text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
ently, expressing the transfer function as a sum using partial fraction expansion gives a
parallel structure:
Np
�
Ns
�
H(z) =
Ckz−k +
e0k + e1kz−1
1 − a1kz−1 − a2kz−2 .
k=1
OSB Figure 6.20 shows a parallel form structure for a sixth-order system. See Section 6.3 for
a more detailed explanation.
k=0
FIR Fil... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
. . . , M .
Using this special property, can we further simplify the tap-delay line filter structure to reduce
the number of multipliers required? The answer is yes. Consider a type I system where the
transfer function H(z) is of even degree and symmetric. Rewrite H(z) as:
H(z) =
h[n]z−n = h[0](1 + z−M )) + h[1](z−... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
is fed into the two input ports of the first stage, while the
output is taken as that of the top branch of the last stage:
General form of a lattice filter
The coefficients k1, k2, . . . , kp+1 are called the k-parameters of the lattice structure. If the
input s[n] is a unit impulse, it can be shown that the intermedia... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
(z) . The following figure shows the overall structure of an all-pole lattice filter.
Note since B0(z) = z−1, the bottom branch of the final lattice section should be connected to
the top branch.
A0(z)
All-pole lattice filter
An important property of all-pole lattice filters is that the systems are stable if and only if... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
magnitudes and pole-
zero plots of an 8th-order bandpass filter under different implementation forms and various
quantization accuracies. Note when the coefficients are quantized to 12 or 10-bits in the direct
form implementation, some poles move to the outside of the unit circle, making the overall
system unstable.
6
... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
20.430 / 2.795 / 6.561 / 10.539
Fields Forces and Flows
in Biological Systems
Fall 2015
Instructors: Mark Bathe, Alan Grodzinsky
11
Textbook:
Fields Forces and Flows in Biological Systems
Garland Science, March 2011
Book cover removed due to copyright restrictions.
Source: Grodzinsky, Alan. Fi... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
with us l
.l::!.2.m& > People
Screenshot removed due to copyright restrictions.
Source: Prof. Paolo Provenzano's website.
•
Degrees
Paolo Provenzano
Assistant Professor
Office: 7-120 Hasselmo Hall
Phone: 612-624-3279
Email: pproyenz@umn ,edu
Provenzano Lab
•
B,S, Mechanical Engineering, University of Wisco... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
PHFKDQLFDOVWLIIQHVV
B,S, Mechanical Engineering, University of Wisconsin, 1998
Employment Opportunities
Administrative Forms
Department Home
Degrees
Provenzano Lab
•
•
Connect with us l
• M,S, Biomedical Engineering (Mechanics), University of Wisconsin, 2000
7KHLQWHUVWLWLDOGLIIXVLRQFRHIILFLHQWRI,J... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
number; Solution procedures
Examples of diffusion-reaction: Diffusion of a ligand through tissue with cell
receptor-ligand interactions; Diffusion-reaction kinetics
Sep 28 More examples of diffusion-reaction
Sep 24 Convective solute transport: examples
Sep 26
Sep 30
Case study: IGF-1 diffusion-reaction within ti... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
pondylitis
• Psoriatic Arthritis
• Psoriasis
1313
Effects of a cell signaling (kinase) blocker (Merck BI-78D)
Several Applications: diabetes; purposely
induce cell death (apoptosis) in tumors
© source unknown. All rights reserved. This content is
excluded from our Creative Commons license. For more
informat... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
information, see http://ocw.mit.edu/help/faq-fair-use/
excluded from our Creative Commons license. For more
information, see http://ocw.mit.edu/help/faq-fair-use/
information, see http://ocw.mit.edu/help/faq-fair-use/
1515
Insulin-like Growth Factor-1 (IGF-1)
• Peptide Growth Factor:
♦ Stimul... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
image © source unknown. All rights reserved. This content is excluded from our
Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
1717
PNAS 2009
• Rett patients express aberrantly high levels of IGFBP3,
which inhibits IGF-1 signaling. Depressed IGF-1 signaling
has indeed b... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
factor-1: detergent binding inhibits
binding protein interactions." Biochemistry 40, no. 37
(2001): 11022-11029.
0
400
800
TIME, minutes
1200 1600
=τ
lag
6
2
L
D
eff
slow reaction, or slow diffusion compared to reaction???
20
Lect Date
Oct 1
8
Oct 5
II. ELECTRICAL SUBSYSTEM
E-fields and transport; Maxwel... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
ed flow and transport
Nov 14
10
11
12
13
14
15
16
17
18
19
20
2121
EM Waves
© Garland Science. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Source: Grodzinsky, Alan. Field, Forces and Flows in Biologic... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
,
see http://ocw.mit.edu/help/faq-fair-use/.
Deep Brain
Stimulation
via B-fields
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Source: Wagner, Tim et al. "Transcranial magnetic stimulation and stroke: a
computer-based human model study." Neuroimage 30, no. 3 (2006): 857-870.
272... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
Connective Tissue
Courtesy of Alan Grodzinsky. Used with permission.
3030
3131 2012
µ-fluidic Chip
© Royal Society of Chemistry. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
7775.
3434
Zeta Potential (particle charge) Instruments
+ (applied electric field) ▬
Measure “ζ” → Infer effective particle charge
© source unknown. All rights reserved. This content is
excluded from our Creative Commons license. For more
information, see http://ocw.mit.edu/help/faq-fair-use/.
© source unkno... | https://ocw.mit.edu/courses/20-430j-fields-forces-and-flows-in-biological-systems-fall-2015/2c91de6fa25b4b73b6450a1a21e40872_MIT20_430JF15_Lecture1.pdf |
18.01 Calculus
Jason Starr
Fall 2005
Lecture 3. September 13, 2005
Homework. Problem Set 1 Part I: (i) and (j).
Practice Problems. Course Reader: 1E1, 1E3, 1E5.
1. Another derivative. Use the 3step method to compute the derivative of f (x) = 1/
is,
Upshot: Computing derivatives by the definition is too much work... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
that for every positive integer n and every pair of numbers a and b,
(a + b)n equals,
a + na n−1b + · · · +
n
a n−k bk + · · · + nabn−1 + b .
n
� �
n
k
This is proved by mathematical induction. First, the result is very easy when n = 1; it just says
that (a + b)1 equals a1 + b1 . Next, make the induction hypoth... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
+ ab
. . . + nab
n + bn+1
Summing in columns gives,
an+1 + (n + 1)anb +
�
. . . + ( k + k−1 )an+1−k bk + ( k+1 +
� �
�
�
�
n
n
n
� �
n
k
)an−kbk+1 +
. . . + (1 + n)ab
n
+ bn+1.
18.01 Calculus
Jason Starr
Fall 2005
Using Pascal’s formula, this simplifies to,
�
an+1 + (n + 1)anb +
. . . +
�
n+1 an+1−... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
+ hn .
Thus the difference quotient is,
f (a + h) − f (a)
h
= na +
n−1
� �
n
2
� �
n
k
a n−2h + · · · +
a n−k hk−1 + · · · + hn−1 .
Every summand except the first is divisible by h. The limit of such a term as h
→
0 is 0. Thus,
lim
→
h 0
f (a + h) − f (a)
h
= na n−1 + 0 + · · · + 0 = na
n−1
.
So f �(x)... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
differentiable functions. If g(a) is nonzero, the
quotient function f (x)/g(x) is defined and differentiable at a, and,
(f (x)/g(x))� = [f �(x)g(x) − f (x)g�(x)]/g(x)2 .
18.01 Calculus
Jason Starr
Fall 2005
One way to deduce this formula is to set q(x) = f (x)/g(x) so that f (x) = q(x)g(x), and the apply
the Lei... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
)
dx
= (1)x n + x
d(xn)
dx
.
By the induction hypothesis, the second term can be replaced,
d(xn+1)
dx
= x + x(nx n−1) = x + nx n = (n + 1)x .
n
n
n
Thus the formula for n implies the formula for n + 1. Therefore, by mathematical induction, the
formula holds for every positive integer n. | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2005/2c949d4dae713c3686f2679eacec4595_lecture_3.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.006 Introduction to Algorithms
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Lecture 6
Hashing II: Table Doubling, Karp-Rabin
6.006 Spring 2008
Lecture 6: Hashing II: Table Doubling,
Karp-Rabin
Lecture Overvi... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/2cc0efa30d154a5ec5e161a0adcfc373_lec6.pdf |
wkax}r}w-rkeepignoreignore≡+product as sumlots of mixing Lecture 6
Hashing II: Table Doubling, Karp-Rabin
6.006 Spring 2008
How fast to grow?
When n reaches m, say
•
m + = 1?
=
= n inserts cost Θ(1 + 2 + + n) = Θ(n2)
rebuild every step
⇒
⇒
· · ·
• m ∗ = 2? m = Θ(n) still (r+ = 1)
rebuild at insertion 2i
n in... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/2cc0efa30d154a5ec5e161a0adcfc373_lec6.pdf |
where and how many
times?)
E.g. s = ‘6.006’ and t = your entire INBOX (‘grep’ on UNIX)
3
Lecture 6
Hashing II: Table Doubling, Karp-Rabin
6.006 Spring 2008
Figure 3: Illustration of Simple Algorithm for the String Matching Problem
Simple Algorithm:
Any (s == t[i : i + len(s)] for i in range(len(t)-len(s)))
- ... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/2cc0efa30d154a5ec5e161a0adcfc373_lec6.pdf |
of code is O(| s |)
for i in range(len(s), len(t)):
ht.skip(t[i-len(s)])
ht.append(t[i])
if hs() == ht(): ...
The second block of code is O(| t |)
Data Structure:
Treat string as a multidigit number u in base a where a denotes the alphabet size. E.g. 256
• h() = u mod p for prime p ≈| s | or | t | (division method)... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/2cc0efa30d154a5ec5e161a0adcfc373_lec6.pdf |
Subclassing and Dynamic Dispatch
6.170 Lecture 3
This lecture is about dynamic dispatch: how a call o.m() may actually invoke the code of different
methods, all with the same name m, depending on the runtime type of the receiver object o.
To explain how this happens, we show how one class can be defined as a subclass... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
date;
...
}
2
Extending a Class by Inheritance
Suppose we want to implement a new kind of account that allows overdrafts. We might call it
AccountPlus, and code it like this:
class AccountPlus extends Account {
int creditLimit;
AccountPlus (String n, int c) {
super (n);
creditLimit = c;
}
boolean checkTrans ... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
whose type is not the variable’s type; it is sufficient that the object
type be the type of a subclass of the variable type. (For now, by the way, we’re using the term
‘type’ to mean classification by class name, to distinguish it from the term ‘class’ which usually
carries the connotation of the code in the class too.... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
general, how variables are declared has no effect
whatsoever on the behavior of the program, if it executes successfully without class cast errors
(more on that below).
Now suppose we want to handle a collection of accounts. We might have a Bank class, imple
mented something like this:
1.
2.
3.
4.
5.
6.
7.
8... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
is said to be polymorphic, meaning ‘many shapes’, since the same piece of code text
can handle different types of account. If the accounts array contains two objects of the class
Account, and a third object of class AccountPlus, the first and second time round the loop the
call to the method checkTrans will execute co... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
print statement will print 0 as the balance. If
the method from AccountPlus is called, it will return true, the posting will occur, and the balance
will print as -50.
The answer depends on the runtime type of the receiver. Although post belongs to the class
Account, since there is no post method in AccountPlus, its... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
Trans fee = new Trans (-1, new Date ());
if (accounts.elementAt (i).checkTrans (fee))
accounts.elementAt (i).post (fee);
}
}
...
}
The class Vector is provided as part of the standard Java library. Unlike arrays, vectors are
not part of the language itself. So there’s no special syntax to access a vector element:... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
4.
5.
6.
7.
}
}
or better:
void chargeMonthlyFee () {
for (int i = 0; i < accounts.size(); i++) {
Trans fee = new Trans (-1, new Date ());
Account acc = (Account) accounts.elementAt (i);
if (acc.checkTrans (fee)) {
acc.post (fee);
}
5
8.
9.
}
}
The (Account) on Statement 4 is called a downcast. At r... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
1.23
1
The phrase (int) in Statement 2 is a typecast or coercion; it ensures the type safety of the
program by actually converting the double created at Statement 1 to an integer, so that d and i
have different values. No such thing happens with a downcast; if the statement
Account acc = (Account) accounts.elementA... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
type, and
interfaces thus contribute to the type hierarchy. Here is a fragment of the type hierarchy that shows
some interfaces implemented by Vector:
Object
AbstractCollection
AbstractList
Collection
AbstractSet
List
Vector
Set
HashSet
The interface names are italicized to distinguish them from the names o... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
cast guarantees the typing property. If the cast
fails (that is, e evaluates to an object of the wrong type), the assignment is aborted; if it succeeds,
the expression e must have evaluated to an object of an appropriate type – that is, a subtype of T.
7 Conclusion
We have distinguished between the declared type of... | https://ocw.mit.edu/courses/6-170-laboratory-in-software-engineering-fall-2005/2d0d2ac53d181170c448b7a78f007a3b_lec3.pdf |
6.895 Essential Coding Theory
September 22, 2004
Lecturer: Madhu Sudan
Scribe: Swastik Kopparty
Lecture 5
1 Algebraic Codes
In this lecture we will study combinatorial properties of several algebraic codes. In particular, we will
introduce:
Reed-Solomon Codes based on univariate polynomials over finite fields.
... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
�
∈
(k
(k
S
−
−
−
−
≈
−
−
n
�
|
|
q
polynomial interpolation theorem:
Theorem 1 Polynomial Interpolation Theorem: Let F be a field. For any �1, �2, . . . �l+1
pairwise distinct, and any y1, y2, . . . , yl+1
p(�i) = yi,
unique polynomial p(x) with degree
F ,
l such that
[l + 1].
F,
�
≤
≤
i
�
�
≤
Proof Idea O... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
yi,
unique poly
≤
[l + 1].
F,
≤
t
�
≤
�
�
≤
This motivates the Reed Solomon Code (1960ish) as defined below:
Given n, q, k, �1, . . . �n, with �i distinct elements of Fq .
•
For c = (c0, c1, . . . ck−1), define polynomial pc(x) = c0 + c1x + . . . ck−1xk−1
•
5-1
The code is specified by the encoding function RS... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
�
�
�
group F� . However, today we don’t bother with that.
q
Any code that meets the projection bound (with d = n
Separable (MDS) code.
k + 1) is called a Maximum Distance
−
2.1 Linear Codes and their Duals
Recall that for any linear code
and the n
−
matrix. This is a code with n
H T GT = 0 and thus GT is the ... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
ynomials which mitigates this problem to some
extent.
Without further ado, we define the multivariate polynomial evaluation map, the Reed Muller code
(1955-1956):
Given m, l, q, our code will be over P l :=
m
polynomials over Fq in m variables of degree1 l
{
}
•
Note that this too is a linear code. To determine the... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
indeed used it to prove the distance of the RS code).
l is zero on at most q
�
q
q
= q
•
Let us just pick some parameters arbitrarily to get a feel for the kinds of codes that we get.
m = 2
1
l = 3 q
•
•
So n = q2 , k =
1
3 q+2
2 ≥
�
1
2 2
18 q2 , d = 3 n = 3 q
2
•
[q2 , 1
�
2
18 q
2 , 3 q2]q codes (note t... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
with relative distance > 2/3 2 The Plotkin bound extends this idea to codes with relative
distance 1/2 and shows that the Hadamard codes are optimal for this distance.
Theorem 3 Plotkin Bound: If there exists a (n, k, n/2)2 code, then k
log2(2n).
�
1n by (vi)j = (
Sketch of Proof Suppose the code consists of words ... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
and x − z. They are both of
relative hamming weight 2/3 + �. Thus they agree in at least 1/3 + 2� of the positions. So the relative distance between y
and z is at most 2/3 − 2�.
5-3
5.1 Proof of the statement about zeroes of a multivariate polynomial
We want to show that for a degree l polynomia... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
of a
nonzero vector with a purely random vector gives a purely random bit. This result shows up in many
other contexts with very different proofs.
5-4
√ | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/2d15b1d620678d017841947ec1205298_lect05.pdf |
Lecture 1
Mean Value Theorem
Theorem 1 Suppose Ω ⊂ Rn , u ∈ C2(Ω), Δu = 0 in Ω, and B = B(y, R) ⊂⊂ Ω, then
u(y) =
1
nωnRn−1
1
Rn
ωn
�
B
udx
�
uds =
∂B
�
∂u
∂Br ∂ν
ds =
�
Br
Δudx = 0. Thus
0 =
ds =
�
∂u
∂Br ∂ν
Proof:By Green’s formula, for r ∈ (0, R),
�
∂u
∂Br ∂r
�
∂u
Sn−1 ∂r
�
∂r Sn−1
(r ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
call u superharmonic.
Also we have �u ≤ 0 =
⇒ u(y) ≥
nωnRn−1 ∂B
1
�
Application: Maximum principle and uniqueness.
1
Theorem 2 Ω ⊂ Rn, u ∈ C2(Ω), Δu ≥ 0, If ∃p ∈ Ω s.t.
u(p) = max u,
Ω
then u is constant.
Proof: Let
M = sup u,
Ω
ΩM = {x ∈ Ω|u(x) = M }.
ΩM is not empty because p ∈ M , ΩM is closed by c... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
constant C = C(n, Ω, Ω�) s.t.
sup u ≤ C inf u.
Ω�
Ω�
Proof: Let y ∈ Ω�, B(y, 4R) ⊂ Ω. Take x1, x2 ∈ B(y, R), we have
�
�
u(x1) =
1
ωnRn
udx ≤
1
ωnRn
u(x2) =
1
ωn(3R)n
udx ≥
1
ωn(3R)n
udx,
B(y,2R)
B(x1,R)
�
B(x2,3R)
udx,
B(y,2R)
�
= ⇒ u(x1) ≤ 3n u(x2),
= ⇒ sup ≤ 3n
.
B(y,R)
inf
B(y,R)
Choose ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
� ⊂ Ω. Then for multiindex α, there exists constant
C = C(n, α, Ω, Ω�) s.t.
|
sup Dα u ≤ C sup u .
|
|
Ω�
Ω
|
Proof: Since ∂ Δ = Δ ∂ , Du is also harmonic. So by mean value theorem and
divergence theorems, we have for B(y, R) ⊂ Ω,
∂xi
∂xi
Du(y) =
1
ωnRn
�
B(y,R)
Dudx =
1
Rn
ωn
�
∂B
u−ν ds →
= ⇒ |Du(y)
... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
, n > 2,
, n = 2.
Note that away from origin, ΔΓ(x) = 0.
3
Theorem 5 Suppose u ∈ C2(Ω), then for y ∈ Ω, we have
�
u(y) =
(x − y) − Γ(x − y)
)dσ +
Γ(x − y)Δudx.
∂u
∂ν
Ω
�
∂Γ
(u
∂Ω ∂ν
Proof: Take ρ small enough s.t. Bρ = Bρ(y) ⊂ Ω. Apply Green’s 2nd formula to u
and v(x) = Γ(x − y), which is harmonic in ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
(0) and ϕ is continuous function on ∂B. Then
�
u(x) =
2 �
−|x |
R2
nωnR
ϕ(x)
ϕ(y)
∂B x−y n ds
|
|
, x ∈ B,
, x ∈ ∂B.
belongs to C2(B) ∩ C0(B) and satisfies Δu = 0 in B.
4 | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2d5342d2f35b2d991e8a284c5ab1e325_lec1.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
(cid:10) 6.642 Continuum Electromechanics
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
(cid:13)
6.642, Continuum Electromechanics, Fall 2004
Prof. Markus Zahn
Lecture 6: Stress Tensors
I. Maxwell Stress Tensor ... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
j
i
⎛
1
∂
⎜
x 2
∂
⎝
i
μ
H H -
K K
ρ
∂μ
∂ρ
H H
K
K
⎞
⎟
⎠
=
∂
x
∂
j
(
μ
)
H H -
j
i
1
2
δ
H H
ij K K
-
μ ρ
⎛
⎜
⎝
μ
∂
∂ρ
⎞
⎟
⎠
=
∂
∂
T
ij
x
j
T = H H -
μ
i
j
ij
1
2
δ
H H
ij K K
-
μ ρ
⎛
⎜
⎝
⎞∂μ
⎟
∂ρ
⎠
II. Air-Gap Magnetic Machine
Courtesy of MIT Press. Used with permission.
6.642, Continuum Electromechanics Lecture 6
P... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
force on a wavelength
⎤
⎥
⎦
f =
z
w
π
k
μ
0
=
w
π μ
k
0
r *
(cid:105) (cid:105)r
⎡
Re H H
⎢
z
⎣
x
⎤
⎥
⎦
⎡
Re -K H
⎢
⎣
(cid:105) (cid:105) r *
x
r
⎤
⎥
⎦
x
s
(cid:105)
⎡
B
⎢
⎢
r
(cid:105)
B
⎣
x
⎤
⎥
⎥
⎦
= k
μ
0
1
sinh kd
⎡
-coth kd
⎢
⎢
⎢
-
⎢
⎣
⎤
⎡
⎥
⎢
⎥
(cid:4)
⎢
⎥ χ
coth kd
⎣
⎥
⎦
1
sinh kd
(cid:4)
⎤χ
⎥
s
⎥
⎦
r
6.642,... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
h kd
r
(cid:4)
χ
⎤
⎥
+ coth kd
⎥
⎦
⎡
= k -
⎢
0
⎢
⎣
μ
(cid:105)
Κ
(cid:105)
Κ
r
jk sinh kd jk
-
s
⎤
coth kd
⎥
⎥
⎦
*
r
⎡
(cid:105) (cid:105)
Re -K H
⎢
r
⎣
x
μ
0
⎤
⎥
⎦
= -Re +
⎡
⎢
⎢
⎣
j k
μ
0
k
*
(cid:105) (cid:105)
K K
r
sinh kd
s
⎛
⎜
⎜
⎝
*
(cid:105) (cid:105)
r
+ K K coth kd
r
⎞
⎟
⎟
⎠
⎤
⎥
⎥
⎦
⎡
= Re -
⎢
⎣
μ
0
*
⎤
(cid:1... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
K = K sin
0
r
ω
r
⎡
⎣
(
t - k z' -
δ
)
⎤
⎦
; z' = z - Ut
6.642, Continuum Electromechanics Lecture 6
Prof. Markus Zahn Page 5 of 8
r
= K sin
0
(
⎡
⎣
ω
r
+ kU t - k z -
)
(
⎡
= Re -jK e
⎣
r
0
(
)
j +kU t
ω
r
kδ
j
e
⎤
⎦
(cid:105)
Κ = -jK e
s
s
0
ωsj
t... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
. Electrostatic Machine
Courtesy of MIT Press. Used with permission.
6.642, Continuum Electromechanics Lecture 6
Prof. Markus Zahn Page 6 of 8
f = w
z
2
π
k
2 k
π
∫
0
T
zx
x=0
dz =
2 w
π
k
2 k
π
∫
0
ε
E E
0 z x
dz
x=0
r
(cid:4)
zE = jk V
(cid:105)
r
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
h kd
⎡
⎢
⎢
⎣
(cid:105)
⎤
+ V coth kd
⎥
⎥
⎦
r
* r
(cid:105) (cid:4)
Re -jk V E = Re -jk
r
x
ε
0
⎤
⎥
⎦
⎡
⎢
⎣
⎡
⎢
⎢
⎣
2
ε
0
(cid:105)
V
*
r
(cid:105)
-V
s
sinh kd
⎛
⎜
⎜
⎝
+ V coth kd
(cid:105)
r
⎞
⎟
⎟
⎠
⎤
⎥
⎥
⎦
f =
z
2
ε
kw
π
k sinh kd
0
⎡
= Re +jk
⎢
⎣
2
ε
0
(cid:105) (cid:105) *
⎤
V V sinh kd
⎥
r
s
⎦
(cid:105) (cid:105)
... | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
V V e e
0
r
0
-jk
δ
⎡
⎣
j
(
-
ω ω
s
r
)
-kU t
⎤
⎦
f =
z
ε
wk
π
0
sinh kd
ω
s
r= + kU
ω
f = -
z
ε
wk
π
0
sinh kd
r
V V sin k
δ
0
s
0
6.642, Continuum Electromechanics Lecture 6
Prof. Markus Zahn Page 8 of 8 | https://ocw.mit.edu/courses/6-642-continuum-electromechanics-fall-2008/2d6f52a2aec3aa1daff32ade842734ad_lec06_f08.pdf |
Color
Wojciech Matusik MIT EECS
Many slides courtesy of Victor Ostromoukhov, Leonard McMillan, Bill Freeman, Fredo Durand
Image courtesy of Chevre on Wikimedia Commons. License: CC-BY-SA. This content is excluded from
our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
1
... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
from our Creative
Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
This image is in the public domain. Source: Wikimedia Commons.
9
Questions?
So far, physical side of colors: spectra
an infinite number of values
(one per wavelength)
© Sinauer Associates, Inc. All rights reserved.... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
wavelength)
Stimulus
Cone responses
Multiply wavelength by wavelength
End up with 3
values (one per
cone type)
Integrate
1 number
1 number
© source unknown. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
1 numbe... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
e
r
G
w
o
l
l
e
Y
e
g
n
a
r
O
d
e
R
s
e
s
n
o
p
s
e
R
r
o
t
p
e
c
e
R
400
500
600
700
Wavelengths (nm)
25
Questions?
26
Plan
• Spectra
• Cones and spectral response
• Color blindness and metamers
• Color matching
• Color spaces
27
Color blindness
• Classical case: 1 type of cone is mi... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
ize, changes the red/green variation to
brightness and blue/yellow variations.
– http://www.vischeck.com/dalton
– http://www.vischeck.com/daltonize/runDaltonize.php
32
Metamers
• We are all color blind!
• These two different
spectra elicit the same
cone responses
• Called metamers
© source unknown. All rig... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
by illuminant
– enables color reproduction with only 3 primaries
39
Questions?
40
Analysis & Synthesis
• Now let’s switch to technology
• We want to measure & reproduce color
as seen by humans
• No need for full spectrum
• Only need to match up to metamerism
41
Analysis & Synthesis
• Focus on additive c... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
Summary
• Physical color
– Spectrum
– multiplication of light & reflectance spectrum
• Perceptual color
– Cone spectral response: 3 numbers
– Metamers: different spectrum, same responses
• Color matching, enables color reproduction with 3 primaries
• Fundamental difficulty
– Spectra are infinite-dimensional (f... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
-green
– Don’t worry, we’ll be able to convert it to any other set of
primaries (Linear algebra to the rescue!)
• Resulting 3 numbers for each input wavelength are
called tristimulus values
54
Now, our interactive
feature!
You are...
THE LAB RAT
55
56
Color Matching Problem
• Some colors cannot be produ... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
• Unfortunately unknown at that time. This would
have made life a lot easier!
61
Recap
• Spectra : infinite dimensional
• Cones: 3 spectral responses
• Metamers: spectra that look the same
(same projection onto cone responses)
• CIE measured color response:
– chose 3 primaries
– tristimulus curves to reprodu... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
be obtained from XYZ by a 3x3 matrix
one example RGB space
69
Other primaries
• We want to use a new set of primaries
– e.g. the spectra of R, G & B in a projector or monitor
• By linearity of color matching,
can be obtained from XYZ by a 3x3 matrix
• This matrix tells us how to match the 3 primary
spectra fr... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
.288284 -0.0282980 0.898611
71
Color gamut
• Given 3 primaries
• The realizable
chromaticities lay in the
triangle in xy
chromaticity diagram
• Because we can only
add light, no negative
light
© source unknown. All rights reserved. This content is
excluded from our Creative Commons license. For more... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
we use linear encoding, we have tons of information
between 128 and 255, but very little between 1 and 2!
• This is why a non-linear gamma remapping of about 2.0
is applied before encoding
• True also of analog imaging to optimize signal-noise
ratio
78
Color quantization gamma
• The human visual system is mo... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
http://ocw.mit.edu/help/faq-fair-use/.
84
MIT OpenCourseWare
http://ocw.mit.edu
6.837 Computer Graphics
Fall 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
Outline
Review
Standard Library
<stdio.h>
<ctype.h>
<stdlib.h>
<assert.h>
<stdarg.h>
<time.h>
1
6.087 Lecture 10 – January 25, 2010
Review
Standard Library
<stdio.h>
<ctype.h>
<stdlib.h>
<assert.h>
<stdarg.h>
<time.h>
2
Review: Libraries
•
linking: binds symbols to addresses.
• static linkage: occ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
the stream to the file.
• returns NULL on error.
• Where can this be used? (redirecting stdin,stdout,stderr)
int fflush (FILE∗ stream)
•
flushes any unwritten data.
•
if stream is NULL flushes all outputs streams.
•
returns EOF on error.
5
<stdio.h>: File operations
int remove(const char∗ filename)
•
removes t... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
_END.
returns non-zero on error.
long ftell (FILE∗ stream)
•
returns the current position within the file. (limitation? long
data type).
•
returns -1L on error.
int rewind(FILE∗ stream)
•
sets the file pointer at the beginning.
•
equivalent to fseek(stream,0L,SEEK_SET);
9
<stdio.h>: File errors
void clearerr (... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
(const char∗ s)
int atoi (const char∗ s)
long atol(const char∗ s)
•
converts character to float,integer and long respectively.
int rand()
•
returns a pseduo-random numbers between 0 and
RAND_MAX
void srand(unsigned int seed)
•
sets the seed for the pseudo-random generator!
13
<stdlib.h>: Exiting
void abort... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
[n-1] in
ascending/descending order.
• function cmp() is used to perform comparison.
16
<assert.h>:Diagnostics
void assert(int expression)
•
used to check for invariants/code consistency during
debugging.
•
does nothing when expression is true.
•
prints an error message indicating, expression, filename
and line... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
suma=sum ( 4 , 1 , 2 , 3 , 4 ) ; / ∗ c a l l e d w i t h
i n t sumb=sum ( 2 , 1 , 2 ) ;
/ ∗ c a l l e d w i t h
f i v e args ∗ /
t h r e e args ∗ /
20
<time.h>
time_t, clock_t , struct tm data types associated with time.
struct tm:
hour since midnight (0,23)
seconds
int tm_sec
int tm_min minutes
int tm_hour... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
(locltime(tp))!
23
<time.h>
size_t strftime (char∗ s,size_t smax,const char∗ fmt,const struct tm∗ tp)
returns time in the desired format.
•
• does not write more than smax characters into the string s.
abbreviated weekday name
full weekday name
abbreviated month name
full month name
day of the month
%a
%A
... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
18.336 spring 2009
lecture 1
02/03/09
18.336 Numerical Methods for Partial Differential Equations
Fundamental Concepts
Domain Ω ⊂ Rn with boundary ∂Ω
�
PDE
b.c.
in Ω
on Γ ⊂ ∂Ω
�
PDE = “partial differential equation”
b.c. = “boundary conditions”
(if time involved, also i.c. = “initial conditions”)
Def.: An e... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/2da291f80e0df5ceca25f76df3c91a5d_MIT18_336S09_lec1.pdf |
·
|α|≤k
(iv) fully nonlinear, if neither (i), (ii) nor (iii).
Def.: An expression of the form
F (Dk u(x), Dk−1 u(x), ..., Du(x), u(x), x) = 0,
x ∈ Ω ⊂ Rn
is called kth order system of PDE,
where F : Rmnk × Rmnk−1 × ... × Rmn × Rm × Ω Rm
and u : Ω → Rm , u = (u1, ..., um).
Typically: # equations = # unknowns , i... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/2da291f80e0df5ceca25f76df3c91a5d_MIT18_336S09_lec1.pdf |
http://ocw.mit.edu
18.336 Numerical Methods for Partial Differential Equations
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/2da291f80e0df5ceca25f76df3c91a5d_MIT18_336S09_lec1.pdf |
Classes &
Interfaces
Java’s Object Oriented System
Justin Mazzola Paluska
Keywords
(cid:122) Class – a template of a data object
(cid:122) Interface – a specification
(cid:122) Instance – an instantiation of a Class or Interface
physically represented in memory
(cid:122) Method – a set sequence of instructions
(cid:1... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/2db153242ca1c747fd6cd7c2b497162a_lecture2a.pdf |
fields
(cid:122) Nested classes
Constructors
(cid:122) Must have the same name of the Class that
they are in
(cid:122) Can have multiple constructors per Class
(cid:122) Handles initialization of your class
(cid:122) Template:
[access] className ([arguments…]) {
//constructor body
}
Example:
Single Constructor
publi... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/2db153242ca1c747fd6cd7c2b497162a_lecture2a.pdf |
0;
}
public void withdraw (int amount) {
balance = balance – amount;
}
public void deposit (int amount) {
balance = balance + amount;
}
}
Field
Constructor
Methods
Accessors
(cid:122) Before we saw the placeholder [access].
(cid:122) There are 4 types of access keywords to
describe which classes have access:
(cid:122... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/2db153242ca1c747fd6cd7c2b497162a_lecture2a.pdf |
() {
return balance;
}
}
Since
CheckingAccount
implements
BankAccount, it
must provide
implementations
for these methods
Abstract Classes
(cid:122) Abstract classes are a mix between
interfaces and classes
(cid:122) can have defined method bodies
(cid:122) can have fields
(cid:122) Helps to capture the idea of ... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/2db153242ca1c747fd6cd7c2b497162a_lecture2a.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.
6.871 - Lecture 1... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
savings,
$10M’s in added revenue
– DuPont
– Manufacturer’s Hanover: Inspector
– The clever (?) paper clip
6.871 - Lecture 1
8
In 1995, in Singapore…
Crime Case Closed Infamous Crimes
Nick Leeson and Barings Bank
The week before Nick Leeson disappeared he had kept
throwing up at work.
Colleagues did not know why ... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
this interesting?
• Applied AI leads to advances in basic
science
– Knowledge acquisition/learning
– Explanation
– Knowledge sharing
6.871 - Lecture 1
12
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
– Writ... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
art of rhetoric
– The syllogisms
• 17th century: Leibniz and the “algebra of
thought”
6.871 - Lecture 1
19
Intellectual Origins
• 19th century: Boole’s logic and The Laws
of Thought
To see this image, please visit:
http://images.google.com/images?q=cgboole.gif
6.871 - Lecture 1
20
Intellectual Origins
• 1... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
• Intelligence?
• Rationality?
• Knowledge?
6.871 - Lecture 1
27
The Physical Symbol System
Hypothesis
• A physical symbol system has the necessary and
sufficient means for general intelligent action
Physical Symbol System consists of:
• A set of symbols
• A set of expressions (symbol structures)
• A set of proc... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
Intro
This lecture will review everything you learned in 6.042.
• Basic tools in probability
• Expectations
• High probability events
• Deviations from expectation
Coupon collecting.
• n coupon types. Get a random one each round. How long to get all
coupons?
• general example of waiting for combinations of eve... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2e00cd3ccdd9e9e9aa3f4b8305a6abef_n4.pdf |
as random choices made when algorithm needs them.
• use for discussing autopartition, quicksort
2
• Proposal algorithm:
– defered decision: each proposal is random among unchosen women
– still hard
– Each proposal among all women
– stochastic domination: X s.d. Y when Pr[X > z] ≥ Pr[Y > z]
for all z.
– Result... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2e00cd3ccdd9e9e9aa3f4b8305a6abef_n4.pdf |
Example: coupon collection/stable marriage.
• Probability didn’t get kth coupon after r rounds is (1 − 1/n)r
≤ e−r/n
•
which is n−β for r = βn ln n
• so probability didn’t get some coupon is at msot n · n−β = n1−β (using
union bound)
• we say “time is O(n ln n) with high probability” because we can
make probabili... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2e00cd3ccdd9e9e9aa3f4b8305a6abef_n4.pdf |
(n)
– Run for time 2T (n), then stop and give default answer
– Probability of default answer at most 1/2 (Markov)
– So, RP .
– If flip default answer, coRP
On flip side, not very strong: balls in bins Pr[> ln n] ≤ 1/ ln n for just one
bin.
• inadequate for union bound.
Can make much stronger by generalizing: Pr[h... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2e00cd3ccdd9e9e9aa3f4b8305a6abef_n4.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.341: Discrete-Time Signal Processing
OpenCourseWare 2006
Lecture 6
Quantization and Oversampled Noise Shaping
Reading: Sections 4.8 - 4.9 in Oppenheim, Schafer & Buck (OSB).
While the title of the course is Discrete-... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
+XM , these 2B+1 levels must cover the range ±XM . This implies that the
spacing � between adjacent quantization levels is therefore
� = XM 2
−B .
The question then arises of how to analyze the error introduced by the process of quantization.
Since a quantizer is generally a highly nonlinear system, we instead choo... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
] is uniform over the range −�/2 to �/2, as illustrated
in OSB Figure 4.52.
The expected value of e[n] is therefore
and its variance is
E{e[n]} = 0,
�2 =
e
�2
.
12
Because it is a white-noise process, its autocorrelation is
and its power spectral density is
The total noise power is therefore
�ee[m] =
�2
1... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
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