text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
unless there is a large difference in the cross-sections
Example: Si transistor, τphonon dominates even though τimpur gets worse with scaling
6
©1999 E.A. Fitzgerald
Example: Electron Mobility in Ge
μ~T-3/2 if phonon dominated
(T-1/2 from vth, T-1 from x-section σ
106
3
105
3
104
3
s
-
t
l
o
v
/
2
m
c
,
e
m
T-3/... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/43fea384070aa4c2f247637091f10d55_lecture_8.pdf |
– Go is the equilibrium generation rate
• Recombination
– intrinsic: across band gap, R=# carriers/vol.-sec
– extrinsic: deep levels due to traps
– Ro is the equilibrium recombination rate, which is balanced by Go
Non-equilibrium intrinsic recombination
Non-equilibrium extrinsic recombination
n-type material
R ... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/43fea384070aa4c2f247637091f10d55_lecture_8.pdf |
6.S096 Lecture 1 – Introduction to C
Welcome to the Memory Jungle
Andre Kessler
Andre Kessler
6.S096 Lecture 1 – Introduction to C
1 / 30
Outline
1 Motivation
1
2 Class Logistics
2
3 Memory Model
3
4 Compiling
4
5 Wrap-up
5
Andre Kessler
6.S096 Lecture 1 – Introduction to C
2 /
30
First Example (Pyth... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
C++?
Speed
Graph of prog
Source: http://benchmarksgame.alioth.debian.org/u64q/which-programs-are-fastest.php.
ram s peed acro ss language im plementations remove d due to copy right r estrictions.
Andre Kessler
6.S096 Lecture 1 – Introduction to C
5 / 30
Why C or C++?
Motivation
Power
C: direct access to memory and m... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
9
10
Topic
Introduction to C: memory and the compiler
Subtleties of C: memory, floating point
Guest lectures: Assembly and Secure C
Transition from C to C++
Object-oriented programming in C++
Design patterns and anti-patterns
Generic programming: templates and more
Projects: putting it all together
Pro... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
To compile: make nothing
Previous step produced an executable named nothing
To run: ./nothing
Surprise! Does nothing.
But you probably have higher aspirations for your programs...
Andre Kessler
6.S096 Lecture 1 – Introduction to C
13
/
30
Class Logistics
Hello, world!
hello.c: takes no a... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
– Introduction to C
17
/
30
Memory Model
Pointers
How do you get at this information about memory?
Through pointers; that is, the & and * operators
int a = 5; The address of a is &a.
int *a ptr = &a; Read declarations from right to left.
See it this way: “*a ptr is declared to be of type i... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
Memory Model
It’s all about the memory
int a = 5;
int *a ptr = &a;
&a
&a ptr
Memory Address
0x7fff6f641914 0x000000000005 a
0x7fff6f641918 0x7fff6f641914 a ptr
Value
Identifier
Note: definitely a 64-bit machine, since the addresses are larger than
232 .
Andre Kessler
6.S096 Lecture 1 – Introduction to C
21... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
.h>
int do_thing( float a, float b ) {
/* do things */
}
void call(void) {
/* do stuff */
do_thing( a, b );
/* do more */
}
int main(void) {
call();
return 0;
}
Andre Kessler
6.S096 Lecture 1 – Introduction to C
25
/
30
What happens when we compile?
Compiling
Three functions main, c... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
/
30
Wrap-up & Friday
Wrap-up
Open lab
Bring your laptops, get a C programming environment working
Test out the automatic grader
Cl
ass on Friday
Will cover floating point arithmetic, memory management, and
headers in more depth.
Q estions?
u
Andre Kessler
6.S096 Lecture 1 – Introduction to C
30
/
... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
Imperative Programming in
Haskell?
Armando Solar-Lezama
Computer Science and Artificial Intelligence Laboratory
MIT
With content from Nirav Dave (used with permission) and
examples from Dan Piponi’s great blog Post
“You could have invented Monads! And Maybe You Already Have”
http://blog.sigfpe.com/2006/08/you-c... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
unit * f = f * unit = f
– lift f * lift g = lift (f . g)
October 7, 2015
L09-4
Random Numbers
• Consider the “function” rand()
– Not really a function, but you can make it a
function
• rand: StdGen->(int, StdGen)
– think of StdGen as the seed that gets updated
(or as some infinitely long list of pre-gener... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
– unit * f = f * unit = f
– lift f * lift g = lift (f . g)
October 7, 2015
L09-8
Monads as a type class
• Monad is a typeclass that requires
– x >>= f
• (>>=) :: something->(a->something)->something
• (equivalent to bind f x)
– return x
• return :: a -> something
• (equivalent to unit x)
– etc.
• So ... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
\y do plusrand y)
return 5 >>= \x (timesrand x >>= \y plusrand y)
MyRand Int
Int->MyRand Int
MyRand Int
Int -> MyRand Int
October 7, 2015
L09-16
IO with Monads
October 7, 2015
L09-17
Word Count Program
Flag to indicate we
are inside a word
wc :: String -> (Int,Int,Int)
wcs :: String -> Bool -> (Int... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
wcs cs True ((nc+1),nw,nl)
September 29, 2010
L09-19
File Handling Primitives
type Filepath = String
data IOMode = ReadMode | WriteMode | ...
data Handle = ... implemented as built-in type
openFile :: FilePath -> IOMode -> Handle
hClose :: Handle -> () -- void
hIsEOF :: Handle -> Bool
hGetChar :... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
By embedding the concept of I/O in a
monad we guarantee that there is a
single sequence of the monadic I/O
operations (no nondeterminism issues)
IO a: computation which does some I/O,
producing a value of type a.
• Unlike other monads, there is no way to
make an IO a into an a
No operation to take a value out o... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
filename
putStr “\n”
no return !
print :: String -> (Int,Int,Int) -> IO ()
September 29, 2010
L09-27
Word Count using monads -
version 1
main = do
(filename:_) <- getArgs
contents <- getFileContents filename
let (nc,nw,nl) = wc contents
print filename (nc,nw,nl)
pure functional
... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
ch h True ((nc+1),(nw+1),nl)
else
wch h True ((nc+1),nw,nl)
September 29, 2010
L09-30
Beyond I/O
• Monadic I/O is a clever way to force
meaningful interactions with the outside
world. This is what most people think of
when they think of monads
• But monads can do more
– A mechanism to structure... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
18.417 Introduction to Computational Molecular Biology
Lecture 7: September 30, 2004
Lecturer: Ross Lippert
Scribe: Mark Halsey
Editor: Rob Beverly
Divide and Conquer: More Efficient Dynamic Programming
Introduction
We have seen both global and local alignment problems in previous lectures. Briefly:
• Global Align... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
n ) time.
7-1
7-2
Lecture 7: September 30, 2004
Space-Efficient Sequence Alignment
The space complexity of the algorithms we have seen previously is proportional to
the number of vertices in the edit graph, i.e. O(nm). Observe however that the only
values needed to compute the alignment scores in column j in the ... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
optimal alignment is simply pref ix(i) + suf f ix(i). pref ix(i) can be
Lecture 7: September 30, 2004
7-3
computed by finding the score si,
, i.e. we compute the score in linear space as shown
earlier for just the first half of the graph. To compute the suf f ix, we rely on the fact
that in a DAG we can flip the dir... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
from Figure 7.2: Iteratively Computing the Optimal Alignment Midpoints
7-4
Lecture 7: September 30, 2004
Block Alignment and the Four-Russians Speedup
The time complexity of the dynamic programming global alignment algorithm we’ve
studied previously was O(n 2). In this section we examine a trick to speedup the
al... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
4
7-5
• Output: The block alignment of v and w with the maximum score
Let �i,j be the alignment score for the (i, j) block. The recurrence for the block
alignment algorithm is:
si−1,j − �block
�
si,j = max si,j−1 − �block
⎧
�
si−1,j−1 + �i,j
⎧
�
where �block is the indel block penalty. Since the indices of the... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
time is: O( logn ).
2 n
t
t
2
LCS and the Four-Russians Speedup
Finally, the path corresponding to the LCS does not necessarily enter and exit through
the corners of blocks. In this section we turn to the more involved problem of allowing
unrestricted entry and exit between blocks in the partitioned edit graph. W... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
4t possible strings, the lookup table requires
2t2t4t4t = 26t space. Setting t = 4
logn as before makes the table of size O(n1.5). This
allows computation of the n1.5 entries in the table to be constructed in O(n1.5log2n)
time. As in the block alignment problem, this time is dominated by the DP and
allows for an O... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
16
RICHARD B. MELROSE
3. Measureability of functions
Suppose that M is a �-algebra on a set X 4 and N is a �-algebra on
another set Y. A map f : X � Y is said to be measurable with respect
to these given �-algebras on X and Y if
(3.1)
f −1(E) ≤ M � E ≤ N .
Notice how similar this is to one of the characterizati... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/447c5870100c43f0ded7978fc946e4f4_section3.pdf |
X then
(3.4)
f�(M) = E ∀ Y ; f −1(E) ≤ M
is always a �-algebra on Y.
�
In particular if f −1(A) ≤ M for all A ≤ G ∀ N then f�(M) is a �-
algebra containing G, hence containing N by the generating condition.
Thus f −1(E) ≤ M for all E ≤ N so f is measurable.
�
Proposition 3.2. Any continuous map f : X � Y between... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/447c5870100c43f0ded7978fc946e4f4_section3.pdf |
�-algebra
in [−⊂, ⊂] the smallest �-algebra containing all open subsets of R and
all sets (a, ⊂] and [−⊂, b); in fact it is generated by the sets (a, ⊂].
(See Problem 6.)
Our main task is to define the integral of a measurable function: we
start with simple functions. Observe that the characteristic function
of a ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/447c5870100c43f0ded7978fc946e4f4_section3.pdf |
22n − 1, set
En,k = {x ≤ X; 2−nk ⊃ f (x) < 2−n(k + 1)},
∗E = {x ≤ X; f (x) → 2n}.
n
18
RICHARD B. MELROSE
These are measurable sets. On increasing n by one, the interval in the
definition of En,k is divided into two. It follows that the sequence of
simple functions
(3.6)
fn =
2−nk�Ek,n
+ 2n�E�
n
is incr... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/447c5870100c43f0ded7978fc946e4f4_section3.pdf |
BUILD IT
Teaching Notes
Charcoal Press
This project is a very low-cost device for forming charcoal briquettes. It is composed of three welded
metal parts and a wooden block. To make it, you will learn to use a band saw, a sander, a drill press,
the OMAX water jet cutter and several hand tools. You will also learn to ... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
Build-It modules were designed to give students experience
with a variety of tools and manufacturing techniques while at the same time
exposing them to some simple, appropriate technologies. These modules
show them both the rapid prototyping equipment that is available at MIT
as well as techniques that are used in ... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
to give the students time to finish the
necessary work before the next session. It is intended that the projects
are done as part of a hands-on tutorial, with plenty of guidance from the
teacher. Some instructions are given in the text of the exercise, but it is
assumed that the teacher will go over basic safety an... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
sure that the guard is set to the right height before beginning
to cut.
Once the initial cut is made, turning the pipe slowly as you cut into it makes
a cleaner, squarer cut.
If you are using a hack saw to cut the pieces, be sure that the pipe is held
firmly in a vise, near the place where you are making the cut. ... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
the $2
charcoal press
Additional Teaching Notes
You may also want to talk with the students
about pipe nomenclature—in the case of
steel pipe, ½” refers to the inner diameter.
You can also explain the difference between
welded pipe, which will have a small ridge
running through it (which can be a problem
if som... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
too great, then the parts will not be
able to be fit together. We will be welding the two parts together, so the fit
need only hold the part in place while we are welding it, and the interference
can be relatively small. Also note that more compliant materials, such as
wood and plastic, can have larger interference... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
fit an endplate configuration
Fig 2b Overhang endplate configuration
4 Session 3
Welding the parts:
Now that the pieces are cut to size you will put them together to make the
final parts. Welding is the process of joining two pieces by melting the
adjacent material and having it re-solidify to form a joint. In some c... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
materials are provided under the Attribution-Non-Commercial 3.0Creative Commons
License, http://creativecommons.org/licenses/by-nc/3.0/. If you choose to reuse or repost the
materials, you must give proper attribution to MIT, and you must include a copy of the non-
commercial Creative Commons license, or a reasonable... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
8.512 Theory of Solids II
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Lecture 3: Properties of t h e Response Function
In this lecture we will discuss some general properties of the response functions X, and som... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/44bb605cb656287b1aaff1a93bcfa8fc_MIT8_512s09_lec03_rev.pdf |
the curved portion of the
KramersKronig
2
contour. Since we have ensured that χ(ω) is analytic in the upper half plane, Cauchy’s residue
theorem guarantees that the integral over the entire contour is 0. As a result, the piece we need,
i.e. the integral from −∞ to ∞ along the real axis, must also be 0. Thus χ(t)... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/44bb605cb656287b1aaff1a93bcfa8fc_MIT8_512s09_lec03_rev.pdf |
dω�χ(q, ω�
�) Pr
�
�
1
ω − ω�
+ iπχ(�q, ω)
(3.9)
where the additional term iπχ(�
we picked up by making a hump over the pole. Thus for fixed �q,
q, ω) is one half of the contribution from the pole at ω�
= ω that
χ�(�
q, ω) = −
χ��(�
q, ω) = −
1
π
1
π
��
Pr
∞
−∞
∞
��
Pr
−∞
�
�
dω� χ��(�
q, ω)
ω... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/44bb605cb656287b1aaff1a93bcfa8fc_MIT8_512s09_lec03_rev.pdf |
Application Specific Integrated Circuit Design
Lecture 5
Vladimir Stojanoviü
6.973 Communication System Design – Spring 2006
Massachusetts Institute of Technology
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts ... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
increases design cost
Can make faster designs but harder to verify and more expensive
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communicat... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
encourseware (http://ocw.mit.edu/), Massachusetts Institute of Technology.
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6.973 Communication System Design
Gate Arrays
Can cut mark costs by prefabricating arrays of
transistors on wafers
Only customize metal layer for each design
Image removed due to copyright restrictions.
Fi... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
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6.973 Communication System Design
Field-Programmable Gate Arrays (FPGA)
Each cell in array contains a programmable logic function
Array has programmable interconnect between logic functions
Arrays mass-pro... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
mask costs
in advanced technologies
Currently a mask set is a couple $M
FPGAs are getting increased attention
We will use this approach
Courtesy of Arvind and Krste Asanovic. Used with permission.
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCou... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
permission.
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
M I T OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System Design
Standard cell characterization
Power Supply Line (V,,)... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
2006.
M I T OpenCourseWare (http://ocw,mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973Communication System Design
The front-end: Verilog to ASIC layout flow
The “push-button” approach
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spr... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
I Fast w t h timincl constraint I
- bad fast path camot be fixed without redesign1
- might have t o add delay into paths t o satisfy hold time
Courtesy of Arvind and Krste Asanovic. Used with permission.
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT Opencour... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
Recursive pattern to match delay
Much less power than grid
More skew and jitter
Skew
Non-uniform loading
Buffer mismatch
Jitter
Supply noise on buffers
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massach... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System Design
Wire capacitance
Capacitance depends on geometry o f surrounding wires and
relative permittivity, &,,of
insulating dielectric
3-9
- silicon dioxide, SiOp E,
... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
atically with length
Simple lumped n model gives reasonable approximation
- Rw is lumped resistance o f wire
- Cw is lumped capacitance (put half at each end)
cw
Delay = Rdriver x -+ (Rdriver + Rw) x
2
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
M I T OpenCou... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
.
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT Opencourseware (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYW].
6.973 Communication System Design
Wire delay scaling, global wires
For wire crossing whole chip
- Re... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
Topic 11 Notes
Jeremy Orloff
11 Argument Principle
11.1 Introduction
The argument principle (or principle of the argument) is a consequence of the residue theorem. It
connects the winding number of a curve with the number of zeros and poles inside the curve. This
is useful for applications (mathematical and otherwi... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
2
Likewise, if
0 is a pole of order then the Laurent series for () near
0 is
() = ( −
0)−()
where () is analytic and never 0 on a small neighborhood of
0. Thus,
0)− ′()
′()
()
( −
= −
0)−−1() + ( −
0)−()
( −
′()
()
= −
−
0
+
Again we have that
0 is a simple pole of ′()∕ () and
(
Res
′()
... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
give a few examples. This is a key concept in the argument principle and you should make
sure you are very comfortable with it.
Example 11.2. Let () = e with 0 ≤ ≤ 2 (the unit circle). Let () = 2. Describe the curve
(cid:253).
Solution: Clearly (cid:253)() = e2 traverses the unit circle twice as goes from 0 to 2.
... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
= ′() so
′()
()
∫
= ∫ (cid:253)
= 2 Ind( (cid:253), 0)
The last equality in the above equation comes from the definition of winding number.
Note that by assumption does not go through any zeros of , so = (()) is never zero and 1∕
in the integral is not a problem.
Here is an easy corollary to the argument pri... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
of = poles of 1 + )
1 = circle of radius 2.
2 = circle of radius 1/2.
3 = circle of radius 1.
Example 11.5. Let () = 2 + Find the winding number of (cid:253) around 0 for each of the following
curves.
1.
2.
3.
answers. () has zeros at 0, −1. It has no poles.
So, has no poles and two zeros inside
, = 2
Like... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
a leash of
length ℎ(cid:253). Here is the analytic proof.
The argument principle requires the function to have no zeros or poles on . So we first show that
this is true of , + ℎ, ( + ℎ)∕ . The argument is goes as follows.
Zeros: The fact that 0 ≤ (cid:240)ℎ(cid:240) < (cid:240) (cid:240) on implies has no zeros on .... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
�
′
= 0. (We showed above that has
Now, it’s easy to compute that
′
=
( + ℎ)′
+ ℎ
−
′
. So, using
′
( + ℎ)′
+ ℎ
= ∫
Ind((cid:253), 0) = ∫
Now equations 5 and 6 tell us , − , = +ℎ, − +ℎ, , i.e. we have proved Rouchés theorem.
Corollary. Under the same hypotheses, If ℎ and are analytic (no poles) th... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
(cid:240)(cid:240) = 2.
1 + shown below.
To apply the corollary to Rouchés theorem we need to check that (for large) (cid:240)ℎ(cid:240) < (cid:240) (cid:240) on
On
1, = , so
(cid:240) ()(cid:240) = (cid:240)3 + (cid:240) ≥ 3,
(cid:240)ℎ()(cid:240) = 2(cid:240)e(cid:240) = 2.
1 + .
So (cid:240)ℎ(cid:240) < (cid... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
, (cid:240)
0
(cid:240)ℎ(cid:240) ≤ (cid:240)−1
(cid:240)−1 + (cid:240)−2
(cid:240)−2 + … + (cid:240)
0
(cid:240) ≤
−1 + −2 + … + <
.
Re(z)Im(z)iR−iRCRC1
11 ARGUMENT PRINCIPLE
7
On (cid:240)(cid:240) = we have (cid:240) ()(cid:240) = , so we have shown (cid:240)ℎ(cid:240) < (cid:240) (cid:240) on the cur... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
function which is a function of a complex variable.
Typically, the complex variable is denoted by and a capital letter is used for the system function.
Let () be such a system function. We will make a standard assumption that () is meromorphic
with a finite number of (finite) poles. This assumption holds in many intere... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
casual and say ‘the system ()’. It is perfectly
11 ARGUMENT PRINCIPLE
8
clear and rolls off the tongue a little easier!
11.3.2 Pole-zero diagrams
We can visualize () using a pole-zero diagram. This is a diagram in the -plane where we put a
small cross at each pole and a small circle at each zero.
Example 1... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
That is, if the unforced system
always settled down to equilibrium.
11.3.4 Closed loop systems
If the system with system function () is unstable it can sometimes be stabilized by what is called a
negative feedback loop. The new system is called a closed loop system. Its system function is given
Re(s)Im(s)xxxG1(s)1i... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
( −
0( −
0) + …
0 it equals ∕() = 1∕.)
which is clearly analytic at
Example 11.13. Set the feedback factor = 1. Assume is real, for what values of is the open
loop system () =
stable? For what values of is the corresponding closed loop system
() stable?
(There is no particular reason that needs to be real in t... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
is a case where feedback
stabilized an unstable system.
− 1
+ 1
Example 11.15. () =
when = 2.
Solution: The only pole of () is in the left half-plane, so the open loop system is stable. The closed
loop system function is
. Is the open loop system stable? Is the closed loop system stable
() =
=
( − 1)∕( + 1)... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
() = 1∕( + 1), with = 1.
Example 11.17. Take () from the previous example. Describe the Nyquist plot with gain factor
= 2.
Solution: The Nyquist plot is the graph of (). The factor = 2 will scale the circle in the
previous example by 2. That is, the Nyquist plot is the circle through the origin with center = 1.
I... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
curve.) Thus, for all large
the system is stable ⇔
1+,
= 0 ⇔ Ind((cid:253), −1) = ,
.
Finally, we can let go to infinity. The assumption that () decays 0 to as goes to ∞ implies
Re(w)Im(w)iR−iRCRC1xz1xz2 11 ARGUMENT PRINCIPLE
12
that in the limit, the entire curve (cid:253) becomes a single point at the origin. ... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
Nyquist plot around -1? Is the closed loop system stable?
Solution: The curve winds twice around -1 in the counterclockwise direction, so the winding number
Ind((cid:253), −1) = 2. Since the number of poles of in the right half-plane is the same as this winding
number, the closed loop system is stable.
Example 11.2... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
ø 3.17.
Example 11.23. What happens when goes to 0.
Solution: As goes to 0, the Nyquist plot shrinks to a single point at the origin. In this case the
winding number around -1 is 0 and the Nyquist criterion says the closed loop system is stable if and
only if the open loop system is stable.
This should make sense, ... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
feel yourself falling you compensate by shifting your weight or tensing your muscles to counteract
the unwanted acceleration.
MIT OpenCourseWare
https://ocw.mit.edu
18.04 Complex Variables with Applications
Spring 2018
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/te... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf |
18.404/6.840 Lecture 23
Last time:
- !"#$%↑ is EXPSPACE-complete
- Thus !"#$%↑ ∉ PSPACE
- Oracles and P versus NP
Today: (Sipser §10.2)
- Probabilistic computation
- The class BPP
- Branching programs
1
Probabilistic TMs
Defn: A probabilistic Turing machine (PTM) is a var... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/44f3286933ae429ff3cd163e8181ee46_MIT18_404f20_lec23.pdf |
error probability.
Significance: Can make the error probability so small it is negligible.
3
NP and BPP
NP
Computation trees
for ! on "
BPP
≥ 1 accepting
Many accepting
Few
rejecting
" ∈ $
" ∉ $
all rejecting
Few accepting
Many rejecting
Check-in 23.1
Which of these are known to be true?
Check all that apply.
(a) B... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/44f3286933ae429ff3cd163e8181ee46_MIT18_404f20_lec23.pdf |
equivalent read-once BPs}
Theorem: !"ROBP ∈ BPP
Check-in 23.2
Assuming (as we will show) that !"ROBP ∈ BPP,
can we use that to show !"BP ∈ BPP by converting
branching programs to read-once branching programs?
(a) Yes, there is no need to re-read inputs.
(b) No, we cannot do that conversion in general.
(c) No, the co... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/44f3286933ae429ff3cd163e8181ee46_MIT18_404f20_lec23.pdf |
1
1
0
!#
0
0
!$
0
0
0
0
Show by example: Input is !" = 0, !# = 1, !$ = 1
The BP follows its execution path.
Label all nodes and edges on the execution path with 1
and off the execution path with 0.
Output the label of the output node 1.
Obtain the labeling inductively by using these rules:
'
!)
0 1
' ∧ !)
' ∧ !)
'#
'"
... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/44f3286933ae429ff3cd163e8181ee46_MIT18_404f20_lec23.pdf |
= −7
1
Check-in 23.3
Revised 4 for 56ROBP: “On input ;", ;#
What is the output for this branching program using
1. Pick a random non-Boolean input assignment.
the arithmetized interpretation if !" = 1, !# = < ?
2. Evaluate ;" and ;# on that assignment.
3. If ;" and ;# disagree then reject.
(1 − <)
(a)
(< + 1)
If they... | https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/44f3286933ae429ff3cd163e8181ee46_MIT18_404f20_lec23.pdf |
Massachusetts Institute of Technology
Department of Materials Science and Engineering
77 Massachusetts Avenue, Cambridge MA 02139-4307
3.21 Kinetics of Materials—Spring 2006
February 22, 2006
Lecture 5: Effects of Capillarity and Stress on Diffusion.
References
1. Balluffi, Allen, and Carter, Kinetics of Material... | https://ocw.mit.edu/courses/3-21-kinetic-processes-in-materials-spring-2006/44fe0b888d0d01732fa446f1df63383c_ls5.pdf |
in solids can affect diffusion in a variety of ways, including via the mobility, its effect on
diffusion potential, and via influencing the boundary conditions for diffusion.
• Even a uniform state of stress may break symmetry and cause jumps in certain directions in the
material to be favored and modify the mobility... | https://ocw.mit.edu/courses/3-21-kinetic-processes-in-materials-spring-2006/44fe0b888d0d01732fa446f1df63383c_ls5.pdf |
3.012 Fund of Mat Sci: Bonding – Lecture 1 bis
WAVE MECHANICS
Photo courtesy of Malene Thyssen, www.mtfoto.dk/malene/
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last Time
1. Players: particles (protons and neutrons in
the nuclei, electrons) and electromagnetic
fields (photons)... | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/451781865b2544b82bfb2234d4808e31_lec01b_bis.pdf |
• Potential energy V
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Polar Representation
Diagram of the Argand plane removed for copyright reasons.
See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 1011, figure B.6.
3.012 Fundamentals of Materials Sci... | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/451781865b2544b82bfb2234d4808e31_lec01b_bis.pdf |
6.626 x 10-34 J s = 2π a.u.)
See animation at http://www.kfunigraz.ac.at/imawww/vqm/movies.html
Select “Samples from Visual Quantum Mechanics” > “Double-slit experiment”
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Time-dependent Schrödinger’s equation
(Newton’s 2nd law for quantu... | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/451781865b2544b82bfb2234d4808e31_lec01b_bis.pdf |
Equation (II)
−
⎡
⎢
⎣
2
h
2
m
2
+∇
r
)(
rV
⎤
r
)(
r
ϕ =
⎥
⎦
r
rE
)(
ϕ
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005) | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/451781865b2544b82bfb2234d4808e31_lec01b_bis.pdf |
Lecture 8
Understanding Transcription
RNA-seq analysis
Foundations of Computational Systems Biology
David K. Gifford
1 Lecture 8 – RNA-seq Analysis
• RNA-seq principles
– How can we characterize mRNA isoform
expression using high-throughput sequencing?
• Differential expression and PCA
– What genes are di... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/45a80bf19f88d0cabfb091b4edbc0f36_MIT7_91JS14_Lecture8.pdf |
to RNA-seq analysis
Short sequencing reads,
randomly sampled from a
transcript
exon 1
exon 2
exon 3
1. Assemble reads into
transcripts. Typical
issues with coverage and
correctness.
2. Map reads to reference
genome and identify
isoforms using constraints
• Goal is to quantify
isoforms and determine
s... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/45a80bf19f88d0cabfb091b4edbc0f36_MIT7_91JS14_Lecture8.pdf |
starting at a specific position Si in a transcript of
length lj is:""
P(S = Si | T = Tj ) = 1
l j
starting position in transcript, Si
Ri
Tj
Transcript length lj
Courtesy of Cole Trapnell. Used with permission.
Slide courtesy Cole Trapnell
13 P(Ri | T=Tj) – Paired end reads
Assume our library fragments have a len... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/45a80bf19f88d0cabfb091b4edbc0f36_MIT7_91JS14_Lecture8.pdf |
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0.01
0.1
1
10
100
1000
10... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/45a80bf19f88d0cabfb091b4edbc0f36_MIT7_91JS14_Lecture8.pdf |
a
r
T
s
t
p
i
r
c
s
n
a
r
T
Transcript categories, by coverage
●●
●●
match
contained
intra−intron
●
novel isoform
repeat
other
●●
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●● ●● ... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/45a80bf19f88d0cabfb091b4edbc0f36_MIT7_91JS14_Lecture8.pdf |
• Differential expression and PCA
– What genes are differentially expressed, and how
can we characterize expressed genes?
• Single cell RNA-seq
– What are the benefits and challenges of working
with single cells for RNA-seq?
18
19 20 21 22 23 Scaling RNA-seq data (DESeq)
i gene or isoform
•
j sample (e... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/45a80bf19f88d0cabfb091b4edbc0f36_MIT7_91JS14_Lecture8.pdf |
parameters in
H1 minus the number of free parameters in H0; in this case
degrees of freedom is 4 – 2 = 2 (H1 has an extra mean and
variance).
• Likelihood ratio test defines a test statistic that follows the Chi
Squared distribution
iT = 2 log
P(
iAK | H1)P(
P
iAK ,
P(H 0) ≈ 1− ChiSquaredCDF (T i | dof )
iBK | H... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/45a80bf19f88d0cabfb091b4edbc0f36_MIT7_91JS14_Lecture8.pdf |
K., Rahul Satija, et al. "Single-cell Transcriptomics Reveals Bimodality in Expression and Splicing in Immune Cells." Nature (2013).
AK Shalek et al. Nature 000, 1-5
(2012) doi:10.1038/nature12172
38 Analysis of co-variation in single-cell mRNA expression levels
reveals distinct maturity states and an antiviral cell... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/45a80bf19f88d0cabfb091b4edbc0f36_MIT7_91JS14_Lecture8.pdf |
2014
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/45a80bf19f88d0cabfb091b4edbc0f36_MIT7_91JS14_Lecture8.pdf |
Course Organization
Spirit of the Undertaking
6.871: Knowledge-Based Systems
Spring 2005
Randall Davis
Logistics
• Info sheet, syllabus
• Personnel:
– Lecturers: Davis (and friends)
• Course notes:
– 1st installment ready now
• You are responsible for what happens in lecture.
• No open laptops.
2
Course Charac... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/45d32f86dec5685e9c9fcbf8b61d804b_lect01_intro1.pdf |
and Barings Bank
The week before Nick Leeson disappeared he had kept
throwing up at work.
Colleagues did not know why but were soon to find out.
The ego of a 28-year-old trader on the Singapore Monetary
Exchange and the greed and stupidity of a 233-years-old bank
had combined to distroy an investment empire and in ... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/45d32f86dec5685e9c9fcbf8b61d804b_lect01_intro1.pdf |
procedure
13
Character of the problems
attacked
• Balancing your checkbook vs.
Getting out of the supermarket
• Telling it what to do vs.
Telling it what to know
– Write down some relevant knowledge
– Advice, not a procedure
Knowledge leads to action,
But knowledge is more fundamental
–The barge story
14
Wh... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/45d32f86dec5685e9c9fcbf8b61d804b_lect01_intro1.pdf |
To see this image please visit:
http://images.google.com/images?q=charles_babbage.full.jpg
21
Image removed due to copyright considerations.
22
Image removed due to copyright considerations.
23
Intellectual Origins
• 2000 years ago
– Aristotle and the art of rhetoric
– The syllogisms
• 17th century: Leibniz an... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/45d32f86dec5685e9c9fcbf8b61d804b_lect01_intro1.pdf |
ationality.
• Knowledge is closely linked to rationality.
• Knowledge is competence-like notion.
30
Character of Knowledge
• Most of what we know knowledge is non
numeric.
• Most of what we know is heuristic.
– What’s certain?
– What’s the alternative?
31
Character of Knowledge
• Empirical, experiential knowl... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/45d32f86dec5685e9c9fcbf8b61d804b_lect01_intro1.pdf |
Chapter 7
Scattering
c(cid:13) B. Zwiebach
In high energy physics experiments a beam of particles hits a target composed of par-
ticles. By detecting the by-products one aims to study the interactions that occur during
the collision.
Collisions can be rather intricate. For example, the particles involved may be not
ele... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/45dbd7038c3e969491eceeae86c44d42_MIT8_06S18ch7.pdf |
(cid:126)2
2M
(7.1.1)
(7.1.2)
(7.1.3)
Figure 7.1: Potential is finite range, or vanishes faster than 1
r as r → ∞
Will be consider solutions with positive energy. The energy is the energy of the particle
far away from the potential, E = (cid:126)2k2
2M . The Schr¨odinger equation then takes the form
(cid:20)
−
(cid:126)... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/45dbd7038c3e969491eceeae86c44d42_MIT8_06S18ch7.pdf |
ϕ(r) satisfies (7.1.4) for any r > a. For r < a,
however, it does not satisfy the equation; ϕ(r) is a solution everywhere only if the potential
vanishes.
Given an incident wave we will also have a scattered wave. Could it be an ψ(r) = eikr
that propagates radially out?
(cid:0)∇2 + k2(cid:1) eikr (cid:54)= 0
fails badl... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/45dbd7038c3e969491eceeae86c44d42_MIT8_06S18ch7.pdf |
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