text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
8.022 (E&M) – Lecture 9
Topics:
RC circuits
Thevenin’s theorem
Last time
Electromotive force:
How does a battery work and its internal resistance
How to solve simple circuits:
Kirchhoff’s first rule: at any node, sum of the currents in = sum
of the currents out (conservation of charge at node... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
hoff’s
law:
EMF supplied by capacitor C: V=Q/C
NB: this is true at any moment in time
tage drop on the resistor: -IR
Vol
Æ Q(t) Æ
V(t)
Q
C
−
IR
0
=
Not useful in this form since I=I(Q)
I=-dQ/dt (- sign because C is losing charge)
Q dQ
+
dt
C
0R
=
Easy integral yields to exponential de... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
[C]=C/V; A=C/s
Æ [RC]=s
Æ [RC]=s
Derive the current:
( )
I t
= −
dQ
dt
= −
Q
0
t
RC
d
dt
−
e
⎛
⎜
⎝
⎞
⎟
⎠
=
Q
0
RC
t
RC
−
e
Same exponential decay as for Q(t)
G. Sciolla – MIT
8.022 – Lecture 9
7
Charging capacitors
Now 3 elements in circuit: EMF, capacitor and resistor
Capacito... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
Details of integration
To solve
dQ
dt
Setting: Q'=
Q CV
−
R
+ − =
V
0 , rewrite as:
Q
C
dQ
dt
= −
Q CV
(
−
RC
)
'
dQ
Q
'
Integrating between t=0 and t:
⇒
= −
dt
RC
Q Q t
=
( )
∫
Q
=
0
t
dQ
'
∫
= −
Q
'
t
t
=
=
0
dt
RC
⇒
ln
Q t CV
( ) -
CV
-
= −
t
RC
⇒
Q
CV
( ) -
t
V
C ... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
=
⎜
⎝
−
t
R C
−
e
⎞
;
⎟
⎠
I
( )
t
=
t
R C
−
e
V
R
Are Kirchhoff’s laws valid at any moment in time?
V
−
Q
C
−
IR
V
= −
V
t
R C
−
e
⎛
1
⎜
⎝
−
⎞
⎟
⎠
−
R
t
R C
−
e
V
R
=
0
O K !
Asymptotic behavior of the capacitor:
At t=0: I=V/R as if C were a short circuit
At t=infinit... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
=V
EMV
when t=15.2
t
G. Sciolla – MIT
8.022 – Lecture 9
13
Verify time constant (E8)
RC circuit with
VEMF = squared 5 V pulses
Variable C initially = 0.3
µF
Variable R initially = 400
Ω
2
R1 = 100 Ω
Display on scope V
C and I(R1)
Verify that time constant is RC
VC(t)
5V
1-e-t/RC
... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
R1
V
+
-
s
R2
+
-
C
Calculate Q(t) on the capac
itor
Solution:
Kirckhoff’s laws will solve it: TEDIOUS!
Use Thevenin’s Theorem
G. Sciolla – MIT
8.022 – Lecture 9
16
8
Thevenin equivalence
Thevenin’s theorem:
Any combination of resistors and EMFs with 2 terminals can be replaced with a
seri... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
i
Æ
R1
V
+
-
R2
A
B
RT
VOC
+
-
+
-
C
Prove that R
T=VOC/I
short
short= current through shorted terminals
There is on y one current go ng through the reduced c rcuit
At t=0, C behaves like a short
At t=0 I
short=VOC/RT
Æ
i
l
with I
i
Æ RT=VOC/I
short
G. Sciolla – MIT
8.022 – Lecture 9
18
9
... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
in:
When we have a messy system or resistors and EMFs, we can reduce it
to a simple R+EMF in series just measuring Ishort
and V
open
:
VOC≡
+
-
RT
Any
unknown
combination
of Rs and EMFs
Careful:
Thevenin works only when the elements in the box follow Ohm’s law,
i.e.
on between V and I
linear relati
G.... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
~ open circuit:
fluorescent light has a (very small) resistance R
τcharge=RC
FL
Thevenin: R
=RT
discharge
T=R//RFL~RFL
C~R
FLC<<
τ
G. Sciolla – MIT
8.022 – Lecture 9
22
11
Norton’s theorem
Any combination of resistors and EMFs with 2 terminals can be replaced with a
parallel of a current generator IN
... | https://ocw.mit.edu/courses/8-022-physics-ii-electricity-and-magnetism-fall-2004/46de97e4978fc003565f683871741451_lecture9.pdf |
Lecture 8
Primitive Roots (Prime Powers), Index Calculus
Recap - if prime p, then there’s a primitive root g mod p and it’s order mod p
is p − 1 = qe1 e2
1 q2 . . . qr . We showed that there are integers gi mod p with order
i − 1 ≡ 0 mod p). Set g = (cid:81) gi -
qei
(counting number of solutions to xq i
exactly
i
(cid... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
ure 37 (Artin’s Conjecture). Let a be a natural number, which is not a square.
Then there are infinitely many primes p for which a is a primite root mod p.
This is an open question. Hooley proved this conditional on GRH, and Heath-
Brown showed that if a is a prime, then there are at most 2 values of a which
fail the co... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
1
(d,p−1)
= (d, p
− 1)
Similarly, if we’re trying to solve the congruence xd ≡ a mod p (a (cid:54)≡ 0 mod p),
we can write a ≡ gl mod p so if x ≡ gk as before then gkd ≡ gl mod p. This
means that gkd−l ≡ 1 mod p ↔ p − 1|kd − l ↔ kd ≡ l mod p − 1 (k is variable),
which has a solution iff (d, p − 1) divides l, in which c... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
+ tp). Since ordp(g + tp) is a multiple of p − 1 and divides
p(p − 1), it’s either equal to p − 1 or equal to p(p − 1) = φ(p2). We’ll show that
there’s exactly one value of t for which the former happens.
2 Since there are p possible values of t(0 ≤ t ≤ p − 1), any of these remaining ones
give a g + tp which is a primi... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
we know it for e.
We know that φ(pe−1) = pe−2(p
gφ(pe−1)
show it for e + 1 - ie.,
(cid:54)≡ 1 mod e. In other words
gφ(p
gφ(p ) (cid:54)≡ 1 mod pe+1.
−
p
e
1). So gφ(pe−1)
e−1
≡
1 mod pe−1 assuming that
) = 1 + bpe−1 with p (cid:45) b. Need to
We know that gpe−2(p−1) = 1 + bpe−1. Raising to power p we get
gpe−1(p−1) = ... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
: otherwise, if n = mm(cid:48) with m and m(cid:48)
coprime and m, m(cid:48) > 2, we’ll show there does not exist a primitive root mod m.
By hypothesis (m, m(cid:48) > 2) we know φ(m) and φ(m(cid:48)) are even. So for (a, n) = 1,
3 we have (a, m) = 1 = (a, m(cid:48)). So aφ(m) ≡ 1 mod m and aφ(m(cid:48)) ≡ 1 mod m(cid:... | https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-2012/4725535fce239230067825c86e89afaa_MIT18_781S12_lec8.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.189 Multicore Programming Primer, January (IAP) 2007
Please use the following citation format:
Saman Amarasinghe, 6.189 Multicore Programming Primer, January
(IAP) 2007. (Massachusetts Institute of Technology: MIT
OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
different machines
Prof. Saman Amarasinghe, MIT.
3
6.189 IAP 2007 MIT
Concurrency and Parallelism
● Concurrency is not (only) parallelism
●
Interleaved Concurrency
Logically simultaneous processing B
Interleaved execution on a single
C
processor
A
● Parallelism
Physically simultaneous processing
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
);
atm.run();
}
Prof. Saman Amarasinghe, MIT.
public void run() {
while(true) {
try {
out.print("AccountID >“ );
String id = in.readLine();
String acc = bnk.get(id);
if (acc == null) throw new Exception();
out.print("Password>“ );
String pass = in.readLine();
if (!acc.is_password(pass))
throw new Exception(... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
new Exception();
out.print("Password>“ );
String pass = in.readLine();
if (!acc.is_password(pass))
throw new Exception();
out.print(“yourbalance is“ + acc.getbal());
out.print("Depositor withdraw amount > “
int val = in.read();
if (acc.getbal() + val > 0)
acc.post(val);
else
throw new Exception();
out.print... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
;
import java.io.*;
public class ATM {
static Bank bnk;
PrintStream out;
BufferedReader in;
ATM(PrintStream out, BufferedReader in) {
this.out = out;
this.in = in;
}
public static void main(String[] args) {
bnk = Bank.getbank();
BufferedReader stdin = new BufferedReader
(new InputStreamReader(System.in));
AT... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
; i++){
atm[i] = new ATMs(i, outdevice(i), indevice(i));
atm[i].start();
}
}
}
}
public void run() {
while(true) {
try {
out.print("AccountID >“ );
String id = in.readLine();
String acc = bnk.get(id);
if (acc == null) throw new Exception();
out.print("Password>“ );
String pass = in.readLine();
if (!acc.is_password... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
89cell
Your account balance is 100
Deposit or Withdraw amount >
Your account balance is 10
-90
Your account balance is 10
Prof. Saman Amarasinghe, MIT.
14
6.189 IAP 2007 MIT
Activity trace II
balance
100
ATM 1
out.print(“yourbalance is“ + acc.getbal());
Your account balance is 100
out.print("Depositor withdra... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
MIT.
16
6.189 IAP 2007 MIT
Synchronization
● All the interleavings of the threads are NOT
acceptable correct programs.
● Java provides synchronization mechanism to restrict
the interleavings
● Synchronization serves two purposes:
Ensure safety for shared updates
– Avoid race conditions
Coordinate actions... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
critical
section at a given time
Once a thread is in the critical section, no other thread
can enter that critical section until the first thread has left
the critical section.
No interleavings of threads within the critical section
Serializes access to section
synchronized int getbal() {
return balance; ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
Amarasinghe, MIT.
23
6.189 IAP 2007 MIT
Synchronizing a block
import java.util.*;
import java.io.*;
public class ATMs extends Thread {
static final int numATMs = 1;
static Bank bnk;
PrintStream out;
BufferedReader in;
int atmnum;
ATMs(int num, PrintStream out, BufferedReader in) {
this.out = out;
this.in ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
account balance is 100
out.print("Depositor withdraw amount >“);
Deposit or Withdraw amount >
-90
int val = in.read();
synchronized(acc)
if (acc.getbal() + val > 0)
acc.post(val);
out.print(“yourbalance is“ + acc.getbal());
Your account balance is 10
ala n c e s h o
u t c o uld
B
b
ATM 2
out.print(“yourbalan... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
();
out.print("Password>“ );
String pass = in.readLine();
if (!acc.is_password(pass))
throw new Exception();
synchronized (acc) {
out.print(“yourbalance is “ + acc.getbal());
out.print("Depositor withdraw amount >“ );
int val = in.read();
if (acc.getbal() + val > 0)
acc.post(val);
else
throw new Exception();
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
)
Waiting for Ben’s account
to be released to perform
BenÆAllysa
synchronized(from)
if (from.getbal() > val)
from.post(-val);
synchronized(to)
Waiting for Allyssa’s account
to be released to perform
DEADLOCKED!
Prof. Saman Amarasinghe, MIT.
29
6.189 IAP 2007 MIT
Avoiding Deadlock
● Cycle in locking graph =... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
synchronized(first) {
synchronized(second) {
if (from.getbal() > val)
from.post(-val);
else
throw new Exception();
to.post(val);
}
}
}
…
Prof. Saman Amarasinghe, MIT.
32
6.189 IAP 2007 MIT
Races
Race conditions – insidious bugs
Non-deterministic, timing dependent
Cause data corruption, crashes
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
� Depends on interleaving of threads
● Usual way to avoid data races:
mutual exclusion
Ensures serialized access of all the shared objects
Prof. Saman Amarasinghe, MIT.
37
6.189 IAP 2007 MIT
Dining Philosophers Problem
● There are 5 philosophers sitting at a round
table.
● Between each adjacent pair of phil... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
MIT.
39
6.189 IAP 2007 MIT
Dining Philosophers Problem: Take I
public void run() {
try {
while(true) {
1
2
3
synchronized(left) {
synchronized(right) {
System.out.println(times + ": Philosopher " + position + " is done eating");
}
}
}
} catch (Exception e) {
System.out.println("Philosopher " + position ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
Semaphores
Blocking & non-blocking queues
Concurrent hash maps
Copy-on-write arrays
Exchangers
Barriers
Futures
Thread pool support
Prof. Saman Amarasinghe, MIT.
43
6.189 IAP 2007 MIT
Potential Concurrency Problems
● Deadlock
Two or more threads stop and wait for each other
● Livelock
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/473fa522f1a7a03c89130c6de55ebc51_lec4concurrency.pdf |
3.044 MATERIALS PROCESSING
LECTURE 9
Example 1: Casting into low conductivity molds
Greatly simplified if:
1. Mold is thick → can neglect air → semi-infinite geometry → erf
2. Mold controls heat loss → Tmold increases → kmold decreases →
3. Superheating is negligible → liquid poured at Tm
gradients are
in mold
D... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/47576d530e3a052c0c3b2451f4076eb6_MIT3_044S13_Lec09.pdf |
− Tm)
ρsHf
(cid:2)
k
mρmcp,m
t
(cid:3) 1
2
s =
2(T0
− Tm)
ρsHf
Boundary Condition: @t = 0, s = 0
Boundary Condition: @s = L, t = tf
3.044 MATERIALS PROCESSING
3
tf ∝ L2
V
A
(cid:2)
L ≈
tf ∝
V
A
(cid:3)
2
⇒ Chvorinov’s Rule
Example 2: Thin castings / Cold molds / Highly conductive molds
out
(cid:9)
(cid:6)
(cid:7)(cid:... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/47576d530e3a052c0c3b2451f4076eb6_MIT3_044S13_Lec09.pdf |
Machine Learning for Healthcare
HST.956, 6.S897
Lecture 5: Risk stratification (continued)
David Sontag
1
Outline for today’s class
1. Risk stratification (continued)
– Deriving labels
– Evaluation
– Subtleties with ML-based risk stratification
2. Survival modeling
2
Where ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
True
positive
rate
Area
under the
ROC curve
Full model
(AUC)
Traditional risk factors
AUC =
Probability that
algorithm ranks
a positive
patient over a
negative patient
Invariant to
amount of class
imbalance
Diabetes
1-year gap
False positive rate
8
... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
Wang J, Cheng YJ, et al. Prevalence and Incidence Trends for Diagnosed
Diabetes Among Adults Aged 20 to 79 Years, United States, 1980-2012. JAMA, 2014.]
© American Medical Association. All rights reserved. This content is excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/help/f... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
from our Creative Commons license.
For more information, see https://ocw.mit.edu/help/faq-fair-use/
15
Intervention-tainted outcomes
• Example from today’s readings:
– Patients with pneumonia who have a history of
asthma have lower risk of dying from pneumonia
– Thus, we le... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
is through
the language of causality:
Patient,
(everything we
know at triage)
Intervention,
(admit to the ICU?)
?
Outcome, (death)
Will admission to ICU lower likelihood of death for patient?
• We return to this in Lecture 14
19
No big wins from... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
mHOSPITAL3) at discharge
0.77(0.75-0.78)
0.75 (0.73-0.76)
0.74 (0.73-0.76)
0.70 (0.68-0.72)
0.76(0.75-0.77)
0.75 (0.74-0.76)
0.73 (0.72-0.74)
0.68 (0.67-0.69)
Length of Stay at least 7 days AUROC (95% CI)
Deep learning 24 hours after admission
Full feature enhanced baseline at 24 hours after admission
Full feature simp... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
0.95(
(0.94-0.96
0.93 (0.92-0.95)
0.93 (0.91-0.94
(
-0.89
(
0.85 (0.81
0.93(
(0.92
(0.89
0.91
(
0.90 (0.88
(
(
(0.83
0.86
-0.94
-0.92
-0.92
-0.88
30-day Readmission, AUROC (95% CI)
Keep2 in mind:
Small wins with deep models may disappear
altogether with dataset shift or non-stationarity
(Jung & S3hah, JBI ‘15)
Deep l... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
.82-0.84
(
((0.75
-0.77
0.76
-0.86
(0.85
0.85(
0.83 (
(0.83
-0.84
0.81
(
(0.80
-0.82
0.74 ((0.73-0.75
1 Area under the receiver operator curve
[Rajkomar et al. ‘18 electronic supplementary material:
h
1
ttps://static-content.spring
/MediaObjects/41
746_2018_29_MOESM1_ESM.pdf]
er.com/esm/art%3A10.1038%2Fs41746-018-0029-... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
/help/faq-fair-use/
25
Survival modeling
• Why not use classification, as before?
– Less data for training (due to exclusions)
– Pessimistic estimates due to choice of window
• What about regression, e.g. minimizing mean-
squared error?
– T is non-negative, may want long tails
– If... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/476c3cdeaf64631d2fb0332832ff6250_MIT6_S897S19_lec5.pdf |
12
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004
A Laguerre Polynomial-Based Bound on the Symbol Error Probability for
Adaptive Antennas With Optimum Combining
Marco Chiani, Senior Member, IEEE, Moe Z. Win, Fellow, IEEE, Alberto Zanella, Member, IEEE, and
Jack H. Winters, Fellow, IEEE
... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
present.
Closed-form expressions for the bit-error probability (BEP) for
coherent detection of binary phase-shift keying (PSK) have been
derived for the case of a single nonfading interferer with Rayleigh
fading of the desired signal in [1] and [2] and with Rayleigh fading
of the desired signal and a single interf... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
J. H. Winters was with the Wireless Systems Research Department, AT&T
Laboratories—Research, Middletown, NJ 07733 USA. He is now with Jack
Winters Communications, LLC, Middletown, NJ 07748-2070 USA
Digital Object Identifier 10.1109/TWC.2003.821165
multiple interferers of arbitrary powers, closed-form expressions
o... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
of the bound can be an issue.
In this letter, starting from an approach similar to that used in
[5], we apply some results on the characteristic polynomial of
a complex Wishart matrix to derive new simple upper bounds
on the symbol error probability (SEP) for coherent detection of
-ary PSK using OC in the presence... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
Filter
A/D
Filter
A/D
OC
γ
represents a unitary transforma
as the elements of
tion. Consequently, the SINR given in (2) can be rewritten as
, since
(4)
D
N A
Filter
A/D
values of the
Desired Signal
Fig. 1. Baseband model of the OC receiver.
By introducing the notation
and
, it is simple to show that t... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
(2)
where the short-term covariance matrix
interference propagation vectors, is given by
, conditioned on all
(3)
It is important to remark that
, and consequently also the
, vary at the fading rate, which is assumed to be much
is a random matrix, and
slower than the symbol rate. Thus,
its eigenvalues are ran... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
the principal diagonal are the eigenvalues of
by
. The elements of the vector
is
is a diagonal matrix whose elements
, denoted
, where
have the same complex Gaussian distribution
with � � �, where �
2Let us define � � �
is the set of the ��2��
complex matrices, and ��
�
,
th elements of
are complex values ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
the characteristic polynomial of a complex Wishart matrix
can be written as
and therefore
(18)
(19)
The derivation of the exact SEP, requiring the evaluation of
statistical expectation of (11) with respect of the eigenvalues
distribution [9], is not a simple task. In the following theorem,
we derive a new upper... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
the absence
of interference reduces to that of MRC.
(or equivalently as
In general, by expanding the Laguerre polynomial, it can be
seen that
(15)
(20)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004
15
10 0
10-1
10-2
10-3
P
E
S
10-4
10-5
10-6
10-7
SIR=10 dB, NA=4, BPSK
0
1... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
the exact
error probability for MRC multiplied by a number greater than
or equal to unity; this number, given by (19), represents an upper
bound on the increase in SEP due to the presence of interfering
signals.
Note also that the definite integral in (14) can be evaluated
in closed form using a canonical decompo... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
SIR [dB]
Fig. 3. SEP versus SIR, with SNR as a parameter ranging from 0 to 15 dB;
� � �, � � �, and BPSK modulation.
P
E
S
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
8 –PSK, NI =3, SIR=15 dB
NA=3
NA=6
NA=3, OC (Bound)
NA=3, MRC (Exact)
NA=6, OC (Bound)
NA=6, MRC... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
are Monte Carlo simulation
results for OC. The results show that, as expected, for small
SNR, the thermal noise is dominant and, therefore, MRC and
OC perform similarly. On the other hand, for sufficiently large
SNR, the role of OC in exploiting the capability of the antenna
array is of increasing importance. This... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
the
8 PSK. The curves are for
two asymptotes (vertical and horizontal) give the values of SIR
and SNR without thermal noise and interference, respectively.
The region below each curve represents the outage domain re
gion in which all points produce an SEP higher than target SEP;
therefore, if the probability dist... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
in space-diversity reception,” Telecommun. Radio Eng., vol. 34/35, pp.
83–85, Oct. 1980.
[2] J. H. Winters, “Optimum combining in digital mobile radio with
cochannel interference,” IEEE J. Select. Areas Commun., vol. SAC-2,
pp. 528–539, July 1984.
[3] V. A. Aalo and J. Zhang, “Performance of antenna array systems ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
multiple antennas,” Bell Labs
Tech. J., vol. 1, no. 2, pp. 41–59, Autumn 1996.
[8] G. J. Foschini and M. J. Gans, “On limits of wireless communications
in a fading environment when using multiple antennas,” IEEE Wireless
Pers. Commun., vol. 6, pp. 311–335, Mar. 1998.
[9] M. Chiani, M. Z. Win, A. Zanella, and J. H.... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
Poor and S. Verdú, “Probability of error in MMSE multiuser de
tection,” IEEE Trans. Inform. Theory, vol. 43, pp. 858–871, May 1997.
[14] M. Z. Win, G. Chrisikos, and J. H. Winters, “MRC performance for
� -ary modulation in arbitrarily correlated Nakagami fading channels,”
IEEE Commun. Lett., vol. 4, pp. 301–303, Oc... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
with unequal fading parameters and SNRs among
branches: An analytical framework,” in Proc. IEEE Wireless Communi
cations and Networking Conf., vol. 3, New Orleans, LA, Sept. 1999, pp.
1058–1064.
[21] M. Z. Win, R. K. Mallik, G. Chrisikos, and J. H. Winters, “Canonical
expressions for the error probability performa... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/4770a22e0087ba5d8439e0be8fd8fdaa_oc_laggure_tw.pdf |
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 9-1
Lecture 9 - Carrier Flow (cont.)
February 23, 2007
Contents:
1. Shockley’s Equations
2. Simplifications of Shockley equations to 1D quasi-neutral
situations
3. Majority-carrier type situations
Reading assignment:
del Alamo, Ch. 5, §§5.3-5... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
Electron current equation:
q
(cid:2) E∇ · (cid:2) = (cid:2) (p − n + ND
(cid:2)
Je = −qn(cid:2)ve
drift
+ qDe∇n
(cid:2)
−)
+ − NA
Hole current equation:
(cid:2)
Jh = qp (cid:2)vh
drift
− qDh∇p
(cid:2)
Electron continuity equation:
∇ · J(cid:2)
(cid:2)
∂t = Gext − U (n, p) + 1
∂n
e
q
Hole continuity equa... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
)
∇ ⇒ ∂
∂x
n+
y
x
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 9-... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
20J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 9-6
Overview of simplified carrier flow formulations
General drift-diffusion
situation
(Shockley's equations)
1D approx.
Quasi-neutral
situation
(negligible volume
charge)
Space-charge
situation
(field independent
of n, p)
Majority-carrier
... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
cy of the concentration of positive and negative species is
negligible in the scale of the charge density that is present.
QN approximation eliminates Gauss’ law from the set:
+
ρ = q(p − n + N
D
− N −
A
) = q(po − no
D+ N − N −
+
A
) + q(p
(cid:5) − n (cid:5))
• Quasi-neutrality in equilibrium:
|
+
po − no + N... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
5)
= (po − no + ND
−)
+ − NA
q
(cid:5)
If, in general define:
Then, in equilibrium:
∂Eo
∂x
and out of equilibrium:
∂E (cid:5)
∂x
= (p (cid:5) − n (cid:5))
q
(cid:5)
Eo computed as in Ch. 4. Here will learn to compute E (cid:5) .
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectroni... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
0J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 9-11
Simplified set of Shockley equations
for 1D quasi-neutral situations
p − n + ND − NA (cid:4) 0
Je = −qnve
drift + qDe
∂n
∂x
Jh = qpv h
drift − qDh
∂p
∂x
∂n = Gext − U + 1 ∂Je
q ∂x
∂t
or
∂Jh
∂p
1
∂t = Gext − U − q
∂x
∂Jt (cid:4) 0
∂x... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
Month YYYY].
6.720J/3.43J Integrated Microelectronic Devices - Spring 2007
Lecture 9-13
� Characteristics of majority carrier-type situations:
• electric field imposed from outside
• electrons and holes drift
• electron and hole concentrations unperturbed from TE
Simplifications:
• neglect contribution of minorit... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
� Equation set for 1D majority-carrier type situations:
n-type
n (cid:4) no (cid:4) ND
p-type
p (cid:4) po (cid:4) NA
Je = −qno[vde(E) − vde(Eo)] Jh = qpo[vdh(E) − vdh(Eo)]
dJe (cid:4) 0,
dx
dJh (cid:4) 0, dJt (cid:4) 0
dx
dx
Jt (cid:4) Je
Jt (cid:4) Jh
Cite as: Jesús del Alamo, course materials for 6.720J Int... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
• In general (low and high fields):
vsat
I = W tqND 1 + vsat L
μe V
which for high fields saturates to:
Isat = W tqNDvsat
I
Isat
0
0
V
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Tec... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
form of continuity equations and consequences.
• Exercises 5.1, 5.2.
• Non-uniformly doped resistor
• Sheet resistance
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded o... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/477625fc6080d062ea9bef7e100acd0b_lecture9.pdf |
Introduction to Algorithms: 6.006
Massachusetts Institute of Technology
Instructors: Erik Demaine, Jason Ku, and Justin Solomon
Lecture 1: Introduction
Lecture 1: Introduction
The goal of this class is to teach you to solve computation problems, and to communicate that
your solutions are correct and efficient.
Pr... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
: if first k contain match, returns match before interviewing student k + 1
– Base case: k = 0, first k contains no match
– Assume for induction hypothesis holds for k = k0, and consider k = k0 + 1
– If first k0 contains a match, already returned a match by induction
– Else first k0 do not have match, so if first k0 + 1... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
2Θ(nc)
21000 ≈ 10301
10281 millenia
Lecture 1: Introduction
Model of Computation
3
• Specification for what operations on the machine can be performed in O(1) time
• Model in this class is called the Word-RAM
• Machine word: block of w bits (w is word size of a w-bit Word-RAM)
• Memory: Addressable sequence of... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
�(1) time
– StaticArray.set at(i, x): write word x to array index i in Θ(1) time
• Stored word can hold the address of a larger object
• Like Python tuple plus set at(i, x), Python list is a dynamic array (see L02)
4
Lecture 1: Introduction
1 def birthday_match(students):
2
’’’
Find a pair of students with th... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
) +
• Quadratic in n is polynomial. Efficient? Use different data structure for record!
How to Solve an Algorithms Problem
1. Reduce to a problem you already know (use data structure or algorithm)
Search Problem (Data Structures)
Static Array (L01)
Linked List (L02)
Dynamic Array (L02)
Sorted Array (L03)
Direct... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/477c78e0af2df61fa205bcc6cb613ceb_MIT6_006S20_lec1.pdf |
Assignments
Problems Sets:
• Problem Set 2 (Scheduling) due
• Problem Set 3 (PDDL Modeling) out soon
Readings:
• Hoffman, Porteous, Sebastia, “Ordered Landmarks in Planning,” Journal of
Artificial Intelligence Research, 22, pp. 215-278, 2004. (Voted most influential
paper during ICAPS 2013).
Karpas, et al., “Temporal ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
St
action1
action2
…
action2
Alone
on Mars
at Ares 3
1
St
action4
Be
Rescued
action3
2
St
action2
2/24/2016
Cognitive Robotics
4
Motivation
Have
enough
food and
water
Get the
Rover
ready for
long trips
Alone
on Mars
at Ares 3
Have
MAV
ready
Be
Rescued
Re-Establish
communication
Drive to
Ares 4
Be at
Are... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
communication
– To get to the rover, I need to exit this the Hab
© 20th Century Fox. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/.
2/24/2016
Cognitive Robotics
10
Fact and Action Landmarks
• We can also consider disjun... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
A →gn B, iff A true one step
before B becomes true for the first time
• Other ordering types exist, which we do not discuss
• Note that:
A →n B ⟹ A →gn B ⟹ A → B
2/24/2016
Cognitive Robotics
15
Outline
• What Landmarks Are
• How Landmarks Are Discovered
• Using Landmarks
– Subgoals
– Heuristic Estimates
– Admissible ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
/2016
Cognitive Robotics
20
Landmark Discovery (1)
Step 1: Find landmarks candidates and orderings
– Start with the goals: every goal is a landmark
– If B is landmark and all actions
that achieve B share A as
precondition, then
• A is a landmark
• A →n B
• Useful restriction: consider only
the case where B is achie... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
t-at-B
t-at-A
t-at-D
2/24/2016
Cognitive Robotics
25
Landmark Discovery (1)
Step 1: Find landmarks candidates and orderings
Example
E
A
B
C
D
Landmark: t-at-D
t-at-A
drive-t-A-E
drive-t-A-B
t-at-E
t-at-B
t-at-A
drive-t-E-D
drive-t-B-C
drive-t-A-E
drive-t-A-B
t-at-D
t-at-C
t-at-E
t-at-B
t-at-A
t-at-E
t-at-D
2/24/2016
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
Robotics
28
Landmark Discovery (1)
Step 2: Verify Landmark Candidates
• For each landmark discovered:
– Remove all the action that can achieve it
– Build relaxed planning graph for π’A and
check if we can find the goals
– If so, remove landmark and ordering from
the landmark graph
o-at-B
t-at-B
o-in-t
t-at-C
p-at-C
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
a SAS+ task < V, A, s0, G >
• The DTG of variable v ∈V (DTGv) represents how the value of
v can change.
• DTGv is a directed graph with nodes Dv that has arc <d, d’> iff:
– d ≠ d’ , and
– ∃ action with v ↦ d’ as effect, and either
– v ↦ d as precondition, or no precondition on v
2/24/2016
Cognitive Robotics
34
Landma... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
↦ d’ ) → ( v ↦ d )
2/24/2016
Cognitive Robotics
38
Landmark Discovery (3)
Find landmarks through forward propagation in relaxed
planning graph
• Propagate information on necessary predecessors
– Label each fact node with itself
– Propagate labels along arcs
•
•
Finds causal landmarks only (preconditions for actions)
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
,d
d
a,b
4
a,c,d
5
a,d
6
a,b,e
e
a,c,d,f
f
a,g
g
a
2/24/2016
Cognitive Robotics
43
Landmark Discovery (3)
Facts
Actions
Facts
Actions
Facts
a
1
a
2
a
1
a,b
b
a,c
c
a,d
d
a,b
4
a,c,d
5
a,d
6
a,b,e
e
a,c,d,f
f
a,g
g
a
2/24/2016
Cognitive Robotics
44
Landmark Discovery (3)
Facts
Actions
Facts
Actions
Facts
a
1
a
2
a
1
a... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
missible Heuristic Estimates
– Enriching the Problem
– Beyond Classical Planning
• Summary
2/24/2016
Cognitive Robotics
47
Using Landmarks
• So what can we do once we have these landmarks?
• We assume that landmarks and orderings are
discovered in a pre-processing phase, and the same
landmark graph is used throughou... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
-B
t-at-B
o-in-t
t-at-C
p-at-C
o-at-C
o-in-p
o-at-E
Partial Plan: drive-t-B
Goal: o-in-t ∨ p-at-C
2/24/2016
Cognitive Robotics
53
Using Landmarks: Subgoals
p
E
C
A
D
B
o
t
Partial Plan: drive-t-B, load-o-t
Goal: t-at-C ∨ p-at-C
o-at-B
t-at-B
o-in-t
t-at-C
p-at-C
o-at-C
o-in-p
o-at-E
2/24/2016
Cognitive Robotics
54
Us... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
Using Landmarks: Subgoals
B
A
D
o
E
p
C
t
o-at-B
t-at-B
o-in-t
t-at-C
p-at-C
o-at-C
o-in-p
o-at-E
Partial Plan: drive-t-B, load-o-t, drive-t-C, unload-o-C, fly-p-C,
load-o-p
Goal: o-at-E
2/24/2016
Cognitive Robotics
58
Using Landmarks: Subgoals
B
A
D
C
t
p
E
o-at-B
t-at-B
o-in-t
t-at-C
o
p-at-C
o-at-C
o-in-p
o-at-E
P... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
holding-A
on-B-C
on-A-B
Partial Plan: pickup-B
Goal: clear-A ∨ on-B-C
2/24/2016
Cognitive Robotics
63
Using Landmarks: Subgoals
on-table-A
on-C-A
clear-C
hand-empty
on-table-B
clear-B
B
C
A
clear-A
holding-B
holding-A
on-B-C
on-A-B
Partial Plan: pickup-B, stack-B-C
Goal: clear-A
2/24/2016
Cognitive Robotics
64
Using ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
on-table-A
on-C-A
clear-C
hand-empty
on-table-B
clear-B
A
B
C
clear-A
holding-B
holding-A
on-B-C
on-A-B
Partial Plan: pickup-B, stack-B-C, unstack-B-C, putdown-B,
unstack-C-A, putdown-C, pickup-A, stack-A-B, unstack-A-B,
putdown-A, pickup-B, stack-B-C, pickup-A, stack-A-B
Goal: ∅
2/24/2016
Cognitive Robotics
68
Using... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
LAMA Approach
• The landmarks that still need to be achieved after reaching
state s via path π are:
L(s, π) = ( L \ Accepted(s, π)) ⋃ ReqAgain(s, π)
• L is the set of all (discovered) landmarks
• Accepted(s, π) ⊂ L is the set of accepted landmarks
• ReqAgain(s, π) ⊆ Accepted(s, π) is the set of required again
landmar... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
and so it is required again
on-B-C
on-A-B
2/24/2016
Cognitive Robotics
76
Multi-path Dependency
I did not achieved A
I achieved A
I need to achieved A
• Suppose state s was reached by paths π1, π2
• Suppose π1 achieved landmark A and π2 did not
• Then A needs to be achieved after state s
• Proof: A is a landmark, ther... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
: assign the right cost to each landmark,
sum over the costs of landmarks (Karpas and Domshlak, 2009)
2/24/2016
Cognitive Robotics
80
Admissible Heuristic Estimates:
Conditions for Admissibility
• Each action shares its cost between all the landmarks it achieves
∀a ∈ A: ∑ cost(a, A) ≤ C(a)
A ∈ L(a | s ,P)
cost(a, A)... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
: can be much worse than the optimal cost
partitioning
2/24/2016
Cognitive Robotics
84
Admissible Heuristic Estimates:
Uniform Cost Sharing
• Advantage: Easy and fast to compute
• Disadvantage: can be much worse than the optimal cost
partitioning
Uniform cost sharing
hL = 2.5
min(0.5)=0.5
min(0.5)=0.5
min(0.5)=0.... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
uristic Estimates:
How we can get better?
• So far:
– Uniform cost sharing is easy to compute, but suboptimal
– Optimal cost sharing takes a long time to compute
• Q: How can we get better heuristic estimates that don’t take a
long time to compute?
A: Exploit additional information - action landmarks
2/24/2016
Cogniti... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
, 2009)
• These temporally extended goals can be expressed in Linear
Temporal Logic (LTL)
• Each LTL formula can be compiled into a finite-state
automaton (FSA)
• Each FSA can be encoded as a single variable in an enriched
planning problem
2/24/2016
Cognitive Robotics
94
Using Landmarks:
Enriching the Problem
A Si... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
is a fact which must be true in every
successful trajectory (possible execution)
• Temporal Landmarks
2/24/2016
Cognitive Robotics
98
Temporal Landmarks
2/24/2016
Cognitive Robotics
99
Landmarks for Temporal Planning
• We can treat a durative action as two “snap” actions:
the start and the end (Fox & Long, 2003)
• ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
-flashlight
2/24/2016
Cognitive Robotics
104
Durative Action: Turn On Flashlight
Duration: 1 seconds
Start:
Condition: have-flashlight
Effect:
Invariant condition:
End:
Condition:
Effect: light
2/24/2016
Cognitive Robotics
105
Causal Landmarks for Flashlight Match Cellar
If we run a casual landmark discovery, we w... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/478e717152ec908cd81662e07167d745_MIT16_412JS16_L7.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.