text stringlengths 16 3.88k | source stringlengths 60 201 |
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6 Young diagrams and q-binomial coefficients.
0 satisfying �1
0 is a sequence � = (�1, �2, . . .) of integers
A partition � of an integer n
i�1 �i = n. Thus all but finitely
and
�i ←
many �i are equal to 0. Each �i > 0 is called a part of �. We sometimes
suppress 0’s from the notation for �, e.g., (5, 2, 2, 1), (... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
, 3) =
}
{
the unique partition (0, 0, . . .) with no parts.) If � = (�1, �2, . . .) and µ =
(µ1, µ2, . . .) are partitions, then define �
µi for all i. This makes
the set of all partitions into a very interesting poset, denoted Y and called
Young’s lattice (named after the British mathematician Alfred Young, 1873... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
the poset L(m, n).
The Young diagram (somtimes just called the diagram) of a partition � is a
left-justified array of squares, with �i squares in the ith row. For instance,
the Young diagram of (4, 3, 1, 1) looks like:
If dots are used instead of boxes, then the resulting diagram is called a
Ferrers diagram. The ad... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
rank-symmetry. To show rank-symmetry, consider the comple
ment � of � in an m
n rectangle R, i.e., all the squares of R except for �.
(Note that � depends on m and n, and not just �.) For instance, in L(4, 5),
the complement of (4, 3, 1, 1) looks like
×
¯
¯
If we rotate the diagram of � by 180∗ then we obtain the... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
|
m
�
�
�
�
Proof. We will give an elegant combinatorial proof, based on the fact
that m+n
is equal to the number of sequences a1, a2, . . . , am+n, where each
m
aj is either N or E, and there are m N ’s (and hence n E’s) in all. We will
n rectangle R with such
associate a Young diagram D contained in an m
a se... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
] = 1 + q + q2 +
+
let q be an indeterminate; and given j
←
qj−1
. Thus [1] = 1, [2] = 1 + q, [3] = 1 + q + q2, etc. Note that [j] is a
polynomial in q whose value at q = 1 is just j (denoted [j]q=1 = j). Next
1, and set [0]! = 1. Thus [1]! = 1, [2]! = 1 + q,
define [j]! = [1][2]
[3]! = (1 + q)(1 + q + q2) = 1 +... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
�
=
1
⎟
−
[4][3][2][1]
[2][1][2][1]
k
k
1
� ⎟
=
4
2
� ⎟
=
5
2
� ⎟
5
3
� ⎟
= [k] = 1 + q + q 2 +
+ q k−1
· · ·
= 1 + q + 2q 2 + q + q
3
4
= 1 + q + 2q 2 + 2q 3 + 2q 4 + q 5 + q 6 .
In the above example,
k
j was always a polynomial in q (and with non
negative integer coefficients). It is not obvious tha... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
⎟
�
=
=
=
=
=
−
−
[k
−
1]![k
[k
−
1]![k
[k
−
1]![k
−
1]!
1
−
1]!
1
−
1]!
1
−
. �
[j
−
[j
−
[j
−
k
j
� ⎟
j]!
1]!
[k
−
1]![k
−
qk−j
[k
j]
−
+
[j
1
[j]
�
−
j] + qk−j [j]
−
j]
[j][k
[k]
−
j]!
�
[k
j]!
−
−
j]! [j][k
j]
−
−
Note that if we put q = 1 in (26) we obtain the w... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
Let pi(m, n) denote the number of elements of L(m, n)
of rank i. Then
pi(m, n)q i =
m + n
m
⎟
.
�
i�0
�
40
(27)
(Note. The sum on the left-hand side is really a finite sum, since pi(m, n) =
0 if i > mn.)
Proof. Let P (m, n) denote the left-hand side of (27). We w... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
the coefficient of qi of both
sides of (29), we see [why?] that (29) is equivalent to
pi(m, n) = pi(m, n
1) + pi−n(m
1, n).
−
−
(30)
�
×
(n
1) rectangle, so there are pi(m, n
i whose Young diagram D fits in an m
Consider a partition �
n rectangle
R. If D does not contain the upper right-hand corner of R, then ... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
that k
is equal to the number of j-dimensional subspaces of a k-dimensional
j
vector space over the field Fq . We will not discuss the proof here since it is
�
not relevant for our purposes.
�
As the reader may have guessed by now, the poset L(m, n) is isomorphic
to a quotient poset Bs/G for a suitable integer s > ... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
wr Sm. However,
we will not discuss the general theory of wreath products here.)
�
6.7 Example. Suppose m = 4 and n = 5, with the boxes of X labelled
as follows.
1
6
2
7
3
8
4
9
5
10
11
12
13
14
15
16
17
18
19
20
42
Then a typical permutation λ in G(4, 5) looks like
16
20
17
19
18
, ... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
. , �m [why?]. There is a unique permutation
�1, . . . , �m of �1, . . . , �m satisfying �1
�m, so the only possible Young
←
diagram D in the orbit λ S is the one of shape � = (�1, . . . , �m). It’s easy
to see that the Young diagram D� of shape � is indeed in the orbit λ S.
For by permuting the elements in the row... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
D� if and only if �
�, respectively. Then there exist D
�� in L(m, n).
and D�
and
√ O
√ O
O
O
O
O
∪
�
and D�
The “if” part of the previous sentence is clear, for if �
�� then D� ∪
D� . The
D�� . So assume there exist D
lengths of the rows of D, written in decreasing order, are �1, . . . , �m, and
similarly for ... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
is by no means ap
m
parent. It was first proved by J. Sylvester in 1878 by a proof similar to
the one above, though stated in the language of the invariant theory of bi
nary forms. For a long time it was an open problem to find a combinato
rial proof that the coefficients of m+n are unimodal. Such a proof would
m
gi... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
is obtained
by taking all the diagrams of size 2 mn. Although the statement of this fact
requires almost no mathematics to understand, there is no known proof that
doesn’t use algebraic machinery. (The several known algebraic proofs are all
closely related, and the one we have given is the simplest.) Corollary 6.10... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
If the elements
, then all the subset
of S are “spread out,” say S =
}
R+ we have fk(S, �) = 0 or 1.
sums of S are distinct. Hence for any �
Similarly, if the elements of S are “unrelated” (e.g., linearly independent over
the rationals, such as S =
), then again all subset sums are
}
distinct and fk (S, �) = 0... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
· · ·
k + 1
2
�
−
�
.
(33)
Conversely, given j1, . . . , jk satisfying (33) we can recover i1, . . . , ik satisfying
(32). Hence fk ([n], �) is equal to the number of sequences j1, . . . , jk satisfying
(33). Now let
�(S) = (jk , jk−1, . . . , j1).
Note that �(S) is a partition of the integer �
and with largest ... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
([n],
�
�
R+, and k
√
√
).
k(n + 1)/2
⊆
⊂
P. Then
Proof. Let S =
with 0 < a1 <
< an. Let T and U
be distinct k-element subsets of S with the same element sums, say T =
< ik and j1 < j2 <
ai1 , . . . , aik }
{
[n]
, so T �, U �
< jk .
.k
· · ·
The crucial observation is the following:
a1, . . . , an}
{
aj... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
, so air = ajr for all r. This
· · ·
contradicts the assumption that T and U are distinct and proves the claim.
�
�
+ aik = aj1 +
jr for 1
�
· · ·
r
· · ·
�
�(S1
{
It is now easy to complete the proof of Theorem 6.11. Suppose that
S1, . . . , Sr are distinct k-element subsets of S with the same element sums.
�)... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
-P
ie k,\ tPOWN', (: 5Vv~a t
-t I
0 4
i
M 'i &s-.
I__
--·ICslll·-----�P
-
�------
- -·
----
6r )
-t
CutJ-4
"\,
C.,~G
Sc
L MA
Q.AMI-o
= 7T 3 N.�
-Jr
K
at~t
T~~~ks~~h~~cs
,~~
-M'
L-K)
-TTclk
\( -L.
A
a -e
-
W~epcPJ,
\
a ee b
e 1
I
C
.olL
h)o; 41
Clr\padlSkt~aic
Is
-t-NI
ON
eL
.... | https://ocw.mit.edu/courses/8-322-quantum-theory-ii-spring-2003/3531bc12f67ad3e90afc38783186e698_83224Lecture3.pdf |
6.092: Java for 6.170
Lucy Mendel
MIT EECS
MIT 6.092
IAP 2006
1
Course Staff
z Lucy Mendel
z Corey McCaffrey
z Rob Toscano
z Justin Mazzola Paluska
z Scott Osler
z Ray He
Ask us for help!
MIT 6.092
IAP 2006
2
Class Goals
z Learn to program in Java
z Java
z Programming (OOP)
z 6.170 problem sets are no... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/35389e5815c6be386befb7bd00974735_lecture1a.pdf |
for making objects
z Java is about objects Æ everything is in a
class
class HelloWorld {
// classname
… <everything> …
}
MIT 6.092
IAP 2006
7
Field
z Object state
class Human {
int age;
}
<class type> <variable name>;
MIT 6.092
IAP 2006
8
Making objects
Human lucy = new Human();
z All object creation requi... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/35389e5815c6be386befb7bd00974735_lecture1a.pdf |
to access methods
MIT 6.092
IAP 2006
14
Constructors
z Constructors are special methods
z no return type
z use them to initialize fields
z take parameters, normal method body (but no
return)
MIT 6.092
IAP 2006
15
Method Body
String firstname(String fullname) {
int space = fullname.indexOf(“ ”);
String wor... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/35389e5815c6be386befb7bd00974735_lecture1a.pdf |
z || = logical or
a. lucy.age >= 21 && lucy.hasCard
b. !someone.name.equals(“Lucy”))
c. (!true || false) && true
MIT 6.092
IAP 2006
21
Arrays
z Objects, but special like primitives
String[] pets = new String[2];
pets[0] = new String(“Fluffy”);
pets[1] = “Muffy”;
// String syntactic sugar
String[] pets = new St... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/35389e5815c6be386befb7bd00974735_lecture1a.pdf |
SI Engine Combustion
Spark discharge
characteristics
Fig.9-39
Schematic of voltage and
current variation with
time for conventional coil
spark-ignition system.
© McGraw-Hill Education. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use.
3
SI engine flame propagation
Entrainment-and-burn model
Rate of entrainment:
(cid:71)(cid:80)(cid:72)
(cid:71)(cid:87)
(cid:36) (cid:54) (cid:3) (cid:36) (cid:88) (cid:11)(cid:20) (c... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
burn rate.
This flame area depends on flame size, combustion chamber shape,
spark plug location and piston position.
(cid:21)(cid:17)
(cid:44)(cid:81)(cid:16)(cid:70)(cid:92)(cid:79)(cid:76)(cid:81)(cid:71)(cid:72)(cid:85)(cid:3)(cid:87)(cid:88)(cid:85)(cid:69)(cid:88)(cid:79)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
-release rate for ten
cycles in a single-cylinder SI engine operating at 1500 rpm, = 1.0, MAP = 0.7
bar, MBT timing 25oBTC
© McGraw-Hill Education. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use.
Cycle-to-cycle chan... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
?
Reactants Products
Premixed
• Premixed flame
– Examples: gas grill, SI engine combustion
• Homogeneous reaction
Knock
– Fast/slow reactions compared with other time
scale of interest
– Not limited by transport process
• Detonation
– Pressure wave driven reaction
Non-premixed
• Diffusion flame
– Exampl... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
modes
Spectrogram
of 4 valve
engine knock
pressure data
(2L I-4 engine;
CR=9.6)
Calculated
acoustic
frequency
of modes
by FEM
SAE Paper 980893
© Society of Automotive Engineers. All rights reserved. This content is excluded from our
Creative Commons license. For more information, see https://ocw.mit.edu/h... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
reacted. The sequence of processes occur extremely rapidly.
11
Knock chemical mechanism
CHAIN BRANCHING EXPLOSION
Chemical reactions lead to increasing number of radicals,
which leads to rapidly increasing reaction rates
R
O
Chain Initiation
... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
R OOH (Isomerization)
R OOH O OOROOH
OOROOH O=ROOH OH
2
Degenerate
Branching
O=ROOH O=R O OH
Branching agent (hydroperoxyl carbonyl species)
Low
temperature
Initiation
RH O2
R HO 2
Propagation
RH HO 2 H O
2 2
R
High
temperature HO2 HO2 H O
2 2
... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
.
Types of hydrocarbons
(See text section 3.3)
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Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use.
14
o
i
t
a
r
n
o
i
s
s
e
r
p
m
o
c
l
a
c
i
t
i
r
C
Knock tendency of
individual
hydroca... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
air-use.
Octane Requirement Increase
Test 1 (no additive)
Test 2 (with additive)
Test 3 (with additive)
Deposit removal
No additive (ORI = 15)
Deposit controlling
additive (ORI = 10)
Clean combustion
chamber only
Clean combustion chamber
and intake valves
)
I
R
O
(
e
s
a
e
r
c
n
i
t
n
e
m
e
r
i
u
q
e
R
e
n... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
2(C2H5)COCH3
Adiabatic cooling of gasoline/ ethanol mixture
Preparing a stoichiometric mixture from air and liquid fuel
)
C
o
(
p
o
r
d
e
r
u
t
a
r
e
p
m
e
T
80
70
60
50
40
30
20
10
0.0
0.2
1.0
Ethanol liquid volume fraction
0.4
0.8
0.6
Note that Evaporation stops when temperature
drops to dew point... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
18.445 Introduction to Stochastic Processes
Lecture 1: Introduction to finite Markov chains
Hao Wu
MIT
04 February 2015
Hao Wu (MIT)
18.445
04 February 2015
1 / 15
About this course
Course description
Course description :
This course is an introduction to Markov chains, random walks,
martingales.
Time a... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/365d85ac35a7c006bc33dd03da633997_MIT18_445S15_lecture1.pdf |
xn+1), we have that
P[Xn+1 = xn+1 | X0 = x0, ..., Xn = xn]
= P[Xn+1 = xn+1 | Xn = xn] = P(xn, xn+1).
Hao Wu (MIT)
18.445
04 February 2015
5 / 15
About this course
Gambler’s ruin
Consider a gambler betting on the outcome of a sequence of
independent fair coin tosses.
If head, he gains one dollar.
If tai... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/365d85ac35a7c006bc33dd03da633997_MIT18_445S15_lecture1.pdf |
is equally likely to be each of the N types.
Hao Wu (MIT)
18.445
04 February 2015
8 / 15
About this course
Coupon collecting
The collector’s situation can be modeled by a Markov chain on the
state space {0, 1, ..., N} :
X0 = 0
Xn : the number of different types among the collector’s first n
coupons.
P[Xn+... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/365d85ac35a7c006bc33dd03da633997_MIT18_445S15_lecture1.pdf |
Then we have that
µn+1 = µnP.
µn = µ0Pn .
E[f (Xn)] = µ0Pnf .
Hao Wu (MIT)
18.445
04 February 2015
12 / 15
About this course
Stationary distribution
Consider a Markov chain with state space Ω and transition matrix P.
Recall that
P[Xn+1 = y | Xn = x] = P(x, y ).
µ0 : the distribution of X0
µn : the distri... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/365d85ac35a7c006bc33dd03da633997_MIT18_445S15_lecture1.pdf |
/deg(x)
0
if y ∼ x
else
.
Theorem
Define
π(x) =
deg(x)
2|E|
, ∀x ∈ V .
Then π is a stationary distribution for the simple random walk on the
graph.
Hao Wu (MIT)
18.445
04 February 2015
15 / 15
MIT OpenCourseWare
http://ocw.mit.edu
18.445 Introduction to Stochastic Processes
Spring 2015
For information abo... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/365d85ac35a7c006bc33dd03da633997_MIT18_445S15_lecture1.pdf |
5 SYMMETRIES OF SCET
Figure 8: SCETI zero-bin from one collinear direction scaling into the ultrasoft region.
there are ultrasoft subtractions for the collinear modes, but no collinear subtractions for the ultrasoft
modes.
It also should be remarked that depending on the choice of infrared regulators, the subtracti... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
) as a way to constrain SCET operators. We will
find that the gauge symmetry formalism is a simple restatement of the standard QCD picture except with
two separate gauge fields. RPI is a manifestation of the Lorentz symmetry which was broken by the choice
of light-cone coordinates, and which acts independently in each... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
ν [γµ, γν ] → h = σ3.
⊥
(5.3)
We can relate this symmetry to the chiral symmetry by noting that under chiral symmetry ξn transforms
as
(cid:18) σ3φn
φn
(cid:18) 0 1
1 0
so ϕn → σ3ϕn .
ξn → γ5ξn =
(cid:19) 1
√
2
(5.4)
(cid:19)
This U (1)A axial-symmetry is broken by fermion masses and non-perturbative instanton effec... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
) ∼ Q(λ2, λ2, λ2)Uu(x).
(5.7)
(5.8)
There is also a global color transformation which for convenience we group together with the Uu. To
avoid double counting, in the collinear transformation we fix Un(n · x = −∞) = 1. We can implement a
collinear gauge transformation on the collinear fields ξn, pl via a Fourier tran... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
of a matrix in momentum space, and the RHS is a number (both
are of course also matrices in color). Then Eq. (5.9) with a sum over repeated indices becomes ξn, p£ →
(Uˆn)p£,q£ ξn,q£ . And if we suppress indices then we have ξn → (Uˆn)ξn.
(5.10)
Finally the ultrasoft fields do not transform under a collinear gauge tr... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
the appropriate representation.
The usoft fields have their usual gauge transformations from QCD.
Usoft Gauge Transformations : Uu(x)
Therefore for the Ultrasoft Gauge Transformations we have
• ξn(x) → Uus(x)ξn(x)
• Aµ
n(x) → Uus(x)Aµ
†
n(x)Uus(x)
• qus(x) → Uus(x)qus(x)
• Aµ
us(x) → Uus
(x)(Aµ
us(x) + i
∂µ)... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
�(x).
For the collinear gauge transformation we have fields in momentum space for labels, and position space
†(−∞) = 1, so the Wilson line transforms only on one side for
representing residual momenta, and Un
collinear transformations. For ultrasoft transformations Wn(x) is actually a local operator with all fields
a... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
�hus
v
v
JSCET = ξnWn
¯
Γhus
v .
¯ ˆ † ˆ
(5.12)
Γhus
v
¯
Now under a collinear gauge transformation JSCET → ξnUnUnWn
= ξnWn
, so the current is
†
hus
collinear gauge invariant. Under an ultrasoft gauge transformation JSCET → ξnU
usΓUus v
=
WnU
¯
, so the current is also ultrasoft gauge invariant. Thus the lead... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
the operators in · D and (1/P)Dn⊥Dn⊥ are O(λ2) and have the correct mass dimension. The latter
will have the correct gauge transformation properties once we include Wns. Nevertheless, nothing so far
rules out the operator
1
P
which is gauge invariant and has the correct λ scaling. To exclude this term we need to c... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
as long as our
result still obeys (5.15). Specifically, there are three sets of transformations which can be made on a set of
light-cone coordinates to obtain another, equally valid, set.
I
nµ → nµ
+ Δ⊥
µ
n¯µ → n¯µ
II
nµ → nµ
n¯µ → n¯µ + ε⊥
µ
III
α µ
→ e n
nµ
−α ¯
n¯µ → e nµ
(5.16)
where ¯n · ε⊥ = n · ε⊥ ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
which were broken by introducing the vectors n and n¯. These generators are defined by
, n¯µM µν } or in terms of our standard light-cone coordinates Q± = J1 ± K2, Q± = J2 ± K1, and
{nµ
K3. Here M µν are the usual 6 antisymmetric SO(3,1) generators.
M µν
µ
2
1
If we start with our canonical basis choice n = (1, 0, ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
with one factor of ¯n/n in both the numerator and denominator.
That is, in one of the combinations
(A · n)(B · n¯),
A · n
B · n
,
A · n¯
B · n¯
(5.18)
where Aµ and Bµ are arbitrary 4-vectors.
In order to derive the complete set of transformation relations we must also determine how pµ trans
forms. Recall that the ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
II
µ
p =⇒ p −
⊥
µ
⊥
nµ
2
ε⊥ · p⊥ − ⊥ n · p .
εµ
2
(5.21)
Summarizing all the type-I and type-II transformations on vectors and fields (using Dµ as a typical vector)
we have
I
n → n + ∆⊥
n¯ → n¯
n · D → n · D + ∆⊥ · D⊥
Dµ
⊥ → Dµ
2 ∆⊥ · D
∆⊥
2 ¯n · D − ¯nµ
µ
(cid:17)
⊥ −
n¯ · D → n¯ · D
4 /∆⊥/¯n
Wn → Wn
(cid:16)... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
n¯/
2
ξn
n ⊥
(5.22)
(5.23)
5.3 Reparamterization Invariance
5 SYMMETRIES OF SCET
is invariant under these transformations. Under a type-I transformation we have
(cid:19)
(cid:18)
(cid:19)
1
in¯ ·
iD/
nD
n/¯
⊥ ξn2
(5.24)
(0)
δIL = δ
nξ
I
(cid:18)
ξnin · D ξn2
n/¯
= ξni∆⊥ · D⊥ /¯n
2
= 0
+ δI
ξniD/ n, ⊥
ξn − ξni∆⊥... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
by using invariance under RPI of type-II. The detailed calculation is given
in [7] with the final result that our Lagrangian L
remains invariant under δII while the term given in
(5.14) does transforms in a way that can not be compensated by any other leading order term in the
Lagrangian. Therefore our SCETI Lagrangi... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
)ξn,p+β(x) .
(5.28)
The set of these β transformations also determines the space of equivalent decompositions I that we mod
out by when constructing pairs of label and residual momenta components (pc, pr) in R3 ×R4/I. Invariance
under this RPI requires the combination
P µ + i∂µ
(5.29)
to be grouped together for ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
into the combination of these derivatives. The unique result which preserves the SCET
gauge symmetries without changing the power counting of the terms is
iDµ ≡ iDµ + Wn
W †
n
n⊥
† ,
in¯ · D ≡ in¯ · Dn + Wnin¯ · DusWn
iDus, µ
n⊥
⊥
(5.31)
(5.32)
where Wn transforms as Wn → UnWn. Stripping off the regular deriv... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
10].
5.4 Discrete Symmetries
After considering the residual form of Lorentz symmetry encoded in reparameterization invariance it is
natural to consider how our SCET fields transform under C, P, and T transformations. In this case we
will satisfy ourselves with the transformations of the collinear field ξn,p. We have ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
shell, and does not change the formulation of the leading order collinear Lagrangians. Therefore the
Lagrangian with multiple collinear directions is
L(0)
SCET =
I
L(0)
us +
(cid:88)
(cid:104)
L(0) +nξ
(cid:105)
.
L(0)
ng
(5.35)
n
44
MIT OpenCourseWare
http://ocw.mit.edu
8.851 Effective Field Theory
Spring 2013
For... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
LECTURE NOTES FOR 18.155, FALL 2004
103
17. Problems
Problem 1. Prove that u+, defined by (1.10) is linear.
Problem 2. Prove Lemma 1.8.
Hint(s). All functions here are supposed to be continuous, I just
don’t bother to keep on saying it.
(1) Recall, or check, that the local compactness of a metric space
X means t... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
∂ > 0 is small enough.
(3) Prove the general case by induction over n.
(a) In the general case, set K ∞ = K ∃ U � and show that the
inductive hypothesis applies to K ∞ and the Uj for j > 1; let
∞ , j = 2, . . . , n be the functions supplied by the inductive
f
j
assumption and put f ∞ =
∞ .
j→2 fj
1
(b) Show th... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
δ-algebra containing the sets
(a, ⊂] ⊃ [−⊂, ⊂]
for all a ≤ R, generates what is called above the ‘Borel’ δ-algebra on
[−⊂, ⊂].
Problem 7. Write down a careful proof of Proposition 1.1.
Problem 8. Write down a careful proof of Proposition 1.2.
Problem 9. Let X be the metric space
X = {0} ∗ {1/n; n ≤ N = {1, 2, . ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
any {Ei}↓
i=1 ⊃ M
with Ei ∃ Ej = π for i ∅= j,
↓
↓
(17.1)
µ
�
i=1
�
µ(Ei)
Ei =
�
i=1
�
LECTURE NOTES FOR 18.155, FALL 2004
105
with the series on the right always absolutely convergenct (i.e., this is
part of the requirement on µ). Define
(17.2)
|µ| (E) = sup
|µ(Ei)|
↓
i=1
�
�
for E ≤ M, with the su... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
. Conclude that the definition of a
measure based on (4.16) is the same as that in Problem 12.
(2) Show that µ± so constructed are orthogonal in the sense that
there is a set E ≤ M such that µ−(E) = 0, µ+(X \ E) = 0.
1
Hint. Use the definition of |µ| to show that for any F ≤ M
and any ∂ > 0 there is a subset F ∞ ≤ M... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
is inner regular on all Borel sets and hence, given ∂ > 0
and E ≤ B(X) there exist sets K ⊃ E ⊃ U with K compact and U
open such that µ(K) ↓ µ(E) − ∂, µ(E) ↓ µ(U ) − ∂.
Hint. First take U open, then use its inner regularity to find K
with K ∞ � U and µ(K ∞) ↓ µ(U ) − ∂/2. How big is µ(E\K ∞)? Find
V ⊥ K ∞\E with V ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
i.e, the inner product is recoverable from the norm, so use the RHS
(right hand side) to define an inner product on the vector space. You
will need the paralellogram law to verify the additivity of the RHS.
Note the polarization identity is a bit more transparent for real vector
spaces. There we have
(x, y) = 1/2(�... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
Prove (8.7), by estimating the integrals.
⎪
�
⎧⎧�
⎧⎧
�
|�|�k,
�
|�|⊥k
⎭
⎭
⎭
⎭
sup
< ∂
x
�D� σ
.
Problem 23. Prove (8.9) where
ϕj (z; x ∞) =
�
∞ ωϕ
ωzj
0
(z + tx∞) dt .
Problem 24. Prove (8.20). You will probably have to go back to first
principles to do this. Show that it is enough to assume u ↓ 0 has
compact... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
!
Problem 26. Prove the generalization of Proposition 8.10 that u ≤
S ∞(Rn), supp(w) ⊃ {0} implies there are constants c� , |�| → m, for
some m, such that
u =
c�D�β .
|�|⊥m
�
108
RICHARD B. MELROSE
Hint This is not so easy! I would be happy if you can show that
u ≤ M (Rn), supp u ⊃ {0} implies u = cβ. To se... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
0, . . . , n .
j=1
�
Problem 29. Consider for n = 1, the locally integrable function (the
Heaviside function),
H(x) =
0
1
x → 0
x > 1 .
Show that DxH(x) = cβ; what is the constant c?
Problem 30. For what range of orders m is it true that β ≤ H m(Rn) , β(σ) =
σ(0)?
Problem 31. Try to write the Dirac measur... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
u weakly in S ∞(Rn) means that for every open
set U � u �N st. uj ≤ U � j ↓ N .
Problem 35. Prove (11.18) where u ≤ S ∞(Rn) and σ, ϕ ≤ S(Rn).
Problem 36. Show that for fixed v ≤ S ∞(Rn) with compact support
S(Rn) � σ ◦∩ v � σ ≤ S(Rn)
is a continuous linear map.
Problem 37. Prove the ?? to properties in Theorem 11.... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
|x| → 1}. Let C0(Bn) ⊃ C(Bn) be
the subspace of functions which vanish at each point of the boundary
and let C(Sn−1) be the space of continuous functions on the unit sphere.
Show that inclusion and restriction to the boundary gives a short exact
sequence
C0(Bn) ψ∩ C(Bn) −∩ C(Sn−1)
(meaning the first map is injecti... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
)
where (gµ)(f ) = µ(f g) for all f ≤ C(Bn). Describe all the measures
with the property that
xj µ = 0 in M (Bn) for j = 1, . . . , n.
Problem 45 (H¨
empty interval.
ormander, Theorem 3.1.4). Let I ⊃ R be an open, non-
i) Show (you may use results from class) that there exists ϕ ≤
ii) Show that any π ≤ C↓(I) may ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
c ≤ C and the ϕj ≤ C↓(Rn) depend on π.
ii) Recall that β0 is the distribution defined by
j=1
�
c
β0(π) = π(0) � π ≤ C↓(Rn);
c
explain why β0 ≤ C
−↓(Rn).
iii) Show that if u ≤ C−↓(Rn) and u(xj π) = 0 for all π ≤ Cc
↓(Rn)
and j = 1, . . . , n then u = cβ0 for some c ≤ C.
LECTURE NOTES FOR 18.155, FALL 200... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
Hadamard regularization]
i) Show that (17.4) just means that for each π ≤ C ↓(R)
↓ dk π
(−1)k
c
z
x+(π) =
dxk (x)x z+k
ii) Use integration by parts to show that
(z + k) · · · (z + 1) 0
�
dx, Re z > −k, z /≤ −N.
(17.5)
z
x+(π) = lim
ξ∗0
��
↓
ξ
π(x)x z dx −
k
j=1
�
Cj (π)∂z+j
�
, Re z > −k, z /≤ −N
for c... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
that if u ≤ C−↓(R) then ˜u(π) = u(π˜), where π˜(x) =
π(−x) � π ≤ C↓(R), defines an element of C −↓(R). What is ˜u
if u ≤ C0(R)? Compute β
d ˜
d
ii) Show that dx u = −
u.
dx
≤ −N and show that d xz = −zx − and
z for z /
z
iii) Define x− = x+
�
z+1
xx− = −x−
.
⎫
iv) Suppose that u ≤ C −↓(R) satisfies the distributiona... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
the dk
dxk
z /≤ −N, the only solution of (17.6) is v = 0.
β0 = 0.
dxk
vii) Conclude that for z /
≤ −N
(17.7)
u ≤ C−↓(R); (x − z)u = 0
�
d
dx
is a two-dimensional vector space.
Problem 50. [Negative integral order] To do the same thing for negative
integral order we need to work a little differently. Fix k ≤ ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
.
z
−)
such that as z ∩ −k the limit is zero.
vi) If you get this far, show that in fact x+ + c(k)xz
z
− also has a
weak limit, uk, as z ∩ −k. [This may be the hardest part.]
vii) Show that this limit distribution satisfies (x dx + k)uk = 0.
viii) Conclude that (17.7) does in fact hold for z ≤ −N as well.
d
[The... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
is Lebesgue measurable, in the sense that f −1(I) ⊃
U ⊃ Rn is measurable for each interval I.
Problem 56. Hilbert space and the Riesz representation theorem. If
you need help with this, it can be found in lots of places – for instance
[6] has a nice treatment.
114
RICHARD B. MELROSE
i) A pre-Hilbert space is a ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
). Show that
4≡v, w� = �v + w�2 − �v − w� + i�v + iw�2 − i�v − iw�
2
2
defines a pre-Hilbert inner product which gives the original
norm.
iv) Let V be a Hilbert space, so as in (i) but complete as well.
Let C ⊃ V be a closed non-empty convex subset, meaning
v, w ≤ C ≥ (v + w)/2 ≤ C. Show that there exists a unique ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
Prove the Riesz Representation theorem, that every continuous
linear functional on a Hilbert space is of the form
uf : H � σ ◦∩ ≡σ, f � for a unique f ≤ H.
Problem 57. Density of C ↓(Rn) in Lp(Rn).
c
i) Recall in a few words why simple integrable functions are dense
in L1(Rn) with respect to the norm �f �L1 =
Rn ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
λ
λU (x) and use v).
vii) Conclude that C↓(Rn) is dense in L1(Rn).
viii) Show that C↓(Rn) is dense in Lp(Rn) for any 1 → p < ⊂.
�
�
�
c
c
y
β
�
σ
Problem 58. Schwartz representation theorem. Here we (well you) come
to grips with the general structure of a tempered distribution.
i) Recall briefly the proof of th... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
it) that if v is a tempered dis
tribution then there is a unique w ≤ S ∞(Rn) such that (1 +
|D|2)N w = v.
� π ≤ S(Rn).
vii) Use the Riesz Representation Theorem to conclude that for each
tempered distribution u there exists N and w ≤ L2(Rn) such
that
(17.10)
u = (1 + |D|
2)N
(1 + |x|
2)N
w.
viii) Use the Four... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
2004
117
ii) Show that if u ≤ C−↓(Rn) and π ≤ C↓(Rn) satisfy
c
supp(u) ∃ supp(π) = ∞
then u(π) = 0.
iii) Consider the space C↓(Rn) of all smooth functions on Rn , with
out restriction on supports. Show that for each N
�f �(N ) =
sup
|�|⊥N, |x|⊥N
|D�f (x)|
is a seminorn on C↓(Rn) (meaning it satisfies �f � ↓ ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
is injective.
C
vi) Show that if v ≤ E ∞(Rn) satisfies (17.11) and f ≤ C↓(Rn) has
f = 0 in |x| < N + ∂ for some ∂ > 0 then v(f ) = 0.
vii) Conclude that each element of E ∞(Rn) has compact support
when considered as an element of C
−↓(Rn).
viii) Show the converse, that each element of C −↓(Rn) with compact
support ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
is a polynomial of degree (at most) 2j − |�|.
(4) Conclude that if π ≤ C↓(Rn+1) is identically equal to 1 in a
c
neighbourhood of 0 then the function
g(θ, Δ) =
1 − π(θ, Δ)
iθ + |Δ|
2
is the Fourier transform of a distribution F ≤ S ∞ (Rn) with
sing supp(F ) ⊃ {0}. [Remember that sing supp(F ) is the com
plement... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
the first variable on Rn, n > 1, and delta in the others.
iv) Show that D E = β, so E is a fundamental solution of Dx1 .
v) If f ≤ C−↓(Rn) show that u = E ξ f solves Dx1 u = f.
vi) What does our estimate on WF(E ξ f ) tell us about WF(u) in
x1
c
terms of WF(f )?
Problem 62. The wave equation in two variables (or o... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
u given by iv),
sing supp(u) ⊃ {(t, x); � (t∞ , x ) ≤ sing supp(f ) with
∞
∞
∞
t ↓ t and t + x = t∞ + x or t − x = t∞ − x }.
∞
viii) Bound WF(u) in terms of WF(f ).
Problem 63. A little uniqueness theorems. Suppose u ≤ C −↓(Rn) recall
that the Fourier transform ˆu ≤ C ↓(Rn). Now, suppose u ≤ Cc
−↓(Rn)
satisfies P ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
where the left hand side is defined by duality “F (s2t, sx) = Fs ”
where
x
t
Fs(π) = s −n−2F (π1/s), π1/s(t, x) = π(
2 , ).
s
s
vi) Conclude that
n
(ωt −
ω2
xj )F (t, x) = G(t, x)
j=1
�
where G(t, x) satisfies
(17.14)
G(s t, sx) = s −n−2G(t, x) in S
2
∞(Rn+1
)
in the same sense as above and has support at m... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
a slab
[t1, t2] × Rn .
LECTURE NOTES FOR 18.155, FALL 2004
121
x) Show that c in (17.15) is non-zero by arriving at a contradiction
from the assumption that it is zero. Namely, show that if c = 0
then u in viii) satisfies the conditions of ix) and also vanishes
in t < T for some T (depending on ϕ). Conclude that... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
summation formula) As in class, let L ⊃ Rn be
an integral lattice of the form
n
L =
v =
⎬
kj v
j , kj ≤ Z
�
j=1
�
where the vj form a basis of Rn and using the dual basis wj (so wj · vi =
ij is 0 or 1 as i ∅= j or i = j) set
β
≥
L
=
w = 2α
⎬
kj w
j , kj ≤ Z
.
�
n
j=1
�
Recall that we defined
(17.18) C↓(TL) =... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
exp(iw · z)F (z) = F (z) for each w ≤ L≥
with equality in S ∞(Rn).
v) Deduce that ˆF , the Fourier transform of F, is L≥ periodic, con
clude that it is of the form
(17.21)
ˆ
F (Δ) = c
β(Δ − w)
w≤L�
�
vi) Compute the constant c.
vii) Show that AL(f ) = F ξ f.
viii) Using this, or otherwise, show that AL(f ) = ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
Y.
iii) If A ≤ B has finite rank (meaning AH is a finite-dimensional
vector space) show that there is a finite-dimensional space V ⊃
H such that AV ⊃ V and AV � = {0} where
V � = {f ≤ H; ≡f, v� = 0 � v ≤ V }.
LECTURE NOTES FOR 18.155, FALL 2004
123
Hint: Set R = AH, a finite dimensional subspace by hypothesis. ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
(Gramm-Schmidt Lemma). Let {vi}i≤N be a sequence in a
Hilbert space H. Let Vj ⊃ H be the span of the first j ele
ments and set Nj = dim Vj . Show that there is an orthonormal
sequence e1, . . . , ej (finite if Nj is bounded above) such that Vj
is the span of the first Nj elements. Hint: Proceed by induction
over N su... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
(assuming the sequence of ej ’s to be infinite) that
the series
j=1
�
↓
≡u, ej �ej
j=1
�
converges in H.
v) Show that if ej is a complete orthonormal basis in a separable
Hilbert space then, for each u ≤ H,
↓
u =
≡u, ej �ej .
j=1
�
Problem 68. [Compactness] Let’s agree that a compact set in a metric
space is... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
ges in C along
the subsequence for each v ≤ H. Show that the subsequnce can
be chosen so that ≡ek , uj � converges for each k, where ek is the
complete orthonormal sequence.
LECTURE NOTES FOR 18.155, FALL 2004
125
Problem 69. [Spectral theorem, compact case] Recall that a bounded
operator A on a Hilbert space H... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
be self-adjoint, deduce that there is a finite-
dimensional subspace M ⊃ H, the sum of eigenspaces with
eigenvalues ±�, containing all the maximum points.
vii) Continuing vi) show that A restricts to a self-adjoint bounded
operator on the Hilbert space M � and that the supremum in
iii) for this new operator is small... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
infinity, i.e. given ∂ > 0 there exists β > 0 such that for all elements
u ≤ B
(17.25)
|y| < β =≥ sup |u(x + y) = u(x)| < ∂ and |x| > 1/β =≥ |u(x)| < ∂.
x≤Rn
Problem 72. [Compactness of sets in L2(Rn).] Show that a subset B ⊃
L2(Rn) is precompact in L2(Rn) if and only if it satisfies the following
two conditions: ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
bly using Problem 71, that if λR is cut-off to a ball of radius R then
λRG(λR ˆun) converges strongly if un converges weakly. Deduce from
this that the weakly convergent subsequence in fact converges strongly
so B is sequently compact, and hence is compact.
Problem 73. Consider the space Cc(Rn) of all continuous fun... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
for all n, m ↓ N, is convergent (in the corresponding
f
sense that there exists f in the space such that f − fn ≤ U
eventually).
(5) If you are determined, discuss the corresponding issue for nets.
Problem 74. Show that the continuity of a linear functional u : C ↓(Rn) −∩
C with respect to the inductive limit topol... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
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