text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
𝑥𝑥" + 𝑘𝑦𝑦"]} 𝑑𝑘𝑥𝑑𝑘𝑦
𝑃𝑆𝐹(𝑥", 𝑦") ≈ [𝑎𝑠𝑖𝑛𝑐 (
𝑥)] [𝑏𝑠𝑖𝑛𝑐 (
𝑦)]
=[𝑎𝑠𝑖𝑛𝑐 (−
𝑎𝑘
𝑓2
𝑥")] [𝑏𝑠𝑖𝑛𝑐 (−
𝑎𝑘
𝑓1
𝑏𝑘
𝑓2
𝑓1
𝑓2
𝑏𝑘
𝑓1
𝑦")]
(10)
3
a/λf1b/λf1ATFPSF
Lecture Notes on Wave Optics (04/07/14)
2.71/2.710 Introduction t... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
ating field.
For example, liquid crystal SLMs control the amplitude and phase of the transmitted
or reflected light. Likewise, TI’s DLP SLM uses arrays of deformable micromirrors
made by MEMS technology to adjust the amplitude and phase of the reflected light.
The basic setup for optical spatial filtering is a tele... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
)
2.71/2.710 Introduction to Optics –Nick Fang
(c) Step and Repeat operation
C. The significance of PSF and ATF
The theory of optical imaging and communication has a lot in common. The above
imaging process might be modeled with an equivalent circuit. (Such analogy has
stimulated research and development for basi... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
object field with point spread function (PSF). Correspondingly, the
Amplitude Transfer Function (ATF) is the Fourier transform of PSF:
𝐸𝑖𝑚𝑎𝑔𝑒(𝑥", 𝑦") = 𝐸𝑜𝑏𝑗𝑒𝑐𝑡 (−
𝐴𝑇𝐹(𝑘𝑥, 𝑘𝑦) = ∫ ∫ 𝑃𝑆𝐹(𝑥, 𝑦) exp(𝑖𝑘𝑥𝑥 + 𝑖𝑘𝑥𝑦) 𝑑𝑥𝑑𝑦
𝑦) ⨂𝑃𝑆𝐹(𝑥, 𝑦)
𝑥, −
𝑓2
𝑓1
𝑓2
𝑓1
(15)
(16)
e.g. ATF o... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
Lecture Notes on Wave Optics (04/07/14)
2.71/2.710 Introduction to Optics –Nick Fang
Simulated intensity pattern of a 5x5 checkerboard illuminated by a light source with different
coherence. (left)100% coherent; (middle)50% coherent; (right) non-coherent.
© Source unknown. All rights reserved. This content is exclu... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
mmRNA: 0.00.05666 mm.05666 mm .002160.1749.08855DLP Projection Image AERIAL IMAGEField = ( 0.000, 0.000) DegreesDefocusing = 0.000000 mmRNA: 1.000.1138 mm0.1138 mm .000840.1876.09420DLP Projection Image AERIAL IMAGEField = ( 0.000, 0.000) DegreesDefocusing = 0.000000 mmRNA: 1.00protrusionphase-shiftscoherent illuminati... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
shadowgraph imaging is important in the
visualization of fluid flows, as it shows phase gradients of the object in a
particular direction. To elaborate that effect, let’s model the transmission
function of the phase mask (e.g. a glass wedge or spiral plate) as following:
AS(𝑥′, 𝑦′) ≈ 1 + 𝑖𝑘(∆𝑛)(𝑥′/𝑎)
(23)
... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
𝑥", 𝑦") ≈ 𝐸𝑜𝑏𝑗𝑒𝑐𝑡(𝑥, 𝑦) + (∆𝑛)
𝑓1
𝑎
𝜕
𝜕𝑥
𝐸𝑜𝑏𝑗𝑒𝑐𝑡(𝑥, 𝑦)
𝐸𝑖𝑚𝑎𝑔𝑒(𝑥", 𝑦") ≈ 𝐸𝑜𝑏𝑗𝑒𝑐𝑡(𝑥, 𝑦) [1 + 𝑖(∆𝑛)
𝑓1
𝑎
𝜕
𝜕𝑥
𝜙(𝑥, 𝑦)]
(26)
(27)
Note that using a mask with phase gradient, the intensity fringes of image are
connected to the index gradient of the fluid flow! Such e... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/4b53a5747f9b58b9b73b2bbcc39945f0_MIT2_71S14_lec16_notes.pdf |
LECTURE 3
Matroids and geometric lattices
3.1. Matroids
A matroid is an abstraction of a set of vectors in a vector space (for us, the normals
to the hyperplanes in an arrangement). Many basic facts about arrangements
(especially linear arrangements) and their intersection posets are best understood
from the more... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
ne space, where independence now
means affine independence.
≤
≤
It should be clear what is meant for two matroids M = (S, I) and M � = (S� , I�)
I
to be isomorphic, viz., there exists a bijection f : S
I� . Let M be a matroid and S a set of points in
if and only if
Rn, regarded as a matroid with independence meani... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
[5] except 123 and 345 (where 123 is short for
and
{
3, 4, 5
1, 2, 3
}
(b) Write I = S1, . . . , Sk for the simplicial complex I generated by S1, . . . , Sk,
{
, etc.).
}
i.e.,
�
�
S1, . . . , Sk =
�
�
T : T
{
= 2S1
∗
∅ · · · ∅
Si for some i
}
2Sk
.
�
Then I = 13, 14, 23, 24 is the set of independent sets of ... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
is isomorphic to the set of nonzero vectors in the vector space F3
2, where
F2 denotes the two-element field.
010
110
011
111
100
101
001
Let us now define a number of important terms associated to a matroid M .
A basis of M is a maximal independent set. A circuit C is a minimal dependent
set, i.e., C is not i... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
subset T
S to be the smallest flat containing T , i.e.,
⊕
∗
T =
F.
flats F ∅T
⎦
This closure operator has a number of nice properties, such as T = T and T �
T
�
T
T .
⊆
∗
∗
3.2. The lattice of flats and geometric lattices
For a matroid M define L(M ) to be the poset of flats of M , ordered by inclusion.
Since... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
ius function of L(M ) and m = rk(M ). Figure 2 shows the
lattice of flats of the matroid M of Figure 1. From this figure we see easily that
ψM (t) = t3
5t2 + 8t
−
4.
Let M be a matroid and x
¯
�
) = 0)
≤
}
then we call x a loop. Thus
M ,
is just the set of loops of M . Suppose that x, y
) = 1. We then call x an... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
M). Thus insofar as intersection lattices L(M )
Then
are concerned, we may assume that M is simple. (Readers familiar with point set
topology will recognize the similarity between the conditions for a matroid to be
simple and for a topological space to be T0.)
�
�
�
Example 3.8. Let S be any finite set and V a vect... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
central arrangement in the vector space V . Define
a matroid M = MA on A by letting B
I(M ) if B is linearly independent (i.e.,
nH : H
{
A has a nonzero normal. Two distinct
Proof. M has no loops, since every H
≤
nonparallel hyperplanes have linearly independent normals, so the points of M are
closed. Hence M i... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
all x, y
(2) If x and y both cover x
≤
L, we have rk(x) + rk(y)
rk(x
y) + rk(x
y).
y, then x
∈
⇒
⊂
⇒
y covers both x and y.
∈
Proof. Assume (1). Let x, y � x
rk(x
y) > rk(x) = rk(y). By (1),
⇒
y, so rk(x) = rk(y) = rk(x
y) + 1 and
∈
∈
rk(x) + rk(y)
rk(y)
y
x
⊆
⊆
(rk(x)
rk(x
1) + rk(x
1
−
y)
⇒
−
⊂
⊂
�... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
is both semimodular and atomic.
≤
≤
To illustrate these definitions, Figure 3(a) shows an atomic lattice that is not
semimodular, (b) shows a semimodular lattice that is not atomic, and (c) shows a
graded lattice that is neither semimodular nor atomic.
We are now ready to characterize the lattice of flats of a matroi... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
maximality
rk(
it follows that
�
y
S but y
of T . Hence M = (A, I) is a matroid, and L ∪= L(M ).
T <
T . But then rk(
T � [why?]. Since L is atomic, there exists y
Conversely, given a matroid M , which we may assume is simple, we need to
show that L(M ) is a geometric lattice. Clearly L(M ) is atomic, since ev... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
any results we prove about geo
metric lattices hold a fortiori for the intersection lattice LA of a central arrangement
A.
T . Then either #(B
Note. If L is geometric and x
y in L, then it is easy to show using semi-
modularity that the interval [x, y] is also a geometric lattice. (See Exercise 3.) In
general, ho... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
gives a “shortening” of the recurrence (2) defining µ.
Normally we take a to be an atom, since that produces fewer terms in (25) than
choosing any b > a. As an example, let L = Bn, the boolean algebra of all subsets
Bn such that x a = ˆ1 = [n],
of [n], and let a =
{
viz., x1 = [n
0, x1] = Bn−1 and
[ˆ
0, x2] = Bn,... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
result
for geometric lattices of rank < n, and let rk(L) = n. Let a be an atom of L in
Theorem 3.9. For any y
L we have by semimodularity that
−
≤
rk(y
∈
a) + rk(y
a)
⇒
→
rk(y) + rk(a) = rk(y) + 1.
⇔
38
R. STANLEY, HYPERPLANE ARRANGEMENTS
Hence x a = ˆ
⇒
From Theorem 3.9 there follows
1 if and only i... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
. Then the characteristic polynomial
ψM (t) strictly alternates in sign, i.e., if
then (
1)n−iai > 0 for 0
−
i
→ →
n.
ψM (t) = antn + an−1tn−1 +
+ a0,
· · ·
Let A be an n-dimensional arrangement of rank r. If MA is the matroid
corresponding to A, as defined in Proposition 3.6, then
(26)
ψA(t) = tn−r ψM (t).
It ... | https://ocw.mit.edu/courses/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004/4b5dc2db132d782df7a9a5ca665c94da_lec3.pdf |
Transient Analysis of First Order RC and RL circuits
The circuit shown on Figure 1 with the switch open is characterized by a particular
operating condition.
Since the switch is open, no current flows in the circuit (i=0) and vR=0. The voltage
across the capacitor, vc, is not known and must be defined. It could be ... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
is closed, current begins to flow in the circuit and we would like
At time
to obtain the form of the voltage vc as a function of time for t>0. Since the voltage across
the capacitor must be continuous the voltage at
is also Vo.
+=
0
t
Our first task is to determine the equation that describes the behavior of this... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
se
st
st
Ae+
=
0
(
RC s
+
)1
st
Ae
0
=
The only non-trivial solution of Equation (0.5) follows from
(
)1
RC s + =
0
This is called the characteristic equation of the system. Therefore s is
And the solution is
s
= −
1
RC
vc t
( )
=
Ae
t
−
RC
t
−
Ae τ
=
(0.4)
(0.5)
(0.6)
(0.7)
(0.8)
The constant A may now be d... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
R -
i(t)
L
+
vL
-
(b)
Figure 6
Our goal is to determine the form of the current i(t).
We start by deriving the equation that describes the behavior of the circuit for t>0. KVL
around the mesh of the circuit on Figure 6(b) gives.
( )
v t
R
+
v t
L
( ) 0
=
(0.11)
Using the current voltage relationship of the resis... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
otakis and Cory
5
RC and RL circuits with multiple resistors.
The capacitor of the circuit on Figure 8 is initially charged to a voltage Vo. At time t=0
the switch is closed and current flows in the circuit. The capacitor sees a Thevenin
equivalent resistance which is... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
and taking into
account the operational characteristics of the inductor at equilibrium. Since under DC
conditions the inductors act as short circuits the corresponding circuit becomes
) current flowing in the circuit we consider the circuit on
t
+=
0
4R
t<0
Io
0.5R
2R
Vs
+
-
2R
Figure 12
6.071/22.071 Spring 2006, ... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
to a step
function for the source voltage Vs as shown on Figure 16.We would like to obtain the
capacitor voltage vc as a function of time. The voltage across the capacitor at t=0 (the
initial voltage) is Vo.
R
i
t=0 + vR -
C
+
vc
-
Figure 15
+
Vs
-
Vs
Figure 16
t
The equation that describes the system is obtained... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
is found by inspection to be
And thus the total solution becomes
cpv
Vs=
t
−
Vs Ae τ
+
cv t
( )
=
The constant A may now be determined by considering the initial condition of the
capacitor voltage. The initial capacitor voltage is Vo and thus A=Vo-Vs.
And the complete solution is
cv t
( )
=
Vs
+
t
−
(
Vo Vs e τ
−
... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
.
tvc = = . We also know that after a long time
0
0
The solution for t1>t>0 is
cv t
( )
=
Vs
(1
−
t
−
e τ
)
(0.32)
For t>t1 the solution is determined by considering as the initial condition, the voltage
across the capacitor at t=t1.
And the solution for t>t1 is
cv t
( 1)
=
Vs
(1
−
1
t
−
e τ
)
cv t
( )
=
Vs
(1
−... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
t → ∞
(final value)
6. The complete solution is:
[
solution = final value + initial value - final value
]
t
−
e τ
vc t
( )
=
vc
(
t
→∞
)
+
⎡
⎣
vc
(
t
=
+
0 )
−
vc
(
t
→∞
)
t
−
e τ
⎤
⎦
i
L
t
( )
=
i
(
L t
→∞
)
+
i
L
⎡
⎣
(
t
=
+
0 )
−
i
(
L t
→∞
)
t
−
e τ
⎤
⎦
6.071/22.071 Spring 2006, Chaniotakis and Cory
13
... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
≤
t
1
For
t
>
t
1
(0.35)
The plot of vc(t) is shown on Figure 22 for 1
R
=
R τ
2, 1 2( 2)
=
τ
and 1 2( 1)
τ=
t
Time constant 1τ
Time constant
2
τ
1
τ=
2
( 1)
Figure 22
6.071/22.071 Spring 2006, Chaniotakis and Cory
14
Problems.
The fuse element is a resistor of resistance Rf which is dest... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/4b6a5fe626d65de20ceb97d8d23f38a4_transient1_rl_rc.pdf |
6.087 Lecture 9 – January 22, 2010
Review
Using External Libraries
Symbols and Linkage
Static vs. Dynamic Linkage
Linking External Libraries
Symbol Resolution Issues
Creating Libraries
Data Structures
B-trees
Priority Queues
1
Review: Void pointers
• Void pointer – points to any data type:
int x; void ∗ px... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
fp )( str1 , str2 );
3
Review: Hash tables
• Hash table (or hash map): array of linked lists for storing
and accessing data efficiently
• Each element associated with a key (can be an integer,
string, or other type)
• Hash function computes hash value from key (and table
size); hash value represents index into a... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
main(), puts(), others in stdio.h
6
Functions and variables as symbols
• Let’s compile, but not link, the file hello.c to create hello.o:
athena% gcc -Wall -c hello.c -o hello.o
1
• -c: compile, but do not link hello.c; result will compile the
code into machine instructions but not make the program
executable
• ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
1
Athena is MIT's UNIX-based computing environment. OCW does not provide access to it.
9
Functions and variables as symbols
• Let’s look at the symbols now:
athena% nm hello
1
• Output:
(other default symbols)
.
. .
0000000000400524 T main
000000000040062c R msg
U puts@@GLIBC_2.2.5
• Addresses for static (allo... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
1
Athena is MIT's UNIX-based computing environment. OCW does not provide access to it.
12
Static linkage
• At link time, statically linked symbols added to executable
• Results in much larger executable file (static – 688K,
dynamic – 10K)
• Resulting executable does not depend on locating external
library files at ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
:
void ∗ dlopen(const char ∗file, int mode);
values for mode: combination of RTLD_LAZY (lazy loading
of library), RTLD_NOW (load now), RTLD_GLOBAL (make
symbols in library available to other libraries yet to be
loaded), RTLD_LOCAL (symbols loaded are accessible
only to your code)
17
Loading shared libraries on ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
if we define puts() in a shared library and
attempt to use it within our programs?
• Symbols resolved in order they are loaded
• Suppose our library containing puts() is libhello.so,
located in a standard library directory (like /usr/lib),
and we compile our hello.c code against this library:
athena% gcc -g -Wall ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
PIC option: create position-independent code, since
code will be repositioned during loading
• Link files using ld to create a shared object (.so) file:
athena% ld -shared -soname libname.so -o
libname.so.version -lc outfile1.o
outfile2.o ...
• If necessary, add directory to LD_LIBRARY_PATH
environment variable, s... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
At high level:
1. traverse tree down to leaf node
2. if leaf already full, split into two leaves:
(a) move median key element into parent (splitting parent
already full)
(b) split remaining keys into two leaves (one with lower, one with
higher elements)
3. add element to sorted list of keys
• Can accomplish in ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
move
closest key from child to parent
if neither child has enough keys, merge both children
if child not a leaf, have to repeat this process
30
Deletion examples
[Cormen, Leiserson, Rivest, and Stein. Introduction to Algorithms, 2nd ed.
MIT Press, 2001.]
Courtesy of MIT Press. Used with permission.
31
Deleti... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
largest child and repeat with
element in new position
36
Inserting data/increasing priority
• Insert element at end of heap, set to lowest priority −∞
• Increase priority of element to real priority:
1. start at element
2. if new priority less than parent’s, we are done
3. otherwise, swap element with parent an... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/4b9152a12a01274abfaf5a3c2686564b_MIT6_087IAP10_lec09.pdf |
18.417 Introduction to Computational Molecular Biology
Lecture 12: October 19, 2004
Lecturer: Ross Lippert
Scribe: Tushara C. Karunaratna
Editor: Peter Lee
Suffix Arrays and BWTs
Notation
We use � to denote the alphabet.
We use S to denote the text string, and n to denote the length of the text.
We use P to denote... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4bcb2459df6be70d5b0da5e8d4742aa0_lecture_12.pdf |
as large as n. Thus, the space
requirement is 5n + 2n = 7n words, which is 28n bytes assuming a machine word size
of four bytes.
12-1
12-2
Lecture 12: October 19, 2004
It is possible to reduce the space requirement to 20n bytes using a technique due to
Kurtz.
Suffix Arrays
Suffix arrays are a more space efficient a... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4bcb2459df6be70d5b0da5e8d4742aa0_lecture_12.pdf |
Construction
The suffix array can be thought of as a left-to-right dump of the leaves of the suffix
tree. Thus, a naive method of computing the suffix array is by first computing the
suffix tree. This naive method takes only O(n) time, but could take a large amount of
space during the intermediate step of constructing the s... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4bcb2459df6be70d5b0da5e8d4742aa0_lecture_12.pdf |
This is because not all permutations are suffix
arrays. There are n! permutations on n integers, whereas there are only |�|n possible
suffix arrays.
Figure 12.2 shows the graph of the suffix array permutation A for the string
acataggagacatacga$. This graph does not show us any obvious structure.
Figure 12.3 shows the gr... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4bcb2459df6be70d5b0da5e8d4742aa0_lecture_12.pdf |
we
require additional O(|�|) storage for the starting positions of each letter in the suffix
array. The following are two important queries done on the Burrows Wheeler string.
Lecture 12: October 19, 2004
12-5
string
i S[A[i] − 1] A−1[A[i] + 1] A[i]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
3
0
9
10
11... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4bcb2459df6be70d5b0da5e8d4742aa0_lecture_12.pdf |
, 2004
• occ(x, i): the number of times character x appears before position i in the
Burrows Wheeler string.
• f ind(x, i): the location of the ith occurence of the character x in the Burrows
Wheeler string.
Pattern lookup
Figure 12.4 gives pseudocode for counting the number of occurences of a pattern P .
The ap... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4bcb2459df6be70d5b0da5e8d4742aa0_lecture_12.pdf |
An example is shown in Table 12.5. This data structure
takes (1 + 4|�|/W )n bytes for storage. Using the data structure, we can perform occ
Lecture 12: October 19, 2004
12-7
queries in O(1 + W ) time. We can perform f ind queries in O(W log n) time using
a simple binary search. There is a trade-off between space a... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/4bcb2459df6be70d5b0da5e8d4742aa0_lecture_12.pdf |
18.034, Honors Differential Equations
Prof. Jason Starr
Lecture 7
2/18/04
1. One more example w/ slope fields. From p. 72 Example 2.4.4. y’=y-y2-0.2sin(t).
Observed that on line
y =
1.2
, y’ is always negative. On line y=0.7, y’ is always positive.
Thus solution curves that enter the region 0.7
0.7
≤
y
≤
1.2
ge... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/4bf2d11f4c44e6399f03782fd83824fb_lec7.pdf |
to
determine equilibrium sol’ns, state line, stability of equilibrium sol’ns:
(1) Find all zeros of f(y).
(3) Draw the state line.
(2) Determine the sign of f(y) b/w these values.
(4) Sketch sol’n curve and determine stability
unstability/ semi stability of equilibria.
18.034, Honors Differential Equations
Prof... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/4bf2d11f4c44e6399f03782fd83824fb_lec7.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.245: MULTIVARIABLE CONTROL SYSTEMS
by A. Megretski �
Solving the H2 optimization problem1
There are several ways to derive a solution to the H2 optimization problem. The path de
veloped below relies on reduction of t... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
) + B2U (t), X(0) = B1, lim X(t) = 0,
t��
(3.5)
and there exist a real matrix-valued function V = V (t), each entry of which is a linear
combination of terms of the form tk e−st with k → {0, 1, . . . }, Re(s) > 0, such that
where
U (t) = �(t)B1 + V (t)D21, lim �(t) = 0,
t��
˙�(t) = �(t)A + V (t)C2, �(0) = 0.
(... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
. when system equations have the form
B1 = I 0
�
, D21 = 0 I
,
�
�
x˙ = Ax + B2u + w1,
y = C2x + w2
(3.8)
(3.9)
with w = [w1; w2]. Then (3.5) follows from (3.8). One way to see this is by applying the
one-sided Laplace transform (denoted by tildes) to (3.8) to get
s˜
x = A˜
x + B2 ˜ w1.
u + ˜
3
Sinc... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
response parameterization, consider the closed
loop response from w to Kx − u, where K is a given matrix.
Theorem 3.2 A matrix function G = G(t) can be achieved as a closed loop impulse
response from w to Kx − u if and only if there exists a real matrix-valued function V =
V (t), each entry of which is a linear com... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
with V ≥ 0) has � ≥ 0
and hence H(t) = H0(t) = KeAtB1. This response can be achieved in (3.10),(3.11) with
V ≥ 0. On the other hand, according to Theorem 3.1, every zero-state response of the
system (i.e. when X(0) = B1 is replaced by X(0) = 0) must coincide with a zero-state
response of (3.10),(3.11).
3.2 Abstrac... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
and, subject to this constraints, minimizes the quadratic integral
�(u(·)) =
�
�
0
|cp(t) + dq(t)|2dt � min .
(3.17)
One motivation for considering abstract H2 optimization is the “full information”
version of the standard H2 output feedback design. According to Theorem 3.1, the closed
loop H2 norm can be expr... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
a given matrix. According to
Theorem 3.2, the closed loop impulse response can be chosen according to
H = �B1 + V D21, � = �A + V C2, �(0) = K.
˙
Let �i, Vi, Ki be the i-th row of �, V , K respectively. The H2 norm of H equals the sum
of the integrals
�
|�i(t)B1 + Vi(t)D21|2dt,
�
0
minimizing which leads to so... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
→ R and at ψ = � (which means that d is left invertible
as well);
(c) d→d > 0, and matrix
H =
�
a − b(d→d)−1d→c
b(d→d)−1b→
c c − c →d(d→d)−1d→ c −a + c →d(d→d)−1b→
→
→
�
has no eigenvalues on the imaginary axis;
(d) d→d > 0 and there exist (unique) matrices β = β→ � 0 and k such that a + bk is a
Hurwitz matri... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
Condition (c) provides a convenient way of checking the non-singularity
imposed by (b). Condition (d) defines a completion of squares procedure, which is a
natural and intuitive way of solving the abstract H2 optimization problem. By comparing
the coefficients on both sides of (3.18), one can derive a quadratic algebra... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
Theorem 3.3 in the abstract setup
defined by (3.19), and absense of sensor singularity is equivalent to satisfying (b) in the
abstract setup defined by (3.19). Hence, in the non-singular case, the corresponding
completion of squares is possible, with matrices β = Pf i, k = Kf i in the control setup,
β = Pse and k = K... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
the definition of Kse and Pse, the minimal estimation error
equals
trace(D12Kf iPse
K → D→
f e 12),
and is achieved when the impulse response G from w to Kf ix − u is given by
where
and
G(t) = �(t)B1 + V (t)D21,
˙�(t) = �(t)A + V (t)C2, �(0) = K,
V (t) = �(t)Kse.
By inspection, this optimal impulse response is ... | https://ocw.mit.edu/courses/6-245-multivariable-control-systems-spring-2004/4c031c3f765729d3087b5558bdb4bb17_lec3_6245_2004.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.080 / 6.089 Great Ideas in Theoretical Computer Science
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Scribe Notes: Introduction to Computational Complexity
Jason Furtado
February 28, 2008
1
Motivating Com... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/4c40dede95d3b2fe07e6aa435b46b471_lec7.pdf |
. Then the computer could just consult the lookup
table whenever a question is asked of it.
The problem with this lookup table is that it would have to be incredibly large. The amount
of storage necessary would be many times larger than the size of the observable universe (which
has maybe 1080 atoms). Therefore, su... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/4c40dede95d3b2fe07e6aa435b46b471_lec7.pdf |
piece of software to find directions. There’s a famous
anecdote where Dijkstra (who worked in a math department) was trying to explain his new result
1
to a colleague. And the colleague said, ”but I don’t understand. Given any graph, there’s at most a
finite number of non-intersecting paths. Every finite set has a min... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/4c40dede95d3b2fe07e6aa435b46b471_lec7.pdf |
the smallest circuit that
computes it? (That is, how many gates are needed?)
As an example, suppose we’re trying to compute the XOR of the n input bits, x1, . . . , xn.
Then how many gates to do we need? (For simplicity, let’s suppose that a two-input XOR gate is
available.) Right, n − 1 gates are certainly sufficien... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/4c40dede95d3b2fe07e6aa435b46b471_lec7.pdf |
SAT and other N P -complete problems? Well, that’s
getting ahead of ourselves, but the short answer is that no one knows!
But in 1949, Claude Shannon, the father of information theory, mathematical cryptography and
several other fields (and who we’ll meet again later), gave a remarkably simple argument for why
such ... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/4c40dede95d3b2fe07e6aa435b46b471_lec7.pdf |
1000, we can say that
almost all Boolean functions with 1000 inputs will need to use at least 21000/2000 NAND gates in
order to compute. That number is larger than 1080, the estimated number of atoms in the visible
universe. Therefore, if every atom in the universe could be used as a gate, there are functions with
... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/4c40dede95d3b2fe07e6aa435b46b471_lec7.pdf |
n3 steps.
Proof: Suppose there were a machine, call it P, that solved the above problem in, say, n2.99
steps. Then we could modify P to produce a new machine P �, which acts as follows given a Turing
machine M and input n:
1. Runs forever if M halts in at most n3 steps given its own code as input.
2. Halts if M ru... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/4c40dede95d3b2fe07e6aa435b46b471_lec7.pdf |
exist constants a, b such that f (n) < ag(n) + b for all n.
The function g(n) is an upper bound on the growth of f (n). A different way to think of Big-O
is that g(n) ”grows” at least as quickly as f (n) as n goes to infinity. f (n) = O(g(n)) is read as
”f (n) belongs to the class of functions that are O(g(n))”.
Big-... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/4c40dede95d3b2fe07e6aa435b46b471_lec7.pdf |
Electricity and Magnetism
• Recap:
– Fundamental Forces
– E.S. Induction
– Coulombs law (qualitative)
• Coulombs Law (quantitative)
• Induction (Demos)
Feb 11 2002
What we learned last time (II)
• Strength of Electrostatic Force (qualitatively):
– If distance gets larger, force gets weaker
– If charge gets bigger, for... | https://ocw.mit.edu/courses/8-02x-physics-ii-electricity-magnetism-with-an-experimental-focus-spring-2005/4c60171da92653f26eb3fffe1105f9fb_2_11_2002_edited.pdf |
for many, many charges?
– 109 e- on glass rod...
• Calculus makes life easier (for a change)!
• Replace sum with integral!
Feb 11 2002 | https://ocw.mit.edu/courses/8-02x-physics-ii-electricity-magnetism-with-an-experimental-focus-spring-2005/4c60171da92653f26eb3fffe1105f9fb_2_11_2002_edited.pdf |
1 Principal Component Analysis in High Dimensions and the Spike
Model
1.1 Dimension Reduction and PCA
When faced with a high dimensional dataset, a natural approach is to try to reduce its dimension,
either by projecting it to a lower dimension space or by finding a better representation for the data.
During this course... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
n are unbiased
estimators for, respectively, the mean and covariance of the distribution.
We will start with the first interpretation of PCA and then show that it is equivalent to the second.
1.1.1 PCA as best d-dimensional affine fit
We are trying to approximate each xk by
xk ≈ µ +
(βk)i vi,
d
(cid:88)
i=1
10
(6)
where v... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
k = 0 we have that the optimal µ is given by
µ∗ =
n1
(cid:88)
n
k=1
xk = µn,
the sample mean.
We can then proceed on finding the solution for (9) by solving
n
(cid:88)
k=1
min
V, βk
V T
V =I
(cid:107)x −
k
µn
− βk(cid:107)2
2 .
V
(9)
Let us proceed by optimizing for βk. Since the problem decouples for each k, we can foc... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
k
2
T
− µn) (xk − µn)
Tµn) V V T
−2 (xk −
T
+ (x − µ ) V V T V V T
Tµn) (xk − µn)
(x
k
(cid:1)
(cid:0)
n
k
= (xk −
µ
− n)
(xk − µn)
Since (xk −
Tµn) (xk − µn) does not depend on V , minimizing (9) is equivalent to
− (xk −
Tµn) V V T (xk − µn) .
max
V T V =I
n
(cid:88)
k=1
(xk − µn) V V T (xk − µn) .
T
(12)
A few more s... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
] where v1, . . . , vd correspond
to the d leading eigenvectors of Σn.
Let us first show that interpretation (2) of finding the d-dimensional projection of x1, . . . , xn that
preserves the most variance also arrives to the optimization problem (13).
1.1.2 PCA as d-dimensional projection that preserves the most variance
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
3)
(cid:13)
(cid:13)
=
n
(cid:88)
k=1
(cid:13)
(cid:13)
T
(cid:13)
2
(cid:1)
V (xk − µn) = Tr V ΣnV ,
(cid:13)
(cid:0)
T
showing that (14) is equivalent to (13) and
that the
two interpretations
of
PCA
are indeed equivalent.
1.1.3 Finding the Principal Components
When given a dataset x1, . . . , xn ∈ Rp, in order to com... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
L
n
n
T
1
n
n. Computing the SVD of X − µn1 takes
meaning that UL correspond to the eigenvectors of Σ
O(min n2p, p2n) but if one is interested in simply computing the top d eigenvectors then this compu-
tational costs reduces to O(dnp). This can be further improved with randomized algorithms. There
O (cid:0)pn log d + ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
In the
next Section we will look into random matrix theory to try to understand better the behavior of the
eigenvalues of Σn and it will help us understand when to cut-off.
1.1.5 A related open problem
We now show an interesting open problem posed by Mallat and Zeitouni at [MZ11]
Open Problem 1.1 (Mallat and Zeitouni [M... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
the procedure were taken in a slightly different order on which step
4 would take place before having access to the draw of g (step 3) then the best basis is indeed
the eigenbasis of Σ and the best subset of the basis is simply the leading eigenvectors (notice the
resemblance with PCA, as described above).
More formally... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
to the ones of Σ.
p
Since EΣn = Σ, if p is fixed and n → ∞ the law of large numbers guarantees that indeed Σn → Σ.
However, in many modern applications it is not uncommon to have p in the order of n (or, sometimes,
even larger!). For example, if our dataset is composed by images then n is the number of images and
p the ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
] and [AGZ10]). This
particular limiting distribution was first established in 1967 by Marchenko and Pastur [MP67] and is
now referred to as the Marchenko-Pastur distribution. They showed that, if p and n are both going
to ∞ with their ratio fixed p/n = γ ≤ 1, the sample distribution of the eigenvalues of Sn (like the
hi... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
104)(cid:0) 1 XX T (cid:1)k(cid:105)
1.2.1 A related open problem
Open Problem 1.2 (Monotonicity of singular values [BKS13a]) Consider the setting above but
with p = n, then X ∈ Rn×n is a matrix with iid N (0, 1) entries. Let
denote the i-th singular value4 of √1 X, and define
n
(cid:18) 1
√ X
n
(cid:19)
,
σi
αR(n) := E... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
2 dF1(λ) =
1
2π
(cid:90)
2
0
1
λ 2
(cid:112) −
(2
λ
λ) λ
=
8
3π
≈
0.8488.
Also, αR(1) simply corresponds to the expected value of the absolute value of a standard gaussian
g
αR(1) = E|g| =
(cid:114)
2
π
≈ 0.7990,
which is compatible with the conjecture.
On the complex valued side, the Marchenko-Pastur distribution also... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
, and β = 1.5:
18
The images suggests that there is an eigenvalue of Sn that “pops out” of the support of the
Marchenko-Pastur distribution (below we will estimate the location of this eigenvalue, and that es-
timate corresponds to the red “x”). It is worth noticing that the largest eigenvalues of Σ is simply
1 + β = ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
understand the behavior of the leading eigenvalue of
Sn =
n1
(cid:88)
n
i=1
x T
ix
1
i = XX T ,
n
5Notice that the Marchenko-Pastur theorem does not imply that all eigenvalues are actually in the support of the
Marchenk-Pastur distribution, it just rules out that a non-vanishing proportion are. However, it is possible ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
rewritten as
1
n
(1 + β)ZT
1
1 Z1v1 + (cid:112)1 + βZT
n
(cid:112)
2 Z1v1 + ZT
1 + βZT
1 Z2v2 = λv1
ˆ
1
n 2 Z2v2 = λv2.
ˆ
1
n
,
(16)
(17)
(17) is equivalent to
1
n
(cid:112)
1 + βZT
2 Z1v1 =
(cid:18)
1
ˆλ I − ZT
n 2 Z2
(cid:19)
v2.
ˆIf λ I − 1 ZT
n
2 Z2 is invertible (this w
on’t be justified here, but it is in [Pau]) t... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
2
(cid:19)
− 1
1
n
(cid:112)1 + βZT
2 Z1
(18)
First observation is that because Z1 ∈ Rn has standard gaussian entries then 1 ZT
n 1 Z1 → 1, meaning
that
(cid:34)
(cid:18)
1 T
1 T
ˆ
ˆ
n 2 Z2
n 1 Z2 λ I − Z
λ = (1 + β) 1 + Z
20
(cid:19)− 1
1
T
n 2 Z1
Z
(cid:35)
.
(19)
(cid:54)
Consider the SVD of Z = U ΣV T where U ∈ Rn... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
−V DV T
(cid:17)
−
1
(cid:0)
V D1/2 U T Z1
(cid:1)
(cid:21)
= (1 + β) +
T
(cid:1)
1
D1/2V T
(cid:16)
V
(cid:104)
ˆ
λ I −D
(cid:105)
V T (cid:17)
−
1
D1
(cid:0)
/2 U T
V
Z1
(cid:21)
(cid:1)
U T Z
= (1 + β)
(cid:20)
1 + (cid:0)U T Z (cid:1)
1
(cid:16)(cid:104)
T
D1/2
ˆ
λ I −D
(cid:105)(cid:17)−
1
D1/2 (cid:0)U T Z1
(cid:... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
1
n
p
−1
(cid:88)
j=1
g2 Djj
j ˆλ − Djj
= (1 + β)
1 +
Because we expect the diagonal entries of D to be distributed
according to the Marchenko-Pastur
distribution and g to be independent to it we expect that (again, not properly justified here, see [Pau])
p−1
1 (cid:88)
p − 1
j=1
g2 D j
j ˆ
j
λ − Djj
→
(cid:90) γ+... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
18)
(cid:19)
γ
β
> γ+.
Another important question is wether the leading eigenvector actually correlates with the planted
perturbation (in this case e1). Turns out that very similar techniques can answer this question as
well [Pau] and show that the leading eigenvector vmax of Sn will be non-trivially correlated with e1... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
and if ξ > 1 then
λ
max
(cid:18) 1
n
√ W + ξvvT
(cid:19)
→ 2,
λmax
(cid:18) 1
n
√ W + ξvvT
(cid:19)
1
→ ξ + .
ξ
(21)
1.3.2 An open problem about spike models
Open Problem 1.3 (Spike Model for cut–SDP [MS15]. As since been solved [MS15]) Let
W denote a symmetric Wigner matrix with i.i.d. entries Wij ∼ N (0, 1). Also, gi... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
.ξ
Remark 1.4 Optimization problems of the type of max {Tr(BX) : X (cid:23) 0, Xii = 1} are semidefinite
programs, they will be a major player later in the course!
(cid:104)
(cid:16)
Since 1 E Tr 11T
n
These observ
ξ 11T + √1 W
n
imply that 1
n
(cid:17)(cid:105)
≈ ξ, by taking X = 11T we expect that q(ξ) ≥
ξ.
ations
≤ ξ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
3.052 Nanomechanics of Materials and Biomaterials Thursday 02/22/07
I
Prof. C. Ortiz, MIT-DMSE
LECTURE 5: AFM IMAGING
Outline :
LAST TIME : HRFS AND FORCE-DISTANCE CURVES .......................................................................... 2
ATOMIC FORCE MICROSCOPY : GENERAL COMPONENTS AND FUNCTIONS.......... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
tip and sample
/ z-piezo
move in unison
-Normal vs. Lateral Force Microscopy
δ2<0
δ3<0
δ4>0
E.
attractive force
keeps tip in
contact with
surface
D.
tip and sample
/ z-piezo
move in unison
retracting
Tip-Sample Separation
Distance, D (nm)
- Conversion of raw data; sensor output, s (Volts) vs. z-
piezo displacement/... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
do not need to be conductive, 4) Sub-nm resolutions have been
achieved on biological samples (detailed
the molecular
conformation, spatial arrangement, structural dimensions, rate dependent
processes, etc.)
information on
-Piezo rasters or scans in the x-y
direction across the sample surface
↓
-Cantilever defl... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
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