
√
2ˇ −1
Z
exp(s2/2)
2ˇ
√
= exp(s 2... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
sb)] = exp(sb)E[exp(saX)] = exp(sb)MX (as).
5
For part (b), we have
MX+Y (s) = E[exp(sX + sY )] = E[exp(sX)]E[exp(sY )] = MX (s)MY (s).
For part (c), by conditioning on the random choice between X and Y , we have
MZ(s) = E[e
sZ] = pE[e
sX ] + (1
−
p)E[e sY ] = pMX (s) + (1
−
p)MY (s).
Example : (Normal... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
, . . . be a sequence of i.i.d. random variables, with mean µ and vari-
ance ˙2 . Let N be another independent random variable that takes nonnegative
N
integer values. Let Y =
i=1 Xi. Let us derive the mean, variance, and mo-
ment generating function of Y .
We have
P
E[Y ] = E[E[Y
|
N ]] = E[N µ] = E[N ]E[X].
F... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
)/(
1
−
(1
−
=
p
p
s
−
s)
−
which we recognize as a moment generating function of an exponential random variable
with parameter p. Using the inversion theorem, we conclude that Y is exponentially
distributed. In view of the fact that the sum of a fixed number of exponential random
variables is far from... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
, if Y1, . . . , Yn is another set of random variables and
MX1,...,Xn (s1, . . . , sn), MY1,...,Yn (s1, . . . , sn) are the same functions of s1, . . . , sn,
in a neighborhood of the origin, then the joint distribution of X1, . . . , Xn is the
same as the joint distribution of Y1, . . . , Yn.
7
Remarks:
(a)... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/1a592ed184fb4c444547f67c9bcdd8ec_MIT6_436JF18_lec13.pdf |
13.811
Advanced Structural
Dynamics and Acoustics
Acoustics
Lecture 5
13.811
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS
Lecture 5
Ewald Sphere Construction
Baffled Piston
Directivity Function
f = ω/2π=kc/2π
Radiating Spectrum
Evanescent
Spectrum
13.811
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS
Lecture 5
circ.m
%
%... | https://ocw.mit.edu/courses/2-067-advanced-structural-dynamics-and-acoustics-13-811-spring-2004/1a6a081bf8c514c5a812db5d121071b2_lect_5_4.pdf |
phi-1)*dphi]' * ones(1,nth);
th=([dth/2:dth:pi/2]'*ones(1,nphi))';
kx=ka*sin(th).*cos(phi);
ky=ka*sin(th).*sin(phi);
kr=ka*sin(th);
ss=rho*k*c*a^2*besselj(1,kr)./kr;
ss=dba(ss);
sm=max((max(ss))');
for i=1:size(ss,1)
for j=1:size(ss,2)
ss(i,j)=max(ss(i,j),sm-40.0)-(sm-40.0);
end
end
xx=ss.*sin(th).*cos(phi);
yy=ss.*sin... | https://ocw.mit.edu/courses/2-067-advanced-structural-dynamics-and-acoustics-13-811-spring-2004/1a6a081bf8c514c5a812db5d121071b2_lect_5_4.pdf |
umber
% k
% rho
Density
% c Speed of Sound
% a Radius of piston of piston
% d Piston separation
Number of pistons
% nd
Array of piston strengths
% qd
clear
figure(1)
hold off
k=10.0;
rho=1000;
c=1500;
a=1.0;
% Half wavelength spacing, d= pi/k
d=pi/k;
nd=10;
ah=(nd-1)*d/2
xd=[-ah:d:ah]';
kxd_0=k/2;
qd=ones(length(xd),1)... | https://ocw.mit.edu/courses/2-067-advanced-structural-dynamics-and-acoustics-13-811-spring-2004/1a6a081bf8c514c5a812db5d121071b2_lect_5_4.pdf |
*pi/(nth-0.5);
phi=[0:dphi:(nphi-1)*dphi]' * ones(1,nth);
th=([dth/2:dth:pi/2]'*ones(1,nphi))';
kx=ka*sin(th).*cos(phi);
ky=ka*sin(th).*sin(phi);
kr=ka*sin(th);
ss=rho*k*c*a^2*besselj(1,kr)./kr;
kx1=reshape(kx,1,size(kx,1)*size(kx,2));
shd=qd'*exp(-i*xd*kx1);
shd=reshape(shd,size(kx,1),size(kx,2));
ss=dba(ss.*shd);
sm=... | https://ocw.mit.edu/courses/2-067-advanced-structural-dynamics-and-acoustics-13-811-spring-2004/1a6a081bf8c514c5a812db5d121071b2_lect_5_4.pdf |
ICS
Lecture 5
Point-Driven Plate Radiation
z
Hankel Transforms
r
θ
y
x
h
Light Fluid Loading
Cylindrical Coordinates
Flexural Wavenumber
x
Hankel Transform
Particle Velocity
13.811
ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS
Lecture 5
z
Radiated Field
r
θ
y
x
h
Inverse Hankel Transform
I = 2 π i Σ Res
Complex C... | https://ocw.mit.edu/courses/2-067-advanced-structural-dynamics-and-acoustics-13-811-spring-2004/1a6a081bf8c514c5a812db5d121071b2_lect_5_4.pdf |
6.087 Lecture 6 – January 19, 2010
Review
User defined datatype
Structures
Unions
Bitfields
Data structure
Memory allocation
Linked lists
Binary trees
1
Review: pointers
• Pointers: memory address of variables
•
’&’ (address of) operator.
• Declaring: int x=10; int ∗ px= &x;
• Dereferencing: ∗px=20;
•
Po... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/1aa34decd30f336826cd087d2f52035b_MIT6_087IAP10_lec06.pdf |
f e r e n t
t y p e ∗ /
4
Structure
• struct defines a new datatype.
• The name of the structure is optional.
struct {...} x,y,z;
•
The variables declared within a structure are called its
members
• Variables can be declared like any other built in data-type.
struct point ptA;
• Initialization is done by speci... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/1aa34decd30f336826cd087d2f52035b_MIT6_087IAP10_lec06.pdf |
i n t
t l y = r e c t . t l . y ;
i n t
/ ∗ nested ∗ /
7
Structure pointers
• Structures are copied element wise.
• For large structures it is more efficient to pass pointers.
void foo(struct point ∗ pp); struct point pt ; foo(&pt)
• Members can be accesses from structure pointers using
’->’ operator.
s t r u c... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/1aa34decd30f336826cd087d2f52035b_MIT6_087IAP10_lec06.pdf |
u c t {
char c ;
/ ∗ padding ∗ /
i ;
i n t
• Why is this an important issue? libraries, precompiled files,
SIMD instructions.
• Members can be explicitly aligned using compiler
extensions.
__attribute__ (( aligned(x ))) /∗gcc∗/
__declspec((aligned(x))) /∗MSVC∗/
10
Union
A union is a variable that may hold obj... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/1aa34decd30f336826cd087d2f52035b_MIT6_087IAP10_lec06.pdf |
��elds vs. masks
CLR=0x1,SND=0x2,NTSC=0x4; struct flag f ;
x|= CLR; x|=SND; x|=NTSC
x&= ~CLR; x&=~SND;
if (x & CLR || x& NTSC)
f .has_sound=1;f.is_color=1;
f .has_sound=0;f.is_color=0;
if ( f . is_color || f .has_sound)
13
6.087 Lecture 6 – January 19, 2010
Review
User defined datatype
Structures
Unions
Bit... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/1aa34decd30f336826cd087d2f52035b_MIT6_087IAP10_lec06.pdf |
s t r u c t node∗ n e x t ;
} ;
s t r u c t node∗ head ; / ∗ b e g i n n i n g ∗ /
Linked list vs. arrays
size
indexing O(n)
inserting O(1)
deleting O(1)
linked-list array
fixed
dynamic
O(1)
O(n)
O(n)
16
Linked list
Creating new element:
s t r u c t node∗ n a l l o c ( i n t data )
{
s t r u c t node∗ p... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/1aa34decd30f336826cd087d2f52035b_MIT6_087IAP10_lec06.pdf |
than the root element and all elements in the right
subtree are assumed to be "greater" than the root element.
20
31802695 Binary tree (cont.)
s t r u c t tnode
{
/ ∗ payload ∗ /
i n t data ;
s t r u c t tnode ∗
s t r u c t tnode ∗ r i g h t ;
l e f t ;
} ;
The operation on trees can be framed as recursive o... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/1aa34decd30f336826cd087d2f52035b_MIT6_087IAP10_lec06.pdf |
about citing these materials or our Terms of Use,visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/1aa34decd30f336826cd087d2f52035b_MIT6_087IAP10_lec06.pdf |
Lecture 8
MOSFET(I)
MOSFET I-V CHARACTERISTICS
Outline
1. MOSFET: cross-section, layout,
symbols
2. Qualitative operation
I-V characteristics
3.
Reading Assignment:
Howe and Sodini, Chapter 4, Sections 4.1-4.3
6.012 Spring 2009
Lecture 8
1
1. MOSFET: layout, cross-section, symbols
active area (thin
int... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/1aadf127cb5e71ec1577aa244e2b3773_MIT6_012S09_lec08.pdf |
the electric
field that drifts the inversion charge fiom the source
to drain
Want to understand the relationship between the drain
current in the MOSFET as a function of gate-to-source
voltage and drain-to-source voltage.
Initially consider source tied up to body (substrate or
6.012Spring 2009
Lecture 8
MOSFE... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/1aadf127cb5e71ec1577aa244e2b3773_MIT6_012S09_lec08.pdf |
VGS − V (y) − VT
for VGS – V(y) ≥ VT. .
Note that we assumed that VT is independent of y.
See discussion on body effect in Section 4.4 of text.
All together the drain current is given by:
[
ID = W • µµµµnCox VGS − V( y) − VT
]•
dV(y)
dy
Simple linear first order differential equation with one
un-known, the cha... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/1aadf127cb5e71ec1577aa244e2b3773_MIT6_012S09_lec08.pdf |
VDS − VT
• VDS
2
for VDS < VGS − VT
Key dependencies:
• VDS↑ → ID↑ (higher lateral electric field)
• VGS↑ → ID↑ (higher electron concentration)
This is the linear or triode region:
It is linear if VDS<<VGS-VT
6.012 Spring 2009
Lecture 8
11
I-V Characteristics (Contd….)
Two important observations
1... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/1aadf127cb5e71ec1577aa244e2b3773_MIT6_012S09_lec08.pdf |
VT
IDsat
=
1 W
2 L
µµµµnCox[VGS − VT ]2
Will talk more about saturation region next time.
6.012 Spring 2009
Lecture 8
14
I-V Characteristics (Contd…….)
Output Characteristics
Transfer characteristics:
6.012 Spring 2009
Lecture 8
15
Output Characteristics
6.012 Spring 2009
Lecture 8
16
Summary... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/1aadf127cb5e71ec1577aa244e2b3773_MIT6_012S09_lec08.pdf |
Lecture Six: More General Operators
1 The Weak definition of a harmonic function
It is sometimes useful to weaken the notion of a harmonic function somewhat. One way
of doing this is with the notion of a weakly harmonic function. Let u be a differentiable
function on some set Ω. We say that u is weakly harmonic on Ω ... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2005/1ac3c08d20609c08ab021ef33149291f_lecture6.pdf |
by n matrix with entries aij (not necessarily constant). An
operator written like this is said to be in divergence form since Lu = div(A�u). Note
that if A is the identity matrix then L is simply the laplacian. We will often be interested
in functions satisfying Lu = 0. Such functions are called Lharmonic. There is... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2005/1ac3c08d20609c08ab021ef33149291f_lecture6.pdf |
|� | ≤
4Λ2
λ2r
2
�
B2r \Br
2
u .
Proof Again we start off by introducing a test function φ with φ ≥ 0 and φ = 0 on the
boundary of B2r . Calculate
�
B2r
�
0 ≤
≤
φ2uLu
φ2 u(� · A�
u)
B2r
�
≤ −
B2r
�
≤ −2
< �(φ2 u), A�u >
φu < �φ, A�u > −
�
B2r
B2r
2
φ2 < �u, A�u >
by Stokes’ theorem. Therefore
�
... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2005/1ac3c08d20609c08ab021ef33149291f_lecture6.pdf |
�u >
(9)
.
�
λ
B2 r
�
λ
B2r
Applying uniform ellipticity and rearranging gives
φ2 < �u, �u > ≤ 2Λ
��
2
u < �φ, �φ >
�1/2 ��
Divide and square to get
B2r
B2r
�
λ2
B2 r
φ2
|�u|
2 ≤ 4Λ2
�
B2r
2u
|�φ|2
.
�
1
2r−|x|
r
if |x| ≤ r;
if r < x ≤ 2r
.
φ(x) =
Then
�
2
|�u| ≤
�
Br
B2r
4Λ2
λ2
4Λ2
λ2r
2
≤... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2005/1ac3c08d20609c08ab021ef33149291f_lecture6.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.727 Topics in Algebraic Geometry: Algebraic Surfaces
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
ALGEBRAIC SURFACES, LECTURE 3
LECTURES: ABHINAV KUMAR
1. Birational maps continued
Recall that the blowup ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/1ac4b82daa4485674ad67c21b5a49e7a_lect3.pdf |
a homogeneous polynomial of degree k. In a
˜
U P1, we have local coordinates x
neighborhood of (p, = [1 : 0]) U
and t = x and π∗f = f (x, tx) = xmfm(1, t) + xm+1fm+1(1, t) +
), giving the
�
desired formula.
⊂ ×
· · ·
∞
∈
y
˜
˜
·
·
→
π∗ : Pic X →
= Pic X⊕Z.
Pic X and Z
Theorem 1. We have maps
→ Pic X, 1 ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/1ac4b82daa4485674ad67c21b5a49e7a_lect3.pdf |
INAV KUMAR
·
·
·
as desired. Moreover, since C (possibly after moving) does not pass through
p, (π∗C) E = 0. Next, taking a curve passing through p with multiplicity
1, its strict transform meets E transversely at one point which corresponds to
the tangent direction of p ∈ C, i.e. C˜ E = 1 and C˜ = π∗C − E. Sinc... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/1ac4b82daa4485674ad67c21b5a49e7a_lect3.pdf |
π is an isomorphism away from E, π : X˜ � E → X � {p
}, so it is clear
i
that OX → π∗O ˜ is an isomorphism except possibly at p, and R π∗OX˜ can only
be supported at p. By the theorem on formal functions, the completion at p
of this sheaf is R� = lim H i(E ,
iπ∗OX˜
n OEn
is the closed subscheme
I n I the
→
defin... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/1ac4b82daa4485674ad67c21b5a49e7a_lect3.pdf |
1) = ⇒
), where E
n
I n/I n+1
∼
2
This implies that the irregularity qX = h1(X, OX ) = q ˜ and geometric genus
X
pg(X) = h2(X, OX ) = pg(X˜ ) are invariant under blowup.
Let X, Y be varieties, X irreducible.
2. Rational maps
Definition 1. A rational map X ��� Y is a morphism φ from an open subset U
of X to Y ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/1ac4b82daa4485674ad67c21b5a49e7a_lect3.pdf |
on the set of zeroes and
poles of φ). If C is an irreducible curve on X, φ is defined on C �(C ∩F ), and we
can set φ(C) = φ(C � (C ∩ F )) (and similarly φ(X) = φ(X � F )). Restriction
gives us an isomorphism between Pic (X) and Pic (X � F ), so we can talk about
the inverse image of a divisor D (or line bundle, or ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/1ac4b82daa4485674ad67c21b5a49e7a_lect3.pdf |
. If p
has no fixed part, then it has finitely many base points (at most (D2)).
4. Properties of Birational Maps between Surfaces
(1) Elimination of indeterminacy
(2) Universal property of blowing up
(3) Factoring birational morphisms
(4) Minimal surfaces
(5) Castelnuovo’s contraction theorem
Theorem 3. Let φ : S... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/1ac4b82daa4485674ad67c21b5a49e7a_lect3.pdf |
fixed component, and defines a rational map
φ1 : X1 ��� Pm which coincides with φ π. If φ1 is a morphism, we are done;
◦
∗ Dn−1 −
otherwise, repeat the process. We obtain a sequence of divisors Dn = πn
�
2
−1 − k2 < Dn
2
2 = Dn
−1, which must terminate.
k E n n = ⇒ 0 ≤ Dn
Theorem 4. Let f : X ��� S be a birationa... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/1ac4b82daa4485674ad67c21b5a49e7a_lect3.pdf |
g1 ∈/ OS�,p. Let
g1 = u
S ,p are coprime and v(p) = 0. Let D be defined on S by
f ∗v = 0. On S we have f ∗u = (f ∗v)x1 (where x1 is the first coordinate function
on S ⊂ An) (because it is true under (f −1)∗): (f −1)∗f ∗u = (f −1)∗f ∗v (f −1)∗x1,
k(S) = k(S�). So f ∗u = f ∗v = 0 on D, and D = f −1(Z) where Z is the sub... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/1ac4b82daa4485674ad67c21b5a49e7a_lect3.pdf |
MIT 6.972 Algebraic techniques and semidefinite optimization
March 2, 2006
Lecturer: Pablo A. Parrilo
Scribe: ???
Lecture 7
In this lecture we introduce a special class of multivariate polynomials, called hyperbolic. These
polynomials were originally studied in the context of partial differential equations. As we w... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/1ae3402ea8aa2ab1edf9c607eeb0ea07_lecture_07.pdf |
vectors x ∈ Rn, the univariate polynomial t �→ p(x − te) has only real roots.
A natural geometric interpretation is the following: consider the hypersurface in Rn given by p(x) = 0.
Then, hyperbolicity is equivalent to the condition that every line in Rn parallel to e intersects this
hypersurface at exactly d points... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/1ae3402ea8aa2ab1edf9c607eeb0ea07_lecture_07.pdf |
It is immediate from homogeneity and the definition above that λ > 0, x ∈ Λ++
arding [G˚
ar59].
Lemma 3. The hyperbolicity cone Λ++ is the connected component of p(x) > 0 that includes e.
Example 4. The hyperbolicity cone Λ++ associated with the polynomial x1x2 · · · xn discussed in Exam
ple 2 is the open positive or... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/1ae3402ea8aa2ab1edf9c607eeb0ea07_lecture_07.pdf |
the γ = 0
case first. It is clear that β �→ p(αie + βv) cannot have a root at β = 0, since p(αie) = (αi)dp(e) = 0. If
β = 0, we can write
p(αie + βv) = 0
⇔
p(αβ−1ie + v) = 0
⇒
αβ−1i < 0 ⇒ β ∈ iR−,
and thus the roots of this polynomial are on the strict negative imaginary axis (we have used v ∈ Λ++ in
the second i... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/1ae3402ea8aa2ab1edf9c607eeb0ea07_lecture_07.pdf |
u, v ∈ Λ++, β, γ > 0 implies that βu + γv ∈ Λ++. The previous result
implies that we can always assume v = e. But then the roots of t �→ p(βu+γe−te) are just a nonnegative
affine scaling of the roots of t �→ p(u − te), since
p(u − t�e) = 0
⇔
p(βu + γe − (βt� + γ)e) = 0,
and u ∈ Λ++ implies that t� > 0, hence βt� + ... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/1ae3402ea8aa2ab1edf9c607eeb0ea07_lecture_07.pdf |
. The corresponding
hyperbolicity cone is the Lorentz or second order cone given by
k=1
k=1
�
Λ+ =
x ∈ Rn+1
| xn+1 ≥ 0,
n
�
2
xk ≤ xn+1
2
�
.
Example 8 (SDP). Consider the homogeneous polynomial
k=1
p(x) = det(x1A1 + · · · + xnAn),
where Ai
degree d. Letting e = (1, 0, . . . , 0), we have
∈ S
d are given... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/1ae3402ea8aa2ab1edf9c607eeb0ea07_lecture_07.pdf |
Rn .
Lemma 10. The hyperbolicity cone Λ+ is basic closed semialgebraic, i.e., it can be described by unquan
tified polynomial inequalities.
The two following results are of importance in optimization and the formulation of interiorpoint
methods.
Theorem 11. A hyperbolic cone Λ+ is facially exposed.
Theorem 12 ([G¨... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/1ae3402ea8aa2ab1edf9c607eeb0ea07_lecture_07.pdf |
semidefinite cone, i.e., it can be represented as the intersection of an affine
subspace and S n
+. As we will see in the next lecture, a special case of the conjecture has been settled
recently.
2
SDP representability
Recall that in the previous lecture, we encountered a class of convex sets in R2 that lacked certa... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/1ae3402ea8aa2ab1edf9c607eeb0ea07_lecture_07.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.341: Discrete-Time Signal Processing
OpenCourseWare 2006
Lecture 5
Sampling Rate Conversion
Reading: Section 4.6 in Oppenheim, Schafer & Buck (OSB).
It is often necessary to change the sampling rate of a discrete-tim... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/1b110cd38e3cbd3efd56f43c9ddf8b2a_lec05.pdf |
0 ≤ i ≤ M − 1
= ⇒ Xd(ejω) =
�
1
T
1
M
M −1
�
i=0
� �
Xc j
−
2πk
T
−
���
2πi
M T
ω
M T
∞
�
k=−∞
= X(e
j(ω−2πi)/M )
= Xd(ejω) =
⇒
M −1
1 �
M
i=0
X(ej(ω−2πi)/M ) .
As an example, the following figure illustrates decimation by M = 2 in the time domain. We
see that re-sampling the continuous signal... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/1b110cd38e3cbd3efd56f43c9ddf8b2a_lec05.pdf |
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