text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
theorem below.
Theorem 6. The signature of Hq(p) is equal to the number of real roots xj of p for which q(xj) > 0,
minus the number of real roots for which q(xj) < 0.
Proof. For simplicity, we assume all roots are distinct (this is easy to change, at the expense of slightly
53
more complicated notation). We have... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
)}xj ∈C\R.
Therefore, the expression above gives a congruence transformation of Hq (p), and we can obtain its sig
nature by adding the signatures of the scalar elements q(xj ) and the 2 × 2 blocks. The signature of the
2 × 2 blocks is always zero (they have zero trace), and thus the result follows.
In particular, n... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
4
Lemma 8. The signature of H1(p) is equal to the number of real roots. The rank of H1(p) is equal to
the number of distinct complex roots of p(x).
Corollary 9. If p(x) has odd degree, there is always at least one real root.
Example 10. Consider p(x) = x3 + 2x2 + 3x + 4. The corresponding Hermite matrix is:
⎡
⎤
3 −2 ... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
polynomials of degree less than or equal
to n (n is even). Then, identifying a polynomial with its n + 1 coefficients (pn, . . . , p0), the set Pn is a
proper cone (i.e., closed, convex, pointed, solid) in Rn+1 .
An equivalent condition for the (nonconstant) univariate polynomial (1) to be strictly positive, is
that ... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
(x) = pn
�
(x − rj )nj
�
(x − ak + ibk )mk (x − ak − ibk )mk ,
j
k
where rj and ak ± ibk are the real and complex roots, respectively. Because p(x) is nonnegative, then
pn > 0 and the multiplicies of the real roots are even, i.e., nj = 2sj .
Notice that (x − a + ib)(x − a − ib) = (x − a)2 + b2 . Then, we can wr... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
criterion) that
required the calculation of only n minors, while for the semidefinite case apparently we needed a much
larger number, 2n − 1.
If the matrix X is positive definite, Sylvester’s criterion requires the positivity of the leading principal
minors, i.e.,
det(X1,1) > 0, det X12,12 > 0,
. . . , det X > 0.
... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
Theorem 16 is the familiar result that A is positive
semidefinite if and only if det A ≥ 0 and TrA ≥ 0.
n−1
k=0 pk tk , that has only real roots.
Lemma 17. Consider a monic univariate polynomial p(t) = tn +
Then, all roots are nonpositive if and only if all coefficients are nonnegative (i.e., pk ≥ 0, k = 0, . . . , n−... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
feasible sets of a semidefinite program are basic closed semial
gebraic.
56
Proof. The condition X � 0 is equivalent to n nonstrict polynomial inequalities in the entries of X. This
can be conveniently shown applying Lemma 17 to the characteristic polynomial of −X, i.e.,
p(λ) = det(λI + X) = λn +
pk (X)λk .
n−1... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
5] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge,
1995.
57 | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
6.087 Lecture 4 – January 14, 2010
Review
Control flow
I/O
Standard I/O
String I/O
File I/O
1
Blocks
• Blocks combine multiple statements into a single unit.
• Can be used when a single statement is expected.
• Creates a local scope (variables declared inside are local
to the block).
•
Blocks can be nested.
... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
/O
File I/O
5
goto
• goto allows you to jump unconditionally to arbitrary part of
your code (within the same function).
• the location is identified using a label.
•
a label is a named location in the code. It has the same
form as a variable followed by a ’:’
s t a r t :
{
i f ( cond )
goto o u t s i d e ; ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
f l a g ) break ;
/ ∗ o u t e r l o o p ∗ /
}
7
6.087 Lecture 4 – January 14, 2010
Review
Control flow
I/O
Standard I/O
String I/O
File I/O
8
Preliminaries
• Input and output facilities are provided by the standard
library <stdio.h> and not by the language itself.
• A text stream consists of a series of l... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
It returns the number of characters printed.
• The format can contain literal strings as well as format
specifiers (starts with %).
Examples:
p r i n t f ( "hello world\n" ) ;
p r i n t f ( "%d\n" , 1 0 ) ; / ∗ f o r m a t : %d ( i n t e g e r ) , argument : 1 0 ∗ /
p r i n t f ( "Prices:%d and %d\n" , 1 0 , 2 0 )... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
0"
bb10 (b:space)
hello
13
printf format specification (cont.)
%[flags ][ width ][. precision ][ modifier]<type>
flag:
output
format
printf ("%d,%+d,%+d",10,−10) 10,+10,-10
printf ("%04d",10)
printf ("%7s","hello")
printf ("%-7s","hello")
0010
bbhello
hellobb
14
printf format specification (cont.)
%[flags ]... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
char str[10]="hello"; /∗make sure the array is large enough∗/
• char str []={ ’h’,’e’,’l’,’l’,0};
Note: use \" if you want the string to contain ".
17
Digression: character arrays
Comparing strings: the header file <string .h> provides the
function int strcmp(char s[],char t []) that compares two strings in
dicti... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
)
scanf("%f",&f)
scanf("%s",str) /∗note no & required∗/
scanf("%20s",str) /∗note no & required∗/
scanf("%20s %20s",fname,lname)
20
String input/output
Instead of writing to the standard output, the formatted data can
be written to or read from character arrays.
int sprintf (char string [], char format [], arg1,a... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
read or EOF on error/end of file.
Note: getchar simply uses the standard input to read a
character. We can implement it as follows:
#define getchar() getc(stdin )
char[] fgets(char line [], int maxlen,FILE∗ fp)
• reads a single line (upto maxlen characters) from the input
stream (including linebreak).
• returns a ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
]="cat")
27
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
MIT OpenCourseWare
http://ocw.mit.edu
6.087 Practical Programming in C
January (IAP) 2010
For information about citing these materials or our Terms of Use,visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
Engineering Systems
Doctoral Seminar
ESD.83-- Fall 2011
Class 5
Faculty: Chris Magee and Joe Sussman
TA: Rebecca Saari
Guest: Professor Mort Webster (ESD)
1
1 Class 5-- Overview
Welcome, Overview and Introductions (5 min.)
Dialogue with Professor Webster (55min)--Redaction
provided by Xin Zhang
Break (10 minutes... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
enarios: Several Views
Match-up of Class 6 with
Framing Questions
Learning Objectives
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
6
6 Scenarios
Introduction to concepts
The Shell approach
The RAND approach (already
introduced in the discussant... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
Challenge our mental models about
the future
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
11
11 Why Scenarios?
Stretch the minds of decisionmakers -
- to think the unthinkable
Rehearse the future --
“The 2-minute drill -- skip the brain.”
A mechanis... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
ussman, Engineering Systems Division, Massachusetts Institute of Technology
15
15 Why Scenarios?
Create a test bed against which to check
the robustness of bundles of strategic
alternatives (where robustness is the ability
of a particular bundle to perform
reasonably well under “plausible” scenarios)
Scenario 1
... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
20 Scenarios: What are the steps?
Schwartz Approach
4. Rank key factors and driving forces
-- According to importance to key decision and degree of
uncertainty
5. Select scenario logics
-- Identifying plots that capture situational dynamics and
communicate effectively
6. Flesh out the scenarios
7. Examine implicatio... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
�� Densification/ Growth management and land use
controls
Intercity HST
And so on……….
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
24
24 Houston Scenarios
The United States of North America
Earth Day 2020
The Balkanization of the World
© 2008 Chri... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
and computers….to trace out causal chains”
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
29
29 The RAND Approach
“Use simulation models to create a
large database of plausible future
scenarios where each entry…
represents one guess of how the
world work... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
33
33 Framing questions for
ESD.83 II
What are the historical roots of the
field of Engineering Systems and
what is their relevance to
contemporary engineering systems
issues and concepts?
What does “practicing” Engin... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
of defendable point
of view of important contributing disciplines
in Engineering Systems Field
Links Across Domains and Methods:
Ability to identify links/connections across
different fundamental domains and
methods relevant to engineering systems
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Divisi... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
Systems Division, Massachusetts Institute of Technology
42
42 Origins and Evolution of
Scenario Planning
• Evolution within Shell (cont.)
– 1980s
• Broadened perspective to socio-political
developments and energy market dynamics
• Recession, sharp rises and collapses in oil
prices, longevity of the USSR and the Cold... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
� revolutionary
Dynamics as Usual
Energy choices
citizen perspective
evolutionary
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
46
46 The Process
Time Frame
Research
Driving Forces
Some are predetermined
Some are “critical uncertainties”
... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
Not having tried to foresee surprising
events, they are at a loss for ways to act when upheaval
continues. They create blind spots for themselves.”
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
49
49 Creating Scenarios
Driving Forces: What moves the plot ... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
53 Combining Driver States and
Selecting Scenario Plots
Choose some combinations of macro-
drivers to serve as the basis for the
scenario plots
Internally consistent
Connections between the states of the different
drivers
Range of possible outcomes
Include not only the “most likely” but also
incorporate b... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
6.825 Techniques in Artificial Intelligence
Logic Miscellanea
• Completeness and Incompleteness
• Equality
• Paramodulation
Lecture 9 • 1
Logic is a huge subject. It includes esoteric mathematical and philosophical
arguments as well as hard-core engineering of knowledge representations and
efficient inference algorit... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
Gödel’s Completeness Theorem: There exists a complete proof
system for FOL
Robinson’s Completeness Theorem: Resolution refutation is a
complete proof system for FOL
Then, Robinson came along and showed that resolution refutation is sound and
complete for FOL.
Lecture 9 • 4
4
Completeness and Decidability
Complete: ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
is the same as
sound). Either there are sentences that are true, but not provable, or there are
sentences that are provable, but not true. It’s not so good either way.
7
Adding Arithmetic
Gödel’s Incompleteness Theorem: There is no
consistent, complete proof system for FOL +
Arithmetic.
Either there are sentences... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
want to talk
about things being equal to each other.
10
Equality
x. x = x
xy. x=y
→
xyz. x=y Æ y = z
y = x
→
∀
∀
∀
x = z
Lecture 9 • 11
One solution is to just treat equality like any other relation and give axioms that
specify how it has to work. So, for instance, equality is an equivalence relation,
which mean... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
�W (y,F(y))
Q(y) ∨W (y,B)
Lecture 9 • 14
We’ll start by looking at a simple example. What if you know that F(x) = B, and
that either Q(y) or W(y, F(y)). It seems like you should be able to substitute B
in for F(y), because F(x) unifies with F(y). And you can! You can conclude
Q(y) or W(y,B).
14
Paramodulation
Need... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
θ
F(x) = B
Q(y) ∨W (y,F(y))
Q(y) ∨W (y,B)
is a literal containing term r
γ[r]
θ= unify(s,r)
where
s = F(x)
t = B
γ[⋅] = W (y,⋅)
r = F(y)
θ= {x / y}
Lecture 9 • 16
So, to see how we can do our simple example with paramodulation, here is the
mapping from the abstract variables in the rule to the actual components of the... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
) = the teacher of x; G(x) = x is a good student)
Lecture 9 • 18
Please do these recitation problems before the next recitation.
18
More Recitation Problems
• Every part is either made by FooCorp or BarCorp.
• All fragile parts are stored in the warehouse of their
manufacturer.
• BarCorp can’t manufacture titanium pa... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
3.052 Nanomechanics of Materials and Biomaterials Thursday 02/15/07
I
Prof. C. Ortiz, MIT-DMSE
LECTURE 4: FORCE-DISTANCE CURVES
Outline :
LAST TIME : ADDITIONAL NANOMECHANICS INSTRUMENTATION COMPONENTS ......................... 2
PIEZOS TUBES : X/Y SCANNING .......................................................... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/264303e0db208ef97529629587899910_lec4.pdf |
L
o
=
= strain (m/m = unitless)
d = strain coefficients or sensitivity (m/Volt)
ij
E = electric field strength (Volt/m)
i
i = direction of applied fie
ld, j = direction of strain
1,2,3 = normal axes ; 4, 5, 6 = shear
+ Poisson's ratio
Δ
L =
o
L d U
31
d
3
where d = wall thickness, U = operating voltage
VA+ B-VC+D
Norma... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/264303e0db208ef97529629587899910_lec4.pdf |
lowers height
tube contracts radially
height increases
tube tilts
- reverse voltage- scan in opposite direction
- same for y-axis positioning
-Coupling - if you tell the piezo tube to move in x-direction, it will also
move a bit in y and z, x is coupled to y and z
Figure by MIT OCW.
-Another approach : individual "pi... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/264303e0db208ef97529629587899910_lec4.pdf |
d
o
t
o
h
P
s
,
t
u
p
t
u
O
r
o
s
n
e
S
0
no
interaction
repulsive
regime
attractive
regime
jump-to-contact
adhesion
z-Piezo Deflection, z (nm)
- Measure sensor output (Volts) vs. z-piezo
displacement/deflection
- See animation on the MIT Server (Force curve animation
from NC State).
5
3.... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/264303e0db208ef97529629587899910_lec4.pdf |
.
Prof. C. Ortiz, MIT-DMSE
www.polymercentre.org.uk
Image removed due
to copyright restrictions.
See Vezenov DV, Noy A, Rosznyai LF,
Lieber CM. 1997. J. Am. Chem. Soc.
Courtesy of The University of Sheffield Polymer Centre. Used with permission.
7
3.052 Nanomechanics of Materials and Biomateria... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/264303e0db208ef97529629587899910_lec4.pdf |
12
8
4
)
N
n
(
e
c
r
o
F
l
a
r
e
t
a
L
0 10 20 30
Normal Force (nN)
-Measure shear / friction coefficient→nanotribology - study of friction and wear.
-Linear dependence of lateral force on normal force between OH-SAM and aggrecan changed upon the
point of full penetration of aggrecan layer by the nanosized probe tip.... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/264303e0db208ef97529629587899910_lec4.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-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
) (cid:2)
(cid:6) (cid:8) (cid:11)
(cid:2) (cid:6) (cid:3)
(cid:15) (cid:16)
(cid:17) (cid:15) (cid:2)
(cid:8) (cid:9) (cid:10)
(cid:2) (cid:6) (cid:3)
(cid:12) (cid:9) (cid:13) (cid:14) (cid:2)
(cid:8) (cid:9) (cid:10)
(cid:14) (cid:3)
(cid:2) (cid:6) (cid:3)
(cid:7) (cid:3)
(cid:11)
(cid:11)
(cid:11)
and that a h... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
cid:21) (cid:18)
(cid:2)
(cid:2)
(cid:7)
(cid:7)
(cid:5)
(cid:4)
(cid:10)
(cid:18)
(cid:7)
(cid:7)
(cid:5)
(cid:4)
(cid:10)
(cid:7)
(cid:7)
(cid:5)
(cid:4)
(cid:10)
(cid:7)
(cid:7)
(cid:5)
(cid:4)
(cid:5)
(cid:6)
(cid:2)
(cid:8)
(cid:9)
(cid:10)
(cid:10)
(cid:3)
The op-amp has the following characteristics:
• It is ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
. (See the class handout for other common config
urations).
The Inverting Amplifier:
(cid:27)
(cid:27)
(cid:21) (cid:9) (cid:5)
(cid:20) (cid:16)
(cid:20)
(cid:16) (cid:13)
(cid:17) (cid:6) (cid:21) (cid:13) (cid:14) (cid:11)
(cid:25)
(cid:16) (cid:13)
(cid:13)
(cid:25)
(cid:26)
(cid:16) (cid:26)
(cid:16) (... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
4)
(cid:30)
(cid:18)
(cid:3)
(cid:2)
The voltage gain is therefore defined by the ratio of the two resistors. The term inverting
amplifier comes about because of the sign change.
The Inverting Summer: The inverting amplifier may be extended to include multiple
inputs:
(cid:27)
(cid:16) (cid:13) (cid:17) (cid:3) (c... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
choff’s current law at the summing junction
i1 + i2 + if =
v2
v1
+
R1 R2
+
Vout
Rf
= 0
or
vout = −
�
�
v1 +
Rf
R2
v2
Rf
R1
which is a weighted sum of the inputs.
The summer may be extended to include several inputs by simply adding additional input
resistors Ri, in which case
vout = −
n � Rf
viRi
i=... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
:18)
(cid:16)
(cid:6)
(cid:4)
(cid:20)
(cid:16)
(cid:30)
(cid:18)
(cid:3)
Then
or
dvout
dt
= −
1
RinC
vin
vout = −
�
t
1
RinC 0
vindt + vout(0)
As above, the integrator may be extended to a summing configuration by simply adding
extra input resistors:
i=1
and if all input resistors have the same value R ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
19)
Amplifiers A1 and A2 are integrators with transfer functions
�
�
�
H1(s) = −
and H2(s) = −
1
R1C1
1
s
�
1
R2C2
" (cid:26)
# " (cid:13)
(cid:22) (cid:9) (cid:2)
(cid:20) (cid:23) (cid:24)
(cid:22) (cid:23) (cid:24)
(cid:8)
(cid:21) (cid:13)
(cid:24)
(cid:9) (cid:27) ! (cid:13)
(cid:22) (cid:9) (cid:2)
... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
–4
(cid:2)
(cid:7)
(cid:17)
(cid:18)
(cid:28)
(cid:31)
(cid:19)
(cid:20)
(cid:22)
(cid:9)
(cid:2)
(cid:2)
(cid:7)
(cid:20)
(cid:23)
(cid:24)
(cid:22)
(cid:23)
(cid:2)
(cid:7)
Using the infinite gain approximation for the op-amp, we set v− = v+ and
R3
R3 + R4
v3 −
R5
R5 + R6
v1 +
R4
R3 + R4
v2 =
R6
R5 + R6
... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
2 + a1s + a0
R6
Kbp = R5
Selection of the output as the output of the summer A3 generates
Hhp(s) = τ1τ2s 2Hlp(s) =
Khps2
s2 + a1s + a0
Khp =
1 + R4/R3
1 + R5/R6
A Band-Stop Filter: The band-stop configuration may be implemented with an addi
tional summer to add the outputs of amplifiers A2 and A (with appropriat... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
)
(cid:18)
(cid:3)
(cid:26) (cid:3)
(cid:18)
(cid:15)
(cid:9) (cid:11) (cid:31)
(cid:9)
(cid:26) (cid:15)
(cid:2)
(cid:19)
(cid:28) (cid:3)
(cid:25) (cid:3)
(cid:25) (cid:20) (cid:23) (cid:24)
With the infinite gain assumption for the op-amps, that is V− = V+ , and with the assumption
that no current flows ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
1 + R1/R3
9–6
(cid:28)
(cid:4)
1.4 First-Order Filter Sections:
Single pole low-pass filter sections with a transfer function of the form
KΩ0
H(s) = s + Ω0
may be implemented in either an inverting or non-inverting configuration as shown in Fig.
11.
(cid:18)
(cid:8)
(cid:25)
(cid:26) (cid:27)
(cid:18)
(cid:3)... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
Its transfer function is
Vout(s)
Vin(s)
�
= 1 +
�
R3
R2
1/R1C
.
s + 1/R1C
9–7
(cid:4)
(cid:9)
(cid:4)
(cid:12)
Classroom Demonstration
Example 2 in the class handout “Op-Amp Implementation of Analog Filters” describes a
state-variable design for a 5th-order Chebyshev Type I low-pass filter with Ωc = 1000 ra... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
(cid:2)
(cid:19)
(cid:2)
(cid:19)
(cid:16) (cid:30) (cid:16) (cid:8) (cid:7)
(cid:3) (cid:3) ( ’ (cid:15) (cid:7)
(cid:3) (cid:30) (cid:10) (cid:10) (cid:7)
(cid:16) (cid:30) (cid:10) (cid:10) (cid:7)
(cid:2)
(cid:19)
& (cid:15) (cid:16) ’ (cid:7)
(cid:10) % (cid:16) &
(cid:10) % (cid:16) &
(cid:10) % (cid... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
1 ≤ t < T2. In order to
process this function in the computer it must be sampled and represented by a finite set of
numbers. The most common sampling scheme is to use a fixed sampling interval ΔT and to
form a sequence of length N : {fn} (n = 0 . . . N − 1), where
fn = f (T1 + nΔT ).
In subsequent processing the fun... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
(cid:3) (cid:18) (cid:15) & (cid:3) (cid:16) (cid:13) (cid:8) (cid:21) (cid:18) (cid:3) $ (cid:3) (cid:3) (cid:3) ’ (cid:3) $ (cid:4) (cid:27) ) (cid:11) (cid:3) (cid:18) (cid:6)
(cid:27)
(cid:8) (cid:30) (cid:6) (cid:14) (cid:21) (cid:29) & (cid:3) (cid:9) (cid:3) (cid:14) (cid:16) (cid:10) (cid:29) (cid:30) & (cid:... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
(ii) At each interval, the output sample yn is computed, based on the history of the input
and output, for example
yn = (fn + fn−1 + fn−2)
1
3
3-point moving average filter, and
yn = 0.8yn−1 + 0.2fn
is a simple recursive first-order low-pass digital filter. Notice that they are algorithms.
(iii) The reconstructor t... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
) (cid:9) (cid:16) (cid:30) (cid:29) (cid:9) (cid:10) & (cid:3)
* & (cid:30) (cid:6) (cid:2) (cid:6) (cid:30) (cid:14) &
(cid:7) (cid:9) (cid:20) & (cid:26) (cid:6) (cid:30) (cid:27)
(cid:6) (cid:30) (cid:3)
(cid:30)
(cid:2)
(cid:2)
(cid:24)
(cid:18) (cid:16) (cid:27)
&
(cid:30) & (cid:29) (cid:6) (cid:13) (c... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
11)
(cid:2) (cid:5) (cid:24) + (cid:3) (cid:4) (cid:6)
+
$ (cid:7) (cid:7) (cid:5) (cid:24) (cid:6) (cid:7) , (cid:7) $ (cid:5) - (cid:6) (cid:2) (cid:5) (cid:24) + (cid:3) (cid:4) (cid:6)
9–10
(cid:24)
(cid:24)
(cid:24)
Note that f (cid:2)(t) is a set of delayed and weighted delta functions, and that the wavefo... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
damental angular frequency. Using this form, the spectrum of the sampled waveform f (cid:2)(t)
may be written
F (cid:2)(jΩ) =
�
∞
−∞
f (cid:2)(t) e−jΩt dt =
∞ �
� ∞
1
ΔT n=−∞
−∞
f (t) ejnΩ0t e
−jΩt dt
=
=
�
∞
1
ΔT n=−∞
�
∞
1
ΔT
n=−∞
F (j (Ω − nΩ0))
� �
F j Ω −
��
2πn
ΔT
The Fourier transform of... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
3.37 (Class 4)
Question: thermal diffusivity?
• Thermal Diffusivity (alpha) = Thermal Conductivity (k) / (density * heat
capacity)
• Combined, or derived parameter
• From Fourier’s 1st law and 2nd law (diffusion equation)
• deltaH = Cp*deltaT
Question: Cold welding of semiconductors, is it gold-to-gold? Typicall... | https://ocw.mit.edu/courses/3-37-welding-and-joining-processes-fall-2002/269cd56a8622f7fff119040dea55e1c9_33704.pdf |
working with test lab (guy inherited business from father, had a
correspondence degree). Had seen a news account of tornado where chickens had all
feathers taken off. They decided low pressure of tornado that did it and tried to
use a vacuum chamber to pluck a chicken ☺.
• Project to lay down weld ... | https://ocw.mit.edu/courses/3-37-welding-and-joining-processes-fall-2002/269cd56a8622f7fff119040dea55e1c9_33704.pdf |
Handout with surface energies
o Water (hydrogen bond)
o Teflon
o Copper
Back to question of gold contact bonds
• Actually get better bond in intermediate donut region of maximum shear
• Diagram on board
Ultrasonic welding:
• Diagram on board
• 20-100kHz is typical bonding frequency
• have a small displacement... | https://ocw.mit.edu/courses/3-37-welding-and-joining-processes-fall-2002/269cd56a8622f7fff119040dea55e1c9_33704.pdf |
6.852: Distributed Algorithms
Fall, 2009
Class 9
Today’s plan
(cid:122) Basic asynchronous network algorithms
− Constructing a spanning tree
− Breadth-first search
− Shortest paths
− Minimum spanning tree
(cid:122) Reading: Sections 15.3-15.5, [Gallager, Humblet,
Spira]
(cid:122) Next lecture:
(cid:122) Synchronizer... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
tree, with a parent
output action.
(cid:122) Starting point: SynchBFS algorithm:
−
i0 floods search message; parent of a node is the first node from which it
receives a search message.
− Try running the same algorithm in asynchronous network.
− Still yields spanning tree, but not necessarily breadth-first tree.
Asyn... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
AsynchSpanningTree
(cid:122) Similar to synchronous BFS
(cid:122) Message broadcast: Piggyback on search
message.
(cid:122) Child pointers: Add responses to search
messages, easy because of bidirectional
communication.
(cid:122) Use precomputed tree for bcast/convergecast
Now the timing anomaly arises.
(cid:122)
(ci... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
ation algorithm (like Bellman-Ford).
− Must inform neighbors of changes.
− Eventually, tree stabilizes to a breadth-first spanning tree.
AsynchBFS
(cid:122) S
−
−
ignature
in receive(m)j,i, m ∈ N, j ∈ nbrs
out send(m)i,j, m ∈ N, j ∈ nbrs
(cid:122) send(m)
i,j
pre: m = head(
eff: remove head of send(j)
send(j))
(cid:12... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
1
2
2
3
AsynchBFS
0
2
6
3
1
1
1
1
2
3
AsynchBFS
0
1
1
6
2
2
3
1
1
2
AsynchBFS
0
0
1
2
2
2
3
1
1
2
AsynchBFS
0
1
1
2
2
3
1
1
2
AsynchBFS
(cid:122) Complexity:
− Messages: O(n |E|)
(cid:122) May send O(n) messages on each link (one for each distance estimate).
− Time: O(diam n (l+d)) (taking pileups into account).
−... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
k-1 also knows its children.
i0 broadcasts newphase message along tree edges, to distance k
processes.
− Each of these sends search message to all neighbors except its parent.
− When any non- i0 process receives first search message, sets parent :=
sender and sends a positive ack; sends nacks for subsequent search ms... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
distance and parent in shortest-paths
tree.
(cid:122) Use Bellman-Ford asynchronously
− Used to establish routes in ARPANET 1969-1980.
− Can augment with convergecast as for BFS, for
termination.
− But worst-case complexity is very bad…
AsynchBellmanFord
(cid:122) Signature
(cid:122) Transitions
− in receive(w)j,i, ... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
k-1
0
2k-2
21
0
20
i0
i1
0
i2
0
ik-1
ik
0
ik+1
0
Exponential time/message
complexity
•
ik sends 2k messages to ik+1, so message complexity is Ω(2n/2) and time
complexity is Ω(2n/2 d).
k are 2 – 1, 2 – 2,…,0.
• Possible distance estimates for i
• Moreover, ik can take on all these estimates in sequence:
k
k
– First, ... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
MST, where each process knows which of its incident edges
belong to the tree.
− Guaranteed to be unique, because of unique weights.
(cid:122) Gallager-Humblet-Spira algorithm: Read this paper!
Recall synchronous algorithm
(cid:122) Proceeds in phases (levels).
(cid:122) After each phase, we have a spanning forest, in... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
cast report); remember direction of best edge.
(cid:122) Leader picks MWOE for fragment.
(cid:122) Send change-root to MWOE’s endpoint, using remembered best edges.
(cid:122) Send connect across MWOE.
(cid:122) There is a unique edge that is the MWOE of two components.
(cid:122) Leader of new component is higher-id end... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
).
− Time bound:
− These problems result from nodes being out-of-synch, at different levels.
− We could try to synchronize levels, but this must be done carefully, so as
not to hurt the time complexity too much.
GHS algorithm
(cid:122) Same basic ideas as before:
(cid:122) Form components, combine along MWOEs.
(cid:1... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
the general theory for MSTs.
Liveness
• Q: Why are merging and absorbing sufficient to ensure that the
construction is eventually completed?
• Lemma: After any allowable finite sequence of merges and absorbs,
either the forest consists of one tree (so we’re done), or some merge or
absorb is enabled.
• Proof:
– Con... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
, since the
component is still searching for its MWOE.
– If j’s level is < i’s, then j doesn’t know if it is in the same or a different
component. So it doesn’t yet respond---waits to catch up to i’s level.
Liveness, again
• Q: Can the extra delays imposed here affect the
progress argument?
• No:
– We can redo the ... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
).
But the level of i is still <
level(C′), when the absorb
occurs.
So mwoe(j) is a different
edge, one whose weight <
weight(i,j).
C′
mwoe(C)
i
C
j
C′
•
•
Claim 2: MWOE for combined
component is not outgoing from a
node in C.
Proof:
–
–
–
(i,j) is the MWOE of C, so there
are no edges outgoing from C
with weigh... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
• Retesting is possible, for accepted edges.
• Reclassify edge as branch as a result of changeroot
message.
Complexity
(cid:122) As for synchronous version.
(cid:122) Messages: O(|E| + n log n)
(cid:122) 4|E| for test-reject msgs (one pair for each direction of
every edge)
(cid:122) n initiate messages per level (bro... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
J Distributed Algorithms
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
Review: step response of 1st order systems we’ve seen
•
Inertia with bearings (viscous friction)
Step input Ts(t) = T0u(t) ⇒ Step response∗
ω(t) =
T0
b
1 − e−
t/τ
, where
τ =
J
b
.
• RC circuit (charging of a capacitor)
³
´
Step input vi(t) = V0u(t) ⇒ Step response
vC (t) = V0
1 − e−
t/τ
, where
τ = RC.
+
vi
−
R
C
+
vC... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
⎧
⎪⎨
·
(Js + b) Ω(s) = KmI(s)
µ
¶
µ
KmKv
R
¶¸
Ω(s) =
Km
R
Vs(s)
Neglecting the DC motor’s inductance (i.e., assuming L/R ≈ 0), we find
⎪⎩
Km
RJ
s +
1
J
b +
µ
KmKv
R
¶
1
R
µ
s +
b
J
s +
1
J
b +
µ
¶
KmKv
R
Ω(s)
Vs(s)
=
I(s)
Vs(s)
=
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
Transfer function for the angular velocity
is of the form... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
system’s response
(both angular velocity and current)
to the step input vs(t) = 30u(t) V.
Substituting the numerical values into the system TF,
whereas the Laplace transform of the input is
we find
Vs(s) ≡ L
vs(t)
h
= L
30u(t)
=
h
i
30
s
.
i
I. Angular velocity
Ω(s) =
15
s (s + 5)
= 15
K1
s
+
K2
s + 5
µ
¶
where
K1 =
1
s... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
) ≡ L
II. Current
vs(t)
h
i
= L
30u(t)
=
h
i
30
s
.
Ω(s)
Vs(s)
I(s)
Vs(s)
=
=
1
1
2
s + 5 [Hz]
1
6
(s + 2 [Hz])
s + 5 [Hz]
⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
I(s) =
5 (s + 2)
s (s + 5)
= 5
K10
s
+
K20
s + 5
µ
I(s) =
2
s
+
3
s + 5
µ
¶
where
K10 =
s + 2
s + 5
i(t) =
2 + 3e−
5t
¶
⇒
s=0
¯
¯
¯
¯
u(t) A.
=
2
5
, K20 =
s + 2
s
=
3
5
⇒
5
−
s=
¯
¯
... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
004 Fall ’07
Lecture 06 – Monday, Sept. 17
DC motor step response (current)
Image removed due to copyright restrictions.
Please see: Fig. 4.1 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.
2.004 Fall ’07
Lecture 06 – Monday, Sept. 17
1st order system response from s-plane rep... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
3
a
4
a
t
5
a
Figure by MIT OpenCourseWare.
2.004 Fall ’07
Lecture 06 – Monday, Sept. 17
DC motor step responses
ω(t) = 3
1 − e−
5t
u(t)
rad/sec
i(t) =
2 + 3e−
5t
u(t) A
ω(∞) = 3 rad/sec
¢
¡
dω
dt
(0+) = 3 × 5 = 15
rad/sec2
2.004 Fall ’07
Lecture 06 – Monday, Sept. 17
¢
i(∞) = 2 A
¡
di
dt
(0+) = ∞
The Final Value t... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
a
0
→
= 1.
2.004 Fall ’07
Lecture 06 – Monday, Sept. 17
The Initial Value theorem: initial slope
The second property of the Laplace transform is the
Initial Value theorem
f (0+) = lims
→∞
sF (s).
Let us use this property to compute the initial slope of the step response, i.e.
the value of the derivative of the step r... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
s(s + z)
s + a
.
We can readily see that
f2(∞) = lims
0sF2(s) = z;
→
f2(0+) = lims
sF2(s) = a;
→∞
df2
dt
(0+) = lims
→∞
s2F2(s) = ∞.
You should verify that these results are consistent with the time—domain solution
for this system.
We can see that the effects of the zero −z on the 1st—order system are (in
comparison to ... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
MIT 6.581/20.482J
FOUNDATIONS OF ALGORITHMS AND COMPUTATIONAL
TECHNIQUES IN SYSTEMS BIOLOGY
Spring 2006
7 February 2006
Tuesday
MOTIVATION/OVERVIEW
There is a disconnect between biology and
computer science.
The biologist will pose the problem statement,
but it may not be amenable for the computer
scientist ... | https://ocw.mit.edu/courses/20-482j-foundations-of-algorithms-and-computational-techniques-in-systems-biology-spring-2006/26a5783a728a1db3e302718e6dc51544_l01.pdf |
to characteristic structure with no additional information
linear, unbranched chains of a unique sequence
sequence folding structure
chemical
biological
(1D) ———→
(3D) ——→ functions ——→ functions ——→ functions
network Á
protein
↑
mRNA
↑
genome (DNA)
x-ray
crystallography
NMR
binding
catalysis
synthesis... | https://ocw.mit.edu/courses/20-482j-foundations-of-algorithms-and-computational-techniques-in-systems-biology-spring-2006/26a5783a728a1db3e302718e6dc51544_l01.pdf |
Convex Optimization — Boyd & Vandenberghe
2. Convex sets
• affine and convex sets
• some important examples
• operations that preserve convexity
• generalized inequalities
• separating and supporting hyperplanes
• dual cones and generalized inequalities
2–1
Affine set
line through x1, x2: all points
x = θx1 + (1 − ... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
x = θ1x1 + θ2x2 + · · · + θkxk
with θ1 + · · · + θk = 1, θi ≥ 0
convex hull conv S: set of all convex combinations of points in S
Convex sets
2–4
Convex cone
conic (nonnegative) combination of x1 and x2: any point of the form
x = θ1x1 + θ2x2
with θ1 ≥ 0, θ2 ≥ 0
x1
x2
0
convex cone: set that contains all con... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
x − xc)T P −1(x − xc) ≤ 1}
with P ∈ Sn
++
(i.e., P symmetric positive definite)
xc
other representation: {xc + Au | �u�2 ≤ 1} with A square and nonsingular
Convex sets
2–7
Norm balls and norm cones
norm: a function � · � that satisfies
• �x� ≥ 0; �x� = 0 if and only if x = 0
• �tx� = |t| �x� for t ∈ R
• �x +... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
finite cone
notation:
• Sn is set of symmetric n × n matrices
• S+
n = {X ∈ Sn | X � 0}: positive semidefinite n × n matrices
X ∈ Sn ⇐⇒
+
z T Xz ≥ 0 for all z
Sn is a convex cone
+
n = {X ∈ Sn | X ≻ 0}: positive definite n × n matrices
• S++
example:
x y
z
y
�
�
∈ S2
+
1
0.5
z
0
1
0
y −1 0
0.5
x
1
... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
0
−1
)
t
(
p
2
1
2
x
0
−1
S
0
π/3
t
2π/3
π
−2
−2
−1
0
x1
1
2
Convex sets
2–12
Affine function
suppose f : Rn → Rm is affine (f (x) = Ax + b with A ∈ Rm×n , b ∈ Rm)
• the image of a convex set under f is convex
S ⊆ Rn convex =⇒
f (S) = {f (x) | x ∈ S} convex
• the inverse image f −1(C) of a convex set... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
) =
Ax + b
cT x + d
,
dom f = {x | c T x + d > 0}
images and inverse images of convex sets under linear-fractional functions
are convex
Convex sets
2–14
example of a linear-fractional function
f (x) =
1
x
x1 + x2 + 1
1
1
2
x
0
C
2
x
0
f (C)
−1
−1
0
x1
1
−1
−1
0
x1
1
Convex sets
2–15
Generalize... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
)
x �Rn
+
y ⇐⇒
xi ≤ yi,
i = 1, . . . , n
• matrix inequality (K = Sn
+
)
X �Sn
+
Y ⇐⇒
Y − X positive semidefinite
these two types are so common that we drop the subscript in �K
properties: many properties of �K are similar to ≤ on R, e.g.,
x �K y,
u �K v =⇒
x + u �K y + v
Convex sets
2–17
Minimum and minim... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
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