text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
1Ex2 )
(cid:20) Ey2 −Ey1
−Ex2 Ex1
(cid:21) (cid:20)−Et1
−Et2
(cid:21)
(28)
Note that the expression given by
inverse (in this case, simply a 2x2 matrix).
1
(Ex1 Ey2 −Ey1 Ex2 ) is the determinant of the partial derivatives matrix, since we are taking its
When can/does this fail? It’s important to be cognizant of edge ca... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5546a6b8d36a2d997929ba1aeb8c5ed3_MIT6_801F20_lec2.pdf |
0
Here, we have two equations and two unknowns. When can this fail?
• When we have linear independence. This occurs when:
– E = 0 everywhere
– E = constant
– Ex = 0
– Ey = 0
– Ex = Ey
– Ex = kEy
• When E = 0 everywhere (professor’s intuition: “You’re in a mine.”)
• When Ex, Ey = 0 (constant brightness).
• Mathematicall... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5546a6b8d36a2d997929ba1aeb8c5ed3_MIT6_801F20_lec2.pdf |
Lecture 1
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 1
1: NON-ABELIAN GAUGE THEORIES
In 8.323, we have studied the quantum theory of:
1.
2.
3.
Scalar fields, spin-0,
Dirac fields, spin- 1
2 ,
Maxw... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/55d5413e73d973bc458b8e1313124c38_MIT8_324F10_Lecture1.pdf |
�†ψ.
i�ψ(t, ⃗x). The conserved charge
x) −→ e
´
Remarks:
1.
2.
All possible phase multiplications ei� form a group, called U (1). U (1) is Abelian, that is, ei�1 ei�2 =
ei�2 ei�1 . A Group is a set closed under a multiplication, satisfying the axioms of (i) associativity, (ii)
existence of an identity and (iii)... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/55d5413e73d973bc458b8e1313124c38_MIT8_324F10_Lecture1.pdf |
under the internal symmetry transformation
L = −iΨ(γµ∂µ − m)Ψ,
U (2) : Ψ(x) −→ U�(x), ¯
Ψ(x) −→ �(x)U†,
¯
where U is a 2 × 2 matrix such that
UU† = 1.
The complex matrices satisfying this condition form the group U (2). We can separate out an overall phase
rotation in these transformations:
where U = ei� =ei��2×... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/55d5413e73d973bc458b8e1313124c38_MIT8_324F10_Lecture1.pdf |
]
[
]
σa σb
,
2
2
= iϵabc
σc
.
2
�
�
a Ψ, and by Noether’s prescription, we
2 + . . ., we have that δΨ = iΛa
For infinitesimal Λa, U =1+iΛa
2
find an associated set of conserved charges Qa, (a = 1, 2, 3), which satisfy the commutation relations
[Qa, Qb] = iϵabcQc (see problem set). That is, the conserved charge... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/55d5413e73d973bc458b8e1313124c38_MIT8_324F10_Lecture1.pdf |
G, where Λa are parameters
labelling points of the manifold, a = 1, 2, . . . , dim G. They satisfy a group multiplication:
where
g(Λ) g(Λ′) = g(Λ′′),
◦
a = fa(Λ, Λ′).
Λ′′
(11)
(12)
Lie groups are tailor-made for describing continuous symmetry transformations in physics: the group product
corresponds to the com... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/55d5413e73d973bc458b8e1313124c38_MIT8_324F10_Lecture1.pdf |
in the expansion, so that the
generators are hermitian for unitary group transformations. ϵa parameterizes the tangent space around g = 1, with
T a a basis for the tangent space. Mathematically, ϵaT a ∈ T1G. Putting (12) into (13), we find T a should satisfy
]
[
T a, T b = if cabT c ,
(14)
[
]
T a, T b
≡ T a T b −... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/55d5413e73d973bc458b8e1313124c38_MIT8_324F10_Lecture1.pdf |
18.336 spring 2009
lecture 17
04/09/09
Conservation Laws
ut + (f (u))x = 0
if u∈C1
⇐⇒
Conservation form
ut + f �(u) ux = 0
� �� �
=c(u)
Differential form
�
� b
d
dt a
Integral form
u(x, t)dx = f (u(a, t)) − f (u(b, t))
f = flux function
Ex.: Transport equation
f (u) = cu
⇒
c(u) = f �(u) = c
Ex.: Burg... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/55d85c9cedf091f71a611ce4a2e63514_MIT18_336S09_lec17.pdf |
)ux = 0
u(x, 0) = u0(x)
Follow solution along line x0 + ct, where c = f �(u0(x0)).
d
dt
u(x + ct, t) = cux(x + ct, t) + ut(x + ct, t)
= (c − f �(u(x + ct, t))) ux(x + ct, t) = 0
�
��
=0
·
�
⇒ u(x + ct, t) = constant = u(x0, 0) = u0(x0).
Ex.:
Transport
Burgers’
Traffic
Characteristic lines intersect
⇒
shocks ... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/55d85c9cedf091f71a611ce4a2e63514_MIT18_336S09_lec17.pdf |
�) ⇔ (∗∗)
Proof: integration by parts.
In addition, (∗∗) admits discontinuous solutions.
Riemann Problem
�
u0(x) =
(uL − uR) s =
·
�
u(x, t) dx
uL x < 0
uR x ≥ 0
� b
d
dt a
= f (uL) − f (uR)
⇒
s =
f (uR) − f (uL)
uR − uL
Image by MIT OpenCourseWare.
Rankine-Hugoniot Condition for shocks
3
Ex.: Burgers’
s ... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/55d85c9cedf091f71a611ce4a2e63514_MIT18_336S09_lec17.pdf |
3
INGREDIENTS FOR SCET
QCD
Here the jet mass is also the mass of the hadronic final state, and the situation which dominates the
2
phenomenology has m
X ∼ QΛQCD.
We have collinear modes for the jet, and ultrasoft modes with p ∼
which are the constituents of the B meson for this inclusive decay. Often the region whe... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
expansion.
For a collinear momentum pµ = (p , p , p , p3) we have p = p + p » p » p = p0 − p so
+
−
3
1
0
3
2
0
1,2
⊥
where the terms in the + . . . are smaller. Keeping only the leading term gives us the spinors
pσ · pp
0
p
= σ3 + . . . ,
u(p) =
v(p) =
(2p
0)1/2
√
2
(2p
0)1/2
√
2
U
σ·p
0 U
p
σ·p
0 V... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
) gives the following relations
/ = 0 ,
nun
nvn = 0 .
/
(3.4)
These can be recognized as the leading term in the equations of motion /
in the collinear limit. We can also define projection operators
pu(p) = /
pv(p) = 0 when expanded
Pn =
/n/¯n
4
=
(cid:18) 1 σ3
1
σ3
1
2
(cid:19)
,
P¯n =
/¯n/n
4
=
(cid:18) 1 −σ3
1
−... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
satisfy the desired spin relations
(3.7)
(3.8)
(3.9)
nξn = 0 ,
/
Pnξn = ξn ,
/¯ = 0 ,
nϕn¯
Pn¯ϕn¯ = ϕn¯ .
(3.10)
ˆ
The label n on ξn reminds us that it obeys these relations and that we will eventually be expanding about
the n-collinear direction. Note that here we denote the collinear field components with a hat,... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
3
p0
− (i(cid:126)σ×(cid:126)p⊥)3
p0
U
(cid:17)
U
(cid:115)
U˜ =
(cid:18)
1 +
p0
2p−
p3
p0
−
(i(cid:126)σ × (cid:126)p⊥)3
p0
(cid:19)
U .
15
(3.11)
(3.12)
3.2 Collinear Fermion Propagator and ξn Power Counting
3
INGREDIENTS FOR SCET
The same derivation gives
vn =
(cid:114)
(cid:19)
(cid:18)
p−
2
˜σ3
V
˜
V
(3.1... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
(cid:19)
,
(cid:88)
s
us
u¯s
n n =
n
/
2
¯n · p .
(3.14)
(3.15)
(3.16)
For later convenience we write down a set of projection operator identities easily derived from n2 = 0,
n¯ · n = 2, and/or hermitian conjugation 㵆 = γ0γµγ0:
PnP¯ = 0 ,
n
Pn
Pn = Pn ,
Pnn¯ = Pn¯ /
n = 0 ,
/
Pn /
n = /
n ,
P¯n = ¯
n /¯
n , /
P †... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
inear Fermion Propagator and ξn Power Counting
Having considered the decomposition of spinors in the collinear limit, we now turn to the fermion propagator
2
in the collinear limit. Here p + i0 = n¯ · p n · p + p , and since both of these terms are ∼ λ2 there is no
⊥
2
16
3.3 Power Counting for Collinear Gluons an... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
(0)|0). At this point we can already identify the λ power counting
for the field ξˆn by noting that if its propagator has the form in Eq. (3.20) then its action must be of the
form
¯ˆ
L(0)
n
=
d4 x L(0)
n
=
n/¯
¯ˆ
d4
ξn
x
'-n" '-n" 2 '
O(λ−4) O(λa)
in · ∂ + . . .
-n
O(λ2)
ˆ ∼ λ2a−2
ξn
" '-n"
O(λa)
.
(3.... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
SCET collinear field ξn, the
further manipulations we will make in section 4 below will not effect its power counting, so we have also
recorded here the fact that the SCET field ξn ∼ λ. Note that this scaling dimension does not agree with
the collinear quark fields mass dimension since [ξˆn] = [ξn] = 3/2. This is simply... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
− τ
i
k2
kµkν
k2
= −
i
k4
(cid:0)k2gµν − τ kµkν(cid:1) ,
(3.23)
where τ is our covariant gauge fixing parameter. From our standard power counting result from the light-
cone coordinate section, we know that k2 = k+k− + k2 = Q2λ2 . So the 1/k4 on the RHS matches up with
⊥
the scaling of the collinear integration measur... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
This result is not so surprising considering that if we are going to formulate a collinear covariant derivative
Dµ = ∂µ + igAµ with collinear momenta ∂µ and gauge fields, then for each component both terms must
have the same λ scaling. Indeed imposing this property of the covariant derivative is another way to derive ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
= p −δ(p − − p '−)δ2(pp ⊥ − pp ' ) ∼ λ−2
(3.27)
Thus the single particle collinear state has |p) ∼ λ−1 for both quarks and gluons. Given the scaling of
the collinear quark and gluon fields, this implies power counting results for the polarization objects. The
collinear spinors un ∼ ξn|p) ∼ λ0 which is consistent wit... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
where Γ = γµ(1 − γ5). Without gluons we can match this QCD current onto a leading order current in
SCET by considering the heavy b field to be the HQET field hv and the lighter u field by the SCET field
ξn. This is shown in Fig. 4 part (a), where we use a dashed line for collinear quarks. The resulting SCET
operator is ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
n ξnΓ
= −gn¯ · AA
n ξnΓ
(cid:17)
(cid:16) −g ¯n · An Γ
n¯ · q
hv
= ξn
+ . . .
(cid:21)
n/
T A hv2
(cid:21)
T Ahv
+ . . .
(3.30)
In the first equality we have used the fact that the incoming b quark carries momentum mbvµ, that
2
k = mbv + q so that k2 − m = 2mbv · q + q2, and that
b
Aµ
n =
µ
µ
n¯
n
+ Aµ
n¯ · An +
... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
we see that in SCET integrating out offshell hard propagators
that are induced by n¯ · An gluons leads to an operator for the leading order current with one collinear
gluon coming out of the vertex, pictured on the RHS of Fig. 4 part (b).
2
Inspecting the final result in Eq. (3.30) we see that, in addition to being a ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
¯n · An collinear gluon field to the
light collinear u quark, as shown below:
Calling the final u quarks momentum p we have kµ = pµ − qµ. However here since both p and q are
n-collinear the propagator momentum kµ also has n-collinear scaling. In particular k2 ∼ λ2 and is not
offshell, it instead represents a propagati... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
gluon emission to k gluon emissions with momenta q1, . . . , qk and
propagators with momenta q1, q1 + q2, . . . ,
pk
i=1 qi yields
(cid:32)
(cid:88)
¯ξn
perm
(−g)k
k!
¯n · Aq1 · · · ¯n · Aqk
[¯n · q1][¯n · (q1 + q2)] · · · [¯n · (cid:80)k
i=1 qi]
(cid:33)
Γhv
(3.32)
Here the sum of permutations (perms) of the {q1, .... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/55e3755fdf069fc9662217e8e036499e_MIT8_851S13_IngrediForScet.pdf |
LOCAL CONVERGENCE OF GRAPHS AND ENUMERATION OF SPANNING
TREES
MUSTAZEE RAHMAN
1. Introduction
A spanning tree in a connected graph G is a subgraph that contains every vertex of G and is
itself a tree. Clearly, if G is a tree then it has only one spanning tree. Every connected graph
contains at least one spanning tree: ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
2
MUSTAZEE RAHMAN
spanned by [−n, n]2 in Z2. There are exponentially many spanning trees in Z[−n, n]2 in terms of
its size. Indeed, let us see that any spanning tree in Z[−n + 1, n − 1]2 can be extended to at least
28n different spanning trees in Z[−n, n]2. Consider the boundary of Z[−n, n]2 which has 8n vertices
of the... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
· 24 = 28n spanning
trees in Z[−n, n]2.
Let sptr(Z[−n, n]2) denote the number of spanning trees in Z[−n, n]2. The argument above shows
that sptr(Z[−n, n]2) ≥ 28nsptr(Z[−n + 1, n − 1]2), from which it follows that sptr(Z[ n, n]2)
≥
24n(n+1). As |Z[−n, n]2| = (2n + 1)2 we deduce that log sptr(Z[−n, n]2
)/|Z[−n, n] | ≥ lo... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
φ(x) = y. In this case we write (G, x) ∼= (H, y). We consider isomorphism classes of rooted graphs,
although we will usually just refer to the graphs instead of their isomorphism class. Given any graph
G we denote Nr(G, x) as the r-neighbourhood of x in G rooted at x. The distance between two
(isomorphism classes of) r... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
H, y)}|. If G is a vertex transitive graph, for example Zd, then
for any vertex ◦ ∈ G we have a random rooted graph (G, ◦) which is simply the delta measure
(cid:3)
supported on the isomorphism class of (G, ◦). The isomorphism class of G consists of G rooted at
different vertices. It is conventional in this case to simp... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
of local weak convergence
should see Aldous and Lyons [1] and the references therein.
Exercise 2.1. Show that the d-dimensional grid graphs Z[−n, n]d converge to Zd in the local weak
limit. Show that the same convergence holds for the d-dimensional discrete tori (Z/nZ)d, where two
vertices x = (x1, . . . , xd) and y = ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
the expected k-step return probability of the SRW on (Gn, ◦n) converges to the
expected k-step return probability of the SRW on (G, ◦). Note that if Nr(G, x) ∼= Nr(H, y) then
4
MUSTAZEE RAHMAN
pk (x) = pk
G
the (k/2)-neighbourhood on the starting point.
H (y) for all 0 ≤ k ≤ 2r since in order for the SRW to return in ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
)
n
G
Gn
(cid:48) )
k
◦n − G(cid:48) ◦
p
( (cid:48))
(cid:12)
(cid:3)
(cid:12)
(cid:48)
n
G
(cid:12)E(cid:2) pk (
(cid:12) = (cid:12)
(cid:12)
(cid:12)E(cid:2) pk
(cid:12)
=
≤ 2P(cid:2) Nk/2(G(cid:48)
G ◦(cid:48)
(
(cid:48)
n
n) − pk
(cid:48)
G (◦(cid:48)); N
n, ◦(cid:48)
(cid:48)
k/2(G
n) (cid:29) Nk/2(G(cid:48), ◦(ci... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
sampling procedure is simple enough that we can calculate the expectations and probabilities that
are of interest to us. We will then relate this model of random d-regular multigraphs to uniform
random d-regular graphs.
The configuration model starts with n labelled vertices and d labelled half edges emanating from
each... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
�)|
n
(1)
E
(cid:105)
→ 1 as n → ∞,
where ◦ is any fixed vertex of Td (note that Td is vertex transitive). Notice that if Nr(Gn,d, v)
contains no cycles then it must be isomorphic to Nr(Td, ◦) due to Gn,d being d-regular. Now
suppose that Nr(Gn,d, v) contains a cycle. Then this cycle has length at most 2r and v lies wit... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
3.
(cid:2)
C(cid:96) be the number of cycles of length (cid:96) in Gn,d. Then limn
→∞
(cid:2)
E C
(cid:3)
(cid:96) =
(
1)(cid:96)
d−
2(cid:96)
.
Proof. Given a set of (cid:96) distinct vertices {v1, . . . , v(cid:96)} the number of ways to arrange them in cyclic
order is ((cid:96)−1)!/2. Given a cyclic ordering, the nu... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
∞. Similarly,
(nd − 2(cid:96) − 1)!!/((nd − 1)!!) = (1 + o(1))(nd)−(cid:96) as n → ∞ and it is at most 3(cid:96)(nd)−(cid:96) if d ≥ 3 (provided
(cid:96)!
(cid:96)
(cid:96)
(cid:96)
6
MUSTAZEE RAHMAN
that (cid:96) is fixed).
(3d − 3)(cid:96) if d ≥ 3.
It follows from these observations that E(cid:2) C(cid:96)
(cid:3) →... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
sptr(Gn)
|Gn
|
= log d
−
(cid:88) pk
G(◦)
k
,
k
≥1
where d is the degree of G and ◦ is any fixed vertex. In this manner we will be able to find expressions
for the tree entropy of Zd and Td and asymptotically enumerate the number of spanning trees in
the grid graphs Z[−n, n]d and random regular graphs Gn,d.
3.1. The Matr... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
G.
(5) If G is connected and has maximum degree ∆ then L has |G| eigenvalues 0 = λ0 < λ1 ≤
· · · ≤ λ G
|
|−1 ≤ 2∆
.
LOCAL CONVERGENCE OF GRAPHS AND ENUMERATION OF SPANNING TREES
7
Let G be a finite connected graph. From part (5) of exercise 2.2 we see that the Laplacian L of
G has n = |G| eigenvalues 0 = λ0 < λ1 ≤ · · ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
x, y}. Prove that sptr(G) = sptr(G \ {x, y}) + sptr(G · {x, y}).
Try to prove the Matrix-Tree Theorem by induction on the number of edges of G, the identity
above, and the expression for the determinant in terms of the cofactors along any row.
It is better for us to express (2) in terms of the eigenvalues of the SRW tr... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
eigenvalues of P . If e is the number of edges in G
−
then we may rewrite (2) as
(3)
sptr(G =
)
(cid:81)
(
x∈V G) deg(x)
2e
n−1
(cid:89)
(1
i
=1
− µi .
)
This formula is derived from determining the coefficient of t in the characteristic polynomial of I −P ,
which equals (det(D))−1det(L − tD). This is a rather tedious ex... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
: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, ◦).
Then
n) converges to
log sptr(G
|Gn|
(cid:104)
h(G, ◦) = E
log deg(◦) −
1
pk
k G(
◦)
(cid:105)
.
(c... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
→ 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 converges to E(cid:2) pk
(cid:12)
(cid:12) (cid:2)
E pk
(cid:12)
(cid:2)
bounded and continuous if 1 ≤ x ≤ ∆. Ther... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
)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
∼
LOCAL CONVERGENCE OF GRAPHS AND ENUME... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
. 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 = (cid:0)(I − P )f, f (cid:1)... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
||P mf ||4 ≤ 2e2
∞
(cid:0)(I − P )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 ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
RAHMAN
Therefore,
(cid:12)
(cid:12)
(cid:12)
and 1 − π(x) ≤
(cid:12)
(cid:12) ≤ (cid:112)2eπ(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... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
1
n
n
G
n
≤
∆
(k + 1)1/4
.
(cid:12)
(cid:12)
≤ ∆ k− /4 and completes the proof of Theorem 3.3.
1
3.2. Tree entropy of T
able to compute the return probability of the SRW on the graph. There is a rich enough theory that
does this for d-regular tree and Zd. We begin
to calculate the tree entropy of a graph we have to be
... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
0
Note that (cid:80)
it can be shown that
h(Td) = log
(cid:104)
(d
2
− 1)d−1
(d − 2d)(d/2)−1
(cid:105)
.
This result was proved by McKay
random d-regular
(cid:2)
the local weak limit we see that E log sptr(Gn,d) (cid:3) = nh(Td) + o(n).
[8]. Since the
graphs Gn,d converge to Td in
A rigorous calculation of the tree ent... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
([0, 1]d) in that any function f
corresponds to an unique f ∈ L2([0, 1]d) such that f (k) = (cid:82) f (x)e2πi x·k dx. This correspondence
preserves the inner product between functions. The Fourier transform maps 1o to the function that
is identically 1 in [0, 1]d. It also transforms the operator log L to the operator ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
holds
(cid:104) (cid:88)
E
F (◦, x)
(cid:105)
=
(cid:104) (cid:88)
E
F (x, ◦)
(cid:105)
.
∼◦ in G
x
x∼◦ in G
Here is an example. Let G be a finite connected graph and suppose ◦ ∈ G is a uniform random root.
Then
(cid:80)
(G, ◦) is unimodular because both the left and right hand side of the equation above equals
(x,y) F ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
1] D. J. Aldous and R. Lyons, Processes on unimodular networks, Electron. J. Probab. 12 #54 (2007), pp. 1454–
1508.
[2] E. A. Bender and E. R. Canfield, The asymptotic number of labelled graphs with given degree sequences, Journal
of Combinatorial Theory Series A 24 (1978), pp. 296–307.
[3] I. Benjamini and O. Schramm, ... | https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-of-data-science-fall-2015/55ff9f23be313f3beefe692dda95aff9_MIT18_S096F15_Ses17.pdf |
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | 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 |
α=α¯=0
[
δ(4)(x − y1)
= ieB
δ2W
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... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
2 − k1. This is an important constraint. To see the implications, we consider k1
q −→ 0, meaning k2 is 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... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
Z2ψ,
SB
= Z2S,
we have that
√ √
phys),
µ = Z3( Z2)2G(
µ
GB
where GB ≡ 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 renorm... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
for connected diagrams with
δ
δ
δJδ1 (z1) δJδ2 (z2)
. . .
δ
δ
δη¯(y1) δη¯(y2)
. . .
δ
δ
δη(x1) 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
.... | https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/565bce007bbabc0e457a758d66c22814_MIT8_324F10_Lecture13.pdf |
to Γµ an external photon line, that is, ϵµΓµ(k, . . .). This is invariant under ϵµ −→
ϵµ + kµ, which is a gauge transformation Aµ −→ Aµ + ∂µΛ.
Only charged particles need to be on-shell. Other photon lines or any other neutral particles (if they
exist) can be off-shell, since they they do not transform under gauge tr... | 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 |
exercising network models of (or helpful in understanding)
the system
Professor C. Magee, 2006
Page 4
A Few “Thought-starter” 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 languag... | 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 |
s
Normal (and nearly so)
Skewed (and heavily skewed)
Skewed (and heavily skewed)
Suggest some normal or nearly 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
... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
12004
/0412004 v2 2
© 2007 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
5
Degree Distributions II
Define as the fraction of nodes in a network with
kp
degree k. This is equivalent to the probability of randomly
picking a node of degree k
k
d f d
i ki
kp
A plot of c... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
not be
very constraining (predictive)
very “
”
Iteration between models and observations is
essential
essential
© 2009 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
8 A Research Process
A Research Process
1. Development of conceptual understanding
(qualitative framework)
(
k... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
and event correlation)
Narrative (time and event correlation)
Numeracy (or quantitative thinking)
Having appropriate intuition about magnitude
Ability to quickly calibrate
Ability to make reasonable estimates about the system relatively
quickly
Knowing the numbers and the way they change over time
C... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
same time of day.
occupy on both trips at precisely the same time of day
© 2008 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
14
Self Observations on Thinking
Self Observations on Thinking
How was your thinking represented?
How did you know you were thinking?
How did you... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
pose to explore the
problem?
Such operations are nearly impossible in language
How difficult was it to “observe” your own
thinking?
thinking?
Most people infer operations by observing the
resulting representation
© 2008 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
17
... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
liers, comparative visual reasoning,
causal chains etc, is essential for
causal chains etc is essential for
effectiveness
Variety of representations and
Variety of representations and
innovation is constantly needed-this
is an important skill (methodology?)
gy )
p
(
© 2008 Chris Magee, Engineering Systems D... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
Systems
Application to Complex Systems
Categories from the small world paper
What do they mean?
Minard/Tufte and statistical thinking
g
/
Review and Discuss the Napoleon March
Graphic
© 2008 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
24 • Napoleon March 1812-13 to Mo... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
with statistical and verbal
descriptions of a data set
© 2008 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
27 Discussion of Rosling Video
Discussion of Rosling Video
Number of “dimensions” or variables
Possible “new observations” from
video (new to you not the world)
video... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
1 km
are needed to see this
QuickTime™ and
TIFF (Uncompressed) de
1 mi
http://www.fakeisthenewreal.org/subway/index.html
© 2008 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
Courtesy of Neil Freeman. Used with permission.
33 Design of Systems Representations
..continued
contin... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
e
P
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
Investme... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
links (k)
Barabasi 2002
© 2008 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
Image by MIT OpenCourseWare.
37
Small multiples (Tufte)
Small multiples (Tufte)
Image removed due to copyright restrictions.
© 2008 Chris Magee, Engineering Systems Division, Massachus... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/569fa3a8ce574e4a6d48a72ab5b7cb6f_MITESD_83F11_lec06.pdf |
, language and mathematics
Levels of thinking..
Operations: patterns and matching (accuracy and
speed, decomposition and holistic approaches)
h )
iti
Appreciate the value of effective visual
d hli ti
d d
g
representation for communication and thinking
b
i f
b ildi
Form basis for building skill at... | 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 |
(w1 | START)
×P (w2 | START, w1)
×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)
... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
Count(the, dog)
Evaluating a Language Model
• We have some test data, n sentences
S1, S2, S3, . . . , Sn
• We could 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 ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
. . . Third, the notion “grammatical in English” cannot be
identified in any way with the notion “high order of statistical
approximation to English”. It is fair to assume that neither
sentence (1) nor (2) (nor indeed any part of these sentences) has
ever occurred in an English discourse. Hence, in any statistical
... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
L(wi) =
Count(wi)
Count()
How close are these different estimates to the “true” probability
P (wi | wi−2, wi−1)?
Linear Interpolation
• Take our estimate Pˆ(wi | wi−2, wi−1) to be
Pˆ(wi | wi−2, wi−1) = �1 × PM L(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.... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
) log Pˆ(w3 | w1, w2)
such that �1 + �2 + �3 = 1, and �i � 0 for all i, and where
Pˆ(wi | wi−2, wi−1) = �1 × PM L(wi | wi−2, wi−1)
+�2 × PM L(wi | wi−1)
+�3 × PM L(wi)
An Iterative Method
Initialization: Pick arbitrary/random values for �1, �2, �3.
Step 1: Calculate the following quantities:
c1 =
c2 =
c3 =
�
w1 ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
vary
• Take a function � that partitions histories
e.g.,
�(wi−2, wi−1) =
1 If Count(wi−1, wi−2) = 0
2 If 1 � Count(wi−1, wi−2) � 2
3 If 3 � Count(wi−1, wi−2) � 5
4 Otherwise
�
�
�
�
�
�
�
�
⎪
• Introduce a dependence of the �’s on the partition:
Pˆ(wi | wi−2, wi−1) = ��(wi−2,wi−1) × PM L(wi | wi−2, wi−1)
+�... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
−2, wi−1)
w PM L(w | wi−1)
w PM L(w)
⎨
⎨
�(wi−2 ,wi−1 )
= �1
�(wi−2 ,wi−1 )
+ �2
�(wi−2 ,wi−1 )
+ �3
= 1
An Alternative Definition of the �’s
• A small change: take our estimate Pˆ(wi | wi−2, wi−1) to be
Pˆ(wi | wi−2, wi−1) =
�1 × PM L(wi | wi−2, wi−1)
+(1 − �1)[�2 × PM L(wi | wi−1) +(1 − �2) × PM L(wi... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
of a
development set.
Discounting Methods
• Say we’ve seen the following counts:
x
the
Count(x) PM L(wi | wi−1)
48
15
11
10
5
2
1
1
1
1
1
• The maximum-likelihood estimates are systematically high
the, dog
the, woman
the, man
the, park
the, job
the, telescope
the, manual
the, afternoon
the, c... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
, w)
Count(wi−1)
�(wi−1) = 1 −
�
w
e.g., in our example, �(the) = 10 × 0.5/48 = 5/48
• Divide the remaining probability mass between words w for
which Count(wi−1, w) = 0.
Katz Back-Off Models (Bigrams)
• For a bigram model, define two sets
A(wi−1) = {w : Count(wi−1, w) > 0}
B(wi−1) = {w : Count(wi−1, w) = 0}
• ... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
�
�
�
�
�
�
⎪
Count�
(wi−2,wi−1,wi )
Count(wi−2,wi−1)
If wi � A(wi−2, wi−1)
�(wi−2,wi−1)PKAT Z (wi |wi−1)
w�B(wi−2 ,wi−1 )
⎨
PKAT Z (w|wi−1)
If wi � B(wi−2, wi−1)
�(wi−2, wi−1) = 1 −
�
w�A(wi−2 ,wi−1 )
Count�(wi−2, wi−1, w)
Count(wi−2, wi−1)
Good-Turing Discounting
• Invented... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
– Syntactic models.
It’s generally hard to improve on trigram models, though!!
Further Reading
See:
“An Empirical Study of Smoothing Techniques for Language
Modeling”. Stanley Chen and Joshua Goodman. 1998. Harvard
Computer Science Technical report TR-10-98.
(Gives a very thorough evaluation and description of a... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
dog
VP � VB
VB � laughs
PROBABILITY
1.0
0.3
1.0
0.1
0.4
0.5
TOTAL PROBABILITY = 1.0 × 0.3 × 1.0 × 0.1 × 0.4 × 0.5
Properties of PCFGs
• Assigns a probability to each left-most derivation, or parse-
tree, allowed by the underlying CFG
• Say we have a sentence S, set of derivations for that sentence
is T (S)... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
PCFG and a sentence S, define T (S) to be
the set of trees with S as the yield.
• Given a PCFG and a sentence S, how do we find
arg max P (T, S)
T �T (S)
• Given a PCFG and a sentence S, how do we find
P (S) =
P (T, S)
�
T �T (S)
Chomsky Normal Form
A context free grammar G = (N, �, R, S) in Chomsky Normal
Form... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
: define P (Nk � wi | Nk ) = 0 if Nk � wi is not in the grammar)
• Recursive definition: for all i = 1 . . . n, j = (i + 1) . . . n, k = 1 . . . K,
λ[i, j, k] =
max
i � s < j
1 � l � K
1 � m � K
{P (Nk � NlNm | Nk ) × λ[i, s, l] × λ[s + 1, j, m]}
(note: define P (Nk � NlNm | Nk ) = 0 if Nk � NlNm is not in the
gra... | https://ocw.mit.edu/courses/6-864-advanced-natural-language-processing-fall-2005/56a788aee71d6c78c483d3b7596e9477_lec3.pdf |
non-terminal
N1 = S (the start symbol)
w.l.g.,
• Define a dynamic programming table
λ[i, j, k] = sum of probability of parses with root label Nk
spanning words i . . . j inclusive
• Our goal is to calculate
⎨
T �T (S) P (T, S) = λ[1, n, 1]
A Dynamic Programming Algorithm for the Sum
• Base case definition: f... | 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 |
i
v
e
D
e
v
i
t
a
l
e
R
20
10
8
6
4
2
1
0.8
0.6
0.4
0.2
Conventional 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 ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
P
Active
Power
100
10
1
0.1
0.01
Passive Power
0.001
1
1994
2004
0.1
0.01
Gate Length (microns)
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
6
6.189 IAP 2007 MIT
Has This Ever Happened Before?
)
2
m
c
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s
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a
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x
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l
F
t
a
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12
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6
4
2
Steam IRO... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
9000
CMOS
Prescott
Steam IRON
5W/cm2
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
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Jayhawk(dual)
T-Rex
Mckinley
IBM GP
Squadrons
IBW RY5
IBM RYZ
Pentium 4
P... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
7
1
Systems and Technology Group
Cell History
● IBM, SCEI/Sony, Toshiba Alliance formed in 20 00
● Design Center opened in March 2001
Based in Austin, Texas
● Single Cell BE operational Spring 2004
● 2-way SMP operational Summer 2004
● February 7, 2005: First technical disclosures
● October 6, 2005: Mercury ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
-time and non-real time worlds
Michael Perrone © Copyrights by IBM Corp. and by other(s) 2007
13
6.189 IAP 2007 MIT
Cell Design Goals
● Cell is an accelerator extension to Power
Built on a Power ecosystem
Used best know system practices for processor design
● Sets a new performance standard
Exploits pa... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/56d952b2cf5c52c89a4018bd081ac92b_lec2cell.pdf |
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