text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
: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×p and V ∈ Rp p
2
(mean... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
1/2V 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:1)
(cid:21)
.
U
(cid:0)
+
(cid:20)
1
1 + ZT
n 1
1
n
1 (cid:0)
n
1
n
(cid:20)
1
Since ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
[Pau])
p−1
1 (cid:88)
p − 1
j=1
g2 D j
j ˆ
j
λ − Djj
→
(cid:90) γ+
x
ˆ
λ
− x
γ
−
dFγ(x).
ˆ
We thus get an equation for λ:
ˆλ = (1 + β)
(cid:20)
1 + γ
(cid:90) +
γ
x
ˆ
λ−
γ
− x
(cid:21)
dF (
γ x)
,
which can be easily solved with the help of a program that computes integrals symbolically (such as
Mathematica) to give (y... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
(cid:105)|2 → 0,
• and if β >
√
γ then
|(cid:104)
vmax, e1
2
(cid:105)| →
1.3.1 A brief mention of Wigner matrices
γ
β2
1
−
1 − γ .
β
Another very important random matrix model is the Wigner matrix (and it will show up later in this
course). Given an integer n, a standard gaussian Wigner matrix W ∈ Rn×n is a symmetric ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
∈ Rn×n sym-
metric, define:
Define q(ξ) as
Q(B) = max {Tr(BX) : X (cid:23) 0, Xii = 1} .
q(ξ) = lim EQ
1
n→∞ n
(cid:18)
ξ
n
11T +
(cid:19)
.
1
√ W
n
What is the value of ξ , defined as
∗
ξ = inf
∗
{ξ ≥ 0 : q(ξ) > 2}.
It is known that, if 0 ≤ ξ ≤ 1, q(ξ) = 2 [MS15].
One can show that 1 Q(B)
≤ λmax(B). In fact,
n
max {Tr(BX... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/4c9fa7ce658a63174f562fdf44b55626_MIT18_S096F15_Ses2_4.pdf |
that a certain semidefinite programming based algorithm for clustering under
the Stochastic Block Model on 2 clusters (we will discuss these things later in the course) is optimal
for detection (see [MS15]).7
∗
Remark 1.5 We remark that Open Problem 1.3 as since been solved [MS15].
7Later in the course we will discuss c... | 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 |
4>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/deflection, δ (nm) to Force, F, versus
tip-sample separation distance, D :
δ=s/m
m= sl... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
dependent
processes, etc.)
information on
-Piezo rasters or scans in the x-y
direction across the sample surface
↓
-Cantilever deflects (δ) in response to
an a topographical feature (hill or
valley)
↓ Feedback loop
-System continuously changes
in
response to an experimental output
(δ= cantilever deflection... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
[
Z
7
1.5
6
5
1.0
4
3
0.5
2
1
0
0
0
0
1
2
2
3
4
x (μm)
X[µm]
4
6
5
8
6
7
10
3D Height image
2D Height image
2D Section Profile
-Select linear region of plot and plot 2D section profile (height along line) z vs. x to get quantitative mathematical
functional form of topography. For example, we can see th... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
● loss of spatial resolution
Noncontact (AC) Mode :
● tip is oscillated near its resonant frequency
without touching the surface
● feedback signal, oscillation amplitude
● nondestructive
● loss of spatial resolution
● difficult in practice
6
3.052 Nanomechanics of Materials and Biomaterials... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
Materials and Biomaterials Thursday 02/22/07
ATOMIC FORCE MICROSCOPY IMAGING : FACTORS AFFECTING RESOLUTION
Prof. C. Ortiz, MIT-DMSE
PIEZO AMPLIFIER, SENSOR AND
CONTROL ELECTRONICS,
MECHANICAL PARAMETERS
Physik Instruments, Nanopositioning 1998
D+ΔD
d
-Z
D
+Z
-X
+Y
+X
electrodes
polarization
~
voltage
app... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
self-image of the tip surface, rather than the object surface.
Mathematical methods of tip deconvolution can be employed for image restoration. The effectiveness of these methods
will depend on the specific characteristics of the sample and the probe tip.
Prof. C. Ortiz, MIT-DMSE
9
3.052 Nanomechanics o... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
stitut für Genetik und
Mikrobiologie, Germany.
Prof. C. Ortiz, MIT-DMSE
Curtesy of Veeco Instruments and
G. Muskhelishvili. Used with permission.
http://people.virginia.edu/~zs9q/zsfig/DNA.html
Courtesy of Zhifeng Shao. Used with permission.
The high resolution of the SPM is able to
discern very subtle features such... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
1
Grain 1
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
0
5
10
15
Distance from Surface (nm)
● AFM can be combined with high resolution force spectroscopy and nanoindentation since cantilever probe tip can be
employed for both imaging and nanomechanical measurements→ nanomechanical measu... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/4ca90db25ed26c7a0cb3c472bc3b5cf6_lec5.pdf |
Queueing Systems: Lecture 3
Amedeo R. Odoni
October 18, 2006
Announcements
• PS #3 due tomorrow by 3 PM
• Office hours – Odoni: Wed, 10/18, 2:30-4:30;
next week: Tue, 10/24
• Quiz #1: October 25, open book, in class;
options: 10-12 or 10:30-12:30
• Old quiz problems and solutions: posted Thu
evening along with ... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/4d1277f5c802d299e369cd47e5833b17_lec7.pdf |
1, 2, …, n
customers in the system
• Epochs = instants immediately following the completion
of a service
M/G/1: Transition probabilities for
system states at epochs (1)
N = number of customers in the system at a random
epoch, i.e., just after a service has been completed
N' = number of customers in the system a... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/4d1277f5c802d299e369cd47e5833b17_lec7.pdf |
have been watching the system for a long
time, T.
ρ, the utilization ratio, is the long-run fraction of
time (= the probability) the server is busy; this means,
assuming the system reaches steady state:
ρ=
amount of time server is busy
T
=
λ⋅T ⋅ E[S]
T
= λ⋅ E[S] =
λ
μ
Idle and Busy Periods; E[B]
Observe a... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/4d1277f5c802d299e369cd47e5833b17_lec7.pdf |
[S] ⋅⎢ ∑ nPn − ∑ Pn ⎥
⎦⎥
⎣⎢n≥1
n≥1
n≥1
⎡
n
E[T2 ] = E[S] ⋅ L − E[S] ⋅ ρ
Derivation of L and W: M/G/1 [3]
• From our “random incidence” result (2.66):
2
σ S
E[T | n] =
1
])2
[
(
S
E
+
]
[
2
S
E
⋅
E[T1 | n] = 0, n = 0
, n ≥ 1
• Thus, giving:
E[T ] =
1 ∑
n
E[T | n] ⋅ P =
1
n ∑
n≥1
2
σ S
])2
S
[
E... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/4d1277f5c802d299e369cd47e5833b17_lec7.pdf |
=
2
1 ρ 2 + λ2 ⋅σ S
+
2λ(1 − ρ )
μ
Wq =
2
ρ 2 + λ2 ⋅σ S
2λ(1 − ρ )
=
Lq =
ρ 2
+
(1
2
2
λ
−
2
⋅σ
S
)
ρ
2 )
ρ 2 (1 + C S
=
2λ(1 − ρ) μ (1 − ρ)
ρ
1
⋅
2 )
(1 + C S
⋅
2
Note : CS =
σ
S
S]
[
E
= μ⋅σ S
Dependence on Variability (Variance)
of Service Times
Expected delay
Demand
ρ = 1.0
Runway E... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/4d1277f5c802d299e369cd47e5833b17_lec7.pdf |
CAP ≅ 40.9 per hour
M/G/1 system with non-preemptive
priorities: background
•
•
•
•
•
•
•
r classes of customers; class 1 is highest
priority, class r is lowest
Poisson arrivals for each class k; rate λk
General service times, Sk , for each class;
fSk(s); E[Sk]=1/μk; E[Sk
FIFO service for each class
Infi... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/4d1277f5c802d299e369cd47e5833b17_lec7.pdf |
Assumptions and Biases
• Every design and system has multiple phenomena
operating at the same time
• This has several consequences
– Parts must be designed and tested separately
and then tested together
– Analytical models and simulations will not be
able to encompass all the important things that
could happen or m... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/4d135fbe38f40639548d7c173392f340_lc1_bkgrnd_whtny.pdf |
in assuring achievement of basic function and
performance
8/24/2006 Background © Daniel E Whitney 1997-2006 4
What’s Basic to MechE
A. Analytical
B. Design
1. Need to consider basic
physical phenomena as part of
every design exercise
2. Must know the limits of the
models
3. Must do geometric reasoning
4. Each phen... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/4d135fbe38f40639548d7c173392f340_lc1_bkgrnd_whtny.pdf |
3.032 Mechanical Behavior of Materials
Fall 2007
Edge dislocation
Dislocation Line
b
b
half-plane
Screw
b
Edge
sheared
unsheared
dislocation line
Figure by MIT OpenCourseWare.
direction of motion
Image source: Callister, W. D., Materials Science and Engineering: An Introduction
Lecture 21 (10.29.07)
3.032 Mechanical B... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/4d447a73d86ca349823bd6eeace9c73b_lec21.pdf |
MA: Butterworth-Heinemann, 2001.
5. Field ion microscopy diffraction
Lecture 21 (10.29.07) | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/4d447a73d86ca349823bd6eeace9c73b_lec21.pdf |
3'008&
9:;:<&
'0=&>=80=&>?&!"#$%&!
!"#$%&
'()*+),-./(&
0.12.(()2.1&
+*&3+*45-)(&
3674(,7&
LECTURE 10
MEASUREMENT AND TIMING
Charles E. Leiserson
!"#$$%&#$'%"()"*+,"-./"01'2#"3,4*56,67"
1
Timing a Code for Sorting
#include <stdio.h>
#include <time.h>
void my_sort(double *A, int n);
void fill(double *A, int n);
... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
500 * 1000;
int step = 20 * 1000;
double A[max];
for (int n=min; n<max; n+=step){
fill(A, n);
Loop over arrays of
increasing length.
Measure time before sorting.
clock_gettime(CLOCK_MONOTONIC, &start);
my sort(A, n);
clock_gettime(CLOCK_MONOTONIC, &end);
double tdiff = (end.tv_sec - start.tv_sec)
+ 1e-9*(en... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
voltage
f = clock frequency
Reducing frequency and voltage results
in a cubic reduction in power (and heat).
But it wreaks havoc on performance measurements!
© 2008–2018 by the MIT 6.172 Lecturers
5
Today’s Lectu... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
iesced System
Experiment (joint work with Tim Kaler)
• Cilk program to count the primes in an interval
• AWS c4 instance (18 cores)
• 2-way hyperthreading on, Turbo Boost on
• 18 Cilk workers
• 100 runs, each about 1 second
e
v
o
b
a
t
n
e
c
r
e
P
m
u
m
n
M
i
i
25%
20%
15%
10%
5%
0%
0
10
20
30
40
50
60
70
80
90
© 200... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
2008–2018 by the MIT 6.172 Lecturers
13
Code Alignment
A small change to one place in the source code can
cause much of the generated machine code to change
locations. Performance can vary due to changes in
cache alignment and ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
Data Alignment
A program’s name can affect its speed!
• [Mytkowicz, Diwan, Hauswirth, and Sweeney, “Producing wrong
data without doing anything obviously wrong,” 2009.]
• The executable’s name ends up in an environment
variable.
• Environment variables end up on the call stack.
• The length of the name affects the st... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
the amount of processor time spent in
user-mode code (outside the kernel) within the
process.
∙ sys is the amount of processor time spent in the
kernel within the process.
© 2008–2018 by the MIT 6.172 Lecturers
19
clock_gettime(CLOCK_M... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
–2018 by the MIT 6.172 Lecturers
22
Interrupting
•
IDEA: Run your program under gdb, and type
control-C at random intervals.
• Look at the stack each time to determine which
functions are usually being executed.
• Who needs a fancy profiler?
• Some people ca... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
01'2#"3,4*56,67"
26
Basic Performance-Engineering Workflow
1. Measure the performance of Program A.
2. Make a change to Program A to produce
a hopefully faster Program A!.
3. Measure the performance of Program A!.
4. If A! beats A, set A = A!.
5. If A is still not fast enough, go to Step 2.
If you can’t measure perfo... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
∙ Arithmetic mean
∙ Energy use or CPU utilization
Fastest/biggest/best
solution
∙ Arithmetic mean
∙ Speedup of wall clock time
© 2008–2018 by the MIT 6.172 Lecturers
30
Summarizing Ratios
Trial
Program A Program B A/B
1
2
... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
80)
(cid:67)(cid:75)
(cid:19)
(cid:75)
(cid:19)
(cid:80)
(cid:16)
n
(cid:67)(cid:19)(cid:67)(cid:20)
(cid:67)(cid:80)
(cid:2477)(cid:3075)
Observation
(cid:30)
The geometric mean of A/B IS the inverse
of the geometric mean of B/A.
(cid:30)
(cid:2632)
(cid:2478)
(cid:3077)
© 2008–2018 by the MIT 6.172 Lecture... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
⋅I + b⋅C ,
where
• I is the number of instructions, and
• C is the number of cache misses.
© 2008–2018 by the MIT 6.172 Lecturers
35
Least-Squares Regression
A least-squares regression can fit the data to the
model
T = a⋅I + b⋅C ,
... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/4d7f4bb31bf1ed90c669a11867d36d36_MIT6_172F18_lec10.pdf |
3.024
“Electronic, Optical, and Magnetic
Properties of Materials”
1 3.024
Objectives and Approach
• How can we understand and predict electrical, optical and magnetic properties?
Emphasis on fundamental physical models in lectures
• Application to real life situations?
Emphasis on real life examples in HW and r... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/4dc248701b5bdb0cc72a6556127b81b6_MIT3_024S13_2012lec1Intro.pdf |
ode?
Linear,
Ohmic
V
• How would you predict it?
• How much voltage or current
can material handle?
• What happens if we shine light
onto a material?
I
Rectification,
Non-linear,
Non-Ohmic
dark
light
V
6 Materials in Modern Electronics
Silicon: Material that enabled the World as we know it
1947
Si pr... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/4dc248701b5bdb0cc72a6556127b81b6_MIT3_024S13_2012lec1Intro.pdf |
useful?
Neurons expressing heat
sensitive proteins
From Prof Beach
.
Magnetic nanoparticles
converting EM waves to heat
Anikeeva Lab @ DMSE
Use magnetic
nanoantennae to
stimulate neurons:
Non-invasive
treatments of
neuro-disorders
11 MIT OpenCourseWare
http://ocw.mit.edu
3.024 Electronic, Optical and Magn... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/4dc248701b5bdb0cc72a6556127b81b6_MIT3_024S13_2012lec1Intro.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
2.161 Signal Processing: Continuous and Discrete
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Massachusetts Institute of Technology
Department of Mechanical Engineering
2.161 Signal Processing - Continuous and Di... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
2))
log(Ωr/Ωc)
= 3.70
7–1
(cid:2)
(cid:17)
(cid:16)
(cid:4)
(cid:5)
(cid:4)
(cid:9)
(cid:9)
(cid:11)
(cid:12)
(cid:9)
(cid:12)
we therefore select N=4. The 4 poles (p1 . . . p4) lie on a circle of radius r = Ωc(cid:2)−1/N = 13.16
and are given by
|pn| = 13.16
(cid:2) pn = π(2n + 3)/8
for n = 1 . . . 4, giving a... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
:
Type 1
|H(jΩ)| 2 =
Type 2
|H(jΩ)| 2 =
1
2 (Ω/Ωc)
1 + �2TN
1
2 (Ωr/Ωc)/TN
1 + �2 (TN
2 (Ωr/Ω))
(1)
(2)
Where TN (x) is the Chebyshev polynomial of degree N . Note the similarity of the form
of the Type 1 power gain (Eq. (1)) to that of the Butterworth filter, where the function
TN (Ω/Ωc) has replaced (Ω... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
magnitude in the interval x
amplitude is 21−N .
|
| ≤ 1.
This “minimum maximum”
In low-pass filters given by Eqs. (13) and (14), this property translates to the following
characteristics:
Stop-Band Characteristic
Filter
Butterworth
Maximally flat
Chebyshev Type 1 Ripple between 1 and 1/(1 + �2) Maximally flat
Ch... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
we write cos−1 (s/jΩc) = γ + jα, then
�
��
s
jΩc
= ±
j
�
s = Ωc (j cos (γ + jα))
= Ωc (sinh α sin γ + j cosh α cos γ)
(6)
(7)
(8)
which defines an ellipse of width 2Ωc sinh(α) and height 2Ωc cosh(α) in the s-plane. The
poles will lie on this ellipse. Substituting into Eq. (16)
�
�
s
jΩc
TN
= cos (N (γ ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
γn =
(2n − 1)π
2N
n = 1, . . . , N
4. Determine the N left half-plane poles
pn = Ωc (sinh α sin γn + j cosh α cos γn)
n = 1, . . . , N
5. Form the transfer function
(a) If N is odd
(b) If N is even
H(s) =
−p1p2 . . . pN
(s − p1)(s − p2) . . . (s − pN )
H(s) =
1
p1p2 . . . pN
1 + �2 (s − p1)(s − p2) . .... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
α = 0.6438, and cosh α = 1.189. Also, γn = (2n − 1)π/6 for n = 1 . . . 6
as follows:
n:
γn:
sin γn:
cos γn:
1
π/6
1/2
√
3/2
Then the poles are
2
π/2
1
0 −
3
5π/6
1/2
√
3/2 −
4
7π/6
−1/2
√
3/2
5
6
3π/2 11π/6
-1 −1/2
√
0
3/2
pn = Ωc (sinh α sin γn + j cosh α cos γn)
�
�
√
p1 = 10 0.6438
×
1
2
+ j1.... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
.438s + 116.5)(s + 6.438)
The pole-zero plot for the Chebyshev Type 1 filter is shown below.
7–6
2.2 The Chebyshev Type 2 Filter
The Chebyshev Type 2 filter has a monotonically decreasing magnitude function in the pass
band, but introduces equi-amplitude ripple in the stop-band by the inclusion of system zeros
on ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
τ /jΩc)
7–7
XXX-6.4380ss-planej10.30-3.219-j10.30jW The poles are found using the method developed for the Type 1 filter, the zeros are
found as the roots of the polynomial TN (τ /jΩc) on the imaginary axis τ = jν. From
the definition TN (x) = cos (N cos−1 (x)) it is easy to see that the roots of the Chebyshev
polyn... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
Determine α:
α =
1
N
sinh−1
�
�
1
ˆ�
=
1
3
sinh−1(8.666) = 0.9520
and sinh α = 1.1024, and cosh α = 1.4884.
The values of γn = (2n − 1)π/6 for n = 1 . . . 6 are the same as the design for the
7–8
Type 1 filter, so that the poles of H(τ ) 2 are
|
|
�
�
pn = Ωc (sinh α sin γn + j cosh α cos γn)
√
�
τ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
giving three filter poles
p1, p2 = −5.609 ± j13.117
p3 = −18.14
The system zeros are the roots of
T3(ν/jΩc) = 4(ν/jΩc)3 − 3(ν/jΩc) = 0
from the definition of TN (x), giving ν1 = 0 and ν2, ν3 = ±j8.666. Mapping these
back to the s-plane gives two finite zeros z1, z2 = ±j23.07, z3 = ∞ (the zero at
∞ does not affect th... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
10) (cid:5) (cid:3) (cid:11)
(cid:9) (cid:3) (cid:12)
(cid:2)
(cid:2) (cid:3) (cid:4) (cid:5) (cid:3) (cid:6)
(cid:2) (cid:7) (cid:5) (cid:8) (cid:3)
(cid:12)
(cid:6) (cid:9) (cid:4) (cid:4) (cid:7) (cid:8) (cid:9) (cid:10) (cid:11)
(cid:3)
(cid:2)
(cid:2) (cid:9) (cid:3) (cid:10) (cid:5) (cid:3) (cid:11)
(ci... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
Butterworth and
the Chebyshev Type 1 filters are all-pole designs and have an asymptotic high-frequency
magnitude slope of −20N dB/decade, in this case -80 dB/decade for the Butterworth design
and -60 dB/decade for the Chebyshev Type 1 design. The Type 2 Chebyshev design has
two finite zeros on the imaginary axis at ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
(N=4)
Chebyshev Type 1 (N=3)
Chebyshev Type 2 (N=3)
0
20
30
40
50
60
Angular frequency (rad/sec)
70
80
90
100
7–11 | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/4dc3c856841a896bbcb8726c3dd4b41b_lecture_07.pdf |
Lecture 9
MOSFET(II)
MOSFET IV CHARACTERISTICS(contd.)
Outline
1. The saturation region
2. Backgate characteristics
Reading Assignment:
Howe and Sodini, Chapter 4, Section 4.4
6.012 Spring 2009
Lecture 9
1
1. The Saturation Region
Geometry of problem
Regions of operation:
• Cutoff: VGS < VT
– No inversion ... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4dd0f37dcb3a9617b66b82a33894ef73_MIT6_012S09_lec09.pdf |
layer
⇒ current saturation.
6.012 Spring 2009
Lecture 9
4
The Saturation Region (contd.)
What happens when VDS = VGS – VT?
Charge control relation at drain:
[
QN (L) = −C ox VGS − VDS − VT
]= 0
No inversion layer at the drain end of channel ???!!!
⇒ Pinchoff.
At pinchoff:
• Charge control equation inaccu... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4dd0f37dcb3a9617b66b82a33894ef73_MIT6_012S09_lec09.pdf |
Experimental finding:
∆L
L
= λ(VDS − VDSsat )
with
Typically,
λλλλ∝
1
L
λλλλ=
0.1 µµµµm • V
−1
L
For L = 1µm, increase of VDS of 1V past VDSsat results in
increase in ID of 10%.
Improved but approximate model for the drain current
in saturation:
I D ≈
W
2L
• µnCox (VGS − VT )2 [1 + λVDS ]
6.012 Spring 20... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4dd0f37dcb3a9617b66b82a33894ef73_MIT6_012S09_lec09.pdf |
⇒ inversion layer charge is reduced!
•
VDS
+
-
VGS
+
-
Metal interconnect
to gate
n+ polysilicon gate
n+ source
0
y
n+ drain
QN(y)
Xd(y)
VBS = 0
x
p-type
Metal interconnect to bulk
Figure by MIT OpenCourseWare.
For the same applied gate-to-source voltage VGS,
application of VBS < 0 reduces the density of electro... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4dd0f37dcb3a9617b66b82a33894ef73_MIT6_012S09_lec09.pdf |
φp − VBS −
−2φφφφp ]
6.012 Spring 2009
Lecture 9
13
Backgate Characteristics (Contd.)
Triode Region VDS ~ 0.1V
6.012 Spring 2009
Lecture 9
14
What did we learn today?
Summary of Key Concepts
• MOSFET in saturation (VDS ≥ VDSsat): pinchoff
point at drain-end of channel
– Electron concentration small, but
– ... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4dd0f37dcb3a9617b66b82a33894ef73_MIT6_012S09_lec09.pdf |
DIRAC’s BRA AND KET NOTATION
B. Zwiebach
October 7, 2013
Contents
1 From inner products to bra-kets
2 Operators revisited
2.1 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Adjoint of a linear operator
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
∗, as well as
)
u
(
=
v
(
u
|
)
v
(
v
|
) ≥
0 for all v, while
v
(
v
|
)
= 0 if and only if v = 0.
for complex constants c1 and c2, but antilinearity in the first argument
u
(
c1v1 + c2v2)
|
= c1(
u
v1)
|
+ c2(
u
v2)
|
,
Two vectors u and v for which
∗
∗
v
c1u1 + c2u2|
(
u
(
= c1 (
v
u1|
= 0 are orthogon... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
.4)
2. Consider the complex vector space of complex function f (x)
functions f (x), g(x) we define
C with x
∈
∈
[0, L]. Given two such
f
g
|
(
) ≡
L
0
f ∗ (x)g(x)dx .
The verification of the axioms is again quite straightforward.
A set of basis vectors
ei}
{
labelled by the integers i = 1, . . . , n satisfyi... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
vector and
also
v
|
) ∈
V as a vector. It is as if we added some decoration
∈
around the vector v to make it clear
| )
by inspection that it is a vector, perhaps like the usual top arrows that are added in some cases. The
label in the ket is a vector and the ket itself is that vector!
Bras are somewhat different... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
. . .
an
, b =
b1
b2
. . .
bn
∗
∗
= a1b1 + a2b2 + . . . a bn
)
∗
n
we had
Now we think of this as
a
(
b
|
=
a
(
|
(
∗ a1, a 2 . . . , a n
∗
∗
,
)
=
b
|
)
and matrix multiplication gives us the desired answer
b1
b2
.
.
.
bn
... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
its action on
u
(
|
arbitrary vectors
as follows
v
|
)
) → (
As required by the definition, any linear map from V to C defines a bra, and the corresponding
)
|
:
u
(
v
|
.
u
v
|
(1.16)
underlying vector. For example let v be a generic vector:
v =
v1
v2
.
.
.
vn
,
A linear map ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
(1.20)
v
α
|
(
This illustrates the point that (i) bras represent dual objects that act on vectors and (ii) bras are
(1.21)
f (v) =
.
)
labelled by vectors.
Bras can be added and can be multiplied by complex numbers and there is a zero bra defined to
give zero acting on any vector, so V ∗ is also a... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
, so v
′
v = 0 and v = v .
′
′
v
−
We can now reconsider equation (1.3) and write an extra right-hand side
−
w
|
v
(
)
w
|
α ∗
=
a1|
1(
)
(
so that we conclude that the rules to pass from kets to bras include
b
α1a1 + α2a2|
(
+ α ∗
b
a2|
2(
α ∗
b
a1|
1(
=
)
)
∗
+ α2
a2|
(
)
b
|
)
v
|
= α1|
a1)
+ α2... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
vector space V . This means that acting on vectors on V it gives vectors on
V , something we write as
Ω : V
V .
→
We denote by Ω
a
|
)
the vector obtained by acting with Ω on the vector
a
|
:
)
The operator Ω is linear if additionally we have
V
a
|
) ∈
Ω
a
|
) ∈
→
V .
+
a
|
)
b
|
Ω
(
)
)
= Ω
a
|
)
+ Ω
b... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
the object
)
a
|
is naturally viewed as a linear operator on V and on V ∗ . Indeed, acting on a vector we let it act as
the bra-ket notation suggests:
(2.36)
Ω =
b
)(
|
,
Ω
v
|
a
) ≡ |
b
) (
v
|
a
) ∼ |
)
, since
b
(
v
|
)
is a number.
Acting on a bra it gives a bra:
w
(
Ω
|
w
≡ (
a
|
b
) (
b
| ∼ (
|
, si... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
now define the ‘matrix elements’
so that the above equation reads
Ωmn
m
≡ (
Ω
|
n
|
)
.
Ωmnan = bm ,
X
n
(2.40)
(2.41)
(2.42)
(2.43)
which is the matrix version of the original relation Ω
=
a
|
)
b
|
. The chosen basis has allowed us to
)
view the linear operator Ω as a matrix, also denoted as Ω, with matr... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
(
n ′
|
)
=
X
m,n
Ωmnδm ′ m δnn ′ = Ωm ′ n ′ ,
(2.46)
as expected from the definition (2.42).
2.1 Projection Operators
Consider the familiar orthonormal basis
an operator Pm defined by
of V and choose one element
i
)}
{|
from the basis to form
m
|
)
This operator maps any vector
V to a vector along
v
|
) ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
.
(2.49)
A hermitian operator P is said to be a projection operator if it satisfies the operator equation
P P = P . This means that acting twice with a projection operator on a vector gives the same as
acting once. The operator Pm is a projection operator since
PmPm =
) (
= 1. The opera... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
|
) (
) (
)
)
)
)
=
m
|
m
v
|
+
n
|
.
n
v
|
Pm,n|
v
n
|
(2.52)
subspace spanned by
extra term
k
|
k
)(
|
with k
m
|
and
n
|
. Similarly, we can construct a rank three projector by adding an
)
m and k = n. If we include all basis vectors we would have the operator
)
=
2
2
|
|
)(
As a matrix P1,...,N has a ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
relation to show that our formula (2.42) for matrix elements
is consistent with matrix multiplication. Indeed for the product Ω1Ω2 of two operators we write
(Ω1Ω2)mn =
m
(
n
Ω1Ω2|
|
N
=
)
=
m
(
Ω1
|
k
|
k
)(
|
)
(
X
k=1
m
(
Ω1 1 Ω2|
n
|
=
n
Ω2|
)
)
N
k
Ω1|
|
k
)(
n
Ω2|
|
)
=
N
(Ω1)mk(Ω2)kn .
(2.56)
X
k=1... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
)
∀
u, v .
Flipping the two sides of (2.57) we also get
from which, taking the ket away, we learn that
v
(
Ω†
|
u
|
)
=
Ωv
(
u
|
)
v
(
Ω†
|
Ωv
≡ (
.
|
(2.57)
(2.58)
(2.59)
(2.60)
(2.61)
Another way to state the action of the operator Ω† is as follows. The linear operator Ω induces a map
of vectors in V a... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
matrix notation we have Ω† = (Ωt)∗ where the superscript t denotes transposition.
(Ω†)ij = (Ωji) ∗ .
Ω†
i
|
(
∗
i
)
|
=
j
(
j
|
→
)
(2.64)
Exercise. Show that (Ω1Ω2)† = Ω Ω by taking matrix elements.
Exercise. Given an operator Ω =
Solution: Acting with Ω on
and then taking the dual gives
†
†
2 1
b
a
)(
|... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
if A† =
A.
−
Exercise: Show that the commutator [Ω1, Ω2] of two hermitian operators Ω1 and Ω2 is anti-hermitian.
There are a couple of equations that rewrite in useful ways the main property of Hermitian oper
ators. Using Ω† = Ω in (2.59) we find
If Ω is a Hermitian Operator:
v
(
Ω
|
u
|
)
=
u
(
Ω
|
v
|
∗ ,
)
... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
a Hermitian Ω we have
=
)
f
Ωg
|
(
)
or explicitly
(Ωf (x)) ∗ g(x)dx =
Ωf
g
|
(
∞
Z
−∞
∞
Z
−∞
(f (x)) ∗ Ωg(x)dx
Verify that the linear operator Ω =
� d
i dx
is hermitian when we restrict to functions that vanish at
.
±∞
An operator U is said to be a unitary operator if U † is an inverse for U , that is,... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
|
)
,
U a
(
U b
|
)
=
a
(
U †U
|
b
|
)
=
.
a
(
b
|
)
(2.75)
(2.76)
(2.77)
(2.78)
Another important property of unitary operators is that acting on an orthonormal basis they give
another orthonormal basis. To show this consider the orthonormal basis
Acting with U we get
a1)
|
,
a2)
|
, . . .
aN
)
|
... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
.
(2.83)
U =
N
X
i=1
U ai)(
|
ai|
,
since
U
aj
)
|
=
N
X
i=1
U ai)(
|
ai|
aj
)
=
U ai)
|
.
In fact for any unitary operator U in a vector space V there exist orthonormal bases
such that U can be written as
N
(2.84)
(2.85)
ai)}
{|
and
bi)}
{|
(2.86)
U =
bi)(
|
ai|
.
X
i=1
ai)
|
any orthono... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
∈
R .
for all values of
,
, . . ., it is
2
1
)
|
)
|
(3.87)
Basis states with different values of x are different vectors in the state space (a complex vector space,
as always in quantum mechanics). Note here that the label on the ket is not a vector! So
,
)
unless x = 0. For quantum mechanics in three dimensi... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
states of particles. We visualize
x
(
x
|
)
∞
the state
as the state of a particle perfectly localized at x, but this is an idealization. We can easily
construct normalizable states using superpositions of position states. We also have a completeness
relation
1 =
dx
x
|
x
)(
.
|
Z
This is consistent with our ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
x
|
x
(
|
.
Given the state
ψ
|
)
of a particle, we define the associated position-state wavefunction ψ(x) by
ψ(x)
x
≡ (
ψ
|
) ∈
C .
(3.92)
(3.93)
(3.94)
This is sensible:
is a number that depends on the value of x, thus a function of x. We can now
do a number of basic computations. First we write any state... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
now introduce momentum states
complete analogy to the position states
p
|
)
that are eigenstates of the momentum operator ˆp in
(3.98)
(3.99)
,
p
|
)
p ′ ) ,
−
p
)(
,
|
Basis states :
p ′
p
(
)
|
1 =
= δ(p
dp
p
|
Z
= p
p
|
)
pˆ
p
|
)
Just as for coordinate space we also have
R .
p
∀
∈
In order t... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
We claim that the last integral is precisely the integral
representation of the delta function δ(p
p ′ ):
−
du ei(p−p ′)u = δ(p
−
1
2π
Z
p ′ ) .
(3.102)
This, then gives the correct value for the overlap
justified using the fact that the functions
p
(
p ′
|
, as we claimed. The integral (3.102) can be
)
fo... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
′ )
)
→
1
2π
Z
′)
dueiu(x−x
,
(3.106)
and back in (3.105) we have justified (3.102).
We can now ask: What is
p
(
ψ
|
? We compute
)
p
(
ψ
|
)
=
Z
dx
p
(
x
|
x
)(
ψ
|
)
=
1
�
~
Z
2π
√
dxe−ipx/ ψ(x)
~
�
˜
= ψ(p) ,
(3.107)
which is the Fourier transform of ψ(x), as defined in (6.41) of Chapter 1. Thus the ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
)
)
|
pˆ
x
(
|
ψ
|
)
=
~
�
d
i dx
Z
dp
x
(
p
|
p
)(
ψ
|
)
The completeness sum is now trivial and can be discarded to obtain
pˆ
x
(
|
ψ
|
)
=
~
�
d
i dx
x
(
ψ
|
)
=
~
�
d
i dx
ψ(x) .
Exercise. Show that
xˆ
p
(
|
ψ
|
)
= i
�
~
d ˜
dp
ψ(p) .
(3.111)
(3.112)
(3.113)
14
... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/4de6d044fa9d7e5b8998c5f8ca984a42_MIT8_05F13_Chap_04.pdf |
1
Introduction
1
INTRODUCTION
These notes provide reading material on the Soft-Collinear Effective Theory (SCET). They are intended
to cover the material studied in the second half of my effective field theory graduate course at MIT.
These latex notes will also appear as part of TASI lecture notes and a review artic... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/4de76f1b11bd063bb34119ac948e42df_MIT8_851S13_Introduction.pdf |
how they cancel between virtual and
real emission diagrams, and how they otherwise signal the presence of nonperturbative physics and the
scale ΛQCD as they do for parton distribution functions.
Finally it should be remarked that later parts of the notes are still a work in progress (particularly
sections marked at... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/4de76f1b11bd063bb34119ac948e42df_MIT8_851S13_Introduction.pdf |
8.701
0. Introduction
0.9 Spin
Introduction to Nuclear
and Particle Physics
Markus Klute - MIT
1
Spin vector, length, and eigenvalues
In quantum mechanics, the spin vector S is quantised in terms of its
length and its components.
Total length is
For components along any axis, e.g. z, eigenvalues can be
with 2s+1 ... | https://ocw.mit.edu/courses/8-701-introduction-to-nuclear-and-particle-physics-fall-2020/4dfc98933086e99ed5278c9422d95e05_MIT8_701f20_lec0.9.pdf |
18.175: Lecture 9
Borel-Cantelli and strong law
Scott Sheffield
MIT
18.175 Lecture 9
1 Outline
Laws of large numbers: Borel-Cantelli applications
Strong law of large numbers
18.175 Lecture 9
2 Outline
Laws of large numbers: Borel-Cantelli applications
Strong law of large numbers
18.175 Lecture 9
3 Borel-C... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/4e8a2305c745bd5a82a5f15cf8bcb8bb_MIT18_175S14_Lecture9.pdf |
Chebyshev implies
P(|Sn − ESn| > δESn) ≤ Var(Sn)/(δESn)2 → 0,
which gives us convergence in probability.
Second, take a smart subsequence. Let
nk = inf{n : ESn ≥ k 2}. Use Borel Cantelli to get a.s.
convergence along this subsequence. Check that convergence
along this subsequence deterministically implies the... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/4e8a2305c745bd5a82a5f15cf8bcb8bb_MIT18_175S14_Lecture9.pdf |
bound fourth moments of An.
E [A4] = n−4E [S 4] = n−4E [(X1 + X2 + . . . + Xn)4].
Expand (X1 + . . . + Xn)4 . Five kinds of terms: Xi Xj Xk Xl and
Xi Xj X 2 and Xi X 3 and X 2X 2 and X 4 .
i
n
The first three terms all have expectation zero. There are 2
of the fourth type and n of the last type, each equal to at
n... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/4e8a2305c745bd5a82a5f15cf8bcb8bb_MIT18_175S14_Lecture9.pdf |
Fubini (interchange sum/integral,
2yP(|Yk | > y )dy ≤
∞
0
k=1
k
∞
t
E (Yk
2)/k 2 ≤
k=1
∞
t
k −2
∞
k=1
0
1(y <k)2yP(|X1| > y )dy =
∞ ∞
t
(cid:0)
0
k=1
k −21(y <k) 2yP(|X1| > y )dy .
(cid:1)
Since E |X1| =
showing that if y ≥ 0 then 2y
P(|X1| > y )dy , complete proof of claim by
k>y k −2 ≤ 4.
S
∞
0
1... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/4e8a2305c745bd5a82a5f15cf8bcb8bb_MIT18_175S14_Lecture9.pdf |
(observe RHS below is finite):
∞
t
≤ 4(1 − α−2)−1m .
−2
P(|Tk(n)−ETk (n)| > Ek(n)) ≤ 4(1−α−2)−1E−2
∞
t
E (Y 2 )m
m
−2
.
n=1
m=1
Since E is arbitrary, get (Tk(n) − ETk(n))/k(n) → 0 a.s.
�
(cid:73)
18.175 Lecture 9
12
MIT OpenCourseWare
http://ocw.mit.edu
18.175 Theory of Probability
Spring 2014
For informatio... | https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/4e8a2305c745bd5a82a5f15cf8bcb8bb_MIT18_175S14_Lecture9.pdf |
LECTURE 12
Derived Functors and Explicit Projective
Resolutions
Let X and Y be complexes of A-modules. Recall that in the last lecture we
defined HomA(X, Y ), as well as Homder
A (X, Y ) := HomA(P, Y ) for a projective
qis
−−→ X, i.e., a projective resolution of X. We also defined the Ext-
complex P
groups Exti
A(X, Y ) ... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/4e977d9bd29ad54aee9bd85d49f67416_MIT18_786S16_lec12.pdf |
map, namely the composition P1 → X1 →
X2, which is killed by the differential since it is a map of chain complexes, and
therefore defines a cohomology class in H 0Hom(P1, X2). So there is some coho-
mology class in H 0Hom(P1, P2) which is a lift of that map through P2, which is
(cid:3)
well-defined and unique up to homoto... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/4e977d9bd29ad54aee9bd85d49f67416_MIT18_786S16_lec12.pdf |
−→ Z[G/H],
which is projective as a complex of Z[G]-modules. This is because both Z[G] and
PH are bounded, so it will be bounded, and inducing up to Z[G] preserves projective
modules as we will still obtain a direct summand of a free module. Alternatively,
we could use the universal property that every map to an acycli... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/4e977d9bd29ad54aee9bd85d49f67416_MIT18_786S16_lec12.pdf |
opic to multiplication by [G : H].
More concretely, suppose we had an H-invariant object and a G-invariant ob-
ject. Taking coset representatives of G/H, we could take the “relative norm” of any
H-invariant element, which would yield a G-invariant element. This is precisely
what our maps are doing above, and explains w... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/4e977d9bd29ad54aee9bd85d49f67416_MIT18_786S16_lec12.pdf |
(X) (i.e., via the image of 1).
We now turn to the problem of constructing explicit projective resolutions of
Z as a G-Module.
Example 12.6. Let G := Z/nZ with generator σ. We claim that the following
is a quasi-isomorphism:
(cid:80)
i σi
Z[G]
1−σ
Z[G]
(cid:80)
i σi
Z[G]
1−σ
Z[G]
0
0
0
(cid:15)
Z
· · ·
· · ·
0
0
· · ·
... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/4e977d9bd29ad54aee9bd85d49f67416_MIT18_786S16_lec12.pdf |
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