text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
to simply think of (G, ◦) as the fixed graph G. So,
for example, Zd is a ‘random’ rooted graph with root at the origin.
Let Gn be a sequence of finite connected graphs of maximum degree at most ∆. Let ◦n denote a
uniform random vertex of Gn. We say Gn converges in the local weak limit if the law of the random
rooted grap... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
(mod n) for exactly one i
and xi = yi for all other i.
Exercise 2.2. Suppose the graphs Gn have maximum degree at most ∆ and converge in the local
weak limit to (G, ◦). Show that
ges in distribution (as integer valued random variables)
to deg(◦). Conclude that E deg(◦n) → E deg(◦) (cid:3).
(cid:2)
deg(◦n
(cid:3)
) conv... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
), (G(cid:48), ◦(cid:48)) on (Ω, Σ, µ) such that (Gn, ◦n) has the law of (G(cid:48)
If Gn converges to (G, ◦) then there is a probability space (Ω, Σ, µ) and G-valued random variables
n, ◦(cid:48)
(G(cid:48)
n), (G, ◦) has the law of (G, ◦),
n) ∼= Nr(G(cid:48), ◦(cid:48))) → 1 as n → ∞. This common
and for every r ≥ 0 ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
:48)
k/2(G
n) (cid:29) Nk/2(G(cid:48), ◦(cid:48)) (cid:3) −→ 0
n, ◦(cid:48)
n) (cid:29) Nk/2(G(cid:48), ◦(cid:48))
as n
→ ∞
.
(cid:12)
(cid:3)(cid:12)
(cid:12)
2.2. Local weak limit of random regular graphs. In this section we will show a classical result
that random d-regular graphs converge to the d-regular tree Td i... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
showed that as n → ∞,
P(cid:2) Gn,d is simple (cid:3) →
1− 2d
e 4
.
Also, conditioned on Gn,d being simple its distribution is a uniform random d-regular graph on
It follows from these observations that any sequence of graph properties An whose
n vertices.
probability under Gn,d tends to 1 as n → ∞ also tends to 1 unde... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
of length at most 2r. It follows from this
where C 2r is the
≤
(cid:3)
(cid:2)
wing lemma
argument that
(cid:2)
E |v ∈ V (Gn,d) : Nr(Gn,d, v) = Nr(Td, ◦)| ≤ 2rdrE C
(cid:2)
, E
if d ≥ 3, and more precisely
(cid:3) ≤ 2r(3d − 3)
shows that E C
follo
∼
2r
(cid:2)
≤2r
(cid:3)
≤2r . The
(cid:3)
C≤2
r
erges to
converges to a... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
{
conclude that
(cid:2)
(cid:3)
E C(cid:96) =
(cid:88)
{v1
,...,v(cid:96)}
(cid:2)
P {v1, . . . , v(cid:96)} forms an (cid:96) − cycle =
(cid:3)
(cid:18) (cid:19)
n ((cid:96)
(cid:96)
−
1)!(d(d
−
2(nd
1))(cid:96)(nd
−
1)!!
− −
2(cid:96)
1)!!
.
Note that (cid:0)n(cid:1) ≤
n(cid:96)/(cid:96)!, and in fact if (cid:96) is ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
to express
log sptr(G
|Gn|
return probabilities of the SRW on (G, ◦). In particular, we shall see that
lim
n→∞
log sptr(Gn)
|Gn|
(cid:104)
= E
log deg(◦) −
(cid:88) pk
G(◦) (cid:105)
.
k
k≥1
The quantity of the r.h.s. is called the tree entropy of (G, ◦). If the limiting graph G is deterministic
and vertex transitive, ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
) = 1 (cid:80)
2
(x,y)(f (x) − f (y))(g(x)
x∼y
− g y)).
(
(2) L is self-adjoint and positive semi-definite: (Lf, g) = (f, Lg) and (Lf, f ) ≥ 0 for all f, g.
(3) Lf = 0 if and only if f is constant on the connected components of f .
(4) The dimension of the eigenspace of L corresponding to eigenvalue 0 equals the number ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
of its roots, which are the eigenvaluesof L, then we can deduce that the coefficient of t is −
(cid:80)
(cid:81)
Exercise 3.2. Let G be a connected finite graph and suppose
{x, y} is an edge of G. Let G \ {x, y}
be the graph obtained from removing the edge {x, y} from G. Let G · {x, y} be the graph obtained
from contracti... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
V (G){|f (x)|}. If G is connected then the largest eigenvalue of P is 1 and it has multiplicity
1 as well. The eigenfuctions for the eigenvalue 1 are constant functions over V (G). Suppose that
∞
∞
∈
−1 ≤ µ1 ≤ µ2 ≤ · · · ≤ µn 1 < µn = 1 are the n eigenvalues of P . If e is the number of edges in G
−
then we may rewrite... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
:80)
k≥1
(4)
log sptr(G)
|G|
=
log 2e
n
(G)
+
(cid:80)
∈ (G) log deg(x)
x V
n
−
(cid:88) 1 (
k
k≥1
(cid:80)
k
x∈V (G) pG(x)) − 1
.
n
Theorem 3.3. Let Gn be a sequence of finite, connected graphs with maximum degree bounded by ∆
and |Gn| → ∞. Suppose that Gn converges in the local weak limit to a random rooted graph (G, ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
converges to 0. Also, deg(◦n) converges in distribution to the degree deg(◦) of (G, ◦) (exercise 2.2).
(cid:3)
→ log x is
The function x
(cid:3)
(cid:2)
converges to E log deg(◦) . Following the discussion is Section 2.1 we conclude that E pk
G(◦) (cid:3) as well. To conclude the proof it suffices to show that
|Gn|−1 con... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
+ 1)1/4
.
(cid:80)
V (G). Let (f, g)π = x V (G) π(x)f (x)g(x) for
Proof. The vector π is a probability
∈
f, g ∈ RV (G). Let P denote the transition matrix of the SRW on G; thus,
G(x) = P k(x, x). Note
pk
that π(x)P (x, y) = 1x y/(2e) = π(y)P (y, x). From this we conclude that (P f, g)π = (f, P g)π. Let
measure on
∼
LO... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
positive and negative values. Apply the inequality
above to the function sgn(f )f 2 and use the inequality |sgn(s)s2 − sgn(t)t2| ≤ |s − t|(|s| + |t|) to
|(|f (x)| + |f ( )|)(cid:3). Straightforward calculations
conclude that ||f ||2 ≤ e (cid:80)
show that
(x,y) K(x, y)(cid:2)|f (x) − f (y)
∞
y
K(x, y)|f (x) − f (y)|2 =... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
)P mf, P m
f
(cid:1)
π
(cid:0)
= 2e2 (P 2m
(cid:1)
− P 2m+1)f, f .
Since ||P g|| ≤ ||g||
∞
∞
, if we sum the inequality abo
ve over 0
≤ m ≤ k we get
(k + 1)||P kf ||4 ≤ 2e2 (cid:88)
∞
k
m=0
||P mf || ≤ 2e2(cid:0)(I − P 2k+1)f, f
∞
(cid:1)
2
π ≤ 2e .
last inequalit
y holds because every eigenvalue of I
−
P m lies in the... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
(x)−1(1 − π(x))(k + 1)−1/4. However, π(x)−1 = 2e/deg(x)
P k(x,x) − 1(cid:12)
π(x)
(cid:12)
(cid:12)
(cid:12)
the degrees in G we have that 2e ≤ ∆n, and this establishes the statement in the lemma.
1. Thus, we conclude that
P k(x,x)
π(x)
− 1
≤ 2e
(cid:12)
(cid:12)
(cid:12) ≤ 2e(k + 1)−1/4. As 2e equals the sum of
Lemma ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
. We begin
to calculate the tree entropy of a graph we have to be
. In order
d and
Zd
Consider the generating function F (t) = k 0 pk (◦)tk. Actually note that pk (
Td
≥
odd because whenever the SRW takes a step along an edge that moves it away from the root it
◦) = 0
k is
Td
if
with the d-regular tree Td.
(cid:80)
mus... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
A rigorous calculation of the tree entropy of Zd requires an excursion into operator theory that is
outside the scope of these notes. We will sketch the argument; for details see Lyons [5] Section 4 or
, λn−1 are the
Lyons [6]. Recall the Matrix-Tree Theorem for finite graphs which states that if λ1, . . .
(cid:80)
n 1
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
xi)). As the Fourier transform
preserves inner products we get that
∈
ˆ
ˆ
(cid:126)
(cid:126)
(cid:90)
h(Zd) =
(cid:16)
log
2d − 2
[0,1]d
d
(cid:88)
i=1
cos(2πxi)
(cid:17)
dx.
4. Open problems
One can consider the space of (isomorphism classes of) doubly rooted graphs (G, x, y) of bounded
degree, analogous to the space... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
of finite
connected graphs is unimodular.
It is not known whether the converse is true: is a unimodular random rooted graph (G, ◦) a local
weak limit of finite connected graphs. This is known to be true for unimodular trees (see Aldous
and Lyons [1]). This is a major open problem in the field.
Here is a problem on tree en... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
14 (2005), pp. 491–522.
[6] R. Lyons, Identities and inequalities for tree entropy, Combin. Probab. Comput. 19 (2010), pp. 303-313.
[7] R. Lyons and Y. Peres, Probability on Trees and Networks, Cambridge University Press, 2014, in preparation.
Current version available at
http://mypage.iu.edu/~rdlyons/.
[8] B. D. McKay... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
Lecture 13
8.324 Relativistic Quantum Field Theory II
Fall 2010
8.324 Relativistic Quantum Field Theory II
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2010
Lecture 13
We continue our analysis of renormalization in quantum electrodynamics from last lecture.
3.1.4: Charge Renormalization
Consider the vertex co... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
2W
δ ¯
η�(x)δη� (y2)
− δ(4)(x − y2)
δ2W
δη¯�(y1)δη� (x)
]
,
J=α=α¯=0
or, equivalently,
1
∂2∂µ ⟨0 T (Aµ(0)ψ�(y1)ψ¯
ξ
|
|
� (y2))
[
0⟩ = eB
δ(4)(x − y1) ⟨0 T (ψ�(x)ψ¯
|
� (y2)) 0⟩ − δ(4)(x − y2) ⟨0 T (ψ�(y1)ψ¯
|
|
� (x))
(4)
]
|
0⟩
.
−→ iqµ
, where qµ ≡ (k2 − k1)µ. We can set x = 0 by applying
(5)
Changing... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
also close to on-shell. Then
= k, k on-shell and
1
S−1(k1) ≈ −
(ik/1 + m − iϵ) + . . .
Z
2
1
S−1(k2) ≈ −
(ik/2 + m − iϵ) + . . .
Z
2
1
(6)
(7)
(8)
Lecture 13
8.324 Relativistic Quantum Field Theory II
Fall 2010
where m here is the physical mass. We then have
or
qδ Γδ (k, k) = −
eB i/q,
Z2
Γδ (k, ... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
≡ DB SB ΓB SB and G(phys) = DSΓ(phys)S. From this, we have that
√
Γδ (k, k) = Z3Z2Γδ
B (k, k)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
and so
√
e = Z3eB .
The dependence of e on Z2 cancels precisely as a result of ΓB ∝
strength renormalization of the photon, has important implications: the ratio is universa... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
δη(x2)
. . . ,
and then setting Jµ = η¯ = η = 0. The resulting expression is most transparently written diagramatically in
momentum space:
pr
.
.
.
p2
p1
kµ
qn
.
.
.
q2
q1
∑
i
= eB
pr
.
..
p2
p1
pr
qn
.
.
.
.
.
.
qi − k − pi + k
.
.
.
.
.
.
p1
q1
2
... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
photon lines or any other neutral particles (if they
exist) can be off-shell, since they they do not transform under gauge transformations.
For Ward identities to be valid, regularization should preserve gauge invariance.
3
MIT OpenCourseWare
http://ocw.mit.edu
8.324 Relativistic Quantum Field Theory II
Fall 201... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
ESD 342 Session 3
Faculty: Magee, Moses, Whitney
February 14, 2006
Professor C. Magee, 2006
Page 1
Point of View & Biases Presentations
• Reverse Alphabetical Order: Please write down or remember
who you follow and come to the front as soon as that person
finishes or as soon as the moderator declares that their 3
m... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/5680ffd3867c58be5719b5af151eb148_lec3_hw2.pdf |
” Project Ideas
• Continue one of last year’s projects-see web site
•
Improve detail of Western Power Grid or some other part of the
electric power grid
• Analyze a software system or a language over time
• Build and analyze a collection of social network data and its time
•
•
dependence
Investigate and model the old... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/5680ffd3867c58be5719b5af151eb148_lec3_hw2.pdf |
Engineering Systems
Engineering Systems
Engineering Systems
Engineering Systems
Doctoral Seminar
Doctoral Seminar
Fall 2011
ESD 83 –– Fall 2011
ESD.83 Fall 2011
ESD 83
ESD.83
Fall 2011
Session 6
Faculty: Chris Magee and Joe Sussman
TA: Rebecca Kaarina Saari
Guest: Professor Stuart Kauffman
Guest: Professor ... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
normal
distributions..and some not likely to be normal
© 2007 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
3 )
%
(
e
g
a
t
n
e
c
r
e
P
6
4
2
0
4
3
2
1
0
0
50
100
150
200
250
0
20
40
60
80
100
120
Heights of Males
Speeds of Cars
Histogram of heights in centimeters of American males... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
th
Histograms
Normal (and nearly so)
Skewed (and heavily skewed)
Reasons for normal vs. skewed?
Power law (skewed)
p k ~
k
Why power laws?
Why power laws?
© 2007 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
6
Power laws are ubiquitous
Low
variability
Ga... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
Development of quantitative model
3. Observe (system)
)
( y
Design a specific version of a known procedure
Develop a new observational procedure
Find, and/or extract and combine data
Find and/or extract and combine data
4. Analyze observations
Use existing models to “reduce” data to model-relevant
D... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
As you think to solve the following puzzle,
observe your thoughts to the best of
observe your thoughts to the best of
your ability
“One morning, exactly at sunrise, a Buddhist monk began to climb a tall
mountain. The narrow path, no more than a foot or two wide, spiraled
around the mountain to a glittering te... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
ignore some facts?
g
y
“Glittering” temple, dried fruit, spiral path?
Did you use other mental operations to explore
the problem?
the problem?
Rotation or “superimposition”, mathematical derivation,
logical rules
How difficult was it to observe your own
”
“
thinking?
Very difficult and ambiguous
© ... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
H
ibl i
b fl
i h
L
thinking?
How might we be more flexible in our
H
ibl i
i ht
b
fl
thinking operations?
Flexibility in Thinking Representations
Flexibility in Thinking Representations
is essential to flexibility in operations
see McKim s book -Thinking Visually and
see McKim’s book Thi... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
Chernoff faces: Eric W. Weisstein. "Chernoff Face." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/ChernoffFace.html
Image by MIT OpenCourseWare.
© 2008 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
22 Examples of Visual Representation &
Application to Comple... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
� Present many numbers in small space
Present many numbers in small space
Make large data sets coherent
Encourage the eye to compare different
p
g
y
pieces of data
Reveal several levels of data detail
Serve a relatively clear purpose
Serve a relatively clear purpose
(description, exploration, tabulatio... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
2008 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
32 Comparative Map abstractions
and scale-Subway Systems
and scale-Subway Systems
scale: 1 km = 7 pixels
QTIar
1 km
are needed to see this
QuickTime™ and
TIFF (Uncompressed) de
1 mi
http://www.fakeisthenewreal.org/subway/index.... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
20
18
16
14
12
10
8
6
4
2
0
1 9 5 2
1 9 5 4
1 9 5 6
1 9 5 8
1 9 6 0
1 9 6 2
1 9 6 4
1 9 6 6
1 9 6 8
1 9 7 0
1 9 7 2
1 9 7 4
1 9 7 6
1 9 5 2
1 9 5 4
1 9 5 6
1 9 5 8
1 9 6 0
1 9 6 2
1 9 6 4
1 9 6 6
1 9 6 8
1 9 7 0
1 9 7 2
1 9 7 4
1 9 7 6
Year
Investment differential (JP-US)
Military differential (US-JP)
Year
Investment d... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
37
Small multiples (Tufte)
Small multiples (Tufte)
Image removed due to copyright restrictions.
© 2008 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
38 Choice of Representation
Choice of Representation
“The form of a representation cannot be divorced
from its ... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
f
b ildi
Form basis for building skill at Systems
F
Representation and Data Visualization
Maps, graphs, matrices, lists, sketches, pictures,
What to think about in choosing representations
What to think about in choosing representations
Understand some basic human capabilities
kill t S
t
Exami... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
6.864: Lecture 3 (September 15, 2005)
Smoothed Estimation, and Language Modeling
Overview
• The language modeling problem
• Smoothed “n-gram” estimates
The Language Modeling Problem
• We have some vocabulary,
say V = {the, a, man, telescope, Beckham, two, . . .}
• We have an (infinite) set of strings, V �
the
a
t... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
P (w3 | START, w1, w2)
×P (w4 | START, w1, w2, w3)
. . .
×P (wn | START, w1, w2, . . . , wn−1)
×P (STOP | START, w1, w2, . . . , wn−1, wn)
For Example
P (the, dog, laughs) = P (the | START)
×P (dog | START, the)
×P (laughs | START, the, dog)
×P (STOP | START, the, dog, laughs)
Deriving a Trigram Probability Mode... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
look at the probability under our model
Or more conveniently, the log probability
n
log P (Si) =
⎧
i=1
n
�
i=1
log P (Si)
n
i=1
P (Si).
⎩
• In fact the usual evaluation measure is perplexity
Perplexity = 2−x where x =
n1
�
W i=1
log P (Si)
and W is the total number of words in the test data.
Some I... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
nor (2) (nor indeed any part of these sentences) has
ever occurred in an English discourse. Hence, in any statistical
model for grammaticalness, these sentences will be ruled out
on identical grounds as equally ‘remote’ from English. Yet (1),
though nonsensical, is grammatical, while (2) is not. . . .
(my emphasis... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
(wi | wi−2, wi−1)
+�2 × PM L(wi | wi−1)
+�3 × PM L(wi)
where �1 + �2 + �3 = 1, and �i � 0 for all i.
• Our estimate correctly defines a distribution:
Pˆ(w | wi−2, wi−1)
w�V
⎨
=
⎨
= �1
w
⎨
= �1 + �2 + �3
w�V [�1 × PM L(w | wi−2 , wi−1) + �2 × PM L (w | wi−1) + �3 × PM L(w)]
PM L (w | wi−2, wi−1) + �2
w PM L (... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
Iterative Method
Initialization: Pick arbitrary/random values for �1, �2, �3.
Step 1: Calculate the following quantities:
c1 =
c2 =
c3 =
�
w1 ,w2 ,w3 �V
�
w1 ,w2 ,w3 �V
�
w1 ,w2 ,w3 �V
Count2 (w1, w2, w3)�1PM L(w3 | w1, w2)
�1PM L(w3 | w1 , w2) + �2PM L(w3 | w2) + �3PM L(w3)
Count2 (w1 , w2, w3 )�2PM L(w3 | w2 ) ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
�’s on the partition:
Pˆ(wi | wi−2, wi−1) = ��(wi−2,wi−1) × PM L(wi | wi−2, wi−1)
+��(wi−2,wi−1)
+��(wi−2,wi−1)
× PM L(wi | wi−1)
× PM L(wi)
1
2
3
where ��(wi−2 ,wi−1 )
2
1
��(wi−2,wi−1)
� 0 for all i.
i
+ ��(wi−2 ,wi−1 ) + ��(wi−2,wi−1)
3
= 1, and
• Our estimate correctly defines a distribution: ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
1) =
�1 × PM L(wi | wi−2, wi−1)
+(1 − �1)[�2 × PM L(wi | wi−1) +(1 − �2) × PM L(wi)]
where 0 � �11, and 0 � �2 � 1.
• Next, define
�1 =
Count(wi−2, wi−1)
� + Count(wi−2, wi−1)
�2 =
Count(wi−1)
� + Count(wi−1)
where � is a parameter chosen to optimize probability of a
development set.
An Alternative Definitio... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
1/48
1/48
(particularly for low count items)
Discounting Methods
• Now define “discounted” counts, for example (a first, simple definition):
Count�(x) = Count(x) − 0.5
• New estimates:
x
the
the, dog
the, woman
the, man
the, park
the, job
the, telescope
the, manual
the, afternoon
the, country
the, street... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
�
PM L(wi)
w�B(wi−1)
PM L(w)
If wi � B(wi−1)
�
�
�
�
�
�
�
�
⎪
PKAT Z (wi | wi−1) =
where
�(wi−1) = 1 −
Count�(wi−1, w)
�
w�A(wi−1)
Count(wi−1)
Katz Back-Off Models (Trigrams)
• For a trigram model, first define two sets
A(wi−2, wi−1) = {w : Count(wi−2, wi−1, w) > 0}
B(wi−2, wi−1) = {w : Count(w... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
were needed within
the Enigma code-breaking effort.
• Define nr = number of elements x for which Count(x) = r.
• Modified count for any x with Count(x) = r and r > 0:
(r + 1)
nr+1
nr
• Leads to the following estimate of “missing mass”:
n1
N
where N is the size of the sample. This is the estimate of the
probabil... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
DT NN
NP ∈ NP PP
NP
PP ∈ P
1.0
0.4
0.4
0.2
0.3
0.7
1.0
Vi ∈ sleeps
Vt ∈ saw
NN ∈ man
NN ∈ woman
NN ∈
DT ∈
IN ∈ with
IN ∈
1.0
1.0
0.7
0.2
telescope 0.1
1.0
the
0.5
0.5
in
• Probability of a tree with rules �i ≥ �i is
i P (�i ≥ �i|�i)
⎩
DERIVATION
S
NP VP
DT N VP
the N VP
the dog VP ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
of example trees.
• If the training data is generated by a PCFG, then as the training data
size goes to infinity, the maximum-likelihood PCFG will converge to the
same distribution as the “true” PCFG.
PCFGs
Booth and Thompson (73) showed that a CFG with rule
probabilities correctly defines a distribution over the s... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
ation:
n = number of words in the sentence
Nk for k = 1 . . . K is k’th non-terminal
N1 = S (the start symbol)
w.l.g.,
• Define a dynamic programming table
λ[i, j, k] = maximum probability of a constituent with non-terminal Nk
spanning words i . . . j inclusive
• Our goal is to calculate maxT �T (S) P (T, S) = λ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
NlNm is in the grammar
prob ≤ P (Nk ≥ NlNm) × λ[i, s, l] × λ[s + 1, j, m]
If prob > max
max ≤ prob
//Store backpointers which imply the best parse
Split(i, j, k) = {s, l, m}
λ[i, j, k] = max
A Dynamic Programming Algorithm for the Sum
• Given a PCFG and a sentence S, how do we find
P (T, S)
�
T �T (S)
• Notatio... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
) = 0 if Nk � NlNm is not in the
grammar)
Initialization:
For i = 1 ... n, k = 1 ... K
λ[i, i, k] = P (Nk ≥ wi|Nk )
Main Loop:
For length = 1 . . . (n − 1), i = 1 . . . (n − 1ength), k = 1 . . . K
j ≤ i + length
sum ≤ 0
For s = i . . . (j − 1),
For Nl, Nm such that Nk ≥ NlNm is in the grammar
prob ≤ P (Nk... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.189 Multicore Programming Primer, January (IAP) 2007
Please use the following citation format:
Michael Perrone, 6.189 Multicore Programming Primer, January (IAP)
2007. (Massachusetts Institute of Technology: MIT OpenCourseWare).
http://ocw.mit.edu (accessed MM DD, YYYY). Li... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
ventional Bulk CMOS
SOI (silicon-on-insulator)
High mobility
Double-Gate
?
Image by MIT OpenCourseWare.
1988 1992 1996 2000 2004 2008 2012
Year
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
4
6.189 IAP 2007 MIT
Power Density – The fundamental p roblem
1000
W/cm2
100
10
1
Nuclea... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
H
e
l
u
d
o
M
14
12
10
8
6
4
2
Steam IRON
5W/cm2
IBM ES9000
Bipolar
Fujitsu VP2000
��IBM 3090S
NTT
Fujitsu M-780
IBM 3090
Start of
Water Cooling
Vacuum
IBM 360
IBM 370
CDC Cyber 205
IBM 4381
IBM 3081
Fujitsu M380
IBM 3033
0
1950
1960
1970
1980
1990
2000
2010
Year of Announcement
Image b... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
© Copyrights by IBM Corp. and by other(s) 2007
8
6.189 IAP 2007 MIT
6.189 IAP 2007
Lecture 2
The Multicore Approach
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
9
6.189 IAP 2007 MIT
Systems and Technology Group
Cell
Courtesy of International Business Machines Corporation. Una... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
low operating voltage with advanced power management
Attacks on the “Memory Wall”
– Streaming DMA architecture
– 3-level Memory Model: Main Storage, Local Storage, Register Files
Attacks on the “Frequency Wall”
– Highly optimized implementation
– Large shared register files and software controlled branching t... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
by IBM Corp. and by other(s) 2007 15
6.189 IAP 2007 MIT
6.189 IAP 2007
Lecture 2
Cell Hardware Components
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007 16
6.189 IAP 2007 MIT
Cell Chip
Courtesy of International Business Machines Corporation. Unauthorized use not permitted.
Michael Perrone... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
�
processor (PowerPC AS 2.0.2)
2-Way hardware multithreaded
L1 : 32KB I ; 32KB D
L2 : 512KB
Coherent load / store
VMX-32
Realtime Controls
–
– Software / hardware managed TLB
– Bandwidth / Resource Reservation
– Mediated Interrupts
Locking L2 Cache & TLB
● Element Interconnect Bus (EIB):
Four 16... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
I/O access
Two queues for DMA commands: Proxy &
SPU
L
o
c
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
L
o
c
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
N
N
96 Byte/Cycle
N
N
NCU
Power Core
(PPE)
L2 Cache
N
N
MFC
AUC
SPU
Local Store
MFC
AUC
SPU
Local Store
Element Interconnect Bus
N
N
C
F
M
U
P
S
C
U
A
e
r
o
S
t
l
a
c
o
L
C
F
M... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
N
L
o
c
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
N
25 GB/sec
XDR DRAM
MIC
96 Byte/Cycle
NCU
Power Core
(PPE)
L2 Cache
Local Store
AUC
SPU
MFC
Local Store
AUC
SPU
MFC
N
N
Element Interconnect Bus
N
N
IOIF0
C
F
M
U
P
S
C
U
A
e
r
o
S
t
l
a
c
o
L
C
F
M
U
P
S
C
U
A
e
r
o
S
t
l
a
c
o
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
16M byte)
–
I/O Device Identifier per page for LPAR
IOST and IOPT Cache – hardware /
software managed
L
o
c
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
N
L
o
c
a
l
S
t
o
r
e
A
U
C
S
P
U
M
F
C
N
25 GB/sec
XDR DRAM
MIC
96 Byte/Cycle
NCU
Power Core
(PPE)
L2 Cache
IIC
IOT
N
N
MFC
AUC
SPU
Local Store
MFC
AUC
SP... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
loading
High frequency design
High bandwidth for memory and IO accesses
Fine tuning for data transfer
PU Data
Staging
via L2
SPU
Data
Staging
Memory
Memor
PU
L2
Memo
Memorry
PU
L2
SPU
SPU
SPU
SPU
SPU
SPU
SPU
SPU
SPU
SPU
SPU
SPU
SPU
SPU
SPU
SPU
l
oads + 2 pr etch
L2 -4 out standingloads +... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
(BE)
6 GFlops (IA32)
12 GFLops (BE)
bioinformatic
smith-waterman
570 Mcups (IA32)
420 Mcups (per SPE)
graphics
transform-light
160 MVPS (G5/VMX)
240 MVPS (per SPE)
security
TRE
AES
TDES
MD-5
SHA-1
communication EEMBC
1.6 fps (G5/VMX)
24 fps (BE)
1.1 Gbps (IA32)
2Gbps (per SPE)
0.12 Gbps (IA32)
0.... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Data streaming / throughput support
Real-time support
● Cell microarchitecture features are exposed to not only its compilers but also its
applications
Performance gains from tuning compilers and applications can be significant
Tools/simulators are provided to assist in performance optimization efforts
Mi... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
..
Finance
Finance
Trade modeling
Trade modeling
Med
Medical Imagin
ical Imaging
g
CT Scan
CT Scan
Ultrasound,
Ultrasound, ……
Industrial
Industrial
Semiconductor / LCD
Semiconductor / LCD
Video Conference
Video Conference
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
30
6.189 ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Cycle accurate SPU simulation (pipeline mode)
Emitter facility for tracing and viewing simulation events
Execution Environment
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
34
6.189 IAP 2007 MIT
SW Stack in Simulation
Execution Environment
Michael Perrone © Copyrights by IBM Corp. and by o... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Management Runtime
Library (32-bit)
std. PPC32
elf interp
32-bit GNU Libs (glibc,etc)
ILP32 Processes
Library (64-bit)
SPE Object Loader
Services
std. PPC64
elf interp
64-bit GNU Libs (glibc)
LP64 Processes
System Call Interface
exec Loader
File System
Framework
Device
Framework
Network
Framework
S... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Extension Libraries
● Standard SPE C library subset
optimized SPE C99 functions including stdlib c lib, math and etc.
subset of POSIX.1 Functions – PPE assisted
Execution Environment
● Audio resample - resampling audio signals
● FFT - 1D and 2D fft functions
● gmath - mathematic functions optimized for gamin... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
provided to demonstrate
system design constructs
● Complex workloads and
demos used to evaluate
and demonstrate system
performance
Terrain Rendering Engine
Geometry Engine
Execution Environment
Physics Simulation
Subdivision Surfaces
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
43
6.189 I... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
pipeline state
● Dynamic analysis (CBE System Simulator)
Generates statistical data on SPE execution
– Cycles, instructions, and CPI
– Single/Dual issue rates
– Stall statistics
– Register usage
– Instruction histogram
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
46
6.189 IAP 2007 MIT
Mi... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Control, PPE Scalar code
● Develop PPE Control, partitioned SPE scalar code
Communication, synchronization, latency handling
● Transform SPE scalar code to SPE SIMD code
● Re-balance the computation / data movement
● Other optimization considerations
PPE SIMD, system bottleneck, load balance
Michael Perrone ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
IB
4X
IB
4X
GbE
BladeCenter Network Interface
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
53
6.189 IAP 2007 MIT
Summary
● Cell ushers in a new era of leading edge processors
optimized for digital media and entertainment
● Desire for realism is driving a convergence between
supercomputin... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
can be used and the
results that may be achieved. Actual environmental costs and performance characteristics will vary depending on individual client configurations and
conditions.
IBM Global Financing offerings are provided through IBM Credit Corporation in the United States and other IBM subsidiaries and divi... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
;
Advanced Micro-Partitioning, eServer, Micro-Partitioning, NUMACenter, On Demand Business logo, OpenPower, POWER, Power Architecture,
Power Everywhere, Power Family, Power PC, PowerPC Architecture, POWER5, POWER5+, POWER6, POWER6+, Redbooks, System p, System
p5, System Storage, VideoCharger, Virtualization Engine. ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
by other(s) 2007
56
6.189 IAP 2007 MIT
(c) Copyright International Business Machines Corporation 2005.
All Rights Reserved. Printed in the United Sates April 2005.
The following are trademarks of International Business Machines Corporation in the United States, or other countries, or both.
IBM
IBM Logo
Power A... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
DD
L
O
R
T
N
O
C
GPR
LS
LS
LS
LS
● RISC like organization
32 bit fixed instructions
Clean design – unified Register file
● User-mode architecture
No translation/protection within SPU
DMA is full Power Arch protect/x-late
● VMX-like SIMD dataflow
Broad set of operations (8 / 16 / 32 Byte)
Graph... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
� DMA is full PowerPC protect/xlate
D P
S F P
● Direct programmer control
DMA/DMA-list
Branch hint
● VMX-like SIMD dataflow
Graphics SP-Float
No saturate arith, some byte
IEEE DP-Float (BlueGene-like)
● Unified register file
●
128 entry x 128 bit
256KB Local Store
Combined I & D
16B/cyc... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
s by IBM Corp. and by other(s) 2007
61
6.189 IAP 2007 MIT
SPE Block Diagram
Floating-Point Unit
Fixed-Point Unit
Permute Unit
Load-Store Unit
Branch Unit
Channel Unit
Result Forwarding and Staging
Register File
Local Store
(256kB)
Single Port SRAM
Instruction Issue Unit / Instruction Line Buffer ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
/SLB misses interrupt PPE
Atomic Cache Facility
●
4 cache lines for atomic updates
2 cache lines for cast out/MMU reload
● Up to 16 outstanding DMA requests in BIU
● Resource / Bandwidth Management Tables
Token Based Bus Access Management
TLB Locking
Legend:
Data Bus
Snoop Bus
Control Bus
X... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Hypervisor)
4K Physical Page Boundary
SPU Master Run Control
SPU ID
SPU ECC Control
SPU ECC Status
SPU ECC Address
SPU 32 bit PU Interrupt Mailbox
MFC Interrupt Mask
MFC Interrupt Status
MFC DMA Privileged Control
MFC Command Error Register
MFC Command Translation Fault Register
MFC SDR (PT Anchor)
MFC AC... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
)
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
66
6.189 IAP 2007 MIT
Memory Flow Controller Commands
DMA Commands
Put - Transfer from Local Store to EA space
Puts - Transfer and Start SPU execution
Putr - Put Result - (Arch. Scarf into L2)
Putl - Put using DMA List in Local Store
Putrl - ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
) 2007
67
6.189 IAP 2007 MIT
SPE Structure
● Scalar processing supported on data-parallel
substrate
All instructions are data parallel and operate on vectors
of elements
Scalar operation defined by instruction use, not opcode
– Vector instruction form used to perform operation
● Preferred slot paradigm
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
Copyrights by IBM Corp. and by other(s) 2007
71
6.189 IAP 2007 MIT
Element Interconnect Bus – Data Topology
● Four 16B data rings connecting 12 bus elements
Two clockwise / Two counter-clockwise
● Physically overlaps all processor elements
● Central arbiter supports up to three concurrent transfers per data... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
3
SPE3
SPE5
SPE5
SPE7
SPE7
IOIF1
IOIF1
Ramp
6
Ramp
Ramp
7
7
Ramp
Ramp
8
8
Ramp
Ramp
9
9
Ramp
Ramp
10
10
Ramp
Ramp
11
11
Controller
Controller
Controller
Controller
Controller
Controller
Controller
Controller
Controller
Controller
Controller
Data
Arbiter
Controller
Controller
Controller
Controll... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
20.110/5.60 Fall 2005
Lecture #5
page
1
Thermochemistry
Much of thermochemistry is based on finding “easy” paths to
calculate changes in enthalpy, i.e. understanding how to work
with thermodynamic cycles.
•
Goal:
To predict H∆ for every reaction, even if it
cannot be carried out in the laboratory
The heat of... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
• ∆
H
f
(° 298.15
K
)
:
The heat of formation is the heat of reaction
to create 1 mole of that compound from its constituent elements in
their most stable forms.
Example (T = 298.15 K)
½ H2 (g,T,1 bar) + ½ Br2 (l,T,1 bar) = HBr (g,T,1 bar)
°
(
H
∆
f HBr T
,
)
=
∆
H
rx
T bar
( ,1
)
=
°
H
HBr
(
T
g,
)
−
(
T
g,
°
H... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
Lecture #5
page
3
∆HI
CH4 (g,T,1 bar) = Cgraphite (s,T,1 bar) + 2H2(g,T,p)
∆HII
2O2 (g,T,1 bar) = 2O2 (g,T,1 bar)
Cgraphite (s,T,1 bar) + ... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
2
∴
In general,
− ∆
H°
4f
,CH
∆
H
rx
=
∑
ν
i
i
∆
°
H
f i
,
(
products
)
−
∆
°
H
f i
,
(
reactants
)
∑
ν
i
i
ν ≡ stoichiometric coefficient
•
H∆
at constant p and for reversible pV process is
= pH q
∆
⇒
If
If
The heat of reaction is the heat flowing into the
reaction from the surroundings
∆
H
rx
∆
H
rx
<
0,
p... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
II)
∆ IIH
Prod. (T2) + Cal. (T2)
=
constant p
Prod. (T1) + Cal. (T1)
________________________
______
________________________
∆
rx TH React. (T) + Cal. (T)
1
1
1
)
(
=
constant p
Prod. (T) + Cal. (T)
1
1
∆
(
TH
rx
1
) =
∆
H
I
+
∆
IH
I
20.110J / 2.772J / 5.601JThermodynamics of Biomolecular SystemsInstructors: ... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
I
)
∆ IU
React. (T1) + Cal. (T1)
adiabatic
=
constant V
Prod. (T2) + Cal. (T2)
II)
∆ IIU
Prod. (T
2) + Cal
. (T )
2
________________________
=
constant V
______
Prod T1
. (
) + Cal. (T
1)
________________________
∆
rx TU React. (T) + Cal. (T) =
1
1
1
(
)
constant V
Prod. (T) + Cal. (T)
1
1
20.110J / 2.772J /... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
pV is from gases.
Ideal gas
Isothermal
∴
H T
1
rx
∆
(
)
=
⇒ ∆
(
)
pV
= ⇒ ∆
T T
∆
R
(
)
nT
=
(
) =
pV T
R
1
1
∆
n
gas
∆
)
(
U T R
1
+
rx
∆
T n
1
gas
∆
(
H T
rx
1
)
= −
T
∫ 2
VT
1
(
C od Cal dT R
Pr
.
.
+
+
∆
T n
1
gas
)
e.g.
4 HCl(g) + O2(g) = 2 H2O(l) + 2 Cl2(g)
T1 = 298.15 K
∆ TU = -195.0 kJ
rx
)
1
(
∆ gasn ... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
calculate ∆Hrx(T2) from the heat capacities of the reactants and
products (assuming no phase change in any component).
Reactants (T2)
∆HI
constant p
R
eactants (T1)
∆
rxH T
2(
→
constant
p
)
Products (T2)
∆HII
constant p
)
Products (T1)
∆
rxH T
1(
→
constant
p
(
(
(
)
)
)
∆
+ ∆
H T H H T H
+ ∆
∆
=
rx
2
rx
I
I... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/56de76a110080853b28006a5ce0b0620_l05.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.