text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
f (b)) ≤ y
is valid for P 1 .
• For P 2 we combine −(x − (cid:5)b(cid:6)) ≤ −1 and x − y ≤ b with multipliers f (b) and
1: (x − (cid:5)b(cid:6))(1 − f (b)) ≤ y. By disjunction, (x − (cid:5)b(cid:6))(1 − f (b)) ≤ y is valid for
conv(P 1 ∪ P 2) = conv(X).
Slide 18
5
(cid:6)
3.5.3 Theorem
(cid:7)
(cid:7)
(cid:5)n A... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
, j} ∈ E is called a clique.
(cid:2)
xi ≤ 1,
i∈U
for any clique U (∗)
is valid.
• A clique U is maximal if for all i ∈ V \ U , U ∪ {i} is not a clique.
• (*) is facet defining if and only if U is a maximal clique.
• U = {1, . . . , k}. Then, ei, i = 1, . . . , k satisfy (*) with equality.
• For each i (cid:10)∈... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
x2 + x3
x1
x1 +
x1 +
x1 + x2
≤ 1
≤ 1
≤ 1
x5 + x6 ≤ 1
+ x6 ≤ 1.
unique optimal solution x 0 = (1/2)(0, 1, 1, 1, 1, 1)(cid:2) . Do maximal clique inequalities
describe convex hull?
Slide 25
6
2
1
3
4
6
5
• x 0 does not satisfy
x2 + x3 + x4 + x5 + x6 ≤ 2. (1)
• Stable sets {2, 4}, {2, 5}, {3, 5}, {3, 6}... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
2, {2, 4}, {2, 5}, {3, 5}, {3, 6}, {4, 6}, and {1} satisfy it with equality.
• 2x1 + x2 + x3 + x4 + x5 + x6 ≤ 2, is valid and defines a facet conv(S).
4.2.1 General principle
(cid:11)
Suppose S ⊂ {0, 1}n, Si = S∩ x ∈ {0, 1}n (cid:7) x1 = i , i = 0, 1, and
is valid for S0 .
(cid:12)
(cid:7)
(cid:5)
n aj xj ≤ a0 (2... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
• P = conv {x ∈ {0, 1}6 | 5x1 + 5x2 + 5x3 + 5x4 + 3x5 + 8x6 ≤ 17}.
• x1 + x2 + x3 + x4 ≤ 3 is valid for P ∩ {x5 = x6 = 0}.
• Lifting on x5 and then on x6, yields x1 + x2 + x3 + x4 + x5 + x6 ≤ 3.
• Lifting on x6 and then on x5, yields x1 + x2 + x3 + x4 + 2x6 ≤ 3.
Slide 29
8
15.083J/6.859J Integer Optimization
Le... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
, 1)(cid:2) , b3 = (1, −1)(cid:2) . Then, L(b1 , b2
) = L(b2 , b3
).
Slide 3
x2
0011
00 4
11
1
0
11
00
00
11
3
0011
01
10
4 x1
01
01
01
b1
0011
2 00000
11
111
000111
b3 000
111
1 000111
01
01
00000
00
11
11111
b2
111
000
0011
11111
00000
111
000
0011
11111
00000
111
000
11
00011111111
00000
... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
• For all x ∈ L: x = Bv with v ∈ Z d .
• det(U ) = ±1, and det(U −1) = 1/ det(U ) = ±1.
• x = BU U −1 v.
• From Cramer’s rule, U −1 has integral coordinates, and thus w = U −1 v is
integral.
Slide 4
Slide 5
• B = BU . Then, x = Bw, with w ∈ Z d, which implies that B is a basis of L.
d
1 , . . . , b
] be bas... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
if and only if −z ∈ A.
Then A contains a non-zero lattice point.
n
n
2.5
Integer normal form
• A ∈ Z m×n of full row rank is in integer normal form, if it is of the form [B, 0],
where B ∈ Z m×m is invertible, has integral elements and is lower triangular.
• Elementary operations:
(a) Exchanging two columns;
(... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
max{i : a1,i > 0}. If k = 1, then
we have transformed A into a matrix of the form (1). Otherwise, k ≥ 2 and by
applying k − 1 operations (c) we transform A to
�
�
�
�
a1,1 A2, . . . , Ak−1 −
a1,2
�
a1,k−1 Ak, Ak, Ak+1, . . . , An
.
a1,k
�
A = A1 −
• Repeat the process to A, and exchange two columns of ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
(j, j) is −1. det(U 2) = −1.
(iii) Adding f ∈ Z times column k to column j, corresponds to multiplying
matrix A by a unimodular matrix U 3 = I + f I k,j . Since det(U 3) = 1, U 3 is
unimodular.
• Performing two elementary column operations corresponds to multiplying the
corresponding unimodular matrices resulting... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
] = AU . Let b ∈ Z m and S = {x ∈ Z n | Ax = b}.
(a) The set S is nonempty if and only if B −1b ∈ Z
m
(b) If S (cid:6)= ∅, every solution of S is of the form
.
Slide 10
x = U 1B −1b + U 2z, z ∈ Z n−m ,
where U 1, U 2: U = [U 1, U 2].
(c) L = {x ∈ Z n | Ax = 0} is a lattice and the column vectors of U 2 constitut... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
z ∈ Z n−m}.
• Let L = {x ∈ Z n | Ax = 0}. By setting b = 0 in part (b) we obtain that
L = {x ∈ Z n | x = U 2z, z ∈ Z n−m}.
Thus, by definition, L is a lattice with basis U 2.
2.10 Example
• Is S = {x ∈ Z 3 | Ax = b} is nonempty
�
�
A =
3
4
6
5
1
5
Slide 12
and b =
�
�
.
3
2
• Integer normal form: [B, ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
and y(cid:2)b /
∈ Z.
• The set S = ∅ if and only if there exists a y ∈ Qm, such that y ≥ 0,
y(cid:2)A ∈ Z and y ∈ Z
.
(cid:2)b /
m
2.12 Proof
• Assume that S = ∅. If there exists y ∈ Qm, such that y (cid:2)A ∈ Z and y ∈ Z,
(cid:2)b /
m
(cid:6)
then y (cid:2)Ax = y (cid:2)
b with y
(cid:2)Ax ∈ Z and y (cid:2)... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
, z ∈ Z n−m}.
•
max c (cid:2)U 2z
s.t U 2z ≥ −x 0
z ∈ Z n−m .
• Different bases give rise to alternative reformulations
max c (cid:2)Bz
s.t. Bz ≥ −x 0
z ∈ Z n−m .
6
MIT OpenCourseWare
http://ocw.mit.edu
15.083J / 6.859J Integer Programming and Combinatorial Optimization
Fall 2009
For information about citi... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/5be17dfa9a150e8db5b9fb76905d2408_MIT15_083JF09_lec03.pdf |
4 SCETI LAGRANGIAN
where
Wn =
(cid:88)
(cid:88) −g)k
(
k p
erm
k!
(cid:32)
n¯
· An(q1) · · · n¯ · An(qk)
[n¯ · q1][n¯ · (q1 + q2)] · · · [n¯ · (cid:80)
k
i=1 qi]
(cid:33)
.
(3.34)
Here Wn is the momentum space version of a Wilson line built from collinear An gluon fields. In position
space the corresponding Wilson li... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
ons, and momentum degrees of freedom. On the way to our final result we introduce the label operator
which provide a simple method to separate large (label) momenta from small (residual) momenta.
4.1 SCET Quark Lagrangian
Lets construct the leading order SCET collinear quark Lagrangian. This desired properties that t... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
⊥ and ¯n · k « n¯ · p corresponds to carrying out a
multipole expansion for the interaction of the ultrasoft gluon with the collinear quark. The LO collinear
quark propagator must be smart enough to give the correct leading order result without further expansions,
irrespective of whether it later emits a collinear g... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
0 ,
n D⊥
ˆ
D⊥
ˆ
n
ˆ
n
ϕn¯
i D/ ⊥ϕn = 0 ,
(4.4)
since ξnPn = 0 and ϕ¯n¯Pn¯ = 0. These simiplifications leave us with the Lagrangian
¯
L = ξˆ
n
n/
2
in · D ξˆn + ϕ¯iD/ ⊥ ξˆn + ξˆ
n
n iD/ ⊥ϕn¯ + ϕn¯
n/
2
in¯ · Dϕn¯ .
(4.5)
So far this is just QCD written in terms of the ξˆn and ϕn¯ fields. However, the field ϕn¯ corr... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
0
(4.6)
in¯ · Dϕn¯ +
ϕn¯ =
1
in¯ · D
n/¯
2
ˆ
iD/ ⊥ξn = 0
n/¯ ˆ
iD/ ⊥ ξn ,
2
where the second line is obtained by multiplying the first by /¯n/2 on the left, and the plus sign in the last
= −i / ¯ Plugging this result back into our Lagrangian, two terms cancel,
line comes from using /¯
and the other two terms g... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
inear and ultrasoft gauge fields, nor the corresponding momentum components. These remaining
steps will be to
2. Separate the collinear and ultrasoft gauge fields.
3. Separate the collinear and usoft momentum components with a multipole expansion.
We then can expand in the fields and momenta and keep the leading piece... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
µ = Aµ + Aµ + · · · .
us
n
(4.9)
Here the ellipsis stand for additional terms involving Wilson lines which only will become relevant when
we formulate power corrections, and hence will be ignorded for our leading order analysis here (they are
given below in Eq.()). The interpretation of Aµ
n will also prove useful... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
us to formulate power corrections in a manner where operators give homogeneous
contributions in λ order by order. For example, consider the denominator of the propagator of a quark
with momentum pn + kus expanded to keep the leading and first subleading terms
1
(pn + kus)2
= −
(pn
1
us)(pn + k−
+
⊥
us) + (pn
+... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
=
(cid:90)
(cid:90)
(cid:90)
dx
dp1 dp2 dk e
ip1x
ik(0)
e−
ip2x ¯
e−
ψ(p1)Aus(k)ψ(p2)
dp1 dp2 dk δ(p1 − p2) ψ(p1)Aus(k)ψ(p2).
¯
(4.12)
We see immediately that this corresponds to a 3-point Feynman rule where the small momentum k is
ignored relative to the large momenta p1 and p2, and that total momentum is not conserv... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
to achieve this. It will allow us to 1) simply derive
the corresponding momentum space Feynman rules, 2) simplify the formulation of gauge transformations
in the effective theory, and 3) incorporate the multipole expansion into propagators rather than vertices.
For the moment we only consider the quark part of the fie... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
so we have a grid as shown in Fig. 5 (for convenience we show
only one of the pµ components). Note that any momentum p has a unique decomposition in terms of
label and residual components. Since pc » pr the spacing between grid points is always much larger than
the spacing between points in a box. This setup has the... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
= 0 does not define a collinear
momentum. Indeed the pc = 0 bin corresponds to the ultrasoft modes. For an ultrasoft momentum p we
simply have
(cid:90)
(cid:90)
d4p →
d4pr .
With this momentum separation we can also label our fields by both components
˜ξn(p) → ˜ξn, p(cid:96)(pr) .
We also have separate conservation l... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
these fields. Altogether, the above steps allow us to rewrite our hatted
collinear field ξˆn(x) as
(cid:90)
ˆξn(x) =
d4p −ip·x ˜
e
(2π)4
ξn(p) =
(cid:90)
(cid:88)
p
(cid:96)(cid:54)=0
(cid:88)
=
p
l=0(cid:54)
e−ip(cid:96)·x ξn, p(cid:96)(x) .
4
d pr e
−ip(cid:96)·x −ipr·x ˜
e
ξn, p (pr)
(cid:96)
(4.21)
We can identify ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
, p£ (x).
Recall that P µ and p only contains the components P ≡ n¯ · P ∼ p ∼ λ0 and P µ ∼ p ∼ λ. Therefore
we have n · P = 0. Also
(4.23)
⊥µ
c
µ
c
−
c
⊥
µ
c
in¯ · ∂ « P ,
µ .
i∂µ « P⊥
⊥
(4.24)
The main advantage of the label operator is that it provides a definite power counting for derivatives. It
is als... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
.26)
where the label operator acts on both fields. Consequently, conservation of label momenta is simply
encoded by this phase factor and is manifest at the level of operators.
Lastly, we must deal with anti-particles and gluons. For the anti-particles, we expand our Dirac field
into two parts
(cid:90)
ψ(x) =
d4p δ(... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
+ ξ−
n, p£
n, −p£
(x)
(4.29)
where pc has either sign, but one picks out particles and one picks out antiparticles. Thus the action of
the fields ξn,p£ and ξn,p£ is that for
¯
n¯ · pc > 0 :
n¯ · pc < 0 :
a particle is destroyed or created
an antiparticle is created or destroyed
The sign convention for the labe... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
destroys a gluon, while for q < 0 it creates a gluon.
With our conventions the action of the label operator on a bunch of labelled fields is
−
c
−
c
P µ(φ† φ† · · ·
q1 q2
φp1 φp2
µ
µ
1 + p2 + ·
· · · ) = (p
· · − q − q − · · · )(φ† φ† · · ·
q1 q2
µ
2
µ
1
φp1 φp2
· · · ).
(4.34)
Thus it gives a minus sign wh... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
result:
(cid:18)
L =
ˆξn
in · D + iD/ ⊥
1
iD/
in¯ · D ⊥
(cid:19) ¯ ˆξn.
n/
2
Changing i∂µ → (Pµ + i∂µ) and ξˆ n → ξn and expanding our derivative operators, we have
in · D = in · ∂ + gn · An + gn · An
µ
+ gA )
n⊥
(cid:123)(cid:122)
(cid:125)
∼λ
P µ
iD = (
⊥
⊥
(cid:124)
+ (i∂µ
(cid:124)
⊥ + gAµ
(cid:123)(cid:122)
∼λ2
... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
+ gAµ
n⊥ ,
⊥
in¯ · Dn = P + gn¯ · An.
Remarks:
(4.39)
(4.40)
• Both terms with covariant derivatives in the (· · · ) in L
are of order λ2 so the leading order La
grangian is order λ4 (recalling that the fields scale as ξn ∼ λ). Since for a Lagrangian with collinear
d4x ∼ λ−4 this gives us an action that is ∼ λ0 a... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
attitude that low energy locality is a desired property
for the effective field theory.
• If we are considering a situation without ultrasoft particles, and without hard interactions that do
not couple to a particular component, then the interaction of collinear fermions alone could equally
29
4.2 Wilson Line Ide... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
· pc
in/
2 (¯n · pc)(n · pr) + (pc⊥)2 + i0
.
(4.42)
The leading order Lagragian is smart enough that it gives the correct propagator in different situations
without having to make further expansions. This is important to ensure that the leading order Lagrangian
strictly give O(λ0) terms, while subleading Lagrangian... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
= (P + gn¯ · An)Wn = 0 .
(4.45)
30
k,uspplr)(,pplr)(+kus,pplr)(,pplr)(+,qqlr)(qr+ql(cid:54)
4.3 Collinear Gluon and Ultrasoft Lagrangians
4 SCETI LAGRANGIAN
With this definition, the action of in¯ · Dn on a product of Wn and some arbitrary operator is
P + gn¯ · An)WnO
in¯ · Dn(WnO) = (
(cid:2)
= (P + gn¯ · An)Wn
... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
P
= Wn
= Wn
1
1
W † .
W †
in¯ · D n .
n
in¯ · Dn
n
The first relation allows us to rewrite L as
(0)
nξ
L(0) = e
nξ
−ix·P ¯ξn
(cid:16)
in · D + i /Dn⊥W †
n
1
P
Wni /Dn⊥
(cid:17) /¯n
2
ξn .
(4.49)
(4.49)
(4.50)
It is also useful to note that we can use the label operator to write a tidy expression for the Wilson line... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
Dµ, Dν ]. Expanding the covariant derivative as we did in the quark sector we keep only
the leading order terms. For a covariant derivative acting on collinear fields the leading order terms are
i
g
iDµ → iDµ =
µ
(P + gn¯ · An) + (P µ + gAµ
n
2
Recall that the field Aµ varys much more slowly than Aµ
n, so we can th... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
− /(cid:48) p/⊥ ⊥
p
n¯
n¯·p n¯·p (cid:48) µ
(cid:21)
¯
n/
2
= ig2 T A T B
n¯·(p−q)
(cid:20)
⊥γν
µ
γ
⊥ − µ
⊥p/⊥
¯n·p ¯nν − p (cid:48)/⊥γ⊥
¯n·p (cid:48) ¯nµ + p (cid:48)/⊥p/⊥
n¯·p n¯·p (cid:48) µn¯ν
n¯
ν
(cid:21)
¯
n/
2
+ ig2 T B T A
n¯·(q+p(cid:48))
⊥γµ
ν
⊥ − γ p/⊥
¯n·p ¯nµ − p (cid:48)/⊥γ⊥
¯n·p (cid:48) ¯nν + p (cid:48... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
collinear gluon Lagrangian is then
L(0)
ng =
1
2g2
(cid:8)
Tr ([i µ, i
D D
µ
(cid:9)
])2
+ τ Tr ([i µ , An µ])2 + 2Tr cn[i us, [i µ, cn]]
Dµ D
Dus
(cid:9)
(cid:8)
(cid:8)
(4.54)
(cid:9)
.
(4.55)
For the Langrangian with only ultrasoft quarks and ultrasoft gluons, at lowest order we simply have
the QCD actions. Using ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
oft modes,
L(0)
= L
(0) + L(0) + L(0)
us
ng
nξ
.
(4.57)
32
μ , Appɂμ , Appɂμ , Aν , Bq 4.4 Feynman Rules for Collinear Quarks and Gluons
4 SCETI LAGRANGIAN
=
−
i
n¯ ·q n·k + q2 + i0
⊥
(cid:18)
µν − (1 − τ )
g
µqν
q
n¯ ·q n·k + q2
⊥
(cid:19)
δ
a,b
= gf abcnµ
(cid:8)n¯ · q1 gνλ − 1 (12 − 1 )[n¯τ
λq1ν + n¯νq2λ](... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
that follow from the collinear quark and gluon
Lagrangians. We do not show the purely ultrasoft interactions which are identical to those of QCD, nor
do we show the purely collinear gluon interactions which are also identical to those of QCD.
The Feynman rules that follow from the leading order collinear quark Lagra... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
size of momenta.
33
a, μb, ν(q, k)b, νc, λa, μq2q1a, μb, νc, λd, ρa, μb, νc, λd, ρ 4.5 Rules for Combining Label and Residual Momenta in Amplitudes
4 SCETI LAGRANGIAN
Instead of this, we need to use a Continuum EFT picture where the EFT modes have propagators that
extend over all momenta, but integrands which obta... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
the case with d-dimensions) is
resolved, given a pair of momenta components (pc, pr) ∈ Rd−1 × Rd . The upshot is that in the simplest
cases the residual momentum can simply be dropped or absorbed into a label momentum in the same
direction (making it continuous), while in the most complicated cases the formalism lea... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
onshell
condition k2 = 0. For the external lines that are collinear it suffices to take label momenta p = n¯ · pc
−
and pc⊥, and a single residual momentum p . This amounts to picking βµ above to contain the full pr
and pµ components. The onshell condition for the collinear particles is then simply p − p − pp 2 = 0.
All... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
particles or collinear loops, and residual momenta for external
+
ultrasoft particle, external collinear particles from p , and from collinear and ultrasoft loops.
r
First we note that if we integrate over all label momenta qc and residual momenta qr that this will be
equal to an integration over all of the qµ momen... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
q+
r ) =
(cid:96)
(cid:88) (cid:90)
q(cid:96)
ddqr F (q− + q− q(cid:96) + q
r , ⊥
(cid:96)
(cid:90)
⊥, qr ) =
+
r
d q F (q−, q⊥, q ) . (4.60)
d
+
In the first step we use the fact that F is constant throughout each box in the grid picture of Fig. 5 so its
the same with the first two arguments shifted by residual momenta... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
the momentum of a collinear propagator. These are referred to as 0-bin restrictions.4 We
will discuss the change needed which handles this complication below. Often the results for collinear loop
integrals are called “naive” if one uses Eq. (4.60). The result from this naive result will be correct if the
added terms... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
for an ultrasoft loop integration. Both
ultrasoft loop integrations and ultrasoft external particles introduce residual momenta into propagators
that can not be absorbed by a rule like that in Eq. (4.59). If we consider a case with an ultrasoft loop
integration, then there will be dependence on the residual momentum... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
(4.63)
(4.63)
where qc = 0 is simply a label to denote the fact that the label momentum qc must be large in order
to correspond to a collinear particle carrying total momentum q. If qc = 0 then the particle would
instead be ultrasoft, and we will often have included another diagram in SCET to account for the differen... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
−
r , q⊥
(cid:96) + q⊥
r , q+
r ) −
(cid:90)
ddqr F 0(q−
r , q⊥
r , q+
r )
q(cid:96)
(cid:90)
ddq F (q−, q⊥, q+) −
(cid:90)
ddqr F 0(q−
r , q⊥
r , q+
r )
(cid:90)
ddq (cid:2)F (q−, q⊥, q+) − F 0(q−, q⊥, q+)(cid:3) .
(4.64)
Here the integrand F 0 is derived from expanding the integrand for F by taking the label momenta ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
scaling for the subtraction is shown
pictorally in Fig. 8. The F 0 term subtracts singularities from F that come from the region where the
collinear momentum behaves like an ultrasoft momentum. In general when the subtraction integration is
non-trivial there will always exist a corresponding ultrasoft diagram where ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
)
(cid:54)
(cid:54)
(cid:54)
(cid:54)
(cid:54)
(cid:54)
(cid:54)
MIT OpenCourseWare
http://ocw.mit.edu
8.851 Effective Field Theory
Spring 2013
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5c136dd2b1e30612a9fdaafd4be06029_MIT8_851S13_SCETLagrania.pdf |
Introduction to Robotics, H. Harry Asada
1
Chapter 6
Statics
Robots physically interact with the environment through mechanical contacts. Mating
work pieces in a robotic assembly line, manipulating an object with a multi-fingered hand, and
negotiating a rough terrain through leg locomotion are just a few examples ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
i from link
i+1 is then given by –fi,i+1. The gravity force acting at the mass centroid Ci is denoted mig, where
mi is the mass of link i and g is the 3x1 vector representing the acceleration of gravity. The
balance of linear forces is then given by
f
i
,1
i
−
f
− +
ii
1,
+
mi
g
=
0
,
i
(cid:34)=
,1
,
n
(6.1.1)
No... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
1, +
Link i
C i
iCi,r
,1−r
i
i
Oi
gim
Link i+1
Actuator i
iτ
z
O
x
y
Figure 6.1.1 Free body diagram of the i-th link
The force
,1−f
i
i
and moment
N
i
,1−
i
are called the coupling force and moment between
. These are
the adjacent links i and i-1. For i=1, the coupling force and moment are
interpreted... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
3x1 vectors. The number of
coupling forces and moments involved is 2(n+1). Therefore two of the coupling forces and
moments must be specified; otherwise the equations cannot be solved. The final coupling force
and moment,
environment. It is this pair of force and moment that the robot needs to accommodate in order ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
forces and moments are so called “constraint forces and moments” merely
joining adjacent links together. Therefore, constraint forces and moments do not participate in
energy formulation. This significantly reduces the number of terms and, more importantly, will
provide an explicit formula relating the joint torques... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
1
f
i
,1
−
i
(6.2.2)
Note that, although we use the same notation as that of a revolute joint, the scalar quantity iτ has
the unit of a linear force for a prismatic joint. To unify the notation we use
joints, and call it a joint torque regardless the type of joint.
iτ for both types of
iτ
,1−f
i
i
,1−N
i
i
Revol... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
T
q
=
q
n
Department of Mechanical Engineering
Massachusetts Institute of Technology
Introduction to Robotics, H. Harry Asada
5
The explicit relationship between the n joint torques and the endpoint force F is given by
the following theorem:
Theorem ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
+nn
exδ
eδϕ
Figure 6.2.2 Virtual displacements of the end effecter and individual joints
(cid:34)
We assume that joint torques
the serial linkage system, while the joints and the end-effecter are moved in the directions
geometrically admissible. Then, the virtual work done by the forces and moments is given by
a... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
coordinates that are complete
and independent. Therefore, for the above virtual work to vanish for arbitrary virtual
displacements we must have:
τ
T=
FJ
This is eq.(6.2.4), and the theorem has been proven.
The above theorem has broad applications in robot mechanics, design, and control. We
will use it repeatedly i... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
1
)
(cid:65)
1
(cid:65)
cos(
cos
+
θ
1
2
(cid:65)
cos(
+
θθ
2
+
θθ
1
2
)
2
1
)
⎞
⋅⎟⎟
⎠
⎛
⎜
⎜
⎝
F
x
F
y
⎞
⎟
⎟
⎠
(6.2.8)
Department of Mechanical Engineering
Massachusetts Institute of Technology
Introduction to Robotics, H. Harry Asada
7
y
Endpoint
Fo... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
. The null space N(J), on the other hand, represents the set of joint
velocities that do not produce any velocity at the end-effecter. If the null space contains a non-
zero element, the differential kinematic equation has an infinite number of solutions that cause
the same end-effecter velocity.
The lower half of F... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
(JT)
mV∈F
Figure 6.3.1 Duality of differential kinematics and statics
The ranges and null spaces of J and JT are closely related. According to the rules of linear
algebra, the null space N(J) is the orthogonal complement of the range space R(JT). Namely, if a
non-zero n-vector x is in N(J) , it cannot also belong to... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
Introduction to Robotics, H. Harry Asada
9
6.4 Closed-Loop Kinematic Chains
The relationship between joint torques and the endpoint force obtained in Theorem 6.1
can be extended to a class of parallel-link mechanisms with closed kinematic-chains. It can also
be extended to multi-fingered hands, leg locomotion, and ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
5
4
=
2
δ
Work
=
(
ττ
3
1
T
)
δθ
⎛
1
⎜⎜
δθ
⎝
3
⎞
−⎟⎟
⎠
(
F
x
T
)
F
y
x
δ
⎛
e
⎜⎜
y
δ
⎝
e
⎞
⎟⎟
⎠
.
(6.4.2)
For any given configuration of the robot, the virtual displacements of the end-effecter are
uniquely determined by the virtual displacements of Joints 1 and 3. In fact, the former is related
to the latter via th... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
reedom robot mechanism with n active joints. Assume that all
the joints are frictionless, and that, for a given configuration of the robot mechanism, there exists
a unique Jacobian matrix relating the virtual displacements of its end-effecter,
, to the
1m
×ℜ∈δ
p
virtual displacements of the active joints,
1n
×ℜ∈δ
... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
by
Department of Mechanical Engineering
Massachusetts Institute of Technology
Introduction to Robotics, H. Harry Asada
δ
Work
=
+
δθτδθτδθτδθτ
4
+
+
1
2
3
4
2
3
1
−
xF
δ
x
e
−
yF
δ
y
e
11
(6.5.1)
Note t... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
force applied to the object, as illustrated in the figure. This internal force is a grasp force
that is needed for performing a task.
2θ
y
1θ
Finger 1
Grasped
Object
External
Endpoint Force
x ,
A y
A
Grasp
Force
Finger 2
4θ
3θ
x
Figure 6.5.1 Two-fingered hand manipulating a grasped object
Department of ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c160fb678fa75191c399a373c6ce648_chapter6.pdf |
Lecture 9 (On-Line Video)
The Hydrogen Atom
Today’s Program:
1. Angular momentum, classical and quantum mechanical.
2. The Hydrogen atom semi-classical approach.
3. The Hydrogen atom quantum mechanical approach.
4. Eigenfunctions and eigenvalues common to Hˆ , Lˆ2 and Lˆ
z .
5. The radial dependence.
6. Energy... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
ivity of free space.
II The Classical Hamiltonian
H r, p
2
p2
e
2 r
Where:
m m
e
mme
p
p me is a reduced mass.
Semiclassical Bohr model that yielded correct energy (but produced generally incorrect results)
relied on the conservation of energy, angular momentum and the balance of forces.... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
electron from the ground state).
Also the Bohr radius also correctly described atomic dimensions.
2
III The Quantum mechanical Hamiltonian
Hˆ rˆ
1, pˆ
2, pˆ
1, rˆ
2
2
pˆ1
2m1
2
pˆ2
2m2
Vˆ
2 rˆ
rˆ
1
Let’s perform the following coordinate transform – break our system ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
m2
, M m1 m2
Since the Center of Mass Hamiltonian commutes with the relative motion Hamiltonian, it
commutes with the whole system Hamiltonian, which means that energy of the center of mass is
conserved and can be used to label states. Then the Hamiltonian above breaks into two
independent problems – one for... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
�2
2 r r
2
r
1 2
r 2
2
2
1
tan sin2 2
1
e2
r
3
IV Finding the eigenfunctions and eigenvalues of the Hamiltonian.
2
1
tan sin2 2
1
e2
r
Hˆ rˆ, pˆ
r
2
1 2
2 r r
1 2
2
2 r r
2
r
2
2
1 ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
�2
1
1
Then we need to solve the following eigenvalue problem for the time-independent Hamiltonian:
2 1 2
2 r r
2 r
Lˆ2
2r
e2
u r,, Eu r,,
r
2
Note that the angular momentum squared operator commutes with the Hamiltonian and recall
that it also commutes with the z-co... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
�
r
m
m
z : Yl , we can look for the
l – the number associated with the eigenvalues of the orbital angular momentum magnitude
operator is called the orbital angular momentum quantum number.
�m – the number associated with the eigenvalues of the z component of the orbital angular
momentum called the magnetic qu... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
��3 n l 1! r l
n 2n n l! na0
2r
na0
2l1
Lnl1
, n l, n 0
Where L2l1 is the generalized Laguerre polynomials. Substituting it into Mathematica, we can
nl1
find the Bohr radius:
2
a0
e2
0.529 A
Example of atomic orbitals – 1s: u100 r,, R10 r
Y0 ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
l
5
Conclusions:
1. The energy levels or the energy eigenvalues En of the hydrogen atom depend only�on n ,
which is called the principal quantum number.
2. Since the energy levels and radial decay rate depend only on the n number this�number is
used to identify an electron shell.
3. For each energy ... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
probability of being near the origin.
9. The spectroscopic notation assigns a letter to each subshell:
l 0 s
l 1 p
l 2 d
l 3 f
6
Discussion
Comparison of the energy level (energy eigenvalues) spacing in free particle, 1D box, 1D
harmonic os... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/5c2116c92a8c7204af60846da432d6b5_MIT3_024S13_2012lec9.pdf |
Introduction to Robotics, H. Harry Asada
1
Chapter 3
Robot Mechanisms
A robot is a machine capable of physical motion for interacting with the environment.
Physical interactions include manipulation, locomotion, and any other tasks changing the state of
the environment or the state of the robot relative to the env... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c238d7e3445337ed40d4af83d9d2239_chapter3.pdf |
at some examples.
Robot mechanisms analogous to coordinate systems
One of the fundamental functional requirements for a robotic system is to locate its end-
effecter, e.g. a hand, a leg, or any other part of the body performing a task, in three-dimensional
space. If the kinematic structure of such a robot mechanism ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c238d7e3445337ed40d4af83d9d2239_chapter3.pdf |
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
i
3
3
3
3
3
3
3
3
3
3
3
3
x
x
x
x
x
x
x
x
x
x
x
x
z
z
z
z
z
z
z
z
z
z
z
z
0
0
0
0
0
0
0
0
0
0
0
0
Figure 3.1.2 Cartesian coordinate robot
Figure by MIT OCW.
Photo removed for copyright reasons.
GMF Robotics model M-100.
Figure 3.1.3 Cylindrical coordinate robot
Figure by MIT OCW.
Departm... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c238d7e3445337ed40d4af83d9d2239_chapter3.pdf |
SCALAR type robot.
Figure by MIT OCW.
Department of Mechanical Engineering
Massachusetts Institute of Technology
Introduction to Robotics, H. Harry Asada
4
The second type, called an articulated robot or an elbow robot, consists of all three
revolute joints, like a human arm. This type of robot has a great degre... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c238d7e3445337ed40d4af83d9d2239_chapter3.pdf |
formed by the five links and, thereby, the two serial link arms must conform to a certain
geometric constraint. It is clear from the figure that the end-effecter position is determined if two
of the five joint angles are given. For example, if angles 1? and 3? of joints 1 and 3 are
determined, then all the link posi... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c238d7e3445337ed40d4af83d9d2239_chapter3.pdf |
to the base frame. The position
and orientation of the moving platform are determined by the six independent actuators. The load
acting on the moving platform is born by the six "arms". Therefore, the load capacity is generally
large, and dynamic response is fast for this type of robot mechanisms. Note, however, tha... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c238d7e3445337ed40d4af83d9d2239_chapter3.pdf |
S
S
S
S
Moving
Moving
Moving
Moving
Moving
Moving
Moving
Moving
Moving
Moving
Platform
Platform
Platform
Platform
Platform
Platform
Platform
Platform
Platform
Platform
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/5c238d7e3445337ed40d4af83d9d2239_chapter3.pdf |
Lecture 02
Voting classifiers, training error of boosting.
18.465
In this lecture we consider the classification problem, i.e. Y = {−1, +1}.
Consider a family of weak classifiers
H = {h : X → {−1, +1}}.
Let the empirical minimizer be
h0 = argmin
1
n
n
Xi=1
I(h(Xi) 6= Yi)
and assume its expected error,
1
2
> ε = Error(h0),... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/5c28ad9411d2f7ad6c2d74583d965522_lecture02.pdf |
final classifier: f = sign(
αtht(x)).
P
Theorem 2.1. Let γt = 1/2 − εt (how much better ht is than tossing a coin). Then
1
n
n
Xi=1
T
I(f (Xi) 6= Yi) ≤
1 − 4γ2
t
Yt=1 p
Proof.
I(f (Xi) 6= Yi) = I(Yif (Xi) = −1) = I(Yi
αtht(Xi) ≤ 0) ≤ e−Yi P
T
t=1 αtht(Xi)
T
Xt=1
Consider how weight of example i changes:
wT +1(i) =
wT (i)... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-statistical-learning-theory-spring-2007/5c28ad9411d2f7ad6c2d74583d965522_lecture02.pdf |
3.46 PHOTONIC MATERIALS AND DEVICES
Lecture 14: Defects and Strain
Lecture
Notes
Perfection
• LRO (Long Range Order)
• SRO (Short Range Order)
Imperfection
• Vibrating atom
• Electronic change
• Chemical impurity
• Point defect
• 1D defect
• 2D defect
• 3D defect
(I, V)
(dislocation)
(grain boundary)
... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/5c303cb9b7be85a43704cb20f73d045d_3_46l14_defects.pdf |
A A + xBB U VA + xV + A A + BB
B
KS = [V ][VB ]x
A
Frenkel Pair
A A → A I + VA
KFP ( ) = [A ][VA ]
A
FP ( ) = [B ][VB ]
K
B
I
I
Anti-Site Defect
A A
+ BB U AB + B
AS = [A ][BA ]
K
B
A
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 14: Defects and Strain
Page 2 of 5
Lecture
... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/5c303cb9b7be85a43704cb20f73d045d_3_46l14_defects.pdf |
e + h U bond
+ ⎤
⎤
−⎡BSi
⎥ + = ⎡
⎥ + p
⎢LiI
⎦ n
⎢⎣
⎦
⎣
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 14: Defects and Strain
Page 4 of 5
Notes
Lecture
T1
T
1
siB−⎡
⎢
⎣
⎤
⎥
⎦
n
i
B−⎡
≡ ⎢
⎣
si
⎤
⎥
⎦
⎡
ILi +
⎤
⎡
⎢
ILi +
⎣
⎥
⎢
⎦
⎣
⎤
⎥
⎦
Consequences:
Lase... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/5c303cb9b7be85a43704cb20f73d045d_3_46l14_defects.pdf |
Copyright
c
Nancy Leveson, Sept. 1999
1980s:
OO design: added inheritance, multiple inheritance, and
polymorphism to ADT.
In process added complexity and increased
some types of connectivity.
Lots of claimed advantages -- so far empirical
evaluation is not supporting them well.
1990s:
Architecture
Patterns ... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5c4493536832111845f291d3d62a6f38_cnotes5.pdf |
veson, Sept. 1999
3. Simplicity
Emphasis on software that is clear, simple, and
therefore easy to check, understand, and modify.
4. Restricted visibility
Locality of information
Copyright
c
Nancy Leveson, Sept. 1999
General Software Design Concepts
Implementations of the general principles
Decomposition
C... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5c4493536832111845f291d3d62a6f38_cnotes5.pdf |
Concepts (4)
Modularity
Separation of concerns:
1. Deal with details of each module in isolation (ignoring
details of other modules)
2. Deal with overall characteristics of all modules and their
relationships in order to integrate them into a coherent
system.
Base on hierarchy and abstraction:
Abstraction hand... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5c4493536832111845f291d3d62a6f38_cnotes5.pdf |
make task of solving it more intractible, i.e., the
design process is not neutral.
2. Expert designers engage in opportunistic behavior:
As solution’s form emerges, problem solving strategy
is adapted to meet new characteristics that are revealed,
i.e., expert designers do not follow a single method.
�
Copyr... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5c4493536832111845f291d3d62a6f38_cnotes5.pdf |
real need for design skills, but does not create
an objective forum for evaluation.
Conclusions:
Life belt has become waterlogged and acting more like a leg iron.
Need to stop pretending that software design is largely a matter
of following a set of well-defined activities. Recognize it as a
creative process that ... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5c4493536832111845f291d3d62a6f38_cnotes5.pdf |
Electricity and Magnetism
• Review
– Electric Charge and Coulomb’s Force
– Electric Field and Field Lines
– Superposition principle
– E.S. Induction
– Electric Dipole
– Electric Flux and Gauss’ Law
– Electric Potential Energy and Electric Potential
– Conductors, Isolators and Semi-Conductors
Feb 27 2002
Today
• Fast s... | https://ocw.mit.edu/courses/8-02x-physics-ii-electricity-magnetism-with-an-experimental-focus-spring-2005/5c48da61d3532b22af63ad3792b99f6b_2_27_2002_edited.pdf |
12
Q1
F1,total
Q2
• Note:
– Total force is given by vector sum
– Watch out for the charge signs
– Use symmetry when possible
Feb 27 2002
Superposition principle
• If we have many, many charges
– Approximate with continous distribution
• Replace sum with integral!
Feb 27 2002
Electric Field
• New concept – Electric Fi... | https://ocw.mit.edu/courses/8-02x-physics-ii-electricity-magnetism-with-an-experimental-focus-spring-2005/5c48da61d3532b22af63ad3792b99f6b_2_27_2002_edited.pdf |
‘complicated’ surfaces and non-constant E:
– Use integral
• Often, ‘closed’ surfaces
Feb 27 2002
Electric Flux
• Example of closed surface: Box (no charge inside)
dA
E
dA
• Flux in (left) = -Flux out (right): ΦE = 0
Feb 27 2002
Gauss’ Law
• How are flux and charge connected?
• Charge Qencl as source of flux through c... | https://ocw.mit.edu/courses/8-02x-physics-ii-electricity-magnetism-with-an-experimental-focus-spring-2005/5c48da61d3532b22af63ad3792b99f6b_2_27_2002_edited.pdf |
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