text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
um
a sequen
1.
′
λ
+
minim
that
Q∗
≠
≠
∗
′
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 the... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
. And furthermore (13.6) implies that EQ∗
≪
] ∈ (A, B).
[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 ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
,k . Then Mn Mmin n,τ
integrable then
)
(
(
)
=
˜
n, i.e. Mn is
∣F ] =
k
is also 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 rec... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
hypotheses.
This advantage
is also seem very clearly in achievable error exponents.
2 δ1 and Q = 1
may b
2 δ0 + 1
e dramatically
2 δ0 + 1
=
2
Theorem 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 la... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
[τ ] < ∞ we have
EP [Sτ ] = EP [τ
]D(P ∥Q)
(13.12)
and
similarly, if EQ[τ ] < ∞ then
To prove these, notice that
EQ[Sτ ] = − EQ[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... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
{τ n
=
}
] =
EQ exp Sτ 1E τ n
∩{ = }
}
{
[
This, however, follows from the fact
definition of Sn.
th
at
E
∩ {τ
= n} ∈ F and
n
]
.
dP
∣Fn
d
Q∣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 = ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/23be1265744c2e9e8962e79985cdda45_MIT6_441S16_chapter_13.pdf |
have
for l0E0 and l1E1 large,
ve d P Z
ha
≲
that
)
( (
( ∥ )
1 Q Z
= )∥ (
= )) ≈
1
( ∥
we
0
1
0
l1E1, therefore l1E1
≲
E0E1 ≤ D(
P ∥Q)D(Q
∥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://... | 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 |
AX-o
sin( Ax)
ax
C O S ( ~ X )i-
ax
= lim
AZ-O
= 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(%) ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/23c2c1b1ab31c9f10745b18e7b0bf131_lec3.pdf |
a sum is the sum of the limits.
u(x + Ax) - u(x)
Ax
Ax-0
(uv)' = ul(x)v(x)+ u(x)vl(x)
Note: we also used the fact that
I >
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
... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/23c2c1b1ab31c9f10745b18e7b0bf131_lec3.pdf |
the red and yellow rectangles. Thus we have:
(Divide by A x and let Ax + 0 to finish the argument.)
[(u+ Au) ( v + Av) - uv]w uAv + vAu
Lecture 3
18.01 Fall 2006
Quotient formula (General)
To calculate the derivative of ulv, we use the notations Au and Av above. Thus,
u(x + Ax)
U(X+ Ax)
Hence,
Therefore.
- ... | 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
−1x||B ≤ c · ||x||V ⇒ ||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 ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
α( ¯Ω),
we had the global Schauder estimate
||u||C2,α( ¯Ω) ≤ 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... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
fficients of L. And, by uniformly
elliptic we see from D 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... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
(∂B) (and not just on T ) then unique solvability
would be equivalent to the unique solvability of ∆ 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 ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/23ec7373e584754f3ddf0157ac29d664_da4.pdf |
es Lu = f on B and has the desired C2,α regularity on B.
4
We now turn to the boundary portion: ∀x0 ∈ T and ρ > 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)∩ ¯... | 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 |
metastable solid with no long-range atomic order
Consider a fictitious A2O3 2-D compound:
A2O3 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... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/244f9ff9b8938d8966cb6ebbf79acc3e_MIT3_071F15_Lecture1.pdf |
imensional
Silica Glass on Graphene." Nano Lett. 12 (2012): 1081-1086.
STEM images of 2-D silica crystal and glass
Sci. Rep. 3, 3482 (2013).
Nano Lett. 12, 1081-1086 (2012).
10
Article: Anne Ju “Shattering records: Thinnest glass in
Guinness book.” Cornell Chronicle. September 12,
2013.
11
Glass formation f... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/244f9ff9b8938d8966cb6ebbf79acc3e_MIT3_071F15_Lecture1.pdf |
cooling curve
during glass transition due
to structural relaxation
Tm
T
15
What is glass (amorphous solid)?
A metastable solid with no long-range atomic order
Liquid: atoms do not have fixed
positions; bonding constraints
relax as temperature rises
Solidification
Melt quenching
Melting
Softening
Cryst... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/244f9ff9b8938d8966cb6ebbf79acc3e_MIT3_071F15_Lecture1.pdf |
00r, g
PDFs of ideal (hard sphere) crystals vs. glasses
g(r)
1st coordination shell
2nd coordination shell
g(r)
1
0
r
r
19
Quantitative description of glass structure
Pair correlation function h(r)
Radial distribution function (RDF): J(r)
J(r)dr gives the number of atoms located between r and... | 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 |
(z1, . . . , zn) =
It is enough 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
∂... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/248fc514e61beb10ea33642006f43e24_18117_lec09.pdf |
q
X there exists open sets Ui, i = 1, . . . , n such 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... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/248fc514e61beb10ea33642006f43e24_18117_lec09.pdf |
can split the tangent space by
Tp
⊗
: Tp
Jp(v
c) = Jpv
c
⊗
⊗
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,... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/248fc514e61beb10ea33642006f43e24_18117_lec09.pdf |
∈
−
Corollary. U is open in 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 ∗
⊗... | 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
2/16/2011 Random networks © Daniel E Whitney 1997-2010
1/46
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 |
susceptibility - an important issue for
sociologists
• Thresholds are used to model these differences
2/16/2011 Random networks © Daniel E Whitney 1997-2010
3/46
and cascades
Diffusion of Pandemic Diseases
• Model assumes disease starts from a point and travels in
two modes: local commuting and international ai... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
348
BLACK
SEA
June 30, 1348
Constantinople
June 30, 1347
AEGEAN
SEA
Seville
December 31, 1347
Ralph's World Civilizations, Chapter 13
Image by MIT OpenCourseWare.
2/16/2011 Random networks © Daniel E Whitney 1997-2010
5/46
and cascades
SIR Model
Susceptible (S)
Recovered (R)
Infected (I)
Transmission
Airborne... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
© Daniel E Whitney 1997-2010
7/46
Using Data and Model to Find Parameters
Likelihood of Epidemic Parameter Values
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.35-0.4
0.3-0.35
0.25-0.3
0.2-0.25
0.15-0.2
0.1-0.15
0.05-0.1
0-0.05
Ro =
0.5
1
0.8
0.6
1.5
2
0.4
IP = 0.2
2.5
2/16/2011
Random networks
a... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
e
s
d
i
r
b
y
h
g
n
i
t
p
o
d
a
s
r
e
m
r
a
f
f
o
r
e
b
m
u
N
300
250
200
150
100
50
0
1
Cumulative number
of adopters
New adopters
"Critical mass"
62
41
253
257
239
203
159
98
61
44
36
36
4 8 12 18 19 25
6
1 6
21
16
14
4
1927 1929 1931 1933 1935 1937 1939 1941 1943 1945
Year
The number of new adopters each year, and t... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
and cascades
Basic Theory
• Network has n nodes
• A pair of nodes is linked (both ways) with probability p
• The number of links in the network m = pn(n-1)/2
– Some fraction of the number if all nodes were linked,
counting each pair of nodes once
• The average nodal degree = z = 2m/n = pn
• The clustering coeff... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
pk (1− p)n−k−1 ≅ e−z z k
⎟
⎝ k ⎠
k!
if n → ∞
p0 = e−1 = 0.3679
p1 = e−1 = 0.3679
p2 = e−1 /2 = 0.1359
p3 = e−1 /6 = 0.0453
z = 10
z = 1.006
• This looks roughly Gaussian for large z, highly peaked for small z
• Standard deviation σ=
z
2/16/2011 Random networks © Daniel E Whitney 1997-2010
14/46
and cascade... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
– In a finite network, the cluster size is
comparable to the size of the network
• Giant clusters appear if the network is dense
enough
• The proven threshold for E-R is z = 1
2/16/2011 Random networks © Daniel E Whitney 1997-2010
16/46
and cascades
Percolation, Cascades, Rumors
• A network consists of nodes th... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
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= 0
∞
k∑ (k −1) pk = z
b
k= 0
∞
∑k(k −1)ρk pk = z
k= 0
Watts rumor cascade model :
ρ ={1 for k ≤ K *
0 for k > K *
k
2/16/2011 Random networks © Daniel E Whitney 1997-2010
19/46
and cas... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
K * +1 ≤ k ≤ 2K* : flip if ≥ 2 neighbors flip
Seed
First Step
K* = 4
Second Step
2/16/2011 Random networks © Daniel E Whitney 1997-2010
23/46
and cascades
Watts’ Cascade Diagram
Theoretical boundary: infinite nodes
Simulation boundary: 10000 nodes
Network is too
densely
connected
Network is
disconnected ... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
Global cascades region: seed of one node can start a
global cascade
– No global cascades region: seed of one node cannot
start a global cascade
2/16/2011 Random networks © Daniel E Whitney 1997-2010
26/46
and cascades
Cascades in Finite E-R Networks Can
Happen in the No Global Cascades Region
D. E. Whitney, ŅD... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
0
0
S = 200 (no TNCs)
S = 210 (no TNCs)
S = 220 (30% TNCs)
S = 220 (70% no TNCs)
S = 230 (90% TNCs)
S = 230 (90% TNCs)
S = 240 (100% TNCs)
S = 270 (100% TNCs)
S = 300 (100% TNCs)
1
3
5
7
9
11
13
15
Step
“Near death” phenomenon
2/16/2011 Random networks © Daniel E Whitney 1997-2010
30/46
and cascad... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
0 i'=0⎝ i ⎠
⎛n − S − F1 −1⎞ k−i−i' (
⎟p
×
⎜
⎝ k − i − i ' ⎠
nSF1
1− pnSF1
F1⎞
⎜
⎝ i' ⎠ F1
⎟pi' (1− pF1)F1−i'
)n−S−F1−1−(k−i−i')
pF1 = z F1 /n reflects available edges from F1
pnSF1
reflects larger p of unflipped nodes
2/16/2011 Random networks © Daniel E Whitney 1997-2010
32/46
and cascades
Theory: Cascad... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
1997-2010
35/46
and cascades
Theory: Ability to Predict Threshold
Seed Size
S Transition: Theory and Simulations
n = 4500, z = 14.56, K* = 5
S Transition: Theory and Simulations
n = 4500, z = 14.56, K* = 4
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
120
130
140
150
160
170
180
Size of Seed, S
S... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
1997-2010
0
36/46
150
200
250
300
Size of Seed, S
At S = 215, Failure Most of the Time
2/16/2011 Random networks © Daniel E Whitney 1997-2010
37/46
and cascades
Occasionally, Success: Why?
Wake-up
“Near death”
2/16/2011 Random networks © Daniel E Whitney 1997-2010
38/46
and cascades
Cause of Wake-up ... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
* NOS = Fj * n /zFj
2/16/2011 Random networks © Daniel E Whitney 1997-2010
40/46
and cascades
Theory and Simulations: Evolution of max
One Short Failures (avg of 20 runs)
If one short exceeds the bound, a TNC almost always occurs.
If one short does not exceed the bound, a TNC almost never occurs.
Variation in one... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
Graphs with Arbitrary Degree
Distributions and Their Applications” arXive:cond-mat/0007235 v2 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_thresho... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
epidemic-modeling
•
• Multiscale mobility networks and the spatial spreading of infectious
diseases.�D.Balcan, V. Colizza, B. Gonsal ves, H. Hu, J. J. Ramasco,
A. Vespignani�Proc Natl Acad Sci U S A 106, 21484-21489 (2009).
• Predictability and epidemic pathways in global outbreaks of infectious
diseases: the SAR... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
011 Random networks © Daniel E Whitney 1997-2010
47/46
and cascades
Subgraph Shapes vs p
p ~ n a , z ~ n a +1
0 n-1 n-1/2
n-1/3
n-1/4
0 n-2 n-3/2
n-4/3
n-5/4
1
n-1
n1/3
n1/2
n-2/3
n-1/2
z
p
a
z for n = 1000: 0 0.001 0.0316
0.1
0.178
1
10
31.6
The threshold probabilities at which different sub... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
) with a statistically
independent probability p. At a critical threshold
pc, long-range connectivity first appears, and this
is called the percolation threshold.” [see
wikipedia reference “percolation threshold”]
• For a square grid, pc = 0.5 for bond percolation
and pc = 0.59274621 for site percolation
2/16/20... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
node
Other stable nodes
Vulnerable cluster
Seed
Vulnerable nodes have a few links to each other (average ~ 1.5) and
more links to stable nodes outside their cluster. Working together,
vulnerable nodes can flip stable nodes but most likely this happens
only when vulnerable nodes co-exist in clusters
2/16/2011 Ra... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
Most theory assumes we are dealing with a sparse network
that has few or no closed loops, especially no small closed
loops
• This is measured by the clustering coefficient, which is
•
small for big random networks where the theory has been
developed
If there is no clustering or small closed loops then it is easy
... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
4
Its excess degree = 3 If its degree = k, its edges are k-times more numerous
The neighbor
The edge
The node
than if its degree = 1 (think of the edge list)
But the fraction of nodes with degree k is pk.
So the likelihood of encountering a node of degree k
by this process is proportional to kpk
Distribution o... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
/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 as z = 1
(See notes)
58/46
continuing
For the case where all nodes have vulnerability = b:
∞
∑
b
k= 0
k(k −1) p = z
k
See notes
For the case where vulnerability is ... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/2494b88060260e2f9661bea27ef072e9_MITESD_342S10_lec13.pdf |
0
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 |
⎞
⎟
⎠
=
μ μ
i
n
+
k T
b
ln
⎛
⎜
⎝
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
Forwar... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
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/Image:Pn-junction-equilibrium.svg.
3.23 Electronic, Optical and Mag... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
2007)
Ohmic to ballistic conductance
What happens when electric field is applied?
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! ... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
3
E
m2
m1
k
m1
e
L
∑
k
1
(cid:61)
E
∂
k
∂
f
+
(E)
=
I+
I-
2
e
h
e
2
h
2
+∞
∞−
∫
(
νf
+
(E)
=
+
I
=
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 Prope... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
to the scattering by phonons
– Estimated mean free path of phonon scattering at R.T. (cid:62) ~1μm
( we do not take inelastic scattering into account)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Image removed due to copyright restrictions.
Please see: Fig. 3 in Javey... | 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)
Phonon dispersions in graphite
Image removed due to copyright restrictions.Please see: Fig. 4 in Mounet, Nicolas, and Nicola Marzari.
"First-principles Determination of the Structural,Vibrational, and Thermodynamic Properties... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
1-d Carbon
Image removed due to copyright restrictions.Please see: Fig. 15 in Mounet, Nicolas, and Nicola Marzari.
"First-principles Determination of the Structural,Vibrational, and Thermodynamic Properties of Diamond, Graphite, and Derivatives."
Physical Review B 71 (2005): 205214.
Images removed due to copyright r... | https://ocw.mit.edu/courses/3-23-electrical-optical-and-magnetic-properties-of-materials-fall-2007/24a5e9d4b751258bbb7778714d00fed7_clean15.pdf |
ite and Graphene."
arXiv:0708.4259v2 [cond- mat.mtrl--sci], 2007.
sci], 2007.
Strong T-dependence of
A'1 mode due to TA-LA
and LO-LA decay channels
Importance of the acoustic
phonon population for the
transport properties.
Courtesy of Nicola Bonini. Used with permission.
Image removed due to copyright restrictions.
Pl... | 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
SSyysstteemmss
LLEECCTTUURREE(cid:1)(cid:1) 1122
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 |
Don't care where
size, // #bytes
PROT_READ | PROT_WRITE, // Read/write
MAP_PRIVATE | MAP_ANON, // Private anonymous
-1, // no backing file
0 // offset (N/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 creat... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
mmap() and other system calls to
expand the size of the user’s heap storage.
●
●
© 2008-2018 by the MIT 6.172 Lecturers
6
Address Translation
virtualaddress
offset
virtual page #
search
frame #
frame #
offset
physical address
page table
page table
If the virtual page does not reside... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
children, but not the
other way around.
B
A
D
C
E
A
A
C
A
C
B
A
B
D
A
C
D
E
A
C
E
invocation tree
views of stack
© 2008-2018 by the MIT 6.172 Lecturers
11
Cactus Stack
A cactus stack supports multiple views in parallel.
B
A
D
C
E
A
A
C
A
C
B
A
B
D
A
C
D
E
A
C
E
invocation tree
views of stack
© 2008-2018 by the... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
the temporary
matrix D obey a
stack discipline.
double *D = malloc(n * n * sizeof(*D));
double *D = malloc(n * n * sizeof(*D));
assert(D != NULL);
#define n_D n
#define X(M,r,c) (M + (r*(n_ ## M) + c)*(n/2))
cilk_spawn mm_dac(X(C,0,0), n_C, X(A,0,0), n_A, X(B,0,0), n_B, n/2);
cilk_spawn mm_dac(X(C,0,1), n_C, X(A,0,0... | 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
16
Worst-Case Recursion Tree
Worst-Case Recursion Tree
n2
(n/2)2
(n/2k)2
(n/2k)2
8
…
(n/2)2
(n/2)2
8
…
(n/2k)2
…
P nodes
Branch fully (8-
way) until we
get to a level k
with P nodes
and then
branch serially
from there on.
Θ(1)
Θ(1)
Θ(1)
We have 8k = P, whi... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
speed for large blocks or small blocks?
A. Small blocks!
Q. Why?
A. Typically, a user program writes all the bytes
of an allocated block. A large block takes so
much time to write that the allocator time has
little effect on the overall runtime. In contrast,
if a program allocates many small blocks, the
allocat... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
6.172 Lecturers
22
SPEED
LIMIT∞PER ORDER OF 6.172
PARALLEL ALLOCATION
STRATEGIES
© 2008-2018 by the MIT 6.172 Lecturers
23
Strategy 1: Global Heap
global heap
∙ Default C allocator.
∙ All threads (processors)
share a single heap.
∙ Accesses are mediated
by a mutex (or lock-free... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
2 Lecturers
26
Strategy 3: Local Ownership
∙ Each object is labeled
with its owner.
∙ Freed objects are
returned to the owner’s
heap.
heap
heap
heap
heap
J Fast allocation and
freeing of local
objects.
L Freeing remote
objects requires
synchronization.
K Blowup ≤ P.
J Resilience to false
sharing.
© ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
a cache-line boundary and
padding out the object to the size of a cache line, but
this solution can be wasteful of space.
An allocator can induce false sharing in two ways:
∙ Actively, when the allocator satisfies memory
requests from different threads using the same cache
block.
∙ Passively, when the program pa... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
6
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 heap i;
if (ui < min(ai - 2S, ai/2)) {
mo... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
which zeros the page while keeping the virtual
address valid.
jemalloc is 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
... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
Example
FROM
TO
head
tail
Remove an item from the queue.
© 2008-2018 by the MIT 6.172 Lecturers
44
Example
FROM
TO
head
tail
Remove an item from the queue.
© 2008-2018 by the MIT 6.172 Lecturers
45
Example
FROM
TO
head
tail
Enqueue adjacent vertices.
© 2008-2018 by the MIT 6.172 Lecturers
46
Example
FROM... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
garbage collector take turns running.
© 2008-2018 by the MIT 6.172 Lecturers
51
Running Collector with Program
FROM
TO
head
tail
If an object O already dequeued in BFS gains a
reference to another object O’, the BFS may not
find O’ and it will be freed.
© 2008-2018 by the MIT 6.172 Lecturers
52
Running... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/24c247c943024dcda40bcc7caa206581_MIT6_172F18_lec12.pdf |
bage Collection Glossary
Garbage Collection Glossary
∙ Stop-the-world: Garbage collector does all of its
work across memory while pausing program.
∙ Incremental: Garbage collector runs incrementally,
allowing pause times to be bounded.
∙ Parallel: Multiple collector threads are running
simultaneously.
∙ Concurre... | 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 |
. . 23
2.2 The density theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Representations of direct sums of matrix algebras . . . . . . . . . . . . . . . . . . . . 24
2.4 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Finite dim... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
3
3.1 Maschke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Duals and ten... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . . . . . . 42
3.10 Problems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Representations of finite groups: further results
47
4.1 Frobenius-Schur indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Frobenius determinant . . . . . . . . . . .... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . . . . . . . . . . . . . 54
4.9 The Mackey formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.10 Frobenius reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.11 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. . . . . . . . 63
4.18 Schur-Weyl duality for gl(V )
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.19 Schur-Weyl duality for GL(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.20 Schur polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
70
4.24.3 Principal series representations . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.24.4 Complementary series representations
. . . . . . . . . . . . . . . . . . . . . . 73
4.25 Artin’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.26 Representations of semi... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
Reflection Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.7 Coxeter elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.8 Proof of Gabriel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.9 Problems
. . . . . ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
101
3
6.6 Adjoint functors
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.7 Abelian categories
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.8 Exact functors
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 Struc... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. In this letter Dedekind
made the following observation: take the multiplication table of a finite group G and turn it into a
matrix XG by replacing every entry g of this table by a variable xg. Then the determinant of XG
factors into a product of irreducible polynomials in
, each of which occurs with multiplicity ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
version of this course. The first author is very indebted to Victor Ostrik for helping him
prepare this course, and thanks Josh Nichols-Barrer and Thomas Lam for helping run the course
in 2004 and for useful comments. He is also very grateful to Darij Grinberg for very careful reading
of the text, for many useful com... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
V , i.e., a linear map preserving the multiplication
V equipped with a homomorphism δ : A
and unit.
⊃
A subrepresentation of a representation V is a subspace U
operators δ(a), a
has an obvious structure of a representation of A.
�
A. Also, if V1, V2 are two representations of A then the direct sum V1
�
→
V which... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
1) with respect to unknown matrices h, e, f .
It is really striking that such, at first glance hopelessly complicated, systems of equations can
in fact be solved completely by methods of representation theory! For example, we will prove the
following theorem.
Theorem 1.1. Let k = C be the field of complex numbers. Th... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
ifically, let us say that Q is of finite type if it has finitely many indecomposable
representations.
We will prove the following striking theorem, proved by P. Gabriel about 35 years ago:
Theorem 1.2. The finite type property of Q does not depend on the orientation of edges. The
connected graphs that yield quivers of ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
G and multiplication
group algebra A = C[G] of a finite group G – the algebra with basis ag, g
law agah = agh. We will show that any finite dimensional representation of A is a direct sum of
irreducible representations, i.e., the notions of an irreducible and indecomposable representation
are the same for A (Maschke’... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
An associative algebra over k is a vector space A over k together with a bilinear
map A
ab, such that (ab)c = a(bc).
A, (a, b)
A
×
⊃
�⊃
Definition 1.4. A unit in an associative algebra A is an element 1
A such that 1a = a1 = a.
�
Proposition 1.5. If a unit exists, it is unique.
Proof. Let 1, 1�
be two units... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
A is commutative if ab = ba for all a, b
A.
�
For instance, in the above examples, A is commutative in cases 1 and 2, but not commutative in
cases 3 (if dim V > 1), and 4 (if n > 1). In case 5, A is commutative if and only if G is commutative.
Definition 1.8. A homomorphism of algebras f : A
f (x)f (y) for all x, ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
the operator of left multiplication by
a, so that δ(a)b = ab (the usual product). This representation is called the regular representation
of A. Similarly, one can equip A with a structure of a right A-module by setting δ(a)b := ba.
⊃
3. A = k. Then a representation of A is simply a vector space over k.
4. A = k x1... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
⊃
�
⊃
Note that if a linear operator θ : V1
1 : V2
V1 (check it!).
linear operator θ−
⊃
⊃
V2 is an isomorphism of representations then so is the
Two representations between which there exists an isomorphism are said to be isomorphic. For
practical purposes, two isomorphic representations may be regarded as “the ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
prove, it is
fundamental in the whole subject of representation theory.
Proposition 1.16. (Schur’s lemma) Let V1, V2 be representations of an algebra A over any field
V2 be a nonzero homomorphism of
F (which need not be algebraically closed). Let θ : V1
representations. Then:
⊃
(i) If V1 is irreducible, θ is inje... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
).
⊃
·
�
Remark. Note that this Corollary is false over the field of real numbers: it suffices to take
A = C (regarded as an R-algebra), and V = A.
Proof. Let ∂ be an eigenvalue of θ (a root of the characteristic polynomial of θ). It exists since k is
V , which
an algebraically closed field. Then the operator θ
is n... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
. A = k. Since representations of A are simply vector spaces, V = A is the only
irreducible and the only indecomposable representation.
2. A = k[x]. Since this algebra is commutative, the irreducible representations of A are its
1-dimensional representations. As we discussed above, they are defined by a single operat... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
normal form theorem (which in particular says
that the Jordan normal form of an operator is unique up to permutation of blocks).
⊃
�
−
9
⇒
⇒
⇒
This example shows that an indecomposable representation of an algebra need not be irreducible.
3. The group algebra A = k[G], where G is a group. A representation of A i... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/24d8b3fa2ce48e48ee6c2d8d5e3562f6_MIT18_712F10_replect.pdf |
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