text stringlengths 16 3.88k | source stringlengths 60 201 |
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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 then
f T Hq (p)f =
n
�... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
q (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, notice that if we want to count the number of roots, we can just use q(x) = 1. The ... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
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 −2
⎦
−2
−2
−2 18
H(p) = ⎣−2
−2
T... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
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 p(x0) > 0 for some x0, and it that has no real roots. Thus, we can use Theorem 6 to write explicit
... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
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 write
�
�
�
(x − ak)2 + b2 �mk ,
(x − rj )2sj
p(x) = pn
k
j
k
Since products of sums of s... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
1,1) > 0, det X12,12 > 0,
. . . , det X > 0.
For positive semidefiniteness, it is not enough to replace strict positivity with the nonstrict inequality;
a simple counterexample is the matrix
�
�
0
0
,
0 −1
for which the leading minors vanish, but is not PSD. As mentioned, an alternative approach is given by
th... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
all coefficients are nonnegative (i.e., pk ≥ 0, k = 0, . . . , n−1).
�
Proof. Since all roots of p(t) are real, this can be obtained from a direct application of Descartes’ rules
of signs; see e.g. [BPR03]. For completeness, we present here a direct proof.
If all roots ti are nonpositive (ti ≤ 0), from the factorizati... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
homogeneous polynomials of degree n − k in the entries of X. For instance, we
have p0(X) = det X, and pn−1(X) = TrX.
Since X is symmetric, all its eigenvalues are real, and thus p(λ) has only real roots. Positive semidef
initeness of X is equivalent to p(λ) having no roots that are strictly positive. It then follows... | 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 |
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 ;
/ ∗ some code ∗ /
goto s t a r t ;
}
o u t s i d e :
/ ∗ o u t s i d e b l o c k ∗ /
5
Spaghetti code
Dijk... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
/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 lines ending with ’\n’.
The standard library takes care of conversion from
’\r\n’−’\n’
• A binary stream con... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
( 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 ) ;
11
printf format specification
The format specification has the following components
%[flags ][ width ][. precision ][ length]<type>
type:
type
d,i
x,X
u
c
s
f
d
e,E
%
meaning
integer
integer (hex)
unsigned intege... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
printf ("%7s","hello")
printf ("%-7s","hello")
0010
bbhello
hellobb
14
printf format specification (cont.)
%[flags ][ width ][. precision ][ modifier]<type>
precision:
format
printf ("%.2f,%.0f,1.141,1.141)
printf ("%.2e,%.0e,1.141,100.00) 1.14e+00,1e+02
printf ("%.4s","hello")
printf ("%.1s","hello")
output... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
<0 if s comes before t
•
the function return a value 0 if s is the same as t
•
the function return a value >0 if s comes after t
• strcmp is case sensitive
Examples
• strcmp("A","a") /∗<0∗/
• strcmp("IRONMAN","BATMAN") /∗>0∗/
• strcmp("aA","aA") /∗==0∗/
• strcmp("aA","a") /∗>0∗/
18
Formatted input
int scan... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
size).
• Returns the number of character written or negative value
on error.
int sscanf(char str [], char format [], arg1,arg2)
• The format specification is the same as scanf;
• The input is read from str variable.
• Returns the number of items read or negative value on
error.
21
File I/O
So far, we have read... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/262cf4e05f039e45c926109c8aa95024_MIT6_087IAP10_lec04.pdf |
or EOF on error.
Note: putchar simply uses the standard output to write a
character. We can implement it as follows:
#define putchar(c) putc(c,stdout)
int fputs(char line [], FILE∗ fp)
• writes a single line to the output stream.
• returns zero on success, EOF otherwise.
int fscanf(FILE∗ fp,char format [], arg1,a... | 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 |
ussman, Engineering Systems Division, Massachusetts Institute of Technology
6
6 Scenarios
Introduction to concepts
The Shell approach
The RAND approach (already
introduced in the discussant
segment)
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
7
7 Sc... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
-- skip the brain.”
A mechanism for continuous
organizational learning
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
12
12 Organizational Learning
Understand the possible long-term
consequences of short-term decisions
Decisionmakers can begin to ident... | 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
16
16
The Shell Approach
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
17
17 Perspective on Scenarios
Scenarios in a corporate environment
Assume that cor... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
clearly, the environment in which
your actions will take place?
How will those actions relate to prevailing forces, trends,
attitudes and influences?
HOW
Invent, and then consider, in-depth several stories of plausible
futures.
THE POINT
Make strategic decisions that will be sound for all plausible futures... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
ation of
the past-- may be OK for prehistoric
humans, but not now
We don‟t anticipate the timing of events
We overestimate our abilities to know
the future (especially experts!)
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
27
27 The RAND Approach
Ou... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
31
31 The RAND Approach
Robust Adaptive Planning--Key Concepts
Multiple highly-differential views of the future
better than point estimates for understanding
the system of interest and its performance
Choose robust strategies that perform well over
a range of plausible futures. Robustness
dominates optimality... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
of new findings in related fields
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
36
36 Learning Objectives
Historical Roots: Understanding of
historical/intellectual roots of key concepts and
principles in engineering systems
ES and observations, data so... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
Massachusetts Institute of Technology
41
41 Origins and Evolution of
Scenario Planning
• Corporate Roots
– Royal Dutch/Shell (30 years)
– Initially utilized in high-level corporate strategy
to improve business decisions in an uncertain
environment
– Needed to move away from forecasts
• Evolution within Shell
– 1970s... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
negotiation
© 2008 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
45
45 Shell Scenarios: Energy Needs, Choices
and Possibilities I
Two Diverging Scenarios
The Spirit of the Coming Age
Energy choices
consumer perspective
revolutionary
Dynamics as Usual
... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
They
take it for granted that some things just can‟t and won‟t
happen; for example, „oil prices won‟t collapse,‟ or „the Cold
War can‟t ever end.‟ 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 a... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/263479e7a374b364f41d8f16533ff0e1_MITESD_83F11_lec05.pdf |
Sussman, Engineering Systems Division, Massachusetts Institute of Technology
53
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 o... | 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 |
teness 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: If KB ² α then KB
`
α
• If it’s entailed, there is a proof
Semi-decidable:
• If there’s a ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
Adding Arithmetic
Gödel’s Incompleteness Theorem: There is no
consistent, complete proof system for FOL +
Arithmetic.
Either there are sentences that are true, but not
provable or there are sentences that are provable,
but not true.
Arithmetic gives you the ability to construct code-
names for sentences within the... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
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 means it’s symmetric, reflexive, and transitive. We can say that in logic
like this.
11
Equality
x. x = x
xy. x=y
→
xyz. x=y Æ y = z
xy. x=y
(P(... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
you can! You can conclude
Q(y) or W(y,B).
14
Paramodulation
Need one more rule to deal with resolution and equality.
is a literal containing term r
γ[r]
θ= unify(s,r)
α∨ (s = t)
β∨γ[r]
(α∨β∨γ[t])θ
F(x) = B
Q(y) ∨W (y,F(y))
Q(y) ∨W (y,B)
Lecture 9 • 15
Here’s the general paramodulation rule. Like resolution, it lets ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
is the
mapping from the abstract variables in the rule to the actual components of the
example.
16
Paramodulation
Need one more rule to deal with resolution and equality.
α∨ (s = t)
β∨γ[r]
(α∨β∨γ[t])θ
F(x) = B
Q(y) ∨W (y,F(y))
Q(y) ∨W (y,B)
P(x) ∨ F(x) = B
Q(y) ∨W (y,F(y))
P(y) ∨ Q(y) ∨W (y,B)
is a literal containin... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/263ad85c152caf1ee707e7cb616f0c99_Lecture9Final.pdf |
part x is stored in
the warehouse of company y; T(x) = part x is made of
titanium; F(x) part x is fragile; use a constant for “the part I
need”.)
Lecture 9 • 19
19 | 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 |
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
Normal Force
Microscopy
(NFM)
mirror
A
B
C D
4-quadrant
position sensitive
photodiode
laser beam
cantileve... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/264303e0db208ef97529629587899910_lec4.pdf |
in y and z, x is coupled to y and z
Figure by MIT OCW.
-Another approach : individual "piezo stacks" with flexures in a "nested design" (Introduction to AFM by
Asylum Research, Inc. (Quicktime Movie)- Pset 2
3
3.052 Nanomechanics of Materials and Biomaterials Thursday 02/15/07
Prof. C. Ortiz, MIT-DMSE
GE... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/264303e0db208ef97529629587899910_lec4.pdf |
iezo
displacement/deflection
- See animation on the MIT Server (Force curve animation
from NC State).
5
3.052 Nanomechanics of Materials and Biomaterials Thursday 02/15/07
Prof. C. Ortiz, MIT-DMSE
HIGH RESOLUTION FORCE SPECTROSCOPY EXPERIMENT: CONVERTED F-D DATA
x-axis conversion
Δδ
Δz... | https://ocw.mit.edu/courses/3-052-nanomechanics-of-materials-and-biomaterials-spring-2007/264303e0db208ef97529629587899910_lec4.pdf |
ASURING MACROMOLECULAR ADHESION : CARTILAGE AGGRECAN
Cartilage aggrecan is a very
unique "bottle-brush"
macromolecule that is largely
responsible for the mechanical
properties and health of
cartilage tissue in our joints.
(*podcasts later on in the
semester on this topic,
unpublished data by L. Han)
Image rem... | 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: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 high-order filter may be implemented by cascading second-order blocks, a... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
)
(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 basically a “three terminal” amplifier, with two inputs and an output. It is a
differential amplifier, that is the output is proportional to the difference in the voltages
ap... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
cid:11)
(cid:25)
(cid:16) (cid:13)
(cid:13)
(cid:25)
(cid:26)
(cid:16) (cid:26)
(cid:16) (cid:16) (cid:13)
(cid:19)
(cid:20)
(cid:6) (cid:21) (cid:18)
In the configuration shown above we note
• Because the gain A is very large, the voltage at the node designated summing junc
is very small, and we approximat... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
6) (cid:13)
(cid:10) (cid:21) (cid:27)
(cid:4) (cid:20) (cid:16) (cid:30) (cid:18) (cid:21) (cid:9) (cid:5) (cid:3) (cid:17) (cid:6) (cid:21) (cid:13) (cid:14) (cid:11)
(cid:20)
(cid:31)
(cid:20)
(cid:31)
(cid:25)
(cid:25)
(cid:16)
(cid:31)
(cid:16)
(cid:2)
(cid:19)
(cid:25)
(cid:26)
(cid:16) (cid:26)
... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
(cid:14) (cid:11)
(cid:25)
(cid:16) (cid:13)
(cid:16) (cid:16) (cid:13)
(cid:13)
!
(cid:16) (cid:26)
(cid:2)
(cid:19)
(cid:20)
(cid:6) (cid:21) (cid:18)
At the summing junction we apply Kirchoff’s current law as before but the feedback current
is now defined by the elemental relationship for the capacitor:
ii... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
30)
(cid:25) (cid:26) (cid:27)
(cid:2)
(cid:19)
(cid:18)
(cid:16)
(cid:31) (cid:15)
(cid:28) (cid:15)
(cid:18)
(cid:3)
(cid:18)
(cid:29)
(cid:18)
(cid:15)
(cid:17)
(cid:3)
(cid:28)
(cid:3)
(cid:2)
(cid:19)
(cid:31) (cid:3)
(cid:8)
(cid:8)
(cid:8)
(cid:2)
(cid:19)
Amplifiers A1 and A2 are integr... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
However, because of the sign inversions in the op-amp circuits
we cannot use the elementary summer configuration described above. Applying Kirchoff’s
Current Law at the non-inverting and inverting inputs of A3 gives
Vin − v+
R5
+
v1 − v+
R6
= 0 and
v3 − v−
R4
+
v2 − v−
R1
= 0.
9–4
(cid:2)
(cid:7)
(cid:17)... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
where
A High-Pass Filter:
the transfer function
where
Klp =
1 + R3/R4
1 + R5/R6
Selection of the output as the output of integrator A1 generates
−Kbpa1s
Hbp(s) = −τ1sHlp(s) = s2 + 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 +... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
) (cid:24)
Consider the input stage:
(cid:26) $
(cid:17)
(cid:3)
(cid:25) (cid:26) (cid:27)
(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 ga... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
τ2
Klp =
1
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)
(ci... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
/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 rad/s
and 1dB ripple in the passband.
The transfer function is
H... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
: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:16) &
(cid:2)
(cid:19)
(cid:2)
(cid:19)
(cid:25)
(cid:20) (cid:23) (cid:24)
This filter was constructed on a bread-board using 741 op-amps, and was demonstrated
to the class, driv... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
(cid:18) (cid:8) (cid:21) (cid:18)
(cid:10) & ( (cid:21) & (cid:13) (cid:29) & (cid:7) * (cid:7) (cid:7) %
(cid:27)
(cid:8) (cid:5) & (cid:10)
" (cid:4) (cid:24) (cid:11)
(i) The sampler (A/D converter) records the signal value at discrete times nΔT to produce
a sequence of samples {fn} where fn = f (nΔT ) (ΔT is ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
11) (cid:7)
* (cid:4) (cid:24) (cid:11)
(cid:10) (cid:9) (cid:27)
(cid:8) (cid:5) & (cid:7) $ (cid:4) (cid:24) (cid:11)
(cid:24)
$
(cid:27)
(cid:29) (cid:6) (cid:27)
(cid:8) (cid:21) (cid:18) & (cid:3) * (cid:27)
(cid:10) (cid:18) (cid:9) (cid:16) (cid:30) (cid:29) (cid:9) (cid:10) & (cid:3)
* & (cid:30) (cid... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
s(t; ΔT ) =
δ(t − nΔT ).
We denote the sampled waveform f (cid:2)(t) as
n=−∞
f (cid:2)(t) = s(t; ΔT )f (t) =
�
∞
f (t)δ(t − nΔT )
n=−∞
(cid:26) (cid:4) (cid:18) (cid: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... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/264d28b72e04d21740367f9c4cc485aa_lecture_09.pdf |
(t; ΔT ) =
�
∞
1
jnΩ0t
e
ΔT n=−∞
where all the Fourier coefficients are equal to (1/ΔT ), and where Ω0 = 2π/ΔT is the fun
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) ... | 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 |
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 metal to build up a part a little at a time.
o Can get significant improvement in properties.
o Can lay down say... | https://ocw.mit.edu/courses/3-37-welding-and-joining-processes-fall-2002/269cd56a8622f7fff119040dea55e1c9_33704.pdf |
Ultrasonic welding:
• Diagram on board
• 20-100kHz is typical bonding frequency
• have a small displacement oscillating back and forth
• microscopic shears
• contamination stays there
• may only have 50% bonded area since did not extrude contaminants, just buries it
• don’t deform the substrates
• most... | 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, but not necessarily breadth-first tree.
AsynchSpanningTree, Process i
(cid:122) Signature
− in receive(“search”)j,i, j ∈ nbrs
− out send(“search”)i,j, j ∈ nbrs
− out parent(j)i, j ∈ nbrs
(cid:122) State
− parent: nbrs U { null }, init null
− reported: Boolean, init false
− for each j ∈ nbrs:
(cid:122) send(j) ∈ ... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
) message complexity.
(cid:122) See book for details.
h = height of tree; may be n
More applications
• Asynchronous broadcast/convergecast:
– Can also construct spanning tree while using it to broadcast
message and also to collect responses.
– E.g., to tell the root when the bcast is done, or to collect aggregated
... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
−
(cid:122)
send(j): FIFO queue of N, init (0) if i =
i
0, else
∅
(cid:122) receive(m)j,i
eff: if m+1 < dist then
dist := m+1
parent := j
for k ∈ nbrs - { j } do
add dist to send(k)
Note: No parent actions---no one
knows when the algorithm is done
AsynchBFS
0
AsynchBFS
0
0
AsynchBFS
0
0
0
AsynchBFS
0
0
1
Asynch... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
: O(D |E|); Time: O(diam D (l+d))
• Termination:
– No one knows when this is done, so can’t produce parent outputs.
– Can augment with acks for search messages, convergecast back to i0.
– i0 learns when the tree has stabilized, tells everyone else.
– A bit tricky:
• Tree grows and shrinks.
• Some processes may partici... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
.
− Time:
− Use simplified analysis:
− Neglecting local computation time l
− Assuming that every message in a channel is delivered in time d
(ignoring congestion delays).
− O(diam2 d)
LayeredBFS vs AsynchBFS
(cid:122) Message complexity:
− AsynchBFS: O(diam |E|), assuming diam is known, O(n |E|) if not
− LayeredBF... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
dist := w + weight(j,i)
parent := j
for k ∈ nbrs - { j } do
add dist to send(k)
AsynchBellmanFord
(cid:122) Termination:
− Use convergecast (as for AsynchBFS).
(cid:122) Complexity:
− O(n!) simple paths from i0 to any other node, which is O(nn).
− So the number of messages sent on any channel is O(nn).
− So message co... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
: Unrestrained asynchrony can cause
problems.
• Return to this problem after we have better
synchronization methods.
• Now, another good illustration of the
problems introduced by asynchrony:
Minimum spanning tree
(cid:122) Assumptions:
− G = (V,E) connected, undirected.
− Weighted edges, weights known to endpoint ... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
.
(cid:122) Phase k+1 (produces level k+1 components):
(cid:122) Leader of each component initiates search for MWOE (broadcast initiate on
tree edges).
(cid:122) Each node finds its mwoe:
− Send test on potential edges, wait for accept (different component) or reject (same
component).
− Test edges one at a time in or... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
) Problems arise:
− Inaccurate information about outgoing edges.
− Less “balanced” combination of components:
− Concurrent overlapping searches/convergecasts:
− When nodes are out of synch, concurrent searches for MWOEs could
interfere with each other (we’ll see this).
− Time bound:
− These problems result from nodes ... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
component ≥ 2k.
• Merging and absorbing are both allowable operations in finding MST,
because they are allowed by 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 abs... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
than i does, 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 ca... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
urs.
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 weight < weight(i,j).
So no edges outgoing from C
with weight < alrea... | https://ocw.mit.edu/courses/6-852j-distributed-algorithms-fall-2009/269e879b44c60a699f32f129f2a67779_MIT6_852JF09_lec09.pdf |
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 (broadcast: only sent on tree
edges)
(cid:122) n report messages per level (convergecast)
(cid:122) 2n test-accept messages per level (one pair per node)
(cid:122) n ... | 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 |
(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
A
s + p
,
p =
1
J
b +
µ
K... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
b = 4kg · m2 · Hz.
We will compute the 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)
... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
input is
we find
Vs(s) ≡ 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... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
ecture 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 representation
• P... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
= 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 theorem: steady-state
We will now learn two additional properties of the Laplace transform, which we
will quote without p... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
initial slope of the step response, i.e.
the value of the derivative of the step response at t = 0+ for the same general
1st—order system with steady state equal to unity, a pole at −a and without a
zero. Since we are interested in the derivative of f (t), the Laplace transform of
interest is
H1(s) = L
df1(t)
dt
·
¸
= ... | https://ocw.mit.edu/courses/2-004-systems-modeling-and-control-ii-fall-2007/26a1e7459044ff2652c63c7c98138e4b_lecture06.pdf |
can see that the effects of the zero −z on the 1st—order system are (in
comparison to a system with the same pole at −a but without the zero)
• amplify the steady—state response by z;
• raise the initial value from zero to A;
• raise the initial slope to infinity.
The infinite initial slope is non—physical; in the case of... | 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 |
→
(3D) ——→ functions ——→ functions ——→ functions
network Á
protein
↑
mRNA
↑
genome (DNA)
x-ray
crystallography
NMR
binding
catalysis
synthesis/
degradation
energy storage/
utilization
gene expression
development
immune
surveillance
control points –
decision
“robustness”
time keepers
oscillators ... | 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 |
2x2 + · · · + 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 conic combinati... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
., 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 + y� ≤ �x� + �y�
notation: � · � is general (unsp... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
}: 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
Convex sets
2–10
Operations that preserve convexity
practical methods for establis... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
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 under f is convex
C ⊆ Rm convex =⇒
f −1(C) = {x ∈ Rn | f (x) ∈ C} convex
examples
• s... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
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
Generalized inequalities
a convex cone K ⊆ Rn is a proper cone if
• K is closed (contains its boundary)
• K is solid (has nonempty interior)
• K is pointed (contains no line)
examp... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
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 minimal elements
�K is not in general a linear ordering : we can have x
��K y and y
��K x
x ∈ S is the minimum element of S with respect to �K if
y ∈ S =⇒
x �K y
x ∈ S is a minima... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
x = a T x0}
where a
= 0 and a
T x ≤ T x0 for all x ∈ C
a
a
x0
C
supporting hyperplane theorem: if C is convex, then there exists a
supporting hyperplane at every boundary point of C
Convex sets
2–20
�
Dual cones and generalized inequalities
dual cone of a cone K:
K ∗ = {y | y T x ≥ 0 for all x ∈ K}
example... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.pdf |
over S for some λ ≻K∗ 0, then x is minimal
λ1
x1
S
x2
λ2
•
if x is a minimal element of a convex set S, then there exists a nonzero
λ �K∗ 0 such that x minimizes λT z over S
Convex sets
2–22
optimal production frontier
• different production methods use different amounts of resources x ∈ Rn
• production set ... | https://ocw.mit.edu/courses/6-079-introduction-to-convex-optimization-fall-2009/26c4c530c9db63a12b898d720dd89a44_MIT6_079F09_lec02.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 3 Ver 2.0
Scheduling and Binary Search Trees
6.006 Spring 2008
Lecture 3: Scheduling and Binary Search Trees
Lecture Overvie... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/26f81dc853b4980d8c0873c0f8875268_lec3.pdf |
= R[1: ] (drop R[0] from R)
Can we do better?
• Sorted list: A 3 minute check can be done in O(1). It is possible to insert new
time/plane rather than append and sort but insertion takes Θ(n) time.
• Sorted array: It is possible to do binary search to find place to insert in O(lg n)
time. Actual insertion however re... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/26f81dc853b4980d8c0873c0f8875268_lec3.pdf |
cheduling and Binary Search Trees
6.006 Spring 2008
Finding the next larger element
next-larger(x)
if right child not NIL, return minimum(right)
else y = parent(x)
while y not NIL and x = right(y)
x = y; y = parent(y)
return(y);
See Fig. 4 for an example. What would next-larger(46) return?
Figure 4: next-larger(x) ... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2008/26f81dc853b4980d8c0873c0f8875268_lec3.pdf |
§ 10. Binary hypothesis testing
10.1 Binary Hypothesis Testing
Two possible distributions on a space X
H0 ∶ X ∼ P
H1 X Q
∼
∶
Where under hypothesis H0 (the null hypothesis)
is distributed according to P , and under H1
(the alternative hypothesis) X is distributed according to Q. A test between two distributions
chooses... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
π , π , π
}
∼
.
0∣0
0∣1
1∣0
1 1
∣
So for any test P
test”
Z∣X there is an associated
(α, β). There are a few ways to determine the “best
• Bayesian: Assume prior distributions P[H0
] = π0 and P[H1
] = π1, minimize the expected error
∗ =
Pb min
tests
+
π0π1 0 π1π0 1
∣
∣
112
• Minimax: Assume there is a prior distributi... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
(
P, Q (HW).
)
1. R(P, Q) is a closed, convex
subset
of [0, 1]2.
2. R(P, Q) contains the diagonal.
1Recall that P is mutually singular w.r.t. Q, denoted by P ⊥ Q, if P [E] = 0 and Q[E] = 1 for some E.
113
R(P,Q)ββα(P,Q)α 3. Symmetry: (α, β
(
) ∈ R P, Q
) ⇔
(
1 α, 1 β
−
−
)
∈ R(
)
P, Q .
Pr
oof.
)
1. For convexity, supp... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
If (α, β
) ∈ R(P, Q), then form the test that
P
whenev
er PZ∣X choses Q, and chooses
Q whenever PZ∣X choses P , which gives 1 α, 1 β) ∈ R(
(
P, Q).
c
ho
−
oses
−
The region R(P, Q) consists of the operating points of all randomized tests, which include
deterministic tests as special cases. The achievable region of dete... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/26fd180f40b6773bf19b659a4c5e8656_MIT6_441S16_chapter_10.pdf |
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