text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
of R themselves we can arrange the
row sizes of S to be in weakly decreasing order. Thus we obtain the Young
diagram D� as claimed. �
· · ·
←
·
·
We are now ready for the main result of this section.
6.9 Theorem. The quotient poset BRmn /Gmn is isomorphic to L(m, n).
Proof. Each element of BR/Gmn contains a unique... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
�1, . . . , �m, and
similarly for D� . Since each row of D is contained in a row of D�, it follows
m, D� has at least j rows of size at least �j . Thus
that for each 1
j of the jth largest row of D� is at least as large as �j . In other
the length ��
j , as was to be proved. �
��
words, �j �
� satisfying D
√ O... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
m+n are unimodal. Such a proof would
m
give an explicit injection (one-to-one function) µ : L(m, n)i �
L(m, n)i+1 for
�
i < 1 mn. (One difficulty in finding such maps µ is to make use of the hypoth
esis that i < 1 mn.) Finally around 1989 such a proof was found by Kathy
O’Hara. However, O’Hara’s proof has the defect ... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
There is an interesting application
of Corollary 6.10 to a number-theoretic problem. Fix a positive integer k.
For a finite subset S of R+ =
R : � > 0
, and for a real number � > 0,
}
define
�
{
√
fk(S, �) = # T
⎠
:
S
k
�
√
�
t = �
⎛
t�T
�
In other words, fk(S, �) is the number of k-element subsets of S whos... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
linearly independent over
the rationals, such as S =
), then again all subset sums are
}
distinct and fk (S, �) = 0 or 1. These considerations make it plausible that
and then choose � appropriately. In
we should take S = [n] =
{
R+ ,
other words, we are led to the conjecture that for any S
we have
1, ≥2, ≥3, ... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
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 part at most n
k. Thus
−
fk ([n], �) = p�−(k+1)(k, n
2
k+1 with at most k parts
2
−
�
k),
�
−
(34)
or equivalently,
=
fk([n], �)q �−(k+1
2 )
n
k
2 )
��(k+1
... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
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}
{
aj1 , . . . , ajk }
{
i1, . . . , ik }
{
and U =
Define T � =
· · ·
j1, . . . , jk}
{
with i... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
ik = 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.
�)
�), . . . , �(Sr
k). Hence r
By the claim,
}
cannot exceed the size of the largest antichain in L(k, n
k). By Theor... | https://ocw.mit.edu/courses/18-318-topics-in-algebraic-combinatorics-spring-2006/351d525546a333048bca0b2f436ea256_young.pdf |
1
I
(FI-.mP
-
2* es I-
N A
[,/' (e.%~ a
-Vk"'
1 I'
tt
-Lrj I, aA ~. Am, ~ (-~Y~'-
, 6 i
So
-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.
... | https://ocw.mit.edu/courses/8-322-quantum-theory-ii-spring-2003/3531bc12f67ad3e90afc38783186e698_83224Lecture3.pdf |
>,.;w·
(A)
C3L~ni 5
+ice
emiss
l,
...
IL \~/S
-1 I 1A\
~ (s
k-(
)A
5.
.~
.;1
. .3
.
..
.- 4
,* W
Et
atce LICq~
. -..., - ..
2-
i , v,," =217~I
C~~~
--
uO3
f\
~ ~
((15,\
~~~~~-,
rG
,
I""-
�---
__
411j;1(4 2h
"I
lg
~~~ (Z.:(w,),
V~
.I
I
--
--'
-3%+
'J4"A3O
tI*- eyf
f
A--C
rv... | 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 |
public static void main(String[] args) {
HelloWorld myHelloWorld = new HelloWorld();
myHelloWorld.shout();
}
}
MIT 6.092
IAP 2006
6
Class
z Template 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 Obje... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/35389e5815c6be386befb7bd00974735_lecture1a.pdf |
() {
myString = new String(“Hello, World!“);
System.out.println(myString);
}
public static void main(String[] args) {
HelloWorld myHelloWorld = new HelloWorld();
myHelloWorld.shout();
}
}
MIT 6.092
IAP 2006
13
Methods
z Process object state
<return type> <method name>(<parameters>) {
<method body>
}
myHe... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/35389e5815c6be386befb7bd00974735_lecture1a.pdf |
92
IAP 2006
18
For Loop
for (int i = 0; i < 3; i++) {
System.out.println(i);
// prints 0 1 2
}
for (<initialize> ; <predicate> ; <increment>) {
execute once
every time
Stop when predicate is false
MIT 6.092
IAP 2006
19
While Loop
int i = 0;
while (i < 3) {
System.out.println(i);
// prints 0 1 2
}
whil... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/35389e5815c6be386befb7bd00974735_lecture1a.pdf |
Integer a = new Integer(3);
z a.add(5);
z a = a.add(5);
// aÆ[3]
// aÆ[3]
// aÆ[8]
MIT 6.092
IAP 2006
24
Break (10 min)
z When we get back, more on Objects from
Corey
MIT 6.092
IAP 2006
25 | 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 |
chematic of entrainment-and-burn model
Fig. 14-12
© 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.
3
SI engine flame propagation
Entrainment-and-burn model
Rate of entrainment:
(cid:7... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
: uT and T
SI Engine design and operating factors
affecting burn rate
(cid:20)(cid:17) (cid:41)(cid:79)(cid:68)(cid:80)(cid:72)(cid:3)(cid:74)(cid:72)(cid:82)(cid:80)(cid:72)(cid:87)(cid:85)(cid:92)(cid:29)
The frontal surface area of the flame directly affects the burn rate.
This flame area depends on flame size... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
cid:70)(cid:82)(cid:80)(cid:83)(cid:82)(cid:86)(cid:76)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:86)(cid:87)(cid:68)(cid:87)(cid:72)(cid:29)
The local consumption of the fuel-air mixture at the flame front
depends on the laminar flame speed SL. The value of SL depends on
the fuel equ... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
turbulence in the combustion chamber (2, 3)
(h) Large scale features of the in-cylinder flow (3)
(i) Flame geometry interaction with the combustion chamber (3)
Cycle distributions
Fig. 9-36 (b)
Fig. 9-33 (b)
Charge variations
Charge and
combustion
duration
variations
Very Slow-burn cycles
Partial burn – sub... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
-use.
15
Heat release rate and pressure wave
• When acoustic expansion is not fast enough
to alleviate local pressure buildup due to heat
release, pressure wave develops
R
q H e a t re le a s e p e r u n it v o lu m e
o v e r s p h e re o f ra d iu s R
a = S o u n d s p e e d
C ritirio n fo r s e ttin g u ... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
100 to 150 bar
– Local T > 2800oK
• High pressure and high temperature lead to
structural damage of combustion chamber
20
10
Knock damaged pistons
From Lichty, Internal Combustion Engines
From Lawrence Livermore website
21
© McGraw-Hill Education. All rights reserved. This content
is excluded from our Creat... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
��
RH
RO
2
RO
R CHO
2
Degenerate Branching
R O
ROOH
RO
OH
R CHO
2O
R CO
2HO
Ignition delay for primary reference fuels
1200 1100 1000
1200 1100 1000
900
900
800
800
T(oK)
700
700
10
10
P = 40 bar
Range of interest
)
s
m
(
l
y
a
e
d
n
g
I
1
1
0.1
0.1
100
9... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
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
M
Degenerate
Branching
H O M OH OH M
2 2
M
NTC regime
Temperature high
enough to shift
formation of RO2 t... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
.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
hydrocarbons
Fig 9-69
Critical compression ratio for
incipient knock at 600 rpm and
450 K coolant temperature for
hydrocarbons
Number of C atoms
© McGraw-Hill Education. All rights reserved. Thi... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
(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
a
t
c
O
0
ACS Vol. 36, #1, 1991
Hours of operation
© American Ch... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
3
(CH3)3COC2H5
(CH3)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 temperatu... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/35c1c5bbf304c0f41e1de0cec0b0ee85_MIT2_61S17_lec10.pdf |
limit.
19
MIT OpenCourseWare
https://ocw.mit.edu
2.61 Internal Combustion Engines
Spring 2017
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | 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 |
|
Definition
A sequence of random variables (X0, X1, X2, ...) is a Markov chain with
state space Ω and transition matrix P if
for all n ≥ 0, and all sequences (x0, x1, ..., xn, 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 20... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/365d85ac35a7c006bc33dd03da633997_MIT18_445S15_lecture1.pdf |
] =
k
N
, E[τ ] = k(N
−
k ).
Hao Wu (MIT)
18.445
04 February 2015
7 / 15
About this course
Coupon collecting
A company issues N different types of coupons. A collector desires a
complete set.
Question :
How many coupons must he obtain so that his collection contains all N
types.
Assumption : each ... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/365d85ac35a7c006bc33dd03da633997_MIT18_445S15_lecture1.pdf |
Notations
µP : measure on Ω
PQ : transition matrix
Pf : function on Ω
Associative
(µP)Q = µ(PQ)
(PQ)f = P(Qf )
Hao Wu (MIT)
18.445
04 February 2015
11 / 15
About this course
Notations
Consider a Markov chain with state space Ω and transition matrix P.
Recall that
P[Xn+1 = y | Xn = x] = P(x, y ).
µ0 : ... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/365d85ac35a7c006bc33dd03da633997_MIT18_445S15_lecture1.pdf |
x.
For x V , deg x
∈
( ) :
the number of neighbors of x.
Definition
Given a graph G = (V , E), we define simple random walk on G to be
the Markov chain with state space V and transition matrix :
�
P(x, y ) =
1/deg(x)
0
if y ∼ x
else
.
Hao Wu (MIT)
18.445
04 February 2015
14 / 15
About this course
Rando... | 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 |
ET
gauge symmetries and reparameterization invariance (RPI) 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 l... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
corresponding
to spin along the direction of motion n of the collinear fields. The relevant generator is
Sz = iEµν [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:1... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
ultrasoft Uu(x) :
i∂µUn(x) ∼ Q(λ2 , 1, λ)Un(x)
i∂µUu(x) ∼ 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 transfo... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
It is convenient to setup a matrix notation for these
convolutions by defining
(Uˆn)p£,q£ ≡ (Un)p£−q£ ,
where the LHS is the (pc, qc) element 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... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
not modify large momenta, but the collinear
transformations do).
For usoft gauge transformations, the field ξn and Aµ
n transform as quantum fields under a background
gauge transformation, which is to say they transform as matter fields with the appropriate representation.
The usoft fields have their usual gauge transf... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
for objects that are built from the fields. An
important case is the Wilson line Wn which is like the Fourier transform of W (x, −∞). In QCD a general
Wilson line with the gauge field along a path will transform on each end as W (x, y) → U (x)W (x, y)U †(x).
For the collinear gauge transformation we have fields in mome... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
��eld can not interact with the collinear gluons while remaining near its mass shell. But recall
that when the offshell collinear gluons are accounted for in matching onto the SCET operator that the
n¯ · An ∼ λ0 gluons generate a Wilson line Wn, so the complete result from tree level matching is
Γhus
v
v
JSCET = ξn... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
n ⊥ ,
⊥
in¯ · Dn = P + gn¯ · An ,
iDµ = i∂µ + gAµ .
us
us
(5.13)
We see that gauge symmetry is a powerful tool in determining the structure of operators. It is reasonable
to ask, is power counting and gauge invariance enough to fix the leading order Lagrangian L
for ξn?
Only the operators in · D and (1/P)Dn⊥Dn⊥ a... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
of the reparameterization symmetry is the
freedom we have in splitting momenta between label and residual components. We will explore these two
in turn.
The only required property of a set of n, ¯n basis vectors is that they satisfy
n 2 = ¯n 2 = 0,
n · n¯ = 2.
(5.15)
Consequently a different choice for n and n¯ ca... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
7)
Thus only Δ⊥ is constrained by the power counting, while large changes are allowed for α and E⊥ . These
RPI transformations are a manifestation of the Lorentz symmetry which was broken by introducing the
vectors n and ¯n. The five infinitesimal parameters Δ⊥ , ε⊥, and α correpsond to the five generators of the
µ
Lo... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
the freedom to make large
transformations. (If we start with a more general choice for n and n¯ that satisfies Eq. (5.15) then the
picture for the Type-III transformation is more complicated than a simple boost.)
The implications of the Type III transformation for SCET operators are very simple, n and n¯ must
appear... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
I
n¯
µ
p⊥ =⇒ p⊥ − Δ⊥ · p⊥ −
2
µ
µ
Δµ
⊥
2
n¯ · p .
(5.20)
The projection relation (n/n¯//4)ξn = ξn also implies that ξn → [1 + ( Δ/ n¯/)/4]ξn. Similar relations can also
be worked out for type-II transformations, for example
⊥
II
µ
p =⇒ p −
⊥
µ
⊥
nµ
2
ε⊥ · p⊥ − ⊥ n · p .
εµ
2
(5.21)
Summarizing all the ... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
⊥
For type-III transformations p⊥
µ does not transform, and neither does Wn.
We can show that our leading order SCET Lagrangian
L(0) = ξnin · D ξn + ξ iD/
nξ
n
n¯/
2
42
1
in¯ · D
iD/
n, ⊥
n¯/
2
ξn
n ⊥
(5.22)
(5.23)
5.3 Reparamterization Invariance
5 SYMMETRIES OF SCET
is invariant under these transforma... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
¯n
2
which is the same transformation as for the second term in (5.24). Consequently, we may replace the
second term with this new term with no violation of power counting, gauge symmetry, or RPI type-I.
This ambiguity is only resolved by using invariance under RPI of type-II. The detailed calculation is given
in [7... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
fferent ways as long as we maintain the power counting. Specifically, a transformation
that takes
P µ → P µ + βµ
i∂µ → i∂µ − βµ
(5.27)
implements this freedom. The transformation on i∂µ is induced by the β-transformation of the fields, for
example
ξn,p(x) → e iβ(x)ξn,p+β(x) .
(5.28)
The set of these β transformat... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
uging of (5.29) would be
iDµ + iDµ
n ⊥
us ⊥ ,
in¯ · Dn + i¯
n · Dus .
(5.30)
However, with the above transformations these combinations do not have uniform transformations under
the gauge symmetries, since Dus does not transform under Un. We can rectify this problem by introducing
our Wilson line Wn into the comb... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
(ξ¯nWn)iD/ ⊥
nξ
us 1
P
(Wn
†iD/ n,⊥ξn) + (ξ¯niD/ n,⊥Wn)
us
iD/ ⊥ (Wn
†ξn).
1
P
(5.33)
The complete set of SCETI Lagrangian interactions up to O(λ2) can be found in Ref. [10].
5.4 Discrete Symmetries
After considering the residual form of Lorentz symmetry encoded in reparameterization invariance it is
natural t... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/366ec7eb7c31fd1b75c13d6837bdb04d_MIT8_851S13_SymmetOfSCET.pdf |
more than one energetic hadron, or more than one energetic jet our list of degrees of
freedom must include more than one type of collinear mode, and hence more than one type of collinear
quark and collinear gluon. When two collinear modes in different directions interact, the resulting particle
is offshell, and does n... | 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 |
gξ ⇒ d(· , K) satisfies the conditions for n = 1
if ∂ > 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
... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
This just means that compact sets are Borel sets if you
follow through the tortuous terminology.)
Problem 6. Show that the smallest 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.... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
X
�
Problem 12. Let (X, M) be a set with a δ-algebra. Let µ : M ∩ R be
a finite measure in the sense that µ(π) = 0 and for 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 converg... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
117!
�
�
Problem 13. (Hahn Decomposition)
1
With assumptions as in Problem 12:
(1) Show that µ+ = 2 (|µ| + µ) and µ− = 2 (|µ| − µ) are positive
measures, µ = µ+ − µ−. 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 s... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
this way with β = 1/m. Show that E =
n
�
Problem 14. Now suppose that µ is a finite, positive Radon measure
on a locally compact metric space X (meaning a finite positive Borel
measure outer regular on Borel sets and inner regular on open sets).
Show that µ is inner regular on all Borel sets and hence, given ∂ > 0
... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
a vector space V . Show that �u� =
(u, u)1/2 for an inner product satisfying (5.1) - (5.4) if and only if the
parallelogram law holds for every pair u, v ≤ V .
Hint (From Dimitri Kountourogiannis)
If � · � comes from an inner product, then it must satisfy the polari
sation identity:
(x, y) = 1/4(�x + y�2 − �x − y... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
.
C
0
Problem 19. Show that exp(− |x|2) ≤ S(Rn).
Problem 20. Prove (7.7), probably by induction over k.
Problem 21. Prove Lemma 7.4.
LECTURE NOTES FOR 18.155, FALL 2004
107
Hint. Show that a set U � 0 in S(Rn) is a neighbourhood of 0 if and
only if for some k and ∂ > 0 it contains a set of the form
σ ≤ S(Rn
)... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
= 0
|t|�↓
where µ = Lebesgue measure and
E + t = {p ≤ Rn ; p ∞ + t , p ∞ ≤ E} .
Problem 25. Prove Leibniz’ formula
�
D x(σϕ) =
�
�
�
�⊥�
� �
D�
xσ · d�−� ϕx
for any C↓ functions and σ and ϕ. Here � and � are multiindices,
� → � means �j → �j for each j? and
�
�
�
�
=
.
�j
�j
�
�
j
�
I suggest induction!... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
D�
xσ(0) = 0 for |�| → m then � σj ≤ S(Rn), σj = 0
m
in |x| → ∂j , ∂j ∪ 0 such that σj ∩ σ in the C norm.
Problem 27. If m ≤ N, m∞ > 0 show that u ≤ H m(Rn) and D�u ≤
H m (Rn) for all |�| → m implies u ≤ H m+m (Rn). Is the converse true?
Problem 28. Show that every element u ≤ L2(Rn) can be written as a
sum
�
�
n... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
for some value of c.
2)−1/2
Problem 34. Recall that a topology on a set X is a collection F of
subsets (called the open sets) with the properties, π ≤ F , X ≤ F and
F is closed under finite intersections and arbitrary unions. Show that
the following definition of an open set U ⊃ S ∞(Rn) defines a topology:
� u ≤ U an... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
that if P (D) is hypoelliptic
then every parametrix F ≤ S(Rn) has sing supp(F ) = {0}.
Problem 39. Show that if P (D) is an ellipitic differential operator of
order m, u ≤ L2(Rn) and P (D)u ≤ L2(Rn) then u ≤ H m(Rn).
Problem 40 (Taylor’s theorem). . Let u : Rn −∩ R be a real-valued
function which is k times continu... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
where continuity is with respect to the
supremum norm, i.e. there must be a constant C such that
|µ(f )| → C sup |f (x)| � f ≤ C(Bn).
x≤Rn
Let M (Bn) be the linear space of such measures. The space M (Sn−1)
of measures on the sphere is defined similarly. Describe an injective
map
M (Sn−1) −∩ M (Bn).
110
RICHAR... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
�
π = π˜ + cϕ, c ≤ C, π˜ ≤ C↓(I) with
c
π˜ = 0.
R
�
iii) Show that if π˜ ≤ C↓(I) and
↓(I) such that dµ = π˜ in I.
c
C
c
dx
iv) Suppose u ≤ C−↓(I) satisfies du = 0, i.e.
˜
�
dx
R π = 0 then there exists µ ≤
u(−
) = 0 � π ≤ C↓(I),
c
dπ
dx
v) Suppose that u ≤ C −↓(I) satisfies dx = c, for some constant c,
show th... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
155, FALL 2004
111
iv) Define the ‘Heaviside function’
↓
H(π) =
π(x)dx � π ≤ C↓(R);
c
show that H ≤ C
v) Compute dx H ≤ C
d
0
�
−↓(R).
−↓(R).
Problem 47. Using Problems 45 and 46, find all u ≤ C −↓(R) satisfying
the differential equation
x = 0 in R.
du
dx
These three problems are all about homogeneous distrib... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
k
j=1
�
Cj (π)∂z+j
�
, Re z > −k, z /≤ −N
for certain constants Cj (π) which you should give explicitly.
[This is called Hadamard regularization after Jacques Hadamard,
feel free to look at his classic book [3].]
iii) Assuming that −k + 1 ↓ Re z > −k, z ∅
= −k + 1, show that
there can only be one set of the const... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
= −zx − and
z for z /
z
iii) Define x− = x+
�
z+1
xx− = −x−
.
⎫
iv) Suppose that u ≤ C −↓(R) satisfies the distributional equation
(x dx − z)u = 0 (meaning of course, x dx = zu where z is a
constant). Show that
dx −
z−1
du
d
z
u
x>0 = c+x−
z
x>0 and u
z
x<0 = c−x−
x<0
⎭
for some constants c±. Deduce that v = u−... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
negative
integral order we need to work a little differently. Fix k ≤ N.
i) We define weak convergence of distributions by saying un ∩ u in
↓(X), where un, u ≤ C−↓(X), X ⊃ Rn being open, if un(π) ∩
C
c
u(π) for each π ≤ C↓(X). Show that un ∩ u implies that
τun ∩ τu for each j = 1, . . . , n and f un ∩ f u if f ≤ C↓(... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
this limit distribution satisfies (x dx + k)uk = 0.
viii) Conclude that (17.7) does in fact hold for z ≤ −N as well.
d
[There are still some things to prove to get this.]
Problem 51. Show that for any set G ⊃ Rn
↓
�
v (G) = inf
v(Ai)
where the infimum is taken over coverings of G by rectangular sets
(products of i... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
. 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 vector space V (over C) with a ‘positive
definite sesquilinear inner product’ i.e. a function
V × V � (v, w) ◦∩ ≡v, w� ≤ C
satisfying
• ≡w, v� = ≡v, w�
• ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
fines 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
v ≤ C minimizing the norm, i.e. such that
�v� = inf �w�.
w≤C
Hint: Use... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
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 |f (x)|dx.
ii) Show that simple functions
N
j=1 cj λ(Uj ) where the Uj are open
�
and bound... | 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 |
) Recall (from class or just show 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
(... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
�
U = 0
⎭
⎭
�
LECTURE NOTES FOR 18.155, FALL 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... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
1)
|v(f )| → C�f �(N ) � f ≤ C↓(Rn).
Show that such a v ‘is’ a distribution and that the map E ∞(Rn) −∩
−↓(Rn) 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 c... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
Δ) ∅= 0 where it
makes sense,
(17.12)
Dk D�
β
1
δ p(θ, Δ)
|�|
=
qk,�,j (Δ)
p(θ, Δ)k+j+1
j=1
�
where qk,�,j (Δ) 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 Fou... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
(H(x)) where H(x) ≤ S ∞(R) is the Heaviside func
tion
H(x) =
1 x > 0
0 x → 0
⎬
.
Hint: Dx is elliptic in one dimension, hit H with it.
iii) Compute WF(E), E = iH(x1)β(x∞) which is the Heaviside in
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 ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
u = E ξf satisfies (D2 −D2
t
x)u =
c
f.
v) With u defined as in iv) show that
supp(u) ⊃ {(t, x); �
(t∞ , x ) ≤ supp(f ) with t + x → t + x and t∞ − x → t − x}.
∞
∞
∞
∞
vi) Sketch an illustrative example of v).
vii) Show that, still with u given by iv),
sing supp(u) ⊃ {(t, x); � (t∞ , x ) ≤ sing supp(f ) with
∞
∞
... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
↓ outside the origin.
iv) Show that F satisfies the heat equation
n
(ωt −
ω2 )F (t, x) = 0 in (t, x) ∅= 0.
xj
j=1
�
120
RICHARD B. MELROSE
v) Show that F satisfies
(17.13)
F (s t, sx) = s −nF (t, x) in S
2
∞(Rn+1
)
where the left hand side is defined by duality “F (s2t, sx) = Fs ”
where
x
t
Fs(π) = s −n−2... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
,S]
(1 + |x|)N |D� u(t, x)| < ⊂ � S > 0, � ≤ Nn+1, N.
ix) Supposing that u satisfies (17.16) and is a real-valued solution
of
n
(ωt −
ω2
xj
)u(t, x) = 0
in Rn+1 , show that
j=1
�
v(t) =
2
u
(t, x)
Rn
�
is a non-increasing function of t.
Hint: Multiply the equation by u and integrate over a slab
[t1, t2] × ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
j )u = ϕ, u = 0 in t < T for some T.
n
j=1
�
What is the largest value of T for which this holds?
xii) Can you give a heuristic, or indeed a rigorous, explanation of
why
c =
exp(−
Rn
|x|2
4
)dx?
�
xiii) Explain why the argument we used for the wave equation to
show that there is only one solution, u ≤ C ↓(Rn+1)... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
functions.
v≤L
�
122
RICHARD B. MELROSE
ii) Show that there exists f ≤ C↓(Rn) such that ALf ∈ 1 is the
c
costant function on Rn .
iii) Show that the map (17.19) is surjective. Hint: Well obviously
enough use the f in part ii) and show that if u is periodic then
AL(uf ) = u.
iv) Show that the infinite su... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
the norm on B being
(17.22)
�B�B = sup{�Bf �H ; f ≤ H, �f �H = 1}.
i) Show that B is complete with respect to this norm. Hint (prob
ably not necessary!) For a Cauchy sequence {Bn} observe that
Bnf is Cauchy for each f ≤ H.
ii) If V ⊃ H is a finite-dimensional subspace and W ⊃ H is a
closed subspace with a finite-d... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
matrix theory). What
might it mean to say in this case that (Id −zA)−1 is meromor
phic in z? (No marks for this second part).
v) Recall that K ⊃ B is the algebra of compact operators, defined
as the closure of the space of finite rank operators. Show that
K is an ideal in B.
vi) If A ≤ K show that
Id +A = (Id +B)(... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
Vj which is orthogonal to all the previous
k ’s.
e
ii) A Hilbert space is separable if it has a countable dense subset
(sometimes people say Hilbert space when they mean separable
Hilbert space). Show that every separable Hilbert space has a
complete orthonormal sequence, that is a sequence {ej } such
that ≡u, ej... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
that a compact subset of a Hilbert space is closed and
bounded.
ii) If ej is a complete orthonormal subspace of a separable Hilbert
space and K is compact show that given ∂ > 0 there exists N
such that
(17.23)
|≡u, ej �|2 → ∂ � u ≤ K.
j→N
�
iii) Conversely show that any closed bounded set in a separable
Hilbert... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
Throughout this problem A will be a
compact operator on a separable Hilbert space, H.
i) Show that if 0 ∅= � ≤ C then
E� = {u ≤ H; Au = �u}.
is finite dimensional.
ii) If A is self-adjoint show that all eigenvalues (meaning E� ∅= {0})
are real and that different eigenspaces are orthogonal.
iii) Show that �A = sup{... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
self-adjoint operator on a sep
arable Hilbert space there is a complete orthonormal basis of
eigenvectors. Hint: Be careful about the null space – it could
be big.
Problem 70. Show that a (complex-valued) square-integrable function
u ≤ L2(Rn) is continuous in the mean, in the sense that
(17.24)
lim sup
ξ∗0 |y|<... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
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:
i) (Equi-continuity in the mean) For each ∂ > 0 there exists β > 0
such that
(17.26)
|u(x + y) − u(x)|2dx < ∂ � |y| < β, u ≤ B.
Rn
�
ii) (Equi-smallness at infinity) For each ∂ > 0 there exi... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
ly 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 functions on
R
n with compact support. Thus each element vanishes in |x| > R for
some R, depending on the function. We want to give this a toplogy in
terms o... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
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 topology defined in (6.16) means
precisely that for each n ≤ N there exists k = k(n) and C = Cn such ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
−1
u(Δ1, Δ∞)dΔ1, � Δ∞ ≤ Rn−1 .
ˆ
R
�
Use Cauchy’s inequality to show that this is continuous as a map on
Sobolev spaces as indicated and then the density of S(Rn) in H m(Rn)
to conclude that the map is well-defined and unique.
Problem 76. [Restriction by WF] From class we know that the product
of two distribution... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
2αi
≡[(A − t − i∂)−1 − (A + t + i∂)−1]π, ϕ� −∩ µα,φ
in the sense of distributions – or measures if you are prepared to work
harder!
Problem 78. If u ≤ S(Rn) and ϕ∞ = ϕR + µ is, as in the proof of
Lemma 12.5, such that
show that
supp(ϕ∞) ∃ Css(u) = ∞
S(Rn) � π ◦−∩ πϕ∞ u ≤ S(Rn)
is continuous and hence (or otherwise... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
2.51)
(17.36)
WFsc(u�v) ⊃ {(x+y, p); (x, p) ≤ WFsc(u)∃(Rn×Sn−1), (y, p) ≤ WFsc(v)∃(Rn×Sn−1)}
∞∞χ∞∞
∞∞χ∞∞|
∗ {(χ, q) ≤ Sn−1 × Bn; χ =
s∞χ∞ + s
|s∞χ∞ + s
, 0 → s ∞ , s → 1,
∞∞
(χ∞ , q) ≤ WFsc(u) ∃ (Sn−1 × Bn
), (χ∞∞
, q) ≤ WFsc(v) ∃ (Sn−1 × Bn)}.
Problem 82. Formulate and prove a bound similar to (17.36) for WFsc(uv... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
standard
fundamental solution for the Laplacian on R3 . Using Problem 83 give
a condition on WFsc(f ) under which u = E � f is defined and satisfies
φu = f. Show that under this condition
f is defined using Prob
lem 84. What can you say about WFsc(u)? Why is it not the case that
φu = 0, even though this is true if u... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
6.801/6.866: Machine Vision, Lecture 19
Professor Berthold Horn, Ryan Sander, Tadayuki Yoshitake
MIT Department of Electrical Engineering and Computer Science
Fall 2020
These lecture summaries are designed to be a review of the lecture. Though I do my best to include all main topics from the
lecture, the lectures will... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
1 Rotation Operations
Relevant to our discussion of quaternions is identifying the critical operations that we will use for them (and for orthonormal
rotation matrices). Most notably, these are:
1. Composition of rotations:
o
p
o
q = (p, q)(q, q) = (pq − q · q, pq + qq + q × q)
2. Rotating vectors:
(cid:48)
o
r
∗
o
q
o... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
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