text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
, and Y1(γ(t)) =
γ�(t)/�γ�(t)�. Then
κgeod(t) =
�Y2(γ(t)), (d/dt)Y1(γ(t))�
�γ�(t)�
.
Lemma 41.2. Take a partial parametrization f : U → M ⊂ R3 which
is compatible with our choice of orientation, and let γ = f (c). Suppose
that we have a moving frame (X1(x), X2(x)) which is positively oriented,
and such that X1(... | https://ocw.mit.edu/courses/18-950-differential-geometry-fall-2008/624f231226f7c15ddce6ba6dfad2db64_ch4_revised.pdf |
equality
κtot = 2πχ(S) −
geod
�
S
κgauss
dvol.
More classical is the case of a geodesic triangle with corners:
�
Corollary 41.4 (proof sketched). Let M ⊂ R3 be a surface, f : U → M a
partial parametrization, and T ⊂ U a curvilinear triangle, whose sides map
to geodesics in M . Let α1, α2, α3 be the angles at ... | https://ocw.mit.edu/courses/18-950-differential-geometry-fall-2008/624f231226f7c15ddce6ba6dfad2db64_ch4_revised.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.013/ESD.013J Electromagnetics and Applications, Fall 2005
Please use the following citation format:
Markus Zahn, Erich Ippen, and David Staelin, 6.013/ESD.013J
Electromagnetics and Applications, Fall 2005. (Massachusetts Institute
of Technology: MIT OpenCourseWare). http://o... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/6254accbbe56e6cdd24d2916bcdbccd3_lec4.pdf |
E0
= ρ
6.013, Electromagnetic Fields, Forces, and Motion Lecture 4
Prof. Markus Zahn Page 1 of 6
i
E ds
=
(cid:118)
∫
C
b
∫
a
I
i
E ds
+
a
∫
b
II
i
E ds
0
= ⇒
b
i
E ds
∫
a
I
(cid:8)(cid:11)(cid:9)(cid:11)(cid:10)
Electromotive Force
(EMF)
=
b
∫
a
II
i
E ds
EMF be... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/6254accbbe56e6cdd24d2916bcdbccd3_lec4.pdf |
i
n r
r
∆ = ∇Φ ∆
i
The gradient is in the direction perpendicular to the equipotential
surfaces.
III. Vector Identity
∇ ×
E 0
=
E = −∇Φ
(
∇ × ∇Φ =
) 0
6.013, Electromagnetic Fields, Forces, and Motion Lecture 4
Prof. Markus Zahn Page 3 of 6
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/6254accbbe56e6cdd24d2916bcdbccd3_lec4.pdf |
� ⇒ ∇ Φ = − ρ
i
2
)
0
ε
0
(
(
2
i
∇ Φ = ∇ ∇Φ =
)
_
i
⎡
⎢
⎣
x
∂
x
∂
_
i
+
y
_
i
z
+
∂
y
∂
∂
z
∂
⎤ ⎡
i
⎥ ⎢
⎦ ⎣
∂Φ
x
∂
_
i
+
x
∂Φ
y
∂
_
i
+
y
∂Φ
z
∂
_
i
z
⎤
⎥
⎦
=
2
2
∂ Φ ∂ Φ ∂ Φ
2
y
∂
∂ 2
2
x
∂
+
+
z
2
VI. Coulomb Superposition Integral
1. Point Charge
E
r
= −
∂Φ
r
∂
=
q
ε
π
r
0
2
4
⇒ Φ =
q
ε
π
4
r
0
+
C
Take referenc... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-fall-2005/6254accbbe56e6cdd24d2916bcdbccd3_lec4.pdf |
Lecture 8
We’ll just list a bunch of definitions. X a topological Hausdorff space, second countable.
Definition. A chart is a trip (ϕ, U, V ), U open in X, V open in C and ϕ : U
� �
V a homeomorphism.
→
If you consider two charts (ϕi, Ui, Vi), i = 1, 2 we get an overlap diagram. Charts are compatible if and
� �
only if... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/626d45058e202e3419b56714fb4c5ad2_18117_lec08.pdf |
),
f
Take X to be an n-dimensional complex manifolds, if we think of X as a C∞ 2n-dimensional then TpX
X, q = f (p)
is well defined. But we showed that TpX has a complex structure. f : X
in the real case dfp : Tp
Tq, but we check that this is also C-linear.
Z holomorphic, then f
g : X Z is as well.
Y holomorphic... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/626d45058e202e3419b56714fb4c5ad2_18117_lec08.pdf |
is the domain of the chart (U0, V, ϕ), V an open set in Cn , ϕ : U0
biholomorphism. Then just apply last lecture version of implicity function theorem to fi
X is an open
Assume df1, . . . , dfk are linearly independent at p. Then there exists a
U0 such that wi = fi for i = 1, . . . , k.
1). →
U , U
V a
(ϕ−
U0.
⊂... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/626d45058e202e3419b56714fb4c5ad2_18117_lec08.pdf |
the implicit function theorem this definition is equivalent to the following weaker definition.
Definition. Y is a k-dimensional submanifold X if for every p
and fi
f1 =
= fl = 0, i.e. locally Y is cut-out by l independent equation.
(U ) where i = 1, . . . , l, l = n
k such that df1, . . . , dfl are linearly independe... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/626d45058e202e3419b56714fb4c5ad2_18117_lec08.pdf |
, zn)
′ n)
∼
(z
′ , z
′ , . . . , z
0
1
= λzi, i = 0, . . . , n.
/
}
0
− {
[z0, z1, . . . , zn]
∼
} →
− {
0
.
(z0, z1, . . . , zn)
∼
1(U ) is open.
We topologize CP n by giving it the weakest topology that makes π continuous, i.e. U
π−
Lemma. With this topology CP n is compact.
⊆
CP n is open if
Proof. Take ... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/626d45058e202e3419b56714fb4c5ad2_18117_lec08.pdf |
7→
where w1 is in the 0th place, and 1 is in the ith place. The overlap diagrams for U0 and U1 are given by
ϕ0
U0 ∩
���������
ϕ0,1
U1
ϕ1
���������
� V1,0
V0,1
We can check that V0,1 = V1,0 =
(z1, . . . , zn), zi = 0
}
{
. Also check that
ϕ0,1 : V0,1
→
V1,0
(z1, . . . , z
n)
This standard atlas gives a com... | https://ocw.mit.edu/courses/18-117-topics-in-several-complex-variables-spring-2005/626d45058e202e3419b56714fb4c5ad2_18117_lec08.pdf |
Lecture 0
Course overview
1) Harmonic functions Δu = 0, i.e.
Dirichlet problem: (Ω ⊂ Rn)
�
�
i uii = 0.
Δu = 0
u = ϕ
, x ∈ Ω,
, x ∈ ∂Ω.
In this course, we will mainly be concerned with the following problems:
2) Heat equation: ut = Δu, u : Rn+1 R.→
Boundary value problem: cylinder domain Ω × [0, T ), Ω ⊂ Rn... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/62906408ea3a1fde43dee67b2c5ee0d0_lec0.pdf |
− f (y)
When α = 1, f is just Lipstitz continuous functions.
| ≤ �f �
⇒ |
For Δu = f in Ω, we will get Interior Estimates
�
u�C2,α(Ω�) ≤ C(�f �Cα(Ω) + �u�C0 (Ω)),
1
�
where Ω� ⊂⊂ Ω, C = C(Ω, Ω�).
Notion of weak solution:
�
Δu = f weakly on Ω if Ω uΔϕ = Ω ϕf, ∀ϕ ∈ Cc
�
2(Ω), here u ∈ L1
loc(Ω).
Regularity th... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/62906408ea3a1fde43dee67b2c5ee0d0_lec0.pdf |
nd order derivation of u belong to Lp, i.e.
�
1
�
(
Ω
1
|D2 u|p) p < ∞ ?
1 < p < ∞
We can get
�
u�W 2,p(Ω�) ≤ C(�f �Lp + �u�Lp ).
We just look at Δ. The next is more general elliptic operators:
�
�
Lu =
aij (x)Dij u +
bi(x)Diu + c(x)u = f.
i,j
i
We also consider the following problems:
�
�
Lu = f
u =... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/62906408ea3a1fde43dee67b2c5ee0d0_lec0.pdf |
understand Perelman’s proof of Poincar´e
conjecture, you have to know this stuff.
∂
∂t
∂
∂t
g = −2Ric
gij ∼ Δg gij + lower terms.
Fundamental Result: (M 3, g) compact 3manifold, then ∃ε > 0 s.t. Ricci flow
system has a smooth solution on M × [0, ε).
(This is called short time existence theorem.)
Examples of harm... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/62906408ea3a1fde43dee67b2c5ee0d0_lec0.pdf |
nωn
y n
|
δij −
i
n(x − yi)(xj − yj )
|x − y n+2
|
}
Definition 2
We have Λt = ΔΛ:
Λ(x, y, t, t0) =
1
(4π t − t0 )n/2
|
|
2
|x−y|
e 4(t0−t) .
Λ +
2
|
x − y|
4(t − t0)2
Λ
Λt =
−
Λ i =
x
1
n
2 (t − t0)
i
x − y
2(t0 − t)
i
Λ
x x⇒
= Λ i i =
= ⇒ ΔΛ = −
(xi − yi)2
4(t0 − t)
1
n
2 (t − t0)
Λ +
Λ +
1
Λ
... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/62906408ea3a1fde43dee67b2c5ee0d0_lec0.pdf |
The four fundamental subspaces
In this lecture we discuss the four fundamental spaces associated with a matrix
and the relations between them.
Four subspaces
Any m by n matrix A determines four subspaces (possibly containing only the
zero vector):
Column space, C(A)
C(A) consists of all combinations of the colum... | https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/62a9db9eeab190694d40afe4734068ca_MIT18_06SCF11_Ses1.10sum.pdf |
:
dim C(AT) = r.
Left nullspace
The matrix AT has m columns. We just saw that r is the rank of AT, so the
number of free columns of AT must be m − r:
dim N(AT) = m − r.
The left nullspace is the collection of vectors y for which ATy = 0. Equiva
lently, yT A = 0; here y and 0 are row vectors. We say “left nullspa... | https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/62a9db9eeab190694d40afe4734068ca_MIT18_06SCF11_Ses1.10sum.pdf |
zero matrix (additive
identity). If we ignore the fact that we can multiply matrices by each other,
they behave just like vectors.
Some subspaces of M include:
2
• all upper triangular matrices
• all symmetric matrices
• D, all diagonal matrices
D is the intersection of the first two spaces. Its dimension is 3;... | https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/62a9db9eeab190694d40afe4734068ca_MIT18_06SCF11_Ses1.10sum.pdf |
THE KODAIRA DIMENSION OF THE MODULI SPACE OF
CURVES
1. Preliminaries
A great reference for background about linear systems, big and ample line bun
dles and Kodaira dimensions is [L]. Here we will only develop a few basics that
will be necessary for our discussion of the Kodaira dimension of the moduli space
of cu... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
aka dimension of a line bundle L on
X is an integer between 0 and dim(X) or it is −→.
Definition 1.3. A line bundle L on a normal, projective variety is called big if its
Iitaka dimension is equal to the dimension of X. A smooth, projective variety is
called of general type if its canonical bundle is big. A singular... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
OE (mD) � 0.
Since D is big by assumption, the dimension of global sections of OX (mD) grows
like mdim(X). On the other hand, dim(E) < dim(X), hence the dimension of global
sections of OE (mD) grows at most like mdim(X)−1 . It follows that
h0(X, OX (mD)) > h0(E, OE (mD)
for large enough m ⊕ N (X, D). The lemma fol... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
large enough
positive number r such that both rA and (r + 1)A are effective. By Kodaira’s
Lemma there is a positive integer m such that mD − (r + 1)A is effective, say
linearly equivalent to an effective divisor E. We thus get that mD is linearly
equivalent to A + (rA + E) proving (2). Clearly (2) implies (3) and (3) ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
c1(Γ ≤ �)
2
− c2(Γ ≤ �))(1 −
c1(Γ)
2
+
2
c1(Γ) + c2(Γ)
12
)
⎞
Expanding (and using the relations we proved in the last unit) we see that this
expression equals
α� 2c1(�) − [Sing] − c1(�) +
2
2
�
2c1(�) + [Sing]
12
⎞
= 13� − 2ν.
We need to adjust this formula to take into account that every element of the locu... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
sum of an ample and
an effective divisor. The G.I.T. construction gives us a large collection of ample
divisors.
For our purposes we need only the following fact:
Lemma 3.1. The divisor class � is big and NEF.
Proof. The shortest proof of this result is based on some facts about the Torelli
map and the moduli spac... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
an effective divisor.
�
4. The moduli space is of general type
In this section we would like to sketch the main steps of the proof of the following
fundamental theorem due to Harris, Mumford and Eisenbud. You can read more
about the details in [HM1] §6.F. The papers [HM2], [H] and [EH5] contain the
proofs.
Theorem... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
do the job for Theorem 4.1.
However, the calculation of these divisor classes are not easy.
The second problem is that even if we show that there are many canonical forms
on M g, this does not necessarily prove that the moduli space is of general type.
The problem is that M g is singular. It is possible that canoni... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
form on V 0/G extends holomorphically to a desingular
ization of V /G.
In view of the Reid-Tai Criterion one has to check whether
ai ∼ 1 holds and
in cases it does not hold verify by hand that the pluri-canonical sections extend
holomorphically to a desingularization. The following theorem characterizes the
stabl... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
− 1 at one point. π is an order 4 automorphism of C1 and is
the identity on C2.
The proof of this result rests on a case by case analysis of the possibilities based
on a lemma that solves the problem for smooth curves.
Lemma 4.4. Let C be a smooth curve. Let π be an automorphism of C of order n.
Let λ be a primiti... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
r) = −1
form a divisor on Mg called the Brill-Noether divisor. Its class is given by the
following theorem:
Theorem 4.5. If g + 1 = (r + 1)(g − d + r), then the class of the Brill-Noether
divisor on Mg is given by
c
⎝
(g + 3)� −
νirr −
g + 1
6
∗g/2⊗
⎛
i=1
i(g − i)νi
�
�
where c is a positive rational consta... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
C lies on a quadric cone. For such curves the two
1
g3 s come together. The Petri divisor is simply the closure of such curves.
1
Exercise 4.7. Calculate the class of the divisor given by the closure of curves
whose canonical model lies in a singular quadric.
Let C be a smooth, non-hyperelliptic curve of genus 6. ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
4.1. When g ∼ 24 and odd, we can use the Brill-Noether
divisor with r = 1. The relevant ratio is that of � and ν0 and is equal to
6 +
12
.
g + 1
When g ∼ 24 this is less than 6.5, hence the canonical class of Mg is big provided
g + 1 is not prime. The Brill-Noether divisors also take care of the cases g = 24, 26... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
.
Brill-Noether theory asks the following fundamental question:
Question 5.1. When can a curve of genus g be represented in Pr as a non-
degenerate curve of degree d?
There is an expected answer to this question. We are asking when does there
exist a degree d line bundle on a curve C of genus g with at least an r+1... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
-constant mero
morphic function has degree 2. For instance, in the case of genus 1, the
Weiestrass p function is such a function.
(3) If S has genus 3, already the story becomes more complicated. If S is hyper-
elliptic, then it does admit a meromorphic funciton of degree 2. However,
not all genus 3 curves are hype... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
(Hint: The canonical image of
a general curve of genus 6 lies on a degree 5 Del Pezzo surface in P5.)
(7) One can carry the analysis a little further. In fact the following is known.
Proposition 5.3. Every Riemann surface of genus g admits a non-constant
meromorphic function of degree ∪ g+3 ⊂. Moreover, a general R... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
case of the Brill-
Noether theorem. If we take r = 1, then we see that the Brill-Noether number is
non-negative if and only if d ∼ ∪ g+3 ⊂.
2
A sketch of the proof. The idea of the proof goes back to Castelnuovo. Let us
consider a g-nodal rational curve and try to calculate the dimension of the space of
rgds on suc... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
will make the Schubert
cycles �r be defined with respect to tangent lines to the rational normal curve.
Note that the semi-stable reduction of such a curve is the normalization of the curve
with g elliptic tails attached at the points that map to the cusps. In particular, the
non-compactness issue disappears.
r
The... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
-space dimension
r + 1 on a genus g curve. Then the sum of the ramification indices satisfy the
following equality
�j (p) = (r + 1)d +
r(r + 1)
2
(2g − 2).
⎛
j,p
Proof of Proposition. The Taylor expansions of order r of the sections in V gives a
map to the bundle of r-jets of sections of L
Taking the r + 1st e... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
· · ·
. . .
· · ·
· · ·
. . .
�
⎝
.
This order is precisely the left hand side of the formula in the proposition.
�
�
�
In particular, when the genus is equal to zero we see that the total ramification
is equal to (r + 1)(d − r). Since the total ramification may not exceed this number
�
it is now easy to conc... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
+ g − 1 = r − d + g.
Since the Brill-Noether number is negative, we must have (r + 1)(r − d + g) ∼ g + 1.
Hence the domain of the map H 0(C, L) ≤ H 0(C, K ≤ L−1), where L is the line
bundle giving the gr
d has dimension at least g + 1. Consequently, the Petri map
cannot be injective. We conclude that for a Gieseker... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
subsection we will briefly sketch the theory of
limit linear series for curves of compact type developped by Eisenbud and Harris
in order to study Brill-Noether theory. Since Joe has written very good accounts
of the theory our treatment will be brief. One of the main uses of the theory is
to describe the closure of... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
0 � OC � OC1 ≥ OC2 � OC1 �C2 � 0
h1(C, OC ) = h1(C1, OC1 ) + h1(C2, OC2 ).
This completes the proof that 1 implies 2.
To see that 2 implies 1, we observe that by the same exact sequence that the
genus of a curves is at least the sum of the genus of its components. If there is a
loop, then by the exact sequence the... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
stable curve is tree-like
if after normalizing the curve at its non-separating nodes one obtains a curve of
compact type. In other words, a tree-like curve differs from curves of compact type
so that the irreducible components may have internal nodes.
The main difficulty. Suppose you have a one-parameter family of cur... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
p the aspects
VY and VZ satisfy
ai(VY , p) + ar−i(VZ , p) ∼ d.
The limit linear series is refined if the following inequalities are equalities for every
i. The limit linear series is crude if one inequality is strict.
Using the Pl¨ucker formulae one may generalize the Brill-Noether theorem to
curves of compact typ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
the marked points and the map that
attaches a fixed genus g − 2 curve, respectively, the pull-back to M0,n is zero while
the pull-back to M2,1 is supported on the Weierstrass divisor.
5.3. Calculating the classes of the Brill-Noether divisors. In this subsection
we complete our discussion of the proof of Theorem 4.1... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
coef
ficients. First, consider the map
atg−2 : M2,1 � Mg
obtained by attaching a fixed genus g − 2 curve with a marked point to curves
of genus 2 with a marked point along their marked points. The theory of limit
linear series shows that the pull-back of the Brill-Noether divisor is a multiple of
the divisor W on M... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
the Weierstrass divisor W , we note that a Weierstrass
point is a ramification point of the canonical linear series. Using this one can exhibit
W as the degenracy locus of a map between vector bundles.
14
Exercise 5.18. Carry this out and complete the calculation of the class of W .
(Hint: See page 338-339... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
the total space of the family is smooth. Contracting the components
with fewer sections (or either of the components when equal numbers of sections
pass through both components), we obtain a P1 bundle with g sections
˜α : C � B.
˜
Since the classes of any two sections differ by a multiple of the fiber class, the
diff... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
the ample cone of Mg . Combining this with our knowledge of
some special effective divisors we could conclude the proof. One may ask the more
detailed questions:
Question 6.1. In terms of the generators of the picard group �, ν1, . . . , ν∗g/2⊗ what
is the ample cone of Mg ? What is the effective cone of Mg ?
Almost... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
-marked points. Let the last
marked point vary keeping the rest fixed.
Show that such curves cover an open
subset of M0,n/Sn and only intersect Θ2 among the boundary divisors.) Show
that the coefficient of Θt has to be non-negative by induction on t. (Hint: Assume
that the effective divisor does not contain any of the... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
component of such a curve
is a P1 whose normalization contains exactly three distinguished points. The one
dimensional strata consist of curves with 3g − 4 + n nodes. Every component
but one of a curve with 3g − 4 + n nodes is a P1 whose normalization has three
distinguished points. The remaining component is eithe... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
the F-curves. As already observed every component
but one of a curve parameterized by a general point on an F-curve corresponds to a
P1 with 3 distinguished points. If the remaining component is a genus 1 curve with
one marked point, then when we separate the curve at this marked point we obtain
a curve of genus g ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
[ACGH] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris. Geometry of algebraic curves.
Vol. I, volume 267 of Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985.
M. Cornalba and J. Harris. Divisor classes associated to families of st... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
359–387.
[GKM] A. Gibney, S. Keel, and I. Morrison. Towards the ample cone of M g,n . J. Amer. Math.
[GH]
[H]
[HM1]
[HM2]
[KL1]
[KL2]
[L]
Soc. 15(2002), 273–294.
P. Griffiths and J. Harris. On the variety of special linear systems on a general algebraic
curve. Duke Math. J. 47(1980), 233–272.
J. Harris. On th... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
Surveys in Mathematics.
Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series.
18 | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/62affca28e907c4bbfb6655e172a04d6_generaltype.pdf |
18.782 Introduction to Arithmetic Geometry
Lecture #16
Fall 2013
10/31/2013
Our goal for this lecture is to prove that morphisms of projective varieties are closed
maps. In fact we will prove something stronger, that projective varieties are complete, a
property that plays a role comparable to compactness in topology. ... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
. Let X ⊆ Am and Y
] = k[y1, . . . , yn], and k[A +n] = k[x1, . . . , xm, y1, . . . , yn], so that we can identity k[Am]
k[An
and k[An] as subrings of k[Am+n] whose intersection is k. The product X × Y is the zero
locus of the ideal I(X)k[An] + I(Y )k[Am] in k[Am+n].
⊆
m
If I(X) = (f1, . . . , fs) and I(Y ) = (g1, . . ... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
the rule
∈
,
(ri1 ⊗ sj1)(ri2 ⊗ sj2) = ri1ri2 ⊗ sj1sj2.
Andrew V. Sutherland
11 In the case of polynomial rings one naturally chooses a monomial basis, in which case this
rule just amounts to multiplying monomials and keeping the variables in the monomials
separated according to which polynomial ring they originally cam... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
⊆ An, and let mP be the corresponding
maximal ideal (x1 − a1, . . . , xn − an) of k[x1, . . . , xn]. Then I(V ) ⊆ mP , and the image of
mP in the quotient k[V ] = k[x1, . . . , xn]/I(V ) is a maximal ideal of k[V ]. Conversely, every
maximal ideal of k[V ] corresponds to a maximal ideal of k[x1, . . . , xn] that contai... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
= φP (u)φP (v) =
(cid:32)
(cid:88)
i∈I
φP (ri)si
(cid:33)
(cid:88)
j∈J
φP (rj)sj = 0
Since S is an integral domain, one of the two sums must be zero, and since
the si are linearly
independent over k, either φP (ri) = 0 for all the ri, in which case P ∈ Xu, or φP (rj) = 0
for all the rj, in which case P ∈ Xv. Th... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
. We are now interested in polynomials in k[x0, . . . , xm, y0, . . . , yn] that
are homogeneous in the xi, and in the yj, but not necessarily both. Another way of say-
ing this is that we are interested in polynomials that are homogeneous as elements of
(k[x0, . . . , xm])[y0, . . . , yn], and as elements of (k[y0, . ... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
indeed a
projective variety. This will be explored on the next problem set.
We may also consider products of affine and projective varieties. In this case we are inter-
ested in subsets of Pm ⊗An that are the zero locus of polynomials in k[x0, . . . , xm, y1, . . . , yn]
that are homogeneous in xi but may be inhomogeneou... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
in X and Y that may have many coordinates; the exact equation
can be explicitly spelled out in the ambient space containing X × Y using generators for
I(X), I(Y ), and the coordinate maps of φ but there is no need to do so). The projection
map X × Y → Y is a closed map, since X is complete, so im(φ) is a closed subset ... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
suppose X is an affine variety of positive dimension and let f be a function in
k[X] that does not lie in k; such an f exists since k(X) has positive transcendence degree.
The morphism f : X → A1 that sends P to f (P ) most then be dominant, because the
dual morphism of affine algebras k[A1] → k[X] is injective; it corresp... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
the set
R = {x ∈ K : (cid:107)x(cid:107) ≤ 1}
is a valuation ring. You also proved that such an R is a local ring.
Definition 16.16. A local ring is a ring R with a unique maximal ideal m. The field R/m
is the residue field of R.
Note that fields are included in the definition of a local ring (the unique maximal ideal
is th... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
⊆ (b). This leads to the following definition.
Definition 16.20. Let R be a valuation ring with fraction field K. The value group of R
is Γ = K×/R×. The valuation defined by R is the quotient map v : K× → Γ.
5
5 The abelian group Γ is typically written additively, and it follows from Lemma 16.19
that it is totally ordered ... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
R with n minimal and
suppose n > 1. We must have a1/a2 (cid:54)∈ R, else the generator a1 = (a1/a2)a2 is redundant.
But then a2/a1 ∈ R and a2 = (a2/a1)a2 is redundant, a contradiction.
Lemma 16.22. A local ring is a valuation ring if and only if it is an integral domain that
is not a field and all of its finitely generat... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
which local rings arise is by localizing an integral domain at one
of its prime ideals.
Definition 16.24. Let R be an integral domain and let p be a prime ideal in R. The
subring of R’s fraction field defined by
is called the localization of R at p.2
Rp := {a/b : a, b ∈ R, b (cid:54)∈ p}
Remark 16.25. As we saw in Lecture... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
one, as you will prove on the problem set.
16.5 Valuative criterion for completeness
We now return to our goal of proving that every projective variety is complete. Let X be
a variety with coordinate ring k[X], and let P be a point in X. We then define the ideal
mP := {f ∈ k[X] : f (P ) = 0}.
Note that we have defined wh... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
is closed,
and we may replace Y with the image of V ⊆ X × Y → Y , since whether the image is
closed or not does not depend on anything outside of its closure. We now replace X with
the image Z of V ⊆ X × Y → X, to which we will apply the hypothesis of the theorem.
We have the following commutative diagram with dominant... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
Y ] (cid:39) k[Z] ⊗ k[Y ]
in k[V ] under the morphism dual to the inclusion V ⊆ Z × Y , and therefore S contains
k[V ] ⊆ k(V ). The intersection of ker Φ with k[V ] is a maximal ideal of k[V ] corresponding
to a point in V . This point must be (P, Q); in fact it suffices to show the second coordinate
is Q, and this is cl... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
orphism
of fields. So k is an integral domain contained in K, and the identity map φ : k → k is
a homomorphism to an algebraically closed field. By Lemma 16.29, there is a valuation
ring S of K/k whose residue field is k. The map k ⊆ S → k is then the identity map.
The preimage of R(cid:48) = Ψ−1(S) ⊆ R under the quotient... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
OP,Z ⊆ R, then X
is complete.
The next lemma is almost trivial, but it is the essential reason why projective varieties
are complete (in contrast to affine varieties), so we consider it separately.
Lemma 16.32. Let R be a valuation ring of F . For any x0, . . . , xn ∈ F × there exists
λ ∈ F × such that λx0, . . . , λxn ∈... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
one
λzi ∈ R×. Let φ : R → k be the quotient map from R to its residue field, and let P be the
projective point (φ(λz0) : φ(λz1) : . . . : φ(λzn)), where we note that at least one φ(λzi) is
nonzero. The point P lies in Z, since for any homogeneous f ∈ I(Z) of degree d we have
f (λz0, . . . , λzn) = λdf (z0, . . . , zn) =... | https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/62ecd19112f0baa33f151c5644f80b06_MIT18_782F13_lec16.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.334 Power Electronics
Spring 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/62f391a56d14d3eed360136016e1c1c1_ch9.pdf |
(cid:20)(cid:27)(cid:17)(cid:23)(cid:19)(cid:24)(cid:45)(cid:18)(cid:25)(cid:17)(cid:27)(cid:23)(cid:20)(cid:45)(cid:29)
Advanced Complexity Theory
Spring 2016
Prof. Dana Moshkovitz
Lecture 20: P vs BPP 1
(cid:54)(cid:70)(cid:85)(cid:76)(cid:69)(cid:72)(cid:29)(cid:3)(cid:36)(cid:81)(cid:82)(cid:81)(cid:92)(cid:80)(cid... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
000000000000000000000
It seems like the former sequence is more random than the latter. We cannot easily find any
patterns in it, for instance. However, by most definitions of randomness, a single string can never
be random.
In [1], Blum and Micali sought to give a useful definition of randomness that could be used in
mod... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
x
←{
[D(G(x)) = 1]
−
Pr
y←{0,1}n
[D(y) = 1] ≤ (cid:15).
We choose C to be a class with certain constraints that we are interested in examining. Picking
C = P is often a good choice. However, for this lecture we will consider C = P/poly, which will
turn out to be more useful.
Notice that if, for a particular G, D, we ha... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
distinguisher algorithms that run fast. These two ‘fast’s might not necessarily be the same.
In [3], Yao showed that the existence of some pseudorandom generators would imply that P = BPP.
This would have profound implications, for instance, for our ability to prove statements that we
1Throughout these notes, all the p... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
, G(z)) = 1].
Since for each input x, we could create the distinguisher D(y) = A(x, y), it follows by assumption
that the above probability is within 1
10 > 1 ,2
we can just accept if the probability we get is more than a half, and reject if it is less than a half.
10 of Pry[A(x, y) = 1]. Then, since 1
2 and 2
10 < 1
3... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
icali generator works as follows: given a one way permutation f , we start using a
random seed to generate a random bit string z. Then, we repeatedly apply f to the current bit
string, and take the first bit to add to the output of our generator. See Figure 1. More formally:
Definition 5. The Blum-Micali generator g : {0... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
{0, 1}k → {0, 1} such
that:
1. f is computable in time 2100k
2. The k + 1 bit string (z, f (z)) looks random to P/poly.
To make the second assumption more precise, we make the definition:
Definition 7. A function f : {0, 1}k → {0, 1} is (cid:15)-easy for size m there exists a circuit C of size
at most m such that
P rx
←{... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
atisfies the Nisan-Wigderson assumption, and n subsets T1, T2, · · · , Tn ⊆
{1, 2, · · · , s} with |T1| = k for each i. Then, for a s-bit string z, and for each i, we will write z|Ti
to refer to the k-bit string taken from the indices in z that are in Ti. Our generator G : {0, 1}s →
{0, 1}n will then be defined by
for al... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
(w) = 1].
w←Hi
For instance, H0 is the distribution of choosing all the bits from y, and so H0 is the uniform
distribution on n bits. Meanwhile, Hn is the distribution of taking all the bits from G(z), and so
Hk is the same distribution that G gives from a uniformly random seed. We have that
and so by the Pigeonhole pr... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
15)
n
.
Then, we can use D to compute f . On a given input z, we compute D(z, 0) and D(z, 1). If the
former is 1 and the latter is 0, then we pick f (z) = 0, while if the former is 0 and the latter is 1
then we pick f (z) = 1. Otherwise, we flip a coin. Relatively simple analysis will show that this
method usually compu... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/6325f3339f40dc4a33e5d6f2cb8525a6_MIT18_405JS16_P_vs_BPP1.pdf |
Lecture 4
8.821/8.871 Holographic duality
Fall 2014
8.821/8.871 Holographic duality
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2014
Lecture 4
Reminder from last lecture
In general geometry, we can identify temperature in this way: we first analytically continue time to imaginary time
t → −iτ , and then from regular... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/633ca57e91c6cbef40c0b84c785b6c28_MIT8_821S15_Lec4.pdf |
vacuum for a QFT in a curved spacetime is not unique. The procedure we described
corresponds to a particular choice. In the Schwarzschild black hole case, it is the “Hartle-Hawking vacuum”;
while in the Rindler case, it is the Minkowski vacuum reduced to the Rindler patch (reduced density matrix
of the Minkowski vacuum... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/633ca57e91c6cbef40c0b84c785b6c28_MIT8_821S15_Lec4.pdf |
Z
here |n(cid:105) represents the n-th excited state, and the partition function is Z = (cid:80)
n e−βEn .
ρT =
1
Z
(cid:88) −βEn
e
|n(cid:105)(cid:104)n|
n
2
Lecture 4
8.821/8.871 Holographic duality
Fall 2014
• In 1960’s, H. Umezawa constructed a quantum field theory at finite temperature. Let us follow his idea:
we c... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/633ca57e91c6cbef40c0b84c785b6c28_MIT8_821S15_Lec4.pdf |
105) as
1
|Ψ(cid:105) = √
Z
e− ωβ a†
2
1a†
2|0(cid:105)1 ⊗ |0(cid:105)2
where a1 and a2 correspond to the annihilation operator in H1 and H2.
• One can show that
where
b1|Φ(cid:105) = b2|Ψ(cid:105) = 0
b1 = cosh θa1 − sinh θa†
2
cosh θ = √
b1 = cosh θa2 − sinh θa†
1
sinh θ =
√
1
1 − e−βω
e− 1
2 βω
1 − e−βω
The above tr... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/633ca57e91c6cbef40c0b84c785b6c28_MIT8_821S15_Lec4.pdf |
cid:126)x)
Dφ((cid:126)x, t)e−SE [φ]
tE <0
3
Lecture 4
8.821/8.871 Holographic duality
Fall 2014
Now we come back to a QFT, say a scalar theory, in Rindler spacetime:
ds2 = −dT 2 + dX 2 = −ρ2dη2 + dρ2
Going to Euclidean signature: T → −iTE, η → −iθ
ds2
E = dT 2
E + dX 2 = ρ2dθ2 + dρ2
With θ ∼ θ + 2π, Euclidean analyti... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/633ca57e91c6cbef40c0b84c785b6c28_MIT8_821S15_Lec4.pdf |
(X, T )e−SE [φ]
=
LHP
ˆ
φ(θ=0,ρ)=φR(X)
φ(θ=−π,ρ)=φL(X)
Dφ(θ, ρ)e−SE [φ]
The above expression can also be reduced in the Rindler notation (Fig. 3)
Ψ [φ(X)] = (cid:104)φ |e−i(−iπ)HR
R
0
|φL(cid:105) =
(cid:88)
n
e−πEn χn [φR] χ∗
n [φL]
Figure 3: Minkowski vacuum wave functional expressed in the Euclidean Rindler coordina... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/633ca57e91c6cbef40c0b84c785b6c28_MIT8_821S15_Lec4.pdf |
http://ocw.mit.edu
8.821 / 8.871 String Theory and Holographic Duality
Fall 2014
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/633ca57e91c6cbef40c0b84c785b6c28_MIT8_821S15_Lec4.pdf |
8.701
Introduction to Nuclear
and Particle Physics
Markus Klute - MIT
1. Fermions, bosons, and
fields
1.1 Quantum field and
matter
1
Quantum fields and matter
Particles come to exist as quantised fields
Extension of quantum mechanics where particles are quantised
We will create and annihilate particles in reactions... | https://ocw.mit.edu/courses/8-701-introduction-to-nuclear-and-particle-physics-fall-2020/63754fd63fb00f3ea745ca8457e5a888_MIT8_701f20_lec1.1.pdf |
Lecture 6: Electromagnetic Power
Outline
1. Power and energy in a circuit
2. Power and energy density in a distributed system
3. Surface Impedance
September 27, 2005
Massachusetts Institute of Technology
6.763 2005 Lecture 6
Power in a Circuit
Power:
Image removed for copyright reasons.
Please see: Figure 2.... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/638707f1794cce4a3fc13dff1cddbd4d_lecture6.pdf |
achusetts Institute of Technology
6.763 2005 Lecture 6
4
Averaged Poynting Vector
For a sinusoidal drive:
and
Energy in a superconductor is dissipated through the normal channel
Massachusetts Institute of Technology
6.763 2005 Lecture 6
Power Loss in a Slab
Image removed for copyright reasons.
Please see: Fig... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/638707f1794cce4a3fc13dff1cddbd4d_lecture6.pdf |
conductor, with λ << δ, to lowest order
Λ(T)
1/σ0(T)
Rs ~ ω2
Ls
For Rs >> ω Ls
For Pb at 2K and 100 MHz, Rs = 10-10 Ohm/
, and Q of cavity = 1010
Massachusetts Institute of Technology
6.763 2005 Lecture 6
7 | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/638707f1794cce4a3fc13dff1cddbd4d_lecture6.pdf |
6.801/6.866: Machine Vision, Lecture 13
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 ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
/”Binary Image Processing”
We can use thresholding and other algorithms such as finding a convex hull to compute elements of an object in a binary image
(black/white) such as:
• Area (moment order 0)
• Perimeter
• Centroid (moment order 1)
• “Shape” (generalization of different-order moments)
• Euler number - In t... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
these methods are oftentimes applied to processed, not raw images.
1.1.2 Binary Template
We will discuss this more later, but this is crucial for the patent on object detection and pose estimation that we will be discussing
today. A binary template is:
• A “master image” to define the object of interest that we are ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/63bf90743360a41d70eda346816d8304_MIT6_801F20_lec13.pdf |
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