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6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
23. Networks — Links and Switches1
This handout presents the basic ideas for transmitting digital data over links, and for connecting
links with switches so that data can pass from lots of sources to lots of destinations. You may ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
copies of it in parallel. This goes by different
names; ‘space division multiplexing’ and ‘striping’ are two of them. Common examples are:
Parallel busses, as in the first four lines of the table.
Switched networks: the telephone system and switched LANs.
Multiple disks, each holding part of a data block, that can ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
pair
Broadcast
µs
1.25 MB/s 100 + µs
6
MB/s 100 + µs
300 MB/s 5
6
MB/s 5
0.2 MB/s 5
KB/s
16
5
MB/s 3
3
ISDN
CAP 16
memory packet
16
32/64 DMA block
1
1
1
16
1
1
packet
packet
?
32
?
packet, 64-1500 B
1
1
1
µs/km 1
µs/km 1
µs/km 1
µs/km 1
µs/km 6 MHz byte or cell
packet, 64-1500 B... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
and decreases it with each trouble-free send. This is called ‘exponential backoff'; when the
3 M. Sachs and A. Varman, Fibre channel and related standards. IEEE Communications 34, 8 (Aug. 1996), pp 40-
49.
4 G. Hoffman and D. Moore, IEEE 1394: A ubiquitous bus. Digest of Papers, IEEE COMPCON ’95, 1995, pp 334-
338.
... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
(backed off) and shortened depend on
the properties of the channel. If the ‘trouble’ signal comes back very quickly and the cost of
trouble is small, senders can shorten their waits aggressively; this happens in the Ethernet, where
collisions are detected in at most 64 byte times and abort the transmission immediate... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
to control acquiring the channel, also uses
backpressure, in the form of carrier sense, to keep a sender from interrupting another sender that
has already acquired the channel; this is called ‘deference’. TCP uses credits to allow the receiver
to control the flow. It also uses backoff to deal with congestion within ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
and this is extra overhead that is serious if the messages are small (for instance,
ATM cells are only 53 bytes, and only 48 bytes of this are payload).
Stop sending messages and send a ‘resync’ request. When the receiver gets this it returns an
absolute rather than an incremental credit. Once the sender gets this i... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
a label, framing is easy: You just label each X with the Y it is part of,
and the position it occupies in that Y. For example, to frame (or encapsulate) an IP packet on the
Ethernet, just make the ‘protocol type’ field of the packet be ‘IP’, and if the packet is too big to
10 F. Bonomi and K. Fendick, The rate-based... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
.13 If a ‘sync’ appears in the middle of a packet, that is taken as
an error, and the next legal symbol is the start of a new packet. A simpler version of this idea
requires at least one transition on every bit (in 10 Mb Ethernet) or byte (in RS-232); the
absence of a transition for a bit or byte time is a sync.
Ce... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
receiver makes a different guess. After at
most 193 tries it will have guessed right. This takes a lot longer than the previous schemes to
acquire sync, but it uses a constant amount of extra bandwidth (unlike escape schemes), and
much less than fixed sync schemes: 1/193 for T-1 instead of 1/5 for FDDI, 1/2 for Ethe... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
at the other end. Of course the
wide wire is more expensive than a single narrow one, and the multiplexers must also be paid
for.
• Traffic aggregated from several links may be more predictable than traffic from a single one.
This happens when traffic is bursty (varies in bandwidth) but uncorrelated on the input li... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
and a sender has an n-bit ‘code’ (10010 in the figure) which it xor’s
with its current data bit. The receiver xor’s the code in again and looks for either all zeros or
all ones. If it sees something intermediate, that is interference from a sender with a different
code. If the codes are sufficiently orthogonal (agre... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
sub-channel appears explicitly in
the packet. If the packets are fixed size, they are often called ‘cells’, as in the Asynchronous
Transfer Mode (ATM) networking standard. Fixed-size packets are used in other contexts,
however, for instance to carry load and store messages on a programmed I/O bus. Variable sized
pa... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
. Granting can be
centralized, as in many I/O busses, or distributed, as in a daisy-chained bus or a token ring
like FDDI.
Flow control means buffering, as we saw earlier, and there are several ways to arrange buffering
around a multiplexer, shown on the left side of figure 1. Having the buffers near the arbitratio... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
lexed to one BI bus, and 4 of these are multiplexed to one internal
processor-memory bus.
Demultiplexing uses the same physical mechanisms as multiplexing, since one is not much use
without the other. There is no arbitration, however; instead, there is addressing, since the input
channel must select the proper outp... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
it15.
Instead, the task is distributed among all the senders. As with link arbitration in general, there are
two ways to do it: scheduling and contention.
14 R. Rettberg et al.: The Monarch parallel processor hardware design. IEEE Computer 23, 18-30 (1990)
15 There are times when this religion is inappropriate. For ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
However, it must keep this clock very close to the clock of the preceding node on the ring to
keep from having to add sync marks or buffer a lot of data.
• Maintaining the single token: with multiple tokens the broadcasting scheme fails. With no
tokens, no one can send. So each node must monitor the ring. When it fi... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
is wasted whenever two packets overlap at the receiver; this is called a ‘collision’.
How often does it happen?
Fig. 3: A dual-attachment ring tolerates failure of one node
• A practical way to solve this problem is to connect all the nodes to a single ‘hub’ in a so-
called ‘star’ configuration, as shown in figure 4... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
one packet long. In this situation exponential backoff gives
an efficiency of 1/e = .37 (see below).
If a node that isn’t sending can tell when someone else is sending (‘carrier sense’), then a
potential sender can ‘defer’ to a current sender. This means that once a sender’s signal has
reached all the nodes without... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
0 Mbits/sec fast Ethernet has the same minimum packet size, and hence a maximum
diameter of 5 µs, 10 times smaller. Gigabit Ethernet has a maximum diameter of .5 µs or 100 m.
However, it normally operates in ‘full-duplex’ mode, in which a wire connects only two nodes
and is used in only one direction, so that two wi... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
.7 at this maximum, including the
successful transmission. The waste, also called the ‘contention interval’, is therefore 1.7r. For
packets of length l the efficiency is l/(l + 1.7r)=1/(1 + 1.7r/l) ~ 1 - 1.7r/l when 1.7r/l is small.
The biggest packet allowed on the Ethernet is 1.5 Kbytes = 20 r, and this yields an ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
which of course is always changing as
well).
All versions of backoff arbitration have the problem that a selfish sender that doesn’t obey the
rules can get more than its share. This isn’t a problem for Ethernet because there are very few
sources of interface chips, and each one has been careful engineered to behave... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
ports.
Fig. 5: An Ethernet with a hub can switch out failed nodes
Handout 23. Networks — Links and Switches
13
Handout 23. Networks — Links and Switches
14
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
A drawback is that the hub is a single point of failure. Since it is ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
view the links as simplex, but usually
they are paired to form full-duplex links so that every input link has a corresponding output link
which sends data in the reverse direction. Often the input and output links are connected to the
same nodes, so that the switch allows any node to send to any other.
A basic swit... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
0 µs 16
10-1000 KB/s
80
MB/s 125
1-100
s many
µs 50K
Handout 23. Networks — Links and Switches
15
Handout 23. Networks — Links and Switches
16
Storage can serve as a switch of the kind shown in figure 6(b). The storage device is the
common channel, and queues keep track of the addresses that input and outpu... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
the slower one can deliver it.
3
3
2
concatenate
4
2
Fig. 7. Composing switches by concatenating.
The only interesting thing about it is the rules for forwarding a single traffic unit:
Can the unit start to be forwarded before it is completely received (‘wormholes’ or ‘cut
through’)17, and
Can parts of two uni... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
, and to watch out for alternating fast and
slow links if you don’t have interleaving. However, a unit shouldn’t be too small on a variable
TDM link because it must always carry the overhead of its address. Thus ATM cells, with 48 bytes
of payload and 5 bytes of overhead, are about the smallest practical units (thou... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
an
end node, and the Internet has 300 million nodes rather than 4.
The new mechanism we need to make this work is routing, which converts an address into a
‘path’, a sequence of decisions about what output link to use at each switch. Routing is done with
a map from addresses to output links at each switch. In addit... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
What spec does a mesh network satisfy? We saw earlier that a broadcast network provides
unreliable FIFO delivery. In general, a mesh provides unreliable unordered delivery, because the
routes can change, allowing one packet to overtake another, even if the links are FIFO. This is
fine for IP on the Internet, which d... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
exists from every sender to every receiver.
19 C. Seitz: The cosmic cube. Communications of the ACM 28, 22-33 (1985)
20 W. Stallings, IPV6: The new Internet protocol. IEEE Communications 34, 7 (Jul 1996), pp 96-109.
21 Actually, this simple configuration can only deadlock if each node fills up with traffic going to th... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
other links up or down arbitrarily; the Autonet does this.
Note that this kind of rule for preventing deadlock may conflict with an attempt to optimize the
use of resources by sending traffic on the least busy links.
Although figure 9 suggests that the resources being allocated are the links, this is a bit
misleadi... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
assuming a global name space N for the nodes; this is
usually coded with some kind of UID such as a LAN address, or by manually assigned addresses
like IP addresses. If the nodes don’t have unique names, life becomes a lot more confusing.
We name links with local names that are relative to the sending node, rather t... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
) -> Bool = RET p1.n = p2.n /\ p1.r <= p2.r
FUNC Nodes(t, p) -> Ns = RET {p' | p' <= p | End(t, p'))
22 T. Anderson et al., High-speed switch scheduling for local area networks. ACM Transactions on Computer
Systems 11, 4 (Nov. 1993), pp 319-352.
23 This is a complicated subject, and our treatment leaves out a lot. A... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
so that an entire subtree will be collapsed into a single node for the purposes of
higher-level routing.
n1
n2
n3
n0
ns = {n1, n2, n3}
n0 IN ns
TYPE L = (L + [n, ll])
FUNC Restrict(t, ns) -> T =
RET (\ n | (\ l | (n IN ns /\ (t(n)(l) IN ns => t(n)(l)) ))
FUNC IsConnected(t, ns) -> Bool =
RET (ALL n1 :IN ns, n2... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
, do not do any
switching of other people’s packets. The end-nodes don’t participate in the neighbor
computation, since that would be an n2 process. Instead, only the routers on the LAN participate,
and there is a separate scheme for the end-nodes. There are two mechanisms needed:
1. Routers need to know what end-n... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
each node how to map a destination node into a link24 on which to send traffic; you
can think of it as the dual of a topology, which for each node maps a link to a destination node.
Then a route is a path that is chosen by sw.
TYPE SW = N -> N -> L
PROC Route(t, sw, n1, n2) -> P = VAR p :IN Paths(t, n1, n2) |
(ALL ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
everyone
knows the topology and the Cost function. For concreteness, we give a popular cost function: the
length of the path.
FUNC IsConsistent(t, sw) -> Bool =
RET ( ALL n1, n2 | Route(t, sw, n1, n2) IN Paths(t, n1, n2) )
FUNC IsBest(t, sw) -> Bool = VAR best := {p :IN Paths(t,n1,n2) | | Cost(p)}.min |
RET ( ALL n... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
mesh network the sw function
in every router has to map each address to a link that leads there. The structure of the address can
make it easy or hard for the router to do the switching, and for all the nodes to learn the topology.
Not surprisingly, there are tradeoffs.
It’s useful to classify addressing schemes as... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/1686c03f438916c36cd791540d12985f_23.pdf |
this method in connection with links, and the issues are the same in networks. One
difference is that the round-trip time may be longer, so that more buffering is needed to support a
given bandwidth. In addition, the round-trip time is usually much more variable, because traffic
has to queue at each switch. Another ... | 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 |
» let new words into them rather casually
» and, therefore, tend to be very large.
» 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... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
the “number”
grammatical category)
» boy, boys, man, men
– Count vs mass
» “too many cats”, “too much water”
» “Wine can be red or 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?)
» c... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
gone for three weeks,}
– Perfect participle (not quite the same thing)
» He {has, will have, had} {cried, been, gone}
» Always the same as the passive participle in English.
Gerunds, BTW
� Note that you can use the imperfective
participle as a so-called “verbal noun”:
Throwing stones at glass houses can be
hazardou... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
:
Jan threw up
made up that story
looked the word up
put me down.
� However, they are identical with the set of
English prepositions.
� So it is appealing to think of these as
prepositions without complements.
Adverbs
� Types
– manner (quickly, rarely, never)
– directional/locative (here, home, downtown)
– te... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
146.
The Penn Treebank Tag
Set
tag
description
example
coord. conjunction
and, but, or
cardinal number
one, two, three
determiner
a, the
existential there
foreign word
a propos
preposition/sub-conj
of, in, by, if
adjective
small
adj., comparative
smaller
adj., superlative
smallest
CC
CD
DT
EX... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
what
possessive wh
whose
noun, plural
cars
WRB
Wh-adverb
how, where
The Major Constituents
� These syntactic categories are may
be thought of as “bigger” versions of
lexical categories:
– Noun phrase (NP)
– Verb phrase (VP, S)
– Prepositional phrase (PP)
– Adjective phrase (AP)
– Adverbial phrase (ADVP)
... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
have said
thus far:
NP fi
(PDT) (D) (Q) AP* NP* N
� Note that
– “X*” is just a shorthand for
Xs fi
Xs fi X
Xs fi Xs X
– “X fi
(Y) Z” is an abbreviation for
X fi Z
X fi Y Z
e
An Approximate Grammar,
Redux
� However, most analyses have more
embedded constituent structure.
� So, a somewhat better set of rules might ... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
general, it is tempting to think that the
grammar of noun phrases can be made
simpler, and that at least some of these
constraints can be explained semantically.
– Exactly how to do so is not always clear.
Preceding the Noun: Odds
and Ends
� Personal pronouns
– can be NPs all by themselves.
NPmin
fi ProP
– and ca... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
NP infC
“the guy to go to in a pinch”
Comments
� Which “NP” are we talking about here?
� Consider
“most baguettes from the Cheese Board”,
This should probably be analyzed as
“[most [baguettes from the Cheese Board]]”
� Also
“a package from overseas delivery”
is okay.
� So, this looks like “NPint”.
Followin... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
ositionals, prototype activity nouns….
� But our rules for NPs lose this distinction.
Solutions?
max
fi
fi
fi
N | NPint NPmin | PP NPmin
(Q) AP* NPmin
((PDT) DP ) NPint
� We can differentiate our grammar rules further.
� E.g., instead of
NPmin
NPint
NP
we could have
NPmin/scc
NP
NP
NP
NP
NP
Nscc | NPint NPmin/sc... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
that we can’t write down the grammar;
it’s that we can’t write down a satisfying one.
The Verb Phrase
� Main clauses, e.g.,
“Pat baked Jan cookies”
are typically analyzed as
[S [NP Pat] [VP [V baked] [NP Jan] [NP cookies]]]
as opposed to
[S [NP Pat] [V baked] [NP Jan ] [NP cookies]]
� I.e., the basic general str... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
baked and Jan iced a chocolate layer cake.
which suggests that [Pat baked] and [Jan
iced] are constituents.
� But the other tests don’t bear this out:
*What was done to the cake was Pat baked.
*Pat baked a cake and so did frost.
The Verb Phrase
� Here are some common structures, and phrases
that conform to them: ... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
E.g., “Jan verbed Pat a book.” «
Pat.” (sometimes)
“Jan verbed a book to
Sentence Level
Constructions
� Sentences are generally regarded as a bigger form of VP,
just as we had different forms of NP.
� But, traditionally, we use the separate symbol “S” anyway.
� Here are some common sentence types:
S fi NP VP
Ja... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
Adjuncts
� For most constituents, there is a syntactically
central part, and some less central parts.
� For example, consider:
“the conservative senator”
– This is a noun phrase whose head is the noun phrase
“conservative senator”.
– This noun phrase in turn has the head “senator”.
– We further say that “senator” ... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
were written, for a given lexical category X,
X, X’, and X’’ (but pronounced “x bar” and “x double
bar”).
X-bar Theory
N’’
Det
that
N’
N
A’’
A’
A
nice
book
In such theories:
Complement is daughter of X’, sister of X.
Adjunct is daughter of X’, sister of X’.
Specifier is daughter of X’’, sister of X’
N’... | https://ocw.mit.edu/courses/6-863j-natural-language-and-the-computer-representation-of-knowledge-spring-2003/169150cf1023769d42ccc43fdde0a3cc_grammarofenglish.pdf |
Jan left.
Oddly, Jan sang folks songs.
So we might add
S fi AA* S
AA fi PP | AdvP
� You can also get these at the end, but then they
are best analyzed as part of the VP:
Jan left in the morning/quickly.
Jan sang folks songs oddly.
Jan quickly left the meeting
So one might add
VP fi AA* VP AA*
An Approximate Grammar, ... | 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 |
aq, your
ipaq?
• Anonymous vs Personal handheld
• Telephone example
• Landline telephones are anonymous
• Cell/Mobile phones are personal
• Tradeoffs
• private state
• can be lost or stolen; should be protected
• setup overhead on user
• daily underhead on user: setup once &
• less dependent on connectivity
• public/... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/16af5502ace4e084c0e06f2757875eb5_l2_python_intro.pdf |
Client Code
import sys
import socket
if len(sys.argv) < 3:
print "usage: socketclient <address> <port>"
sys.exit(2)
s = socket.socket(socket.AF_INET, socket.SOCK_STREAM)
s.connect( (sys.argv[1], int(sys.argv[2]) ) )
print "connected. type stuff."
while True:
data = sys.stdin.readline()
if len(data) == 0:
print "closin... | 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 |
even need to
tell you this.
14.2 How to Attack a Problem
We’ll now discuss several 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 interes... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
the general problem!)
– Whatever your solution is, its running time must of course fall within the lower and upper
bounds you find.
• Is there a useful diagram?
1 A comparison-based sorting algorithm is a sorting algorithm which makes no assumptions about the items being
sorted except that there is a well-defined notion ... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
Is it similar to a known problem? Can we modify the known algorithm to suit it to this version?
• Can the problem be solved with convolution? If so, the fast Fourier transform will improve
efficiency.
• Is there any further way to take advantage of the given information?
• Can randomization help? It is often better to fi... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
, here is a
solution:
Algorithm:
Let M be the set of men and let W be the set of women; let s and t be objects representing
a source and a sink. Create a bipartite directed graph∗ G with vertex set M ∪ W ∪ {s, t}.
Draw an edge from s to each man and an edge from each woman to t. Next, for each man
m and each woman w, d... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
O (cid:161)n3(cid:162). Actually,
assuming each person has a relatively small list of acceptable spouses, we ought to be more
granular about this bound: we have
(cid:195)
E = O
n +
(cid:181)size of the ith
person’s list
(cid:182)(cid:33)
,
n
(cid:88)
i=1
and the running time is at most
(cid:195)
(cid:195)
T = O
n ·
n +... | https://ocw.mit.edu/courses/6-046j-design-and-analysis-of-algorithms-spring-2012/16e5c6a1dea0d1c210b3597e2eb4786a_MIT6_046JS12_lec14.pdf |
.3 Recommended Reading
There are many great references and exercise books for algorithmic problem solving and mathemat-
ical problem solving in general. Two particular selections are:
• George Pólya, How to Solve It
• The Princeton Companion to Mathematics, VIII.1: “The Art of Problem Solving.”
Another good source of a... | 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 |
Data
Memory
wdata
I-Fetch
(IF)
Decode, Reg. Fetch
(ID)
Execute
(EX)
Memory
(MA)
s
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 Instruc... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
Store instructions
- Write back
September 28, 2005
Pipelined MIPS Datapath
without jumps
E
IR
M
IR
6.823 L6- 7
Arvind
W
IR
31
RegDst
RegWrite
we
rs1
rs2
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
Septemb... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
3 L6- 10
Arvind
Freeze earlier pipeline stages until the data
becomes available ⇒ interlocks
If data is available somewhere in the datapath
provide a bypass to get it to the right stage
Speculate about the hazard resolution and kill
the instruction later if the speculation is wrong.
September 28, 2005
Feedback to Re... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
IF3
IF3
stalled stages
t5
t6
t7
. . . .
ID2 EX2 MA2 WB2
IF3
ID3 EX3 MA3 WB3
IF4
ID4 EX4 MA4 WB4
IF5
ID5 EX5 MA5 WB5
time
t0
I1
IF
ID
EX
MA
WB
t1
I2
I1
t2
I3
I2
I1
Resource
Usage
September 28, 2005
. . . .
t4
I3
I2
t5
I3
I2
t6
t3
I4
I3
I3
I2
nop nop nop I2
I1
t7
I5
I4
I3
nop nop nop I2
I1
I5
I4
I3
nop nop nop I2
I5
I... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
IR
we
rs1
rs2
rd1
ws
wd rd2
GPRs
Imm
Ext
A
B
ALU
Y
MD1
MD2
we
addr
rdata
Data
Memory
wdata
wdata
Should we always stall if the rs field matches some rd?
not every instruction writes a register ⇒ we
not every instruction reads a register ⇒ re
September 28, 2005
6.823 L6- 15
Arvind
IR
31
Cdest
R
Source & Destination R... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
ALR
...
⇒ on
⇒ off
Cstall
re1 = Case opcode
ALU, ALUi,
LW, SW, BZ,
JR, JALR
J, JAL
re2 = Case opcode
ALU, SW
...
⇒ on
⇒ off
⇒ on
⇒ off
stall = ((rsD =wsE).weE +
(rsD =wsM).weM +
(rsD =wsW).weW) . re1D +
((rtD =wsE).weE +
(rtD =wsM).weM +
(rtD =wsW).weW) . re2D
th e full sto ry !
T his is n ot
September 28, 200... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
28, 2005
20
Five-minute break to stretch your legs
Complications due to Jumps
PCSrc (pc+4 / jabs / rind/ br)
stall
6.823 L6- 21
Arvind
0x4
Add
Add
nop
Jump?
E
IR
I1
M
IR
PC
104
addr
inst
Inst
Memory
IR
I2
Note fetching the
next instruction
before decode is
speculation ⇒ kill
I1
I2
I3
I4
096
100
104
304
ADD
J 200
... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
304: ADD
t5
t6
t7
. . . .
t2
t3
t4
t1
ID1 EX1 MA1 WB1
IF2
ID2 EX2 MA2 WB2
IF3 nop nop nop nop
IF4
ID4 EX4 MA4 WB4
time
t0
I1
IF
ID
EX
MA
WB
t1
I2
I1
t2
I3
I2
I1
Resource
Usage
September 28, 2005
t5
t6
t7
. . . .
t4
t3
I4
I5
nop I4
I2
I1
I5
nop I4
I2
I1
I5
nop I4
I2
I5
nop I4
I5
nop ⇒ pipeline bubble
Pipelining Cond... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
?
zero?
M
IR
I1
E
IR
I2
A
ALU
Y
I1
I2
I3
I4
096
100
104
304
ADD
BEQZ r1 200
ADD
ADD
September 28, 2005
If the branch is taken
- kill the two following instructions
- the instruction at the decode stage
is not valid
⇒ stall signal is not valid
Pipelining Conditional Branches
stall
PCSrc (pc+4/jabs/rind/br)
6.823 L6- ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
5
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 = Case opcodeE
BEQZ.z, BNEZ.!z ⇒ nop
...
⇒
Case opcodeD
J, JAL, JR, JALR ⇒ nop
⇒ IM
...
Give priority
to the older
instruction,
i.e., execute
... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
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
Memory
IR
D
we
rs1
rs2... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
WB2
ID3 EX3 MA3
IF3
ID4 EX4
IF4
ID5
IF5
Each stall or kill introduces a 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
. ... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
34
Arvind
stall = ( ((rsD =wsE).weE + (rsD =wsM).weM + (rsD =wsW).weW).re1D
+((rtD =wsE).weE + (rtD =wsM).weM + (rtD =wsW).weW).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 A... | https://ocw.mit.edu/courses/6-823-computer-system-architecture-fall-2005/16eb29d3b9c087566a1a28aca412bf02_l06_pipeline.pdf |
E
IR
M
IR
W
IR
31
PC
addr
D
inst
IR
Inst
Memory
we
rs1
rs2
rd1
ws
wd rd2
GPRs
Imm
Ext
A
B
ALU
Y
BSrc
MD1
MD2
we
addr
rdata
Data
Memory
wdata
wdata
R
Is there still
a need for the
stall signal ?
September 28, 2005
stall = (rsD=wsE). (opcodeE=LWE).(wsE≠0 ).re1D
+ (rtD=wsE). (opcodeE=LWE).(wsE≠0 ).re2D
Why an Instructio... | 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| = - int p(t) dt + c ln|x| = - t^2 + c
Exponentiate: |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 ,... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
the process for so long that the insulation of the cooler starts to break
down!
Systems and signals analysis:
The system is the cooler.
The output signal = system response is x(t) , the temperature in the cooler.
The input signal is the external temperature T_ext(t) .
Note that the right-hand side is k times the ... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
standard form, but not reduced
standard form.) Here's a *trick*. Multiply both sides by t :
t^2 x' + 2t x = t^2
The left hand side is now the derivative of a product:
(d/dt) (t^2 x) = t^2
We can solve this by integrating:
t^2 x = t^3/3 + c
so x = t/3 + c t^{-2}
[In the first lecture, I posed this (with a dif... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
20 + 2t) e^{t/3}
Integrate:
e^{t/3} x = 60 e^{t/3} + \int 2t e^{t/3} dt
Um. Parts: \int u dv = uv - \int v du
u = 2t , dv = e^{t/3} dt
du = 2dt , v = 3 e^{t/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 so... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/17138787c3f8592b01dbdc312765cc7a_MIT18_03S10_c04.pdf |
does. It is then
called a TRANSIENT. All solutions come to look more and more alike as
time goes on. This is a funnel!
I graphed the solutions 42 + 6t and x , and some others along with T_ext .
If the temperature in the cooler is more than 60 degrees at the start,
then it declines at first, crosses the nullcline x... | 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 |
0
This will be a terse introduction to general relativity. It will be logically complete, and 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 ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
∂x�β ∂xλ
)
We can write the transformation law for gµν in matrix form:
G� = R−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
transfor... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
(x�) = φ(x)
•
Vector
A�
µ(x�) = ∂xα
∂x�µ
Aα(x)
• Operator
∂xα ∂
� ≡ ∂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 correct... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
xσ ∂x�ν∂x�µ
Note that the inhomogeneous part is symmetric in µ
↔
ν. So we can assume 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... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
µ − ∂λgµν
Conversely, this works!
Extra term in Γ�:
∂xσ �
1 λβ ��
�
� ������
µ ∂x�λ gσν + ∂µ
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 ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
�
·
This Rα
βµν automatically transforms as a proper tensor; the “hard” part is to show that the
derivatives on Aα all cancel so we get this form.
�µAβ = ∂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 ... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
∂µ∂σgβν − ∂ν ∂β gµσ − ∂ν ∂σgβµ
= g ασRσβγδ
1 �
2
1
2
with Rαβγδ = 2
�
�
1 ∂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 h... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
γ Aν
=
αβγ �ν Aµ + Rν
(1)
−�αRν
µβγ �αAν
(4)
(3)
µβγ Aν − Rν
(2)
µβγ �αAν
(2) cancels against (4), (3) will go away by the symmetry of Rν
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 vect... | https://ocw.mit.edu/courses/8-952-particle-physics-of-the-early-universe-fall-2004/1723f9a7383f25b91c34216de12ab80e_89521.pdf |
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