text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
We could generalize SW to N -> N -> SET L, and then
PROC Route(t, sw, n1, n2) -> SET P = RET {p :IN Paths(t, n1, n2) |
(ALL p' | p' <= p /\ p'.r # {} ==>
p'.r.last IN sw(End(t, p'{r := p'.r.reml})(n2))}
Flat
Local —
Hierarchical
Source routing
Circuits = distributed source routing:
route once, keep state in rout... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
=
RET ( ALL n1, n2 | Routes(t, sw, n1, n2) <= Paths(t, n1, n2) )
FUNC IsBest(t, sw) -> Bool = VAR best := {p :IN Paths(t,n1,n2) | | Cost(p)}.min |
RET ( ALL n1, n2 | (ALL p :IN Routes(t, sw, n1, n2) | Cost(p) = best) )
Addressing
In a broadcast network addressing is simple: since every node sees all the traffic, all... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
by allocating a fixed bandwidth to a path or ‘circuit’
from a sender to a receiver. The telephone system works this way, and it does not allow traffic to
flow unless it can commit all the necessary resources. A variation that is proposed for ATM
networks is to allocate a maximum bandwidth for each path, but to overc... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
An Overview of the
Grammar of English
Outline
� Grammatical, Syntactic and Lexical
Categories
– Parts of Speech
� Major Constituents
– Noun Phrases
– Verb Phrases
– Sentences
� Heads, Complements and Adjuncts
Grammatical Categories
� The dimensions
– along with constituents can vary, and
– to which the gramm... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
» Major ones are noun, verb, adjective and adverb.
– Closed classes
» change very little
� Indeed, to a closed class is viewed as language change.
» include “function” words, i.e., terms of high
grammatical significance
» Examples are prepositions, pronouns, conjunctions.
What Are They?
� Traditional grammar tel... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
white.”, “Tigers have
stripes.”
Verbs
� Types
– auxiliary (closed)
» List: do, have
– modal (closed)
» List: can, might, should, would, ought, must,
may, need, will, shall (dare?)
» copula (List: be)
– main (open)
Verbs (con’t)
� Verbs have lots of forms:
– Finite forms:
»Can be the only verb in a sentence
»Tends... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
use the imperfective
participle as a so-called “verbal noun”:
Throwing stones at glass houses can be
hazardous.
� This is called a gerund.
– It looks like a verb internally, but a noun
externally.
� Note there is an “more nominal” form:
The throwing of stones at glass houses …
– This uses the same base form, but i... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
downtown)
– temporal (now, tomorrow, Friday)
– WH-adverbs (when, where, why)
� The different subtypes have very
different syntactic properties.
� Traditionally, there is another subtype:
– degree (very, extremely, so, too, rather)
� Most linguists prefer to have a degree
modifier or intensifier word class, rather
... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
LS
MD
NN
NNS
NNP
proper noun, sing.
Jan, Mt. Etna
NNPS
proper noun, pl.
Giants
PDT
POS
PP
PP
RB
RBR
RBS
RP
predeterminer
all, both
possessive ending
's
personal pronoun
I, me, you, he
possessive pronoun
your, one's
adverb
oddly, ever
adverb, comparative
quicker
adverb, superlative
quickes... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
oun Phrase: Preceding
the Noun
� We can build NPs by preceding a N with
– one or more APs:
small apple, very small apples, small green apples
– one or more NPs (nominal compounds):
heavy [cigar smoker]
[Cuban cigar] smoker
[gas meter] [turn-off valve]
– quantifiers, determiners, predeterminers:
a book , the b... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
| PP NPmin
(Q) AP* NPmin
((PDT) DP ) NPint
Noun and PP Compounds
� We allow NPs to be modified by PPs,
especially particles:
“up elevator button”
“elevator up button”
and more speculatively:
“a special [up] to the roof button”
“those in the bag deals”
A Possible “Determiner
Phrase”
� DP fi
� E.g.:
D |
NPmax... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
such as:
NPmin
fi ProperN
Odds and Ends (con’t)
� Gerundive phrases can also be nouns. E.g.:
I enjoy watching television.
Watching television rots your brain.
� So we could just add:
NPint
fi GrvP
� However, recall that, in English, gerunds
are identical with imperfective participles.
– Moreover, below, we will in... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
“arms akimbo” , “I alone”, “attorneys general”
� And a more general post-nominal adjective
construction:
– “love false or true”, “children 8 years old or younger”
And, Finally, Coordination
� Conjunction:
Dorothy, the tin woodman, and the
scarecrow
So add
NP fi NP+ Conj NP
� Note this allows
“a pig in a poke an... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
There’s More Like This
� Other grammatical categories of the lexical items
need to “shine through” to the NPs.
� E.g.:
“Most little girls like ice cream.”
“*That little boy like ice cream.”
“*Most little girls likes ice cream.”
“*Those little boy likes ice cream.”
So, would we would have to differentiate our NPs ... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
Note: These are generally revealing, but don’t
always agree with each other, leaving lots to
debate about the particulars.
Constituent Structure
Analysis Examples
� Substitution
Pat [baked Jan cookies] fi Pat [did so], Pat [ran]
Pat baked [Jan cookies] fi Pat baked [???].
� Question and fragment response
What did ... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
on
the verb.
– Traditionally, we have the
transitive/intransitive distinction.
– But here we see that particular verbs
subcategorize for a variety of different
structures.
– This is the principle area in which syntax has to
come to grips with the properties of individual
words.
Solutions?
� We really only have one ... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
” the NP, which is now part of the S.
� There are other constructions that similarly leave
“gaps”:
Whichever toy you pick Eli will want to play with.
� Dealing with gaps is a major cottage industry.
And We Have the Second
Half of Our NP Problem
� We noted that NPs had to export the
“number” (and “person”) propert... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
part of the same constituent as the verb.
– Sometimes these are called “distant complements” (but
this usage doesn’t seem widespread).
Projections and Syntactic
Categories
� Above, we stipulated quite a few NP syntactic
categories.
� However, it might be that we can get away with
fewer if we understood the relati... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
many heads as elements they
coordinate.
� There is some disagreement as to what is
the head of a given constituent type.
– E.g., some linguists have argued that phrases
like “the little girl” are really determiner
phrases, rather than noun phrases.
Note
� We posited (deep) cases only for
(possibly distant) comple... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
iPaq Intro, Python,
and Connectivity
Feb 13, 2006
Larry Rudolph
1
Pervasive Computing MIT 6.883 SMA 5508 Spring 2006 Larry Rudolph
Administration
• iPaq’s and Mobile Phones very similar
• both use python, bluetooth, internet
• This week:
• Ipaq comments, Python, Network
• Problem set, due in one week
• On your own, w... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/16af5502ace4e084c0e06f2757875eb5_l2_python_intro.pdf |
overhead on user
• daily underhead on user: setup once &
• less dependent on connectivity
• public/private keys easy to use once setup
forget
Pervasive Computing MIT 6.883 SMA 5508 Spring 2006 Larry Rudolph
Connectivity
• Ipaq: 802.11 (WiFi) or Bluetooth
• Mobile: GPRS (edge) or Bluetooth
Pervasive Computing MIT 6.88... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/16af5502ace4e084c0e06f2757875eb5_l2_python_intro.pdf |
asive Computing MIT 6.883 SMA 5508 Spring 2006 Larry Rudolph
Online Tutorials
• Tutorials
• http://www.python.org/doc/tut/tut.html
• http://diveintopython.org/
• http://www.intelinfo.com/newly_researched_free_tra
ining/Python.html
• use google or go to python.org
Pervasive Computing MIT 6.883 SMA 5508 Spring 2006 Larr... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/16af5502ace4e084c0e06f2757875eb5_l2_python_intro.pdf |
Lecture 14
Interlude: Problem Solving
Supplemental reading in CLRS: None
This lecture was originally given as a pep talk before the take-home exam. In note form, this chapter
will be light reading, a break in which we look back at the course material as veterans.
14.1 What to Bring to the Table
In most technical underg... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
tips that might help you to crack a problem open. We’ll keep the following
concrete example in the back of our mind:
Problem 14.1 (Bipartite Matching). In a group of n heterosexual people, each woman has a list of
the men she is interested in marrying, and each man has a list of the women he is interested in
marrying. ... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
makes no assumptions about the items being
sorted except that there is a well-defined notion of “less than” and that, for objects a and b, we can check whether a < b in
constant time. The fact we are using here is that any comparison-based sorting algorithm takes at least Ω(n lg n) time. The
proof is as follows. Suppose... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
• Can randomization help? It is often better to find a fast algorithm with a small probability of
error than a slower, correct algorithm.
Remember that a hard problem usually cannot be solved in one sitting. Taking breaks and chang-
ing perspective help. In the original setting of a pre–take-home exam pep talk, it was i... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
for each man
m and each woman w, draw an edge from m to w if m is on w’s list and w is on m’s list.
Give all edges capacity 1. The graph G is now a flow network with source s and sink t (see
Figure 14.1).
Note that there is a bijection between valid matchings and integer flows in G. In one
direction, given a matching, we... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
)size of the ith
person’s list
(cid:182)(cid:33)(cid:33)
.
n
(cid:88)
i=1
∗ A graph G = (V , E) is called bipartite if the vertex set V can be written as the union of two disjoint sets
V = V1 (cid:116) V2 such that every edge in E has one endpoint in V1 and one endpoint in V2.
14.2.5 Reflect and Improve
• Can we achieve... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
of 5
MIT OpenCourseWare
http://ocw.mit.edu
6.046J / 18.410J Design and Analysis of Algorithms
Spring 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
1
Pipeline Hazards
Arvind
Computer Science and Artificial Intelligence Laboratory
M.I.T.
Based on the material prepared by
Arvind and Krste Asanovic
6.823 L6- 2
Arvind
Technology Assumptions
• A small amount of very fast memory (caches)
backed up by a large, slower memory
• Fast ALU (at least for integers)
• Multipo... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
e
c
r
u
o
s
e
R
time
IF
ID
EX
MA
WB
t0
I1
t1
I2
I1
t2
I3
I2
I1
t3
I4
I3
I2
I1
t4
I5
I4
I3
I2
I1
t5
t6
t7
. . . .
I5
I4
I3
I2
I5
I4
I3
I5
I4
I5
September 28, 2005
6.823 L6- 4
Arvind
Write
-Back
(WB)
Pipelined Execution:
ALU Instructions
6.823 L6- 5
Arvind
0x4
Add
PC
addr
inst
IR
Inst
Memory
IR
IR
IR
31
A
B
ALU
Y
MD1... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
rd1
ws
wd rd2
GPRs
Imm
Ext
OpSel
ALU
Y
A
B
MD1
MD2
MemWrite
WBSrc
we
addr
rdata
Data
Memory
wdata
wdata
R
ExtSel
BSrc
F
D
0x4
Add
PC
addr
inst
IR
Inst
Memory
September 28, 2005
How Instructions can Interact
with each other in a pipeline
6.823 L6- 8
Arvind
• An instruction in the pipeline may need a
resource being u... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
stage
3
stage
4
• Detect a hazard and provide feedback to previous
stages to stall or kill instructions
• Controlling a pipeline in this manner works provided
the instruction at stage i+1 can complete without
any interference from instructions in stages 1 to i
(otherwise deadlocks may occur)
September 28, 2005
Inte... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
nop nop I2
I1
t7
I5
I4
I3
nop nop nop I2
I1
I5
I4
I3
nop nop nop I2
I5
I4
I3
I5
I4
I5
nop ⇒ pipeline bubble
Interlock Control Logic
stall
ws
Cstall
rs
rt
?
6.823 L6- 14
Arvind
0x4
Add
nop
IR
IR
IR
31
PC
addr
inst
IR
Inst
Memory
we
rs1
rs2
rd1
ws
wd rd2
GPRs
Imm
Ext
A
B
ALU
Y
MD1
MD2
we
addr
rdata
Data
Memory
wdata
wd... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
(rs) op imm
rt ← M [(rs) + imm]
ALU
ALUi
LW
SW M [(rs) + imm] ← (rt)
BZ
cond (rs)
true: PC ← (PC) + imm
false: PC ← (PC) + 4
PC ← (PC) + imm
r31 ← (PC), PC ← (PC) + imm
PC ← (rs)
J
JAL
JR
JALR r31 ← (PC), PC ← (rs)
rs, rt
rs
rs
rs, rt
rs
rs
rs
rs
rd
rt
rt
31
31
September 28, 2005
6.823 L6- 17
Arvind
Deriving the ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
31
PC
addr
inst
IR
Inst
Memory
...
M[(r1)+7] ← (r2)
r4 ← M[(r3)+5]
...
September 28, 2005
we
rs1
rs2
rd1
ws
wd rd2
GPRs
Imm
Ext
A
B
ALU
Y
MD1
MD2
we
addr
rdata
Data
Memory
wdata
wdata
R
Is there any possible data hazard
in this instruction sequence?
Load & Store Hazards
6.823 L6- 19
Arvind
...
M[(r1)+7] ← (r2)
r4 ←... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
addr
inst
Inst
Memory
IRSrcD
nop
IR
nop
I2
I1
I2
I3
I4
096
100
104
304
ADD
J 200
ADD
ADD
kill
September 28, 2005
Any
interaction
between
stall and
jump?
IRSrcD = Case opcodeD
⇒ nop
⇒ IM
J, JAL
...
Jump Pipeline Diagrams
6.823 L6- 23
Arvind
time
t0
IF1
(I1) 096: ADD
(I2) 100: J 200
(I3) 104: ADD
(I4) 304: ADD
t5
... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
28, 2005
Branch condition is not known until
the execute stage
what action should be taken in the
decode stage ?
6.823 L6- 25
Arvind
Pipelining Conditional Branches
stall
PCSrc (pc+4 / jabs / rind / br)
0x4
Add
Add
nop
PC
108
addr
inst
Inst
Memory
IRSrcD
nop
IR
I3
?
BEQZ?
zero?
M
IR
I1
E
IR
I2
A
ALU
Y
I1
I2
I3
I4
09... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
D
) . !((opcodeE=BEQZ).z + (opcodeE=BNEZ).!z)
Don’t stall if the branch is taken. Why?
Instruction at the decode stage is invalid
September 28, 2005
Control Equations for PC and IR
Muxes
6.823 L6- 28
Arvind
PCSrc = Case opcodeE
BEQZ.z, BNEZ.!z ⇒ br
...
⇒
Case opcodeD
⇒ jabs
J, JAL
JR, JALR ⇒ rind
...
⇒ pc+4
IRSrcD... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
.
t6
t5
t4
I5
nop I5
nop nop I5
I2
I1
nop nop I5
I2
nop nop I5
nop ⇒ pipeline bubble
Reducing Branch Penalty
(resolve in decode stage)
6.823 L6- 30
Arvind
• One pipeline bubble can be removed if an extra
comparator is used in the Decode stage
PCSrc (pc+4 / jabs / rind/ br)
0x4
Add
Add
nop
E
IR
PC
addr
inst
nop
Inst
M... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
bubble in the pipeline
⇒ CPI > 1
A new datapath, i.e., a bypass, can get the data from
the output of the ALU to its input
t0
t1
IF1
time
(I1) r1 ← r0 + 10
(I2) r4 ← r1 + 17
(I3)
(I4)
(I5)
September 28, 2005
t3
t4
t5
t2
ID1 EX1 MA1 WB1
IF2
t6
t7
. . . .
ID2 EX2 MA2 WB2
IF3
ID3 EX3 MA3 WB3
IF4
ID4 EX4 MA4 WB4
IF5
ID5 EX... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
).re2D )
ws = Case opcode
we = Case opcode
ALU
⇒ rd
ALUi, LW ⇒ rt
JAL, JALR ⇒ R31
ALU, ALUi, LW ⇒(ws ≠ 0)
JAL, JALR ⇒ on
⇒ off
...
ASrc = (rsD=wsE).weE.re1D
Is this correct?
No because only ALU and ALUi instructions can benefit
from this bypass
Split weE into two components: we-bypass, we-stall
September 28, 2005
B... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
signal ?
September 28, 2005
stall = (rsD=wsE). (opcodeE=LWE).(wsE≠0 ).re1D
+ (rtD=wsE). (opcodeE=LWE).(wsE≠0 ).re2D
Why an Instruction may not be
dispatched every cycle (CPI>1)
6.823 L6- 37
Arvind
• Full bypassing may be too expensive to
implement
– typically all frequently used paths are provided
– some infrequentl... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
18.03 Class 4, Feb 10, 2010
First order linear equations: integrating factors
[1] First order homogeneous linear equations
[2] Newtonian cooling
[3] Integrating factor (IF)
[4] Particular solution, transient, initial condition
[5] General formula for IF
Definition: A "linear ODE" is one that can be put in the "s... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
|x| = e^c e^{ - int p(t) dt } |x| = e^c e^{-t^2}
Eliminate the absolute value and reintroduce the lost solution:
x = C e^{- int p(t) dt} x = C e^{-t^2}
In the example, we chose a particular anti-derivative of k , namely kt.
That is what I really have in mind to do in ... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
The input signal is the external temperature T_ext(t) .
Note that the right-hand side is k times the input signal, not the input
signal itself.
What constitutes the input and output signals is a matter of the
interpretation of the equation, not of the equation itself.
Question 4.1: k large means
1. good insulatio... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
integrating:
t^2 x = t^3/3 + c
so x = t/3 + c t^{-2}
[In the first lecture, I posed this (with a different righthand side)
as a flashcard problem, but I did it just after describing the calculation
of an integrating factor for a *reduced* equation. The reduced equation is
x' + 2x/t = 1 , and this has integrating... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
/3}
e^{t/3} x = 60 e^{t/3} + 6 t e^{t/3} - 18 e^{t/3} + c
= ( 42 + 6 t ) e^{t/3} + c
Solve for x:
x = ( 42 + 6t ) + c e^{-t/3}
That's the general solution. Remember, you can check it easily.
u is an "integrating factor."
[4] We still should finish the IVP process:
32 = x(0) = 42 + c so c = -10 :
x = 42 + ... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
5] Let's compute an integrating factor for the general first order linear
equation (*) : we are to solve u' = up .
This is a separable equation: du/u = p dt
ln|u| = int p dt
The constant of integration is in the indefinite integral.
|u| = e^{int p dt}
Now there is a choice of sign. Pick one and go with it; say ... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
The first part of this course will cover the foundational material of homogeneous big bang
cosmology. There are three basic topics:
1. General Relativity
2. Cosmological Models with Idealized Matter
3. Cosmological Models with Understood Matter
1
General Relativity
References:
• Landau and Lifshitz, Volume 2: Th... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
adequate
for out later purposes, but a lot of good stuff is left out (astrophysical applications, tests, black
holes, gravitational radiation, . . . ).
1.1 Transformations and Metrics
We want equations that are independent of coordinates. More precisely, we want them to be
invariant under “smooth” reparameterizatio... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
−1G(R−1)T
From linear algebra, we can insure G� is diagonal with ±1 (or 0) entries. The signature, e.g.
�
1
�
−1
−1
−1
, is determined.
There are residual transformations that leave this form of gµν intact. They are the Lorentz
transformations!
Generalizing gµν , dxµ, we define tensors of more general kinds
Tµ1... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
� ≡ ∂x�ν = ∂x�ν ∂xα
∂ν
∂
Is there an invariant derivative?
�
(R−1)α
�
µAα
� A�
∂ν
µ =
=
�
Aβ
�
∂xα ∂
∂xβ
∂x�ν ∂xα ∂x�µ
∂xα ∂xβ
∂x�ν ∂x�µ
��
�
good
∂2xβ
Aβ
∂αAβ +
∂x�ν ∂x�µ
�
��
�
�
bad
(hard to use
not a tensor)
Add correction term: �ν Aµ ≡ ∂ν Aµ − Γλ
νµAλ:
��
ν A�
ν Sβ
µ = Sα
? Sα
=
µ ∂αAβ ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
Γλ = Γλ
βα
αβ
consistently. (The antisymmetric “torsion” part is a tensor on its own!)
Given Γ, we can take covariant derivatives as
�αTµ1...µm
ν1...νn
= ∂αT
µ1...µm
ν1...νn
−Γλ T
αµ1 λµ2...µm
ν1...νn
−. . .−Γλ T
αµm µ1...λ
This gives a tensor. We use the Leibniz rule in products.
1.3 Covariant Derivativ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
g �∂�
2
�
�
∂xσ
∂x�ν gλσ
�
� ������
∂xσ
ν ∂x�λ gσµ +
�
+ �∂�
�
�
∂ν
�
∂xσ
∂x�µ gλσ
�
∂x�
� �
− �∂�
λ
∂x�µ
�
�
�
�
��
��
∂� ∂xσ
gσν − ��λ����
gσµ
∂x�ν
The boxed terms give the desired inhomogeneous terms; the others cancel.
1.4
Invariant Measure
d4 x
�
= det R
�
�
d4 x� =
∂x�µ
∂xν
��
... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
= ∂µAβ − Γλ
νβ Aλ
�µ
(�ν Aβ ) = ∂µ
µβ �ν Aσ
(�ν Aβ ) −
����� −Γσ
σ �σAβ
Γµν
�
�
��
⇒
symm etric
drop it
νλAσ) − Γσ
(Γσ
− ∂µ νλAσ − Γσ Aσ − Γσ
ρ − Γα Γρ
νρ µβ
νλ∂µ
∂µ∂ν Aβ − ∂µ
= ����
Γσ
µν ∂ν Aσ + Γρ
µβ Γνρ
σ Aσ
µλ∂ν Aσ + Γρ Γσ
µβ
νρAσ
so Rα = ∂µΓα
βµν
νβ − ∂ν Γα
µβ + Γα Γνβ
µρ
4
Symmetry p... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
α∂δgβγ + ∂β∂γ gαδ − ∂α∂γ gβδ − ∂β∂δgαγ
⇒ Rαβγδ = −Rβαγδ
Rαβγδ = −Rαβδγ
Rαβγδ = Rγδαβ
Rαβγδ + Rαγδβ + Rαδβγ = 0
(e.g. Look at the coefficient of ∂α∂δgβγ :
+1 in Rαβγδ
−1 in Rαγδβ
0 in Rαδβγ )
Since these are tensor identities, they hold in any frame!
Also notable is the Bianchi identity;
�αRµνβγ + �β Rµνγα + �γ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
ν
generates the Bianchi identity. This identity is the gravity analogue of
αβγ + Rν
βγα + Rν
γαβ = 0, so (1)
in electromagnetism, or � · B = 0, � × E = − ∂B (existence of vector potential).
∂t
∂αFβγ + ∂βFγα + ∂γ Fαβ = 0
5
1.6
Invariant Actions
Since we have an invariant measure � √
expressions inside (L ).
gd... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
∂α
√
g
[To prove ∂αg = ggµσ∂αgµσ , use expansion by minors and expansion for inverse matrix. Check
on diagonal matrices!]
�√
√
√
d4x ∂µ(
gaµ) is semi-trivial: it is a
d4
x
g �µ
Aµ
gAµ �
= �
Aµ
So �µ
boundary term.
1
g ∂µ
= √
. Thus �
�
d4 x�µAµ�ν Aν gives dynamics.
This supports a gauge transformation
√
g... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
) −
�
√
d4
gjµAµ
x
�
��
�
coupling to current
√
gL
δ
∂µ
δ∂µAν
√
gL
δ
δ∂µAν
�√
�
ggµγ g νδ (∂γ Aδ − ∂δAγ )
= −∂µ
gF µν )
·
=
exercise!
−
√
g �µ
F µν
√
= −∂µ (
√
= −
gjν
Equation of motion: �µF µν = jν
√
�
√
�
or ∂µ
µγ
g νδFγδ =
gjν
gg
gjν �
= 0 ⇒
√
g �ν jν
As a consistency condition we have: ∂ν ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
the field equation for gravity is varying
√
ggαβδRαβ is a total derivative.
we can use the trick that
To prove this relatively painlessly, we can adopt a system of locally geodesic coordinates
�
⇒
�
∂αgβγ = 0 .
√
ggαβRαβ . However,
Then
�
∂µ
δΓµ
g αβ δRαβ = g αβ
�
= ∂µ g αβ δΓµ
≡ ∂µωµ
�
αβ − ∂αδΓµ
βµ
α ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
g
�
1
gαβ δgαβ
2
c. Matter
S =
δS =
�
√
gΛ
� �
√
δ
gΛ
δgαβ − ∂µ δ∂µgαβ
gΛ
√
δ
�
δgαβ
We define the energy-momentum tensor by
√
gΛ
√
δ
gΛ
δgαβ − ∂µ δ∂µgαβ
δ
√
g
2
=
Tαβ
We will now see that this makes sense with both examples and conservation laws.
i. Examples:
8
•
Scalar field:
1
2
1
Λ = g αβ∂... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
δ
αγ
gg g βδFαβ Fγδ
�
= −
1
2
2
Tµν = √
g
√
αγ
gg g βδFαβ Fγδ + 2
gµν
� �
�
1
√
−
δ
4
gg g βδFαβ Fγδ
αγ
√
gg βδFµβ Fµδ
�
= −Fµβ Fνδ g βδ +
1
gµν g αγ g βδFαβ Fγδ ⇒
4
gµν Tµν = 0 (�)
In flat space we have:
�
· 2 B2 − E2
�
1
4
T00 = E2 +
� 1 �
= E2 + B2
2
�
As an exercise, check the Poynting vector a... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
(Killing equations)
αρ
Γν
��
�
�
1
g νβ
= gαµ∂α�ν + gαµ
·
2
gµβ �
+ g αν ∂α�µ + g αν 1
·
2
= gαµ∂α�ν + g αν ∂α�µ + gαµg ρν ∂ρgαβ �ρ
= gαµ∂α�ν + g αν ∂α�µ − ∂ρg αβ �ρ
�
�
�ρ
∂αgβρ + ∂ρgαβ − ∂β gαρ
�
�ρ
∂αgβρ + ∂ρgαβ − ∂β gαρ
where the last step follows from differentiating: gαµ gµβ = δα
gαµ ∂λgµβ = 0... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
��x the coupling constant) by looking at the
situation with nearly flat space and only T00 = ρ significant. (For now, of course, we ignore the
cosmological term.) The stationary action condition gives
�
κ Rαβ − gαβ R = Tαβ
or
1
2
�
1
2
2κRαβ = Tαβ −
1
2
T gαβ
(g00 ≈ c 2 � gij )
Focus on R00:
2κR00 =
ρ
2
in ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
about quantum field
theory and the standard model that are not assumed elsewhere in the course. Don’t worry if not
everything is clear (or even meaningful) to you at this stage. Ask me if you’re curious!
Central material
1. The notion of local Lorentz invariance, vierbeins, and R recipe (Appendices 1-2).
2. The ide... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
, for example, we can form the Dirac equation
(γa ea
µDµ + m) ψ = 0
11
But Dµ needs discussion. We want invariance under local Lorentz transformations. This requires
(exercise!)
DµS(Λ(x)) = S(Λ(x))Dµ
a typical gauge invariance. We solve this problem “as usual” by introducing a gauge potential
ωab(x) ∈ so(3, 1)
... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
metrizing in µ
↔
ab should be “more primitive”
ν. Thus
To solve for ω we go through a slight rigamarole, reminiscent of what we did to get Γ from �g = 0
∂µe a
ν − ∂ν e a
µ = ωµ
ac ecν − ων
ac ecµ
+eaρ
�
�
ν − ∂ν ea
a
∂µe
µ
�
�
a
a
−eaµ ∂ν eρ − ∂ρeν = −��e�
=
aceaρecν
ωµ
− �����
ac
� eaρecµ
ων
���������� ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
�
2ωµ
or
ef =
ωµ
�
ef ν
2
∂µe e
ν − ∂ν e e
µ + eaµe eρ∂ρe a
�
ν − (e ↔ f )
12
�
�
�
Now we can construct a curvature by differentiating (say) a space-time scalar, which is a local
Lorentz vector field
(DµDν − Dν Dµ)φa = R a
µν
b φb
This leads to
R
a
a
c
µν b = ∂µων b − ∂ν ωµ b + ωµ cων b − ων cωµ b
a
a
a
c... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
ativity?
Ordinary spin-1 gauge fields are in danger of producing wrong-metric particles or “ghosts”. This
is because covariant quantization conditions (commutation relatives (?) ) for the different polar
izations:
[a† , aν ] = −gµν
if normal for the space-like pieces are abnormal for the time-like and vice versa. Gau... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
want to relax
this to incorporation additional symmetry. Some important physical ideas have arisen (or have
natural interpretations) along this line.
Weyl wanted to unify electromagnetism with gravity. He postulated
and the symmetry
�αgµν = sαgµν
�
(x) = λ(x)gµν (x)
g
µν
s�
α(x) = sα + ∂αλ
This was the histori... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
Bluetooth Tutorial
Larry Rudolph
1
Pervasive Computing MIT 6.883 SMA 5508 Spring 2006 Larry Rudolph
from bluetooth import *target_name = "My
Phone"target_address = Nonenearby_devices =
discover_devices()for address in nearby_devices:
if target_name == lookup_name( address ):
target_address = address breakif
targe... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/174043c6531c12d2db27fb724842c42f_r2_bluetooth_tut.pdf |
: socket.bind( ( "", free_port ) ) break
except BluetoothError: print "couldn't bind
to ", free_port# listen, accept, and the rest of
the program...
Asynchronous
from bluetooth import *from select import *class
MyDiscoverer(DeviceDiscoverer): def
pre_inquiry(self): self.done = False def
device_discovered(self, a... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/174043c6531c12d2db27fb724842c42f_r2_bluetooth_tut.pdf |
Introduction to Simulation - Lecture 8
1-D Nonlinear Solution Methods
Jacob White
Thanks to Deepak Ramaswamy Jaime Peraire, Michal
Rewienski, and Karen Veroy
Outline
• Nonlinear Problems
– Struts and Circuit Example
• Richardson and Linear Convergence
– Simple Linear Example
• Newton’s Method
– Derivation of Newton
... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/176914834032f0c2438eb23cf185a65f_lec8a.pdf |
1
x
(cid:32)
x
1
2
(cid:12)
(cid:14)
(cid:11)
y
2
(cid:16)
2
y
1
(cid:12)
(
(cid:72)
L
o
(cid:16)
L
1
)
x
2
f
2
x
(cid:32)
x
1
(
(cid:72)
L
o
(cid:16)
L
2
)
(cid:16)
x
0
x
2
(cid:16)
L
1
(cid:16)
L
2
(cid:166)
f
1
x
f(cid:14)
2
x
(cid:32)
0
(cid:166)
f
1
y
(cid:14)
f W
(cid:14)
2
y
(cid:32)
0
Nonlinear
problems
Strut ... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/176914834032f0c2438eb23cf185a65f_lec8a.pdf |
2, ….
x
0
k
(cid:12)
x
(cid:11)
1k
(cid:14) (cid:32)
W x
(cid:12)1
(cid:11)
k
f x (cid:14) (cid:124)
0
until
Ask
• Does the iteration converge to correct solution ?
• How fast does the iteration converge?
Richardson
Iteration
Definition
Richardson Iteration Definition
k
k
1
(cid:14) (cid:32)
x
(cid:14)
An iteration st... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/176914834032f0c2438eb23cf185a65f_lec8a.pdf |
(cid:14)
10
Start with
x (cid:32)
0
0
x
1
x
2
x
3
x
4
(cid:32)
(cid:32)
(cid:32)
(cid:32)
x
0
x
1
x
2
x
3
(cid:14)
(cid:14)
(cid:14)
(cid:14)
(
f x
0
(
f x
1
(
f x
2
(
f x
3
) 10
(cid:32)
)
40
(cid:32)
) 130
(cid:32)
)
(cid:32)
400
No convergence !
Richardson
Iteration
Iteration Equation
Exact Solution
k
x
*
x
1
(ci... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/176914834032f0c2438eb23cf185a65f_lec8a.pdf |
14) (cid:16)
*
x
(cid:32)
x
Computing Differences
k
f x
(
f x
(
(cid:16)
(cid:16)
(cid:14)
x
)
*
k
*
)
1
(cid:14)
(cid:32)
(cid:167)
(cid:168)
(cid:169)
(cid:11) (cid:12) (cid:11)
f x
(cid:4)
(cid:119)
x
(cid:119)
(cid:183)
(cid:184)
(cid:185)
k
x
(cid:16)
x
(cid:12)*
Richardson
Iteration
If
1
(cid:14)
(cid:11) (cid:... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/176914834032f0c2438eb23cf185a65f_lec8a.pdf |
) and update
Another approach
Newton’s method
From the Taylor series about solution
0
(cid:32)
*
f x
(
)
(cid:17)
k
f x
(
)
(cid:14)
df
dx
k
(
x
*
) (
x
(cid:16)
k
x
)
Define iteration
Do k = 0 to ….
(cid:170)
1
(cid:14)
(cid:16) (cid:171)
(cid:172)
(cid:32)
x
x
k
k
df
dx
k
(
x
)
1
(cid:16)
(cid:186)
(cid:187)
(cid:18... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/176914834032f0c2438eb23cf185a65f_lec8a.pdf |
:16)
x
* 2
)
Dividing through
k
1
(cid:14)
(
x
(cid:16)
*
x
)
(cid:32)
[
df
dx
k
(
x
1
(cid:16)
)]
2
d f
2
d x
( )(
x x
(cid:4)
k
(cid:16)
x
* 2
)
Suppose
df
dx
(cid:170)
(cid:171)
(cid:172)
x
( )
1
(cid:16)
(cid:186)
(cid:187)
(cid:188)
2
d f
2
d x
x
( )
(cid:100)
L
for all
x
then
x
k
1
(cid:14) (cid:16)
*
x
(cid:100... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/176914834032f0c2438eb23cf185a65f_lec8a.pdf |
:100)
L x
(cid:74)
0
(cid:16)
*
x
x
1
(cid:16)
*
x
or x
2
(cid:16)
*
x
2
(cid:100)
(cid:74)
x
1
(cid:16)
*
x
(cid:100)
3
(cid:74)
x
0
(cid:16)
*
x
x
(cid:159) (cid:16)
3
*
x
4
(cid:100)
(cid:74)
x
2
(cid:16)
*
x
(cid:100)
7
(cid:74)
x
0
(cid:16)
*
x
Newton’s Method
Convergence
Theorem
If
L
is bounded (
df
dx
bounded... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/176914834032f0c2438eb23cf185a65f_lec8a.pdf |
16)
k
x
(cid:31)
(cid:72)
x
a
(cid:14)
(cid:72)
x
r
k
1
(cid:14)
x
SMA-HPC ©2003 MIT
Summary
• Nonlinear Problems
– Struts and Circuit Example
• Richardson and Linear Convergence
– Simple Linear Example
• 1-D Newton’s Method
– Derivation of Newton
– Quadratic Convergence
– Examples
– Global Convergence
– Convergence ... | https://ocw.mit.edu/courses/6-336j-introduction-to-numerical-simulation-sma-5211-fall-2003/176914834032f0c2438eb23cf185a65f_lec8a.pdf |
TUPLES, LISTS,
ALIASING,
MUTABILITY, CLONING
(download slides and .py files and follow along!)
6.0001 LECTURE 5
6.0001 LECTURE 5
1
LAST TIME
functions
decomposition – create structure
abstraction – suppress details
from now on will be using functions a lot
6.0001 LECTURE 5
2
TODAY
have seen variable types... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/1776670e271578eeb99fc25975f20586_MIT6_0001F16_Lec5.pdf |
()
for t in aTuple:
nums( )
words( )
?
?
if not already in words
i.e. unique strings from aTuple
?
nums = nums + (t[0],)
if t[1] not in words:
words = words + (t[1],)
min_n = min(nums)
max_n = max(nums)
unique_words = len(words)
return (min_n, max_n, unique_words)
6.0001 LECTURE 5
6
LISTS
ordered sequence of info... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/1776670e271578eeb99fc25975f20586_MIT6_0001F16_Lec5.pdf |
• list elements are indexed 0 to len(L)-1
• range(n) goes from 0 to n-1
6.0001 LECTURE 5
10
OPERATIONS ON LISTS - ADD
add elements to end of list with L.append(element)
mutates the list!
L = [2,1,3]
L.append(5)
what is the dot?
L is now [2,1,3,5]
• lists are Python objects, everything in Python is an object
•... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/1776670e271578eeb99fc25975f20586_MIT6_0001F16_Lec5.pdf |
and mutates L = [1,3,7]
6.0001 LECTURE 5
13
CONVERT LISTS TO STRINGS
AND BACK
convert string to list with list(s), returns a list with every
character from s an element in L
can use s.split(), to split a string on a character parameter,
splits on spaces if called without a parameter
use ''.join(L) to turn a ... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/1776670e271578eeb99fc25975f20586_MIT6_0001F16_Lec5.pdf |
� singer, rich
he is known by many names
all nicknames point to the same person
• add new attribute to one nickname …
Justin Bieber
singer
rich
troublemaker
• … all his nicknames refer to old attributes AND all new ones
The Bieb
singer
JBeebs
singer
rich
rich
troublemaker
troublemaker
6.0001 LECTURE 5
18
ALIASES
... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/1776670e271578eeb99fc25975f20586_MIT6_0001F16_Lec5.pdf |
ocw.mit.edu
6.0001 Introduction to Computer Science and Programming in Python
Fall 2016
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/1776670e271578eeb99fc25975f20586_MIT6_0001F16_Lec5.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.438 Algorithms for Inference
Fall 2014
1 Course Overview
This course is about performing inference in complex engineering settings, providing a
mathematical take on an engineering subject. While driven by applicati... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/177720360c220f677921a76b6cc33174_MIT6_438F14_Lec1.pdf |
control and achieve a desired trajectory for spacecraft. Formally,
such scenarios are well modeled by an undirected Guassian graphical model
shown in Figure 2. An efficient inference algorithm for this graphical model is
the Kalman filter, developed in the early 1960’s.
Figure 1: Navigation feedback control in the pre... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/177720360c220f677921a76b6cc33174_MIT6_438F14_Lec1.pdf |
ords (among
27 = 128 possible 7-bit sequences), each codeword corresponding to one of 16
possible 4-bit messages. The 16 possible codewords can be described by means
of constraints on the codeword bits. These constraints are represented via the
graphical model in Figure 4, an example of a factor graph. Note that th... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/177720360c220f677921a76b6cc33174_MIT6_438F14_Lec1.pdf |
‘structure’ (in the form of frequency response) of each of
these time sgements (the so-called cepstral coefficient vector or the “features”).
Speech has structure that is captured through correlation in time, i.e., what one
says now and soon after are correlated. A succinct way to represent this correla
tion is via a... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/177720360c220f677921a76b6cc33174_MIT6_438F14_Lec1.pdf |
This is also known as a Markov random field (MRF). An example of an
MRF is shown in Figure 6. The loopy belief propagation algorithm provides an
efficient inference solution for such scenarios.
1.2
Inference, Complexity, and Graphs
Here we provide the key topics that will be the focus of this course: inference proble... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/177720360c220f677921a76b6cc33174_MIT6_438F14_Lec1.pdf |
|X| many
values of x1, so overall, the number of operations needed scales as |X|2 . In
general, if we are thinking of N variables, then this starts scaling like |X|N . This
is not surprising since, without any additional structure, a distribution over N
variables with each variable taking on values in X requires st... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/177720360c220f677921a76b6cc33174_MIT6_438F14_Lec1.pdf |
computational complexity of MAP estima
tion is thus N · |X|.
Thus, independence or some form of factorization enables efficient computation
of both posterior beliefs (marginalization) and MAP estimation. By exploiting fac
torizations of joint probability distributions and representing these factorizations via
graphi... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/177720360c220f677921a76b6cc33174_MIT6_438F14_Lec1.pdf |
18.03 Class 3, Feb 8, 2010
First order linear equations; systems and signals perspective
[1] First order linear ODEs
[2] Bank Accounts; rate and cumulative total
[3] Systems and signals language
[4] RC circuits
[1] If I had to name the most important general class of differential
equations it would be "linear equ... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17828cb4e899b98aa351a2d6d6f8da2e_MIT18_03S10_c03.pdf |
mathematically:
With Delta t = 1/12 , the statement at the end of the month will read:
x( t + Delta t ) = x(t) + I x(t) Delta t
+ [deposits - withdrawals between t and t+Delta t]
I has units (year)^{-1} . These days I is typically very small, say
1% = 0.01 . You don't get 1... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17828cb4e899b98aa351a2d6d6f8da2e_MIT18_03S10_c03.pdf |
__
| / \ /
| / \ /
_____|/______\______/_________________________________
| \ /<---- here
\__/
Answer: when the slope is positive, not necessarily when Q is positive: (1).
So (assuming q(t) is continuous)
x ( t + Delta t ) ~ x(t) + I x(t) Delta t + q(t) Delta t
Now su... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17828cb4e899b98aa351a2d6d6f8da2e_MIT18_03S10_c03.pdf |
x(0)
|
|
|
V
______________
| |
--------------> | Bank | -------------->
q(0) |______________| x(t)
We will develop a theory of linear equations, complete with an algorithm
for solving them. It's important to recognize them when you see them.
Question 3.2. Which of the following are linear ODE's?
(a... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17828cb4e899b98aa351a2d6d6f8da2e_MIT18_03S10_c03.pdf |
say the positive direction in the circuit is clockwise (ie to the
right over the top, for digital clock users). So if current is flowing
counterclockwise along the wire, an ammeter would give a negative reading.
The system is powered by a variable power source, which creates a "voltage
increase" across it. This wha... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17828cb4e899b98aa351a2d6d6f8da2e_MIT18_03S10_c03.pdf |
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