text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
to �. Note the two
following easy facts:
(F1) λ(φ, �) = 0 unless φ
� (mod 2).
|
|
�
(F2) λ(φ, �) is the coefficient of � in the expansion of (D + U )α(Ø) as a
linear combination of partitions.
Because of (F2) it is important to write (D + U )α as a linear combination
of terms U iDj , just as in the proof of Theor... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/23b76f41bf5e26f7221734e3499e73fb_notes2.pdf |
i,j
=
bij (φ)(DU iDj + U i+1Dj ).
�
i,j
In the proof of Theorem 8.3 we saw that DU i = U iD + iU i−1 (see equation
(43)). Hence we get
bij (φ + 1)U iDj =
bij (φ)(U iDj+1 + iU i−1Dj + U i+1Dj ).
(46)
�
i,j
�
i,j
As mentioned after (44), the expansion of (D + U )α+1 in terms of U iDj is
unique. Hence equating... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/23b76f41bf5e26f7221734e3499e73fb_notes2.pdf |
i
bi0(φ)
f ��.
�
��i
Since by Lemma 8.5 we have bi0(φ) = α
i (1
·
even, the proof follows from (F2). �
�
3 5
·
· · ·
(φ
−
i
−
1)) when φ
−
i is
Note. The proof of Theorem 8.6 only required knowing the value of
i0(φ). However, in Lemma 8.5 we computed bij (φ) for all j. We could have
b
carried out the proof... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/23b76f41bf5e26f7221734e3499e73fb_notes2.pdf |
(i) = Yi , the number of partitions of i. (The function p(i) has
been extensively studied, beginning with Euler, though we will not discuss
its fascinating properties here.)
|
|
8.8 Theorem. The eigenvalues of Yj−1,j are given as follows: 0 is an
j, the numbers
eigenvalue of multiplicity p(j)
1); and for 1
p(... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/23b76f41bf5e26f7221734e3499e73fb_notes2.pdf |
enspace of A for the eigenvalue 0, so 0 is an eigenvalue of multiplicity
at least p(j)
p(j
1).
⊕
−
−
Case 2. Let v
⊕
ker(Ds) for some 0
s
j
1. Let
�
�
sU j−1−s(v) + U j−s(v).
−
�
v =
j
�
±
RYj−1,j , with v� =
Note that v�
j−1
Using equation (43), we compute
−
⊕
≥j
±
−
sU j−1−s(v) and v� = U ... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/23b76f41bf5e26f7221734e3499e73fb_notes2.pdf |
1
p(j)
−
p(j
−
1) + 2
(p(s)
p(s
−
1)) = p(j
−
1) + p(j)
−
�
s=0
eigenvalues of A. (The factor 2 above arises from the fact that both +≥j
and
vertices, we have found all its eigenvalues. �
s are eigenvalues.) Since the graph Yj−1,j has p(j
s
1) + p(j)
≥j
−
−
−
−
An elegant combinatorial consequenc... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/23b76f41bf5e26f7221734e3499e73fb_notes2.pdf |
+
·
4m + 5m + 7m . When m = 1 we get 30, the number of edges of the graph Y6,7
[why?].
·
82 | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/23b76f41bf5e26f7221734e3499e73fb_notes2.pdf |
Engineering Risk Benefit Analysis
1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82,
ESD.72, ESD.721
DA 1. The Multistage Decision Model
George E. Apostolakis
Massachusetts Institute of Technology
Spring 2007
DA 1. The Multistage Decision Model
1
Why decision analysis?
A structured way for ranking decision op... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/23bb56e577f168fe62b4e4a37e7dd81c_da1.pdf |
.3
strong,
L3: $100,000, new product and the market is mild,
P[m] = P[L3/N] = 0.5
L4: -$100,000, new product and the market is weak,
P[w] = P[L4/N] = 0.2
DA 1. The Multistage Decision Model
6
Building the decision tree
Decision
Options
N
O
Payoff
depends on
market
Payoffs
L2 $300K
L3 $100K
L4
-$100K
L1 $150K
DA ... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/23bb56e577f168fe62b4e4a37e7dd81c_da1.pdf |
O has the largest EMV, therefore it should
be chosen.
DA 1. The Multistage Decision Model
11
Calculation of the EMV (cont’d)
s, $90
m, $50
EMV[N]=$120
w, -$20
EMV[O]=$150
L2 $300
L3 $100
L4
-$100
$150
L1 $150
Best Decision: O
DA 1. The Multistage Decision Model
12
A New Decision
• The DM considers the possibility of... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/23bb56e577f168fe62b4e4a37e7dd81c_da1.pdf |
*
150,000
300,000
100,000
-100,000
150,000
300,000
100,000
-100,000
150,000
300,000
100,000
-100,000
150,000
300,000
100,000
-100,000
DA 1. The Multistage Decision Model
Figure by MIT OCW.
15
New inputs
• The earnings must be reduced by the survey cost of
$20K: L1 = $130K, L2 = $280K, L3 = $80K,
L4 = -$120K
• The pr... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/23bb56e577f168fe62b4e4a37e7dd81c_da1.pdf |
.143
1.000
P(L4/w) = 0.583
1.000
DA 1. The Multistage Decision Model
19
The updated decision tree
s
D
.34
S
C
.42
m
D
.24
D
w
D
S
C
1
D
O
N
O
N
O
N
O
N
C
C
C
C
C
C
C
C
1
.706
.294
0
1
.143
.714
.143
1
0
.417
.583
1
.3
.5
.2
L1
L2
L3
L4
L1
L2
L3
L4
L1
L2
L3
L4
L1
L2
L3
L4
130,000
280,000
80,000
-120,000
130,000
280,000... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/23bb56e577f168fe62b4e4a37e7dd81c_da1.pdf |
N
C
C
150,000
O
120,000
N
C
1
.706
.294
0
1
.143
.714
.143
1
0
.417
.583
1
.3
.5
.2
L1
L2
L3
L4
L1
L2
L3
L4
L1
L2
L3
L4
L1
L2
L3
L4
130,000
280,000
80,000
-120,000
130,000
280,000
80,000
-120,000
130,000
280,000
80,000
-120,000
150,000
300,000
100,000
-100,000
DA 1. The Multistage Decision Model
22
Figure by MIT OCW.
... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/23bb56e577f168fe62b4e4a37e7dd81c_da1.pdf |
.
2. The DM must select one of the initial acts Aj.
3. The Aj may be viewed as “learning experiments”
providing, at specified costs, opportunities for obtaining
partial or complete information about present
uncertainties.
4.
Following the probabilistic results a1, a2, …, of the initial
decision, the DM must select t... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/23bb56e577f168fe62b4e4a37e7dd81c_da1.pdf |
§ 13. Hypothesis testing asymptotics II
Setup:
H0 ∶ X n ∼ P
∶
test PZ
X n
∣
n
X
X
n
→
n
H1 ∶ X ∼ QX
}
{
0, 1
n
(i.i.d.
)
sp
ecification:
n
)
1 − α = π(
∣0
1
≤ −nE0
2
n
( )
β = π0
∣1
≤ 2−nE1
Bounds:
• achievability (Neyman Pearson)
α = 1 − π1∣0 = PX n[Fn > τ ],
β = π0∣1 = QX n[Fn > τ
]
• converse (strong)
where
∀(α, β) a... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
be finite and also T
≠ const since P ≠ Q):
ψP (λ) = log EP [eλT ]
= log ∑ P (x 1−λQ x λ
)
( ) =
x
θλ − ψP
(λ)
∗ ( )
ψP θ
= sup
λ
R
∈
138
log
∫
dP 1
)
(
−λ
(
dQ
λ
)
P
≪
≪
Q and Q
( )
Note that since ψP (0) = ψP (1) =
1. Furthermore,
λ
[
(
assuming
ψP λ continuous everywhere on 0, 1 (
)
(
arguments).
on 0, 1 it follows f... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
P θ . Note that
) =
is increasing, θ E1 θ is decreasing.
( ) =
0
)
(
ψP 1
ψ
P
(
↦
Remark 13.1 (R´enyi divergence). R´enyi defined a family of divergence indexed by λ ≠ 1
Dλ(P ∥Q) ≜
1
λ − 1
log EQ [(
λ
)
] ≥ 0.
dP
dQ
( ∥ ) = −
which generalizes Kullback-Leibler
( − )
λ 1 Dλ Q P
pro
( )
and 1, and the slope at endpoints i... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
13.1. The idea is to apply the large deviation theory to iid sum n
∑
k 1 Tk. Specifi-
=
cally, let’s rewrite the bounds in terms of T :
• Achievability (Neyman Pearson)
let τ = −
nθ,
(n)
π
0
1
∣
=
P [
n
∑
k
1
=
T
≥
k nθ
]
π
n
( )
∣1
0
= Q [
n
∑
1
k
=
Tk
< nθ]
• Converse (strong)
let γ = 2−nθ,
π1∣0 + 2−nθπ0∣1 ≥ P [∑
1
k=... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
θ
( ) −
(
)
] =
( + )
ψP λ 1
)
(
thus E0, E1
verse:
Con
(
( ))
)
(
E0 θ , E1 θ
bound we have:
in (13.1) is achievabl
e.
We want to show that any achievable (
0, E1 pair must be below the curve
in the above Neyman-Pearson test with parameter θ. Apply the strong converse
E
)
2−nE0 + 2−nθ2−nE1 ≥ 2−nψ∗
1 + θ) ≤
⇒ min(E0, E... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
≤
(
∥
)
(13.3)
Proof. The first part is verified trivially. Indeed,
we have
if
we
fix
λ and let
θ
(λ) ≜ E
P
λ
[T
], then from (11.13)
D(Pλ∥P
) = ψP
∗ (θ) ,
whereas
Also
from
)
( ∥
D Q P .
D(Pλ∥Q) = EPλ
[
log
dPλ
dQ
] = EPλ
[log
dPλ
dP
dP
dQ
] =
D Pλ P
∥ )
(
−
EPλ T
[ ] =
∗ θ
( ) −
ψP
θ .
(11.12) we know that as λ ranges i... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
≠
∗
′
D(Q∗∥Q) =
min
D Q′ P E0
(
∥ )≤
∶
Q′
( ∥ )
D Q Q D P Q
∥ ) ≤
(
′
On the other hand, since E0 ≤ D(Q∥P ) we
also
have
D(Q∗∥P
) ≤ D(Q∥P ) .
Therefore,
EQ∗[T ] = EQ∗[
log
∗
dQ
dP
dQ
dQ∗
] = D(Q∗∥P ) − D(Q∗∥Q) ∈ [−D(P ∥Q), D(Q∥P )] .
(13.6)
Next, we have from Corollary 12.1 that there exists a unique Pλ with the follow... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
).
[T
≪
141
(
∥
)
(
interpretation of (13.3) is as follows: As λ increases from 0 to 1, or equivalently,
Note: Geometric
−
θ increases from D P Q to D Q P , the optimal distribution traverses down the curve. This
curve is in essense a geodesic connecting P to Q and exponents E0,E1 measure distances to P and
Q. It may ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
a martingale. If collection Mn is uniformly
n-measurable and E M
n
F
}
{
[
• For more details, see [C¸ 11, Chapter V].
[Mτ
] = [
]
E M0 .
E
142
Different realizations of Xk are informative to different levels, the total “information” we receive
instead of fixing the sample size n, we can make n a stopping
follows a rando... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
13.3. Assume bounded LLR:2
∣
log
(x)
P
)
(
Q x
∣
≤
∀
c0, x
where c0 is some positive constant. If the error
pr
ob
abilities
satisfy:
π1∣0 ≤ 2−l0E0,
π0∣
1
l1E1
≤
2−
for large l0, l1, then the following inequality for the exponents
holds
2This assumption is satisfied for discrete distributions on finite spaces.
E0E1 ≤ D(P ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
τ ]D(Q∥P ) .
Mn = Sn − nD(P
∥Q
)
is
clearly a martingale w.r.t.
Fn. Consequently,
is also a martingale. Thus
or, equivalently,
˜Mn ≜ Mmin
(τ,n)
E ˜Mn
[
] = [
E ˜M0
] =
0 ,
E[Smin τ,n
)
(
] = E[min(τ, n)]D(P ∥Q) .
(13.13)
This holds for every n
∣
Smin(n,τ )
can
nc and thus
us, we
(13.13) and interchange expectation and ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
�Fn
(13.15)
=
exp
{Sn} by the very
We now proceed to the proof. For achievability we apply (13.14) to infer
π1∣0 = P[Sτ ≤ −A]
= EQ[exp{Sτ }1{S
≤
e A
−
τ ≤ −A
}]
Next, we denot τ0 = inf{n ∶ Sn ≥ B and observe that τ
from (13.12):
}
≤
τ0, whereas expectation of τ0 we estimate
[τ ] ≤ EP [τ0] = EP [Sτ0] ≤ B + c0 ,
EP
where... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
�P ), as l0, l1 → ∞
145
MIT OpenCourseWare
https://ocw.mit.edu
6.441 Information Theory
Spring 2016
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.01 Single Variable Calculus
Fall 2006
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Lecture 3
18.01 Fall 2006
Lecture 3 (presented by Kobi Kremnizer):
Derivatives of Products, Quotients, Sine, and
Cosine
Derivative ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/23c2c1b1ab31c9f10745b18e7b0bf131_lec3.pdf |
0
So, we know the value of -sin x and of -cos x at x = 0. Let us find these for arbitrary x.
d
dx
d
dx
d
-sin x = lirn
dx
AX-0
sin(x + Ax) - sin(x)
ax
18.01 Fall 2006
Lecture 3
Recall:
-
=
lirn
Ax-0
lim [
AX-o
sin x cos Ax + cos x sin Ax - sin(%)
Ax
1
sin x(cos Ax - 1) cos x sin Ax
Ax
Ax
+
= AX-o... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/23c2c1b1ab31c9f10745b18e7b0bf131_lec3.pdf |
lirn u(x + Ax) = u(x)
Ax-0
(true because u is continuous)
This proof of the product rule assumes that u and v have derivatives, which implies both functions
are continuous.
Lecture 3
18.01 Fall 2006
Figure 1: A graphical "proof" of the product rule
An intuitive justification:
We want to find the difference in... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/23c2c1b1ab31c9f10745b18e7b0bf131_lec3.pdf |
the derivative of ulv, we use the notations Au and Av above. Thus,
u(x + Ax)
U(X+ Ax)
Hence,
Therefore.
- - - - -
-
-
u(x)
V ( X )
u + A u
v + A v
u
v
-- (" +
- u(v +
(common denominator)
(v + Av)v
( A u ) ~- u(Av)
(v + Av)v
--
(cancel uv - uv)
u
u'v - uv'
u 2 | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/23c2c1b1ab31c9f10745b18e7b0bf131_lec3.pdf |
Lecture 15
April 8th, 2004
The Continuity Method
Let T : B1 → B2 be linear between two Banach spaces. T is bounded if
||T || = sup
x∈B1
||T x||B2
||x||B1
< ∞ ⇔ ||T x||B2 ≤ c · ||x||B1 for some c > 0.
Continuity Method Theorem.
Let B be a Banach space , V a normed space, L0, L1 : B → V
bounded linear operators. Assume ∃... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
Ls
−1|| ≤ c.
As an application we see that
||T x1 − T x2||B ≤ (t − s)c · (||L0|| + ||L1||)||x1 − x2||,
1
and for t close enough to s (precisely for t ∈ [s −
1
c(||L0||+||L1||) , s +
1
c(||L0||+||L1||) ]) we therefore have
a contraction mapping! Therefore T has a fixed point by the previous theorem which essentially
mea... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
c(γ, Λ, Ω, n)(cid:0)||u||C0(Ω) + ||f ||Cα(Ω)(cid:1).
C. Under the assumptions of B, when c(x) ≤ 0
2
||u||C2,α ( ¯Ω) ≤ c(sup
∂Ω
|u| + sup
Ω
|f |).
D. The above applies to the Dirichlet problem
Lu = f on ¯Ω,
u = ϕ on ∂Ω
and in particular when ϕ = 0 we get very simply
||u||C2,α ( ¯Ω) ≤ c · ||Lu||Cα ( ¯Ω).
T heorem.
Let Ω... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
above
||u||C2,α ( ¯Ω) = ||u||C2,α (B(Ω)) ≤ c · ||Ltu||Cα( ¯Ω),
with c independent of t (depends just on L). Note Cα( ¯Ω) is a Banach space and in particular a
vector space. The Continuity Method thus applies.
Strangely enough, we are now back to solving Dirichlet’s problem for ∆ in domains.
Our methods so far were good... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
� on B which we have! Therefore this Theorem is
a slight generalization.
Proof. As was just outlined the crucial problem lies in the (possible) absence of regularity of ϕ
on part of the boundary. So we approximate ϕ by a sequence {ϕk} ⊂ C3( ¯B) such that both
||ϕk − ϕ||C0( ¯B) −→ 0 and ||ϕk − ϕ||C2,α( ¯B) −→ 0. Solve L... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
0 such that B(x0, ρ) ∩ ∂B ⊆ T
we have the usual boundary Schauder estimates (for smooth enough functions) which give us
||ui − uj||C2,α(B(x0,ρ)∩ ¯B) ≤ c · (cid:0)||ui − uj||C0(B) + ||ϕi − ϕj||C2,α(B(x0,ρ)∩ ¯B)(cid:1). This means that in fact
2,α(B(x0,ρ)∩ ¯B)
ui −−−−−−−−−−−→
C
u and in particular u ∈ C2,α at x0. ∀x0 ∈ T... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
MIT 3.071
Amorphous Materials
1: Fundamentals of the Amorphous State
Juejun (JJ) Hu
1
What is glass (amorphous solid)?
Mechanical properties
Brittle, fragile, stiff
Optical properties
Transparent, translucent
“A room-temperature
malleable glass”
(As60Se40)
Video courtesy of
IRradiance Glass Inc. ... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/244f9ff9b8938d8966cb6ebbf79acc3e_MIT3_071F15_Lecture1.pdf |
O3 crystal
A2O3 glass
Short-range order
is preserved (AO3
triangles)
Long-range order
is disrupted by
changing bond
angle (mainly) and
bond length
Structure lacks
symmetry and is
usually isotropic
Zachariasen's Random Network Theory (1932)
6
Glass consists of a continuous atomic network
Figure of ... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/244f9ff9b8938d8966cb6ebbf79acc3e_MIT3_071F15_Lecture1.pdf |
2012).
10
Article: Anne Ju “Shattering records: Thinnest glass in
Guinness book.” Cornell Chronicle. September 12,
2013.
11
Glass formation from liquid
V, H
Liquid
When the system is kept
in thermal equilibrium:
First-order liquid-solid
phase transition
Discontinuity of
extensive thermo-
dynamic p... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/244f9ff9b8938d8966cb6ebbf79acc3e_MIT3_071F15_Lecture1.pdf |
range order but
lack long range order
16
What is glass (amorphous solid)?
A metastable solid with no long-range atomic order
n
o
i
t
u
b
i
r
t
s
d
i
d
e
z
i
l
a
m
r
o
N
Si-O-Si bond-bending
constraint is relaxed at
the forming temperature
of silica glass
Si-O-Si bond angle
distribution in silica glass
mea... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/244f9ff9b8938d8966cb6ebbf79acc3e_MIT3_071F15_Lecture1.pdf |
��
20
()()1hrgr2()4()Jrrgr22()4()14()Grrgrrhr Summary
V
Supercooled
liquid
Liquid
Glass
Crystal
Tm
T
The amorphous state is
metastable
Amorphous structures
possess short-range
order and lack long-range
order
Amorphous materials can
be obtained from liquid by
melt quench
Melt... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/244f9ff9b8938d8966cb6ebbf79acc3e_MIT3_071F15_Lecture1.pdf |
Lecture 9
We have a manifold CP n . Take
a homogenous polynomial. Then
P (z0, . . . , P zn) =
α
cαz
=m
α
�
|
|
1. P (λz) = λmP (z), so if P (z) = 0 then P (λz) = 0
2. Euler’s identity holds
zi
∂P
∂zi
n
i=0
�
= mP
Lemma. The following are equivalent
Cn+1
1. For all
, dPz = 0
z
0
2. For all z
Cn+1
0
, P... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/248fc514e61beb10ea33642006f43e24_18117_lec09.pdf |
to show that X0 = γ−
−
P (1, z1, . . . , zn). X0 is the set of all points such that p = 0. It is enough to show that p(z) = 0 implies
dpz = 0 (showed last time that this would then define a submanifold)
1(X) is a complex n
Suppose dp(z) = p(z) = 0. Then
p(1, z1, . . . , zn) = 0 =
∂P
∂zi
(1, z1, . . . , zn) = 0
i = ... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/248fc514e61beb10ea33642006f43e24_18117_lec09.pdf |
that
1. Ui is biholomorphic to a connected open subset of Cn
2. p
U1
∈
3. q
∈
4. Ui
Un
Ui+1 =
.
∅
∩
Theorem. If X is a connected complex manifold and f
a local maximum then f is constant.
(X) then if for some p
∈
∈ O
X, f : X R takes
→
|
|
Corollary. If X is compact and connected
(X) = C.
O
This implies ... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/248fc514e61beb10ea33642006f43e24_18117_lec09.pdf |
⊗
⊗
C. Also, we can introduce a complex conjugation operator
C
→
Tp
⊗
C
c
v
⊗
7→
v
⊗
c¯
⊗
T 1,0 if Jpv = +√
where v
p
∈
T 1,0 iff ¯v
If v
p
∈
We can also take Tp∗
1l.
⊗
∈
p
−
J ∗l =
p
√
−
Check that l
J ∗l(v) = l(Jpv) = √
p
∈
−
Tp
C = T 1,0
p
T 0,1
p
⊗
T 0,1 if Jpv =
p
⊕
√
1v and v
−
−
T 0,1 and... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/248fc514e61beb10ea33642006f43e24_18117_lec09.pdf |
X and p
U . Then if f
(U then dfp
(Tp∗)1,0 .
∈
∈ O
∈
Corollary. (U, z1, . . . , zn) a coordinate patch then (dz1)p, . . . , (dzn)p is a basis of (Tp∗)1,0 and (dz¯1)p, . . . , (dz¯n)p
is a basis of (Tp∗)0,1 .
From the splitting above we get a splitting of the exterior product
Λk(Tp ∗
⊗
C)
=
Λl,m(Tp ∗
C)
⊗
l+m=... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/248fc514e61beb10ea33642006f43e24_18117_lec09.pdf |
Random Networks and Percolation
• Percolation, cascades, pandemics
• Properties, Metrics of Random Networks
• Basic Theory of Random Networks and Cascades
• Watts Cascades
• Analytic Model of Watts Cascades
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and cascades
Types of Percolation Models
•
•... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
7-2010
3/46
and cascades
Diffusion of Pandemic Diseases
• Model assumes disease starts from a point and travels in
two modes: local commuting and international air travel
• Disease follows SIR or SIS model, with parameters that
need to be estimated for each outbreak
• Procedure is to run the model with different... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
46
and cascades
SIR Model
Susceptible (S)
Recovered (R)
Infected (I)
Transmission
Airborne
Values of R0 of well-known infectious diseases[1]
Disease
Measles
Pertussis Airborne droplet
Diphtheria Saliva
Smallpox Social contact
Polio
Rubella
Mumps
HIV/AIDS Sexual contact
SARS
Influenza
(1918 pandemic... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
6
1.5
2
0.4
IP = 0.2
2.5
2/16/2011
Random networks
and cascades
© Daniel E Whitney 1997-2010
8/46
Adoption of Innovations - Rogers
Innovativeness and Adopter Categories
Adopter categorization on the basis of innovativeness
Innovators 2.5%
Early
adopters
13.5%
Early
majority
34%
Late
majority
34%
Laggards 16%
x -... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
Image by MIT OpenCourseWare.
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and cascades
What’s Interesting About Random
Networks
• They represent one extreme of networks
– Another is regular structures like grids or arrays
– Another is “designed” networks with rational but not
necessa... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
loops
• To get a big connected cluster, we need z >1 for the graph as
a whole and z > 2 for a connected cluster because z of a tree
is ~2
– For a tree, m = n −1, z = 2m /n,∴ z ≈ 2
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and cascades
Degree Distribution of ER Random Network
•For large n the... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
number = 0.5771
2/16/2011 Random networks © Daniel E
Whitney 1997-2010
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and cascades
Percolation and Cascades
• Both terms are ~synonymous with emergence of a
“giant cluster”
– In an infinitely large random network, the size
of a connected cluster is a non-zero % of the
total number of nodes
– In a finit... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
• Derived by Newman and others using generating
functions
• Recreates and extends the Molloy-Reed criterion
• Extended by Watts
• Assumes graphs (or vulnerable subgraphs) are
trees and networks are of infinite size
All nodes vulnerable
∞
∑k(k −1) pk = z Molloy-Reed criterion
Vuln with pr = b
Vuln is fct of k
k... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
k ≤ 2K* : flip if ≥ 2 neighbors flip
Seed
First Step
K* = 4
Second Step
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and cascades
Watts’ Cascade Diagram
Theoretical boundary: infinite nodes
Simulation boundary: 10000 nodes
Network is too
densely
connected
Network is
disconnected
K* = ⎣1/... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
and cascades
Cascades in Finite E-R Networks Can
Happen in the No Global Cascades Region
D. E. Whitney, ŅDynamic theory of cascades on finite clustered random networks with a
threshold ruleÓPhysical Review E. E 82, 066110 (2010)
2/16/2011 Random networks © Daniel E Whitney 1997-2010
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and cascades
Simulati... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
k
pk = ∑ pS i,S pnS k − i,n − S)
)
(
(
i= 0
pk
= ∑
k
⎛S⎞
⎜ ⎟
0 ⎝ i ⎠
p i 1 − p S−i ⎛N −1 − S⎞
p k−i 1 − p N −1−S−(
(
⎟
⎜
⎝ k − i ⎠
)
)
(
i=
k−i
)
Any unflipped node:
n-S of them
Any unflipped node:
n-S-F1 of them
First flipped set
= F1
Seed = S
Network = n
2/16/2011 Random networks © Daniel E Whitney 1... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
-2010
33/46
and cascades
Theory and Simulations: Cascade in “Global
Cascades” Region
n = 4500, z = 11.5, S = 1, K* = 9
Theory and Simulation (avg of 10 runs)
4000
3500
3000
2500
2000
1500
1000
500
0
T = theory
S = simulation
Flip by Step (T)
Flip Total (T)
Flip by Step (S)
Flip Total (S)
1
3
5 ... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
of Seed, S
S Transition: Theory and Simulations
n = 4500, z = 11.5, K* = 4
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Simulations
Theory
Simulations
Theory
2/16/2011
0
50
70
Random networks
and cascades
150
130
110
Size of Seed, S
90
© Daniel E Whitney 1997-2010
0
36/46
150
200
250
300
Siz... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
flip =
N OS
n
If each node in Fj flips one node, the cascade is self - sustaining.
So 1 =
NOSzFj
n
or NOS = n /zFj
or Fj * NOS = Fj * n /zFj
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and cascades
Theory and Simulations: Evolution of max
One Short Failures (avg of 20 runs)
If one short ex... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
7 May
2001
[Watts] “A Simple Model of Global Cascades on Random Networks” PNAS
April 30, 2002, pp 5766-5771
[Wikipedia] http://en.wikipedia.org/wiki/Percolation_theory and
http://en.wikipedia.org/wiki/Percolation_threshold
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and cascades
More Referenc... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
V. Colizza, A. Barrat, M. Barthelemy,
A. Vespignani�BMC Medicine 5, 34 (2007)
• Seasonal transmission potential and activity peaks of the new influenza
A(H1N1): a Monte Carlo likelihood analysis based on human
mobility�D. Balcan, H. Hu, B. Goncalves, P. Bajardi, C. Poletto, J. J.
Ramasco, D. Paolotti, N. Perra, M.... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
threshold probabilities at which different subgraphs appear in a random graph. For
pn 3 / 2 → 0 the graph consists of isolated nodes. For p ~ n−3 / 2 trees of order 3 appear,
while for p ~ n−4 / 3 trees of order 4 appear, but not many. At p ~ n −1 trees of all orders
are present and at the same time cycles of all or... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
In a random network with n
nodes each node has n neighbors.
It must be linked to at least one
for there to be a chance of a giant
cluster. So pc = 1/n. But z = pn
so this is the same as zc = 1.
See Albert and Barabasi “Stat Mech of Complex
Networks” for a detailed derivation
There is no proof or formula for pc ... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
1E+12
1E+10
1E+08
1E+06
10000
100
1
loops z = 3
loops z = 6
0
10
20
30
40
50
60
n - number of nodes
Diestel, R., Graph Theory, 3rd edition online at <http://www.math.uni-
hamburg.de/home/diestel/books/graph.theory/index.html> page 298
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and c... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
) Simple percolation has b = 1 for all nodes
– C) Watts rumors cascade model has
b = 1 for k ≤ K *
b = 0 for k > K *
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and cascades
Derivation of Cascade Conditions (Newman)
Pick an edge leading from a node and follow it to
a neighboring node.
What is t... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
⎥
⎣ z1 ⎦
This diverges when ⎢
= 1
⎤
⎡z2 ⎥
⎣ z1 ⎦
Generalizeable
Cascade condition
Molloy-Reed criterion
∞
∑k(k −1) pk
k= 0
or
z
= 1
∞
∑ k(k −1) pk = z
k= 0
2/16/2011 Random networks © Daniel E Whitney 1997-2010
and cascades
(from prev slide)
For E - R
k 2
= k 2 = z
2
So, for E - R this is the same a... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
2/16/2011 Random networks © Daniel E Whitney 1997-2010
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and cascades
MIT OpenCourseWare
http://ocw.mit.edu
ESD.342 Network Representations of Complex Engineering Systems
Spring 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
3.23 Fall 2007 – Lecture 15
ANHARMONICITY
Image from Wikimedia Commons, http://commons.wikimedia.org/wiki/... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
⎜
⎝
N
D
n
i
⎞
⎟
⎠
V
∴ =
bi
k T
b
q
ln
⎛
⎜
⎝
d
N N
a
2
n
i
⎞
⎟
⎠
Qualitative Effect of Bias
• Forward bias (+ to p, - to n) decreases depletion region, increases
diffusion current exponentially
• Reverse bias (- to p, + to n) increases depletion region, and no current
flows ideally
Forward Bias
Reverse Bias
qVbi‐q|Va... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
: Fig. 29.5 in Ashcroft, Neil W., and Mermin, N. David. Solid State Physics.
Belmont, CA: Brooks/Cole, 197.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Semiconductor solar cells
Image removed due to copyright restrictions. Please see http://commons.wikimedia.org/wiki... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
Image removed due to copyright restrictions. Please see Fig. 1.7.2 in Datta, Supriyo. Electronic Transport in Mesoscopic Systems.
New York, NY: Cambridge University Press, 1995.
• If we reduce the length conductance grows
indefinitely!
• Experiment shows limiting value Gc.
• This resistance comes from contacts
Elec... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
=
e
L
I
=
I
+
−
−
I
=
∑
k
e
2
h
∞+
[
∞−
∫
+
EfEf
-)
(
(
)]
dE
=
-
μ
2
)
=
μ
1
−
e
2
e
2
h
V
m2
Figure by MIT OpenCourseWare.
f
+
(E)dE
conductance quantum
G
=
Id
Vd
=
22
e
h
Nch
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Conductance from transmission
• Predominant... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
Nicola Bonini. Used with permission.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Courtesy of Nicola Bonini. Used with permission.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Courtesy of Nicola Bonini. Used with per... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Phonons 4
Courtesy of Nicolas Mounet. Used with permission.
2500
2000
1500
1000
500
)
1
-
g
k
.
1
-
K
.
J
(
P
C
0
0
500
1000
1500
2000
2500
Temperature (K)
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic ... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
q, ω
q', ω'
q - q' ±G, ω−ω'
q + q' ±G, ω+ω'
q, ω
q', ω '
phonon decay
phonon absorption
Anharmonic decay channels of E2g mode in graphene
Image removed due to copyright restrictions.
Please see Fig. 4b in Bonini, Nicola, et al. "Phonon Anharmonicities in Graphite and Graphene." arXiv:0708.4259v2 [cond-mat.mtrl-sci], 2... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
66..117722
PPeerrffoorrmmaannccee
EEnnggiinneeeerriinngg
ooff SSooffttwwaarree
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PPaarraalllleell SSttoorraaggee
AAllllooccaattiioonn
JJuulliiaann SShhuunn
© 2008-2018 by the MIT 6.172 Lecturers
SPEED
LIMIT∞PER ORDER OF 6.172
1
SPEED
LIM... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
/A)
);
The Linux kernel finds a contiguous, unused region in
the address space of the application large enough to
hold size bytes, modifies the page table, and creates
the necessary virtual-memory management structures
within the OS to make the user’s accesses to this area
“legal” so that accesses won’t result in ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
page does not reside in
physical memory, a page fault occurs.
© 2008-2018 by the MIT 6.172 Lecturers
7
physical memory
frame 0
offset
frame 1
frame 2
frame 3
⋮
Address Translation
virtualaddress
virtual page #
offset
search
frame #
physical memory
frame 0
offset
frame 1
frame 2
frame 3
⋮
frame #
o... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
invocation tree
views of stack
© 2008-2018 by the MIT 6.172 Lecturers
12
Heap-Based Cactus Stack
A heap-based cactus stack allocates frames off the heap.
A
B
C
E
D
13
© 2008-2018 by the MIT 6.172 Lecturers
Space Bound
Theorem. Let S1 be the stack space required by a
serial execution of a Cilk program. The stack... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
/2);
cilk_spawn mm_dac(X(C,1,1), n_C, X(A,1,0), n_A, X(B,0,1), n_B, n/2);
cilk_spawn mm_dac(X(D,0,0), n_D, X(A,0,1), n_A, X(B,1,0), n_B, n/2);
cilk_spawn mm_dac(X(D,0,1), n_D, X(A,0,1), n_A, X(B,1,1), n_B, n/2);
cilk_spawn mm_dac(X(D,1,0), n_D, X(A,1,1), n_A, X(B,1,0), n_B, n/2);
mm_dac(X(D,1,1), n_D, X(A,1,1), n_A, X(... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
(P1/3n2).
© 2008-2018 by the MIT 6.172 Lecturers
17
Interoperability
Problem: With heap-based linkage, parallel functions
fail to interoperate with legacy and third-party serial
binaries. Our implementation of Cilk uses a less
space-efficient strategy that preserves interoperabi... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
Remark. Modern 64-bit processors provide about
Remark.
248 bytes of virtual address space. A big server
might have 240 bytes of physical memory.
.
© 2008-2018 by the MIT 6.172 Lecturers
21
Fragmentation Glossary
Fragmentation Glossary
∙ Space overhe... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
blocks in parallel,
contention can be a significant issue.
© 2008-2018 by the MIT 6.172 Lecturers
25
Strategy 2: Local Heaps
heap
heap
heap
heap
∙ Each thread allocates
out of its own heap.
∙ No locking is necessary.
J Fast — no
synchronization.
L Suffers from memory
drift: blocks allocated
by o... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
2008-2018 by the MIT 6.172 Lecturers
32
How False Sharing Can Occur
How False Sharing Can Occur
A program can induce false sharing having
different threads process nearby objects.
∙ The programmer can mitigate this problem by
aligning the object on a cache-line boundary and
padding out the object to the size of a... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
© 2008-2018 by the MIT 6.172 Lecturers
36
Hoard Deallocation
Let ui be the in-use storage in heap i, and
let ai be the storage owned by heap i.
Hoard maintains the following invariant for
all heaps i:
ui ≥ min(ai - 2S, ai/2),
where S is the superblock size.
free(x), where x is owned by thread i:
put x back in ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
a popular choice for parallel systems
due to its performance and robustness.
●
SuperMalloc is an up-and-coming contender. (See
paper by Bradley C. Kuszmaul.)
© 2008-2018 by the MIT 6.172 Lecturers
39
Allocator Speeds
Allocator Speeds
Allocator
Allocator
Default
Hoard
jemal... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
TO
head
tail
Enqueue adjacent vertices.
© 2008-2018 by the MIT 6.172 Lecturers
46
Example
FROM
TO
head
tail
Enqueue adjacent vertices.
Place forwarding pointers in FROM vertices.
© 2008-2018 by the MIT 6.172 Lecturers
47
Example
FROM
TO
head
tail
Update the pointers in the removed item to refer
to its adja... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
urers
53
Baker’s Algorithm
Baker’s Algorithm
∙ Program follows forward pointer if there is one.
∙ Whenever the program accesses an object not in
the TO space, mark object as explored and copy it
over to the TO space.
∙ Whenever the program allocates an object, put it in
the TO space.
∙ Requires a read barrier ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
On Bounding Time and Space for Multiprocessor Garbage
Collection” (PLDI 1999), and “A Parallel, Real-Time Garbage
Collector” (PLDI 2001) by Cheng and Blelloch
© 2008-2018 by the MIT 6.172 Lecturers
57
Summary
Summary
∙ malloc() vs. mmap()
∙ Cactus stacks... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
Introduction to representation theory
Pavel Etingof, Oleg Golberg, Sebastian Hensel,
Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina
February 1, 2011
Contents
1 Basic notions of representation theory
1.1 What is representation theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . . . . . . . 13
1.9 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.10 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.11 The tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
sums of matrix algebras . . . . . . . . . . . . . . . . . . . . 24
2.4 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Finite dimensional algebras
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1
2.6 Characters of representations . . . . . . .... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . 34
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Duals and tensor products of representations
. . . . . . . . . . . . . . . . . . . . . . 36
3.5 Orthogonality of characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Unitary repre... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . . . . . . 47
4.2 Frobenius determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Algebraic numbers and algebraic integers
. . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Frobenius divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5 B... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . 57
4.12 Representations of Sn
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.13 Proof of Theorem 4.36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.14 Induced representations for Sn
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2
4.15 The ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . . . . . . . 66
4.22 Polynomial representations of GL(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.23 Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.24 Representations of GL2(Fq) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . . . . . . . . . . 78
5.2
Indecomposable representations of the quivers A1, A2, A3 . . . . . . . . . . . . . . . . 81
5.3
Indecomposable representations of the quiver D4
. . . . . . . . . . . . . . . . . . . . 83
5.4 Roots
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 Morphisms of functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Equivalence of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.5 Representable functors . . . . . . . . . . . . . . . ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
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