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18.409 An Algorithmist’s Toolkit
September 22, 2009
Lecturer: Jonathan Kelner
Scribe: Dan Iancu (2009)
Lecture 4
1 Random walks
Let G = (V, E) be an undirected graph. Consider the random process that starts from some vertex v ∈ V (G),
and repeatedly moves to a neighbor of the current vertex chosen uniformly at random.
... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/100377025e8520aab9f61d8585e71cc5_MIT18_409F09_scribe4.pdf |
8)
π(i) =
(cid:88)
u∈
V (G)
u∈
V (G)
d(u
)
(cid:80)
v∈V (G) d(v) (cid:80)
= u∈V (G) d(u)
v∈V (G) d(v)
(cid:80)
= 1.
We next show that, if the random walk follows the distribution π at time t, then it has the same
distribution at time t + 1. This is expressed using matrix notation in the following claim.
4-1
Claim 2 W ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/100377025e8520aab9f61d8585e71cc5_MIT18_409F09_scribe4.pdf |
6}. Assume without loss of generality that the random walk starts at time t0 = 1 at vertex
6. Then, at time t, the current vertex is odd if and only if t is odd. Therefore, the walk does not converge
to any distribution.
3
Lazy Random Walks
There is an easy way to fix the above periodicity problem. We introduce a m... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/100377025e8520aab9f61d8585e71cc5_MIT18_409F09_scribe4.pdf |
is an eigenvector of N , with eigenvalue λ. Let q = D1/2 · v. Then,
N · v = λ · v = D−1/2 · W · D1/2 · v = D−1/2 · W · q.
Multiplying by D1/2 on the left we obtain
Therefore, q is an eigenvector of W with eigenvalue λ.
W · q = λ · D1/2 · v = λ · q.
Observe that, by Claim 2, W has eigenvector D · 1, with eigenvalue... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/100377025e8520aab9f61d8585e71cc5_MIT18_409F09_scribe4.pdf |
�2 Convergence
Define the spectral gap to be
λ := 1 − μ(cid:3)
2.
For probability distributions p, q, we define their (cid:4)2 distance to be
(cid:4)p − q(cid:4)2 =
� �
(p(i) − q(i))2 .
i
The following theorem gives a bound on the rate of convergence of the lazy random walk to the stationary
distribution π.
The... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/100377025e8520aab9f61d8585e71cc5_MIT18_409F09_scribe4.pdf |
μk
i
· vi
�
n
i=2
(cid:4)pk − π(cid:4)2 = (cid:4)
ci · μk
i
n
�
i=2
�
n
�
�
�
· vi(cid:4)2 = �
n
�
2 · μ2
ci
i
�
�
�
�
k ≤ μk
2
n
�
2
ci
i=2
i=2
≤ μk
2
(vi
T p0)2 ≤ μk
2 = (1 − λ)k .
i=1
Using a similar argument, we can also show an analogous bound for (cid:4)∞ convergence.
Theorem 6 F... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/100377025e8520aab9f61d8585e71cc5_MIT18_409F09_scribe4.pdf |
of the connection between conductance and the rate of convergence is as
follows. If a graph has high conductance, it is well-connected. Therefore, a large amount of probability mass
can very quickly move from one part of the graph to another.
7
Introduction to Monte Carlo methods
Assume that we want to estimate π ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/100377025e8520aab9f61d8585e71cc5_MIT18_409F09_scribe4.pdf |
Theorem 8 (One version of the Chernoff bound) The probability that R fails to (cid:5)-approximate E[R]
is
Pr [|R − E[R]| ≥ (cid:5)E[R]] ≤ 2e −np(cid:2)2/12 = 2e −E[R](cid:2)2/12 .
Some notes on the above bound:
• The bound is near tight.
• It is necessary for the trials to be independent, in order for the bound to ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/100377025e8520aab9f61d8585e71cc5_MIT18_409F09_scribe4.pdf |
if we have a big dartboard, and a small
circle.
In particular, if p is exponentially small, then we need exponentially many trials to expect a constant number
of successes.
We can also run into trouble if it is hard to throw darts at all. That is, if it is hard to draw samples
uniformly at random from the ambient ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/100377025e8520aab9f61d8585e71cc5_MIT18_409F09_scribe4.pdf |
DIVISOR CLASSES ON THE MODULI SPACE OF CURVES
1. The cohomology of the moduli space of pointed genus zero curves
In this section we discuss the Chow rings of the moduli spaces of n-pointed
genus zero curve M0,n. Recall that we are working over the complex numbers C.
The cohomology and Chow groups of M0,n turn out t... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
λ is the class of the exceptional divisor.
Now we introduce the generators of the Chow ring. Let S be a subset of
{1, . . . , n} with the property that both S and its complement have at least two
elements. We will denote the number of elements of S by #S. Given such a set
we can define the class ζS on M0,n as the cl... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
≥S
i,l≥S,j,k /
⎪
≥S
1
(3) For two subsets S and T
ζS ζT = 0
unless S ⊗ T, T ⊗ S, S ⊗ T c or T c ⊗ S.
�
Example 1.3. Since M0,4 = P1, the classes of the three boundary divisors �{1,2},
�{1,3} and �{1,4} are linearly equivalent. If we specialize the statement of the
theorem to n = 4 we recover the cohomology of ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
�T contain the point represeting a curve C in their
intersection if and only if there are two nodes on C that divide C into C1, C2 and
∅ , C ∅ where the labeling on C1 is S and the labeling on C ∅ is T . Observe that
C1
unless the conditions S ⊗ T, T ⊗ S, S ⊗ T c or T c ⊗ S are satisfied �S and �T are
disjoint, henc... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
that one can give a very simple presentation of the coho
mology ring of D5 realizing it as the blow-up of P2 in four points. Sending the
divisors ζi,5 to the classes of the four exceptional divisors E1, . . . , E4 and ζi,j to
H − Ek − El (where {k, l} is disjoint from {i, j, 5}) for the remaining i, j gives
a ring ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
of n − 3 blow-ups along explicit smooth centers.
Set X1 = M0,n × M0,4. If S is a subset of {1, . . . , n}, we can embed the divisors
�S into X1 by first mapping �S by the universal section corresponding to the i-th
point to M0,n+1, then following it with the map to X1. Let X2 be the blow-up of
X1 along �S where #Sc ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
homology.
Remark 1.9. Observe that M0,n is an affine variety. Fixing three of the points
at 0, 1 and → we can view this space as the complement of hyperplanes in Cn−3 .
Hence, M0,n is affine of dimension n − 3. Recall that the homology of an affine
manifold vanishes above half its real dimension.
Theorem 1.10. Let X be ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
the canonical divisor of M0,n.
In fact, we can do better than the previous corollary.
Proposition 1.13. Let n ∼ 4. Fix three distinct indeces i, j, k. The second coho
mology group of M0,n has basis ζ{j,k}, ζS where i ⊕ S and #S � n − 3.
Proof. We can give an elementary proof of this result that does not depend on t... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
forgetting a point other than i, j, k.
It immediately follows that all the coefficients of the relation have to be zero. �
Remark 1.14. Note that the following proposition implies that the rank of the
second cohomology group is
2n−1 −
n2 − n + 2
.
2
2. The second homology group of the moduli space of curves
Orig... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
isors. Let ζirr be the class of
the divisor of curves �irr that contain a non-separating node. Let 0 � h � g be
an integer and let S be a subset of {1, . . . , n}. Let ζh,S be the class of the divisor
�h,S of curves that contain a node which separates the curve into two components
of genus h with marked points pi f... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
among these classes are generated by
(2) If g = 2, there is the additional relation
ζh,S = ζg−h,Sc .
(3) If g = 1, there are the following two additional relations
5� = 5ξ + ζirr − 5ζ0 + 7ζ1.
� = ξ − ζ0, 12ξp = ζirr + 12
ζ0,S .
p≥S,#S�2
⎪
Since Theorem 1.2 already determines the genus zero case we will omit it ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
´e duality and the long exact sequence of
cohomology
Hc
k (Mg,n, Q) � H k (Mg,n, Q) � H k(ζMg,n, Q) � Hc
k+1(Mg,n, Q)
we conclude the following proposition.
Proposition 2.2. The map H k(Mg,n, Q) � H k (ζMg,n, Q) is an isomorphism when
k < d(g, n) and injective when k = d(g, n), where d(g, n) is defined by
n − 4
... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
boundary of Mg,n. The map
H k (Mg,n, Q) � ≥i≥I H k(Xi, Q)
is injective if k � d(g, n).
Sketch. This proposition follows from the fact that the map
H k (Mg,n, Q) � H k(ζMg,n, Q)
is a morphism of Hodge structures. Since the map is an injection in the claimed
range and H k(Mg,n, Q) is pure of weight k, the cohomolog... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
eth decomposition, Proposition 2.3 and the fact
that H 1(Mg,n, Q) = 0 for every g and n. There are many ways of proving the last
statement. It follows, for example, from the fact that Mg,n is simply connected.
We will see an elementary proof in the next section.
2.1. The relations among tautological classes. In thi... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
,S≤{x,y} otherwise
if g = 2h, n = 0
⎨
Finally, we need to know the pull-backs of tautological classes by the morphism
ath,S : Mg−h,n−S≤{x} � Mg,n
obtained by attaching a fixed curve of genus h and marking S ∞ {y} to curves in
Mg−h,n−S≤{x} by identifying x and y.
Exercise 2.8. Show that the following relations hold:
(1... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
obtained the claimed relations in The
orem 2.1. Recall that the Hodge class ω is the first chern class of the Hodge bundle.
⎧
⎧
⎧
�
Lemma 2.9 (Mumford’s relation). On any Mg,n there is the following relation
� = 12ω − ζ + ξ.
Proof. It suffices to prove the formula when n = 0. The general case follows
by pulling-back ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
0.
�
Now all the relations follow when we observe that on M2 we have the relation
10ω = ζ0 + 2ζ1.
To prove this relation, for instance, consider the following test families.
(1) To a fixed genus 1 curve attach a fixed point of a genus 1 curve at a variable
point.
(2) On a genus 1 curve identify a variable point wit... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
,n is the morphism that identifies the two points
x, y. Since by induction H 2(Mg−1,n≤{x,y}, Q) is tautological ��d may be expressed
as a linear combination of tautological classes. Moreover, since the morphism is
symmetric under exchanging x and y, the expressions of divisors involving x and y
need to be symmetric.... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
the points x and y. The classes of these two pull-backs have to
coincide. This gives a relation that shows that ��(d − dt) must be identically zero.
Since by Proposition 2.4, the map �� is injective, we conclude that d is tautological.
To conclude the proof then one needs to analyze the cases of genus 1 and 2 in
gr... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
section we have the
following Reduction Lemma.
⎧
�
Lemma 3.2 (Reduction Lemma). Let k be an odd integer. Suppose that
H q (M g,n, Q) = 0
for all odd q � k, and for all g and n such that q > d(g, n), then
for all odd q � k and all g and n.
H q (M g,n, Q) = 0
In other words, as long as all the odd cohomology for j... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
m or k = l. In this case either the genus is smaller than
g or if the genus is equal to g the number of marked points is smaller than n. A
�
double induction concludes the proof.
Proof of vanishing of the first cohomology. By the Reduction Lemma to prove that
the first cohomology groups of M g,n vanish we need to ch... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
cohomology
group of P1−{0, 1, →} has rank 2. In contrast we saw above that all odd cohomology
groups of M 0,n vanish.
To emphasize the point, observe that the Euler characteristic of M0,n is given
by the formula
α(M0,n) = (−1)(n−3)(n − 3)!.
To prove this formula consider the map M0,n � M0,n−1 given by forgetting ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
omorphic to P1, hence their third cohomology clearly vanishes.
The moduli spaces M0,5 and M1,2 both have complex dimension 2 or real dimen
sion 4. By Poincar´e duality we conclude that the dimension of H 3 is equal to the
dimension of H 1 . Since H 1 vanishes we conclude that H 3 vanishes.
To show the vanishing of ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
i
Y = Y d ⊗ Y d−1 ⊗ · · · ⊗ Y 1 ⊗ Y 0
11
so that Yi = Y i\Y i−1 is empty or of pure dimension i for every i, then by the exact
sequence of cohomology with compact supports the Euler characteristic of
Y with
cohomology with compact supports is the sum of those of Yd and Y d−1. Repeating
the process and using Poin... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
of M1,2, M1,3, M0
,5
denote the space obtained by taking the quotients of M0,4 and M0,5 under the
operation of interchanging the labeling of two marked points. To calculate the
Euler characteristics of the latter two we note that we have morphisms from M0,4
and M0,5 to these spaces. Both morphisms have degree 2 sinc... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
ramified
over the points 0, 1, −1, → of the sphere and it has the extra automorphism coming
from rotating the sphere by ψ along the 0 − → axis (multiplication by −1). In the
other case the elliptic curve can be realized as ramified over the cube roots of unity
and →. Its automorphism group has order 6 and it can be g... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
p2 is not a 2-torsion point and not the special point considered in the
previous case. In this case the fiber is an elliptic curve with two points removed.
Adding up the various Euler characteristics we conclude that α(M1,3) = 0. This
information together with an enumeration of the strata of M 1,3 suffices to calculate... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
group of the moduli functor
In this section we will determine the Picard group of the moduli functor following
[AC1]. A very good introduction to Picard groups of moduli functors is contained
in [Mum].
Let Mg,n denote the moduli functor of genus g stable curves with n marked
points. Let (C � S, π1, . . . , πn) den... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
node. The class ζi is the class of the divisor of curves that
contain a node that separates the curve to a subcurve of genus i and genus g − i.
Similarly ω, ξ1, . . . , ξn, ζirr, ζh,S are elements of P ic(Mg,n). Recall that ω is the
Hodge class. The class ξi is the class of the cotangent line at the i-th marked
poi... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
ω, ξ1, . . . , ξn and the classes of boundary divisors. The Picard group
P ic(Mg,n) is freely generated by ω and ξ1, . . . , ξn.
Sketch of the proof of Theorem 4.1. We first remark that P ic(Mg) is torsion free
and contains P ic(Mg) as a finite index subgroup. To see that P ic(Mg) is torsion
free one uses Teichm¨ulle... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
, ζ∪g/2∈ .
The point is to show that it may be expressed as an integral linear combination.
The strategy is to construct two different sets of one-parameter families of curves
F1, . . . , F∪g/2∈+2 and G1, . . . , G∪g/2∈+2 such that their intersection matrices with
respect to
ω, ζirr, ζ1, . . . , ζ∪g/2∈
are non-sin... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
zero.
All the intersections with the boundary divisors vanish unless i = 1, h or h + 1.
The degree of ζ1 on Fh is 1 if h > 1, 0 if g − h − 1 > h = 1 and −1 if g = 3
and g − h − 1 = h = 1. The degree of ζh on Fh is −1 if g − h − 1 > h = 1 or if
g − h − 1 = h = 1, 0 if g − h − 1 > h = 1 and −2 if g − h − 1 = h > 1.
... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
genus is 2m + 2, the intersection matrix for
the families
Kh, C, CE, F1, . . . , Fm
has determinant (−1)m+1(h + 1) if m ∼ h ∼ 2. Again taking h = 2 and h = 3 gives
�
two relatively prime determinants.
5. The Tautological ring of Mg
In this course we will not have time to discuss the tautological ring. In this se... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
a rank g locally free sheaf called the Hodge bundle
E. The Hodge bundle is defined by E = ψ1��Mg,1 /Mg . The chern classes ωl = cl (E)
also define classes in Al (Mg ). Ths subring of the Chow ring generated by these
classes is called the tautological ring.
= ψ1�K l+1
One of the first things to observe is that the coho... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
2] E. Arbarello and M. Cornalba. Calculating cohomology groups of moduli spaces of curves
via algebraic geometry. Inst. Hautes Etudes Sci. Publ. Math. (1998), 97–127 (1999).
´
[Fab] C. Faber. A conjectural description of the tautological ring of the moduli space of curves.
In Moduli of curves and abelian varieties,... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
J. Math. 25(2001),
237–243.
[GV2] T. Graber and R. Vakil. Relative virtual localization and vanishing of tautological classes
on moduli spaces of curves. Duke Math. J. 130(2005), 1–37.
[Har1] J. Harer. The second homology group of the mapping class group of an orientable surface.
Invent. Math. 72(1983), 221–239.
... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
Picard groups of moduli problems. In Arithmetical Algebraic Geometry
(Proc. Conf. Purdue Univ., 1963), pages 33–81. Harper & Row, New York, 1965.
R. Vakil. The moduli space of curves and its tautological ring. Notices Amer. Math. Soc.
50(2003), 647–658.
[V]
17 | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/10078cf3d80a1cdf4ff766a80eee8b32_picard.pdf |
18.409 An Algorithmist’s Toolkit
10/8/2009
Lecturer: Jonathan Kelner
Lecture 9
At the end of the previous lecture, we began to motivate a technique called Sparsification. In this lecture,
we describe sparsifiers and their use, and give an overview of Combinatorial and Spectral Sparsifiers. We
also define Spectral Spa... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
in G(cid:2). Say the answer is a cut with value S(cid:2); then our algorithm
will output the estimate S = S(cid:2)/p for the original graph G.
Denoting the number of edges between S and S ¯ by e(S) = pc, we have the following concentration result
due to Chernoff’s inequality:
So our result will be close to the corre... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
a slightly modified version of the Chernoff bound:
9-1
Theorem 2 (Chernoff Bound) Let X1, . . . , Xn be random variables so that Xi ∈ [0, 1], and let X = Xi.
Then,
(cid:2)
P r[|X − E[X]| ≥ (cid:2)X] ≤ 2e −Θ(1)(cid:2)2E[X]
Proof The only difference here is that the random variables Xi are no longer discrete variables... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
to be small. In this way, we
can approximate cut problems while throwing away more edges which are present in only cuts of high size.
Thus, a natural choice for we would be the size of the smallest cut containing e. Unfortunately, we
do not know we; however, it is possible to approximate it quickly. The final result ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
the instances x ∈ [−1, 1]n by normalization. We now have a good definition for a spectral version of
sparsification:
Definition 3 A Spectral Sparsifier G(cid:2) of a graph G is one for which the relation
(1 − (cid:2))x T LG� ≤ x T LGx ≤ (1 + (cid:2))x T LG� x
for all x ∈ [0, 1]n
9-2
It is clear from this definition t... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
not a spectral sparsifier of G. Let
(cid:4)
x =
0
1
. . . n/2 − 1 n/2 − 1
. . . 1
0
(cid:5)
.
Then, we have that
since each vertex contributes Θ(
x T LG� x = Θ(nk3)
(cid:2)
k
=1 k2) to the sum. On the other hand,
i
x T LGx = Θ(nk3) + (
n − 1)2
2
If k is constant, we get that we need (cid:2) = Θ(1/n) for ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
possibly affect the graph orders.
We try again with another definition:
M (cid:5) N if the ith eigenvalue of M is ≥ the ith eigenvalue of N for all indices i
This is better in that it is basis independent - but it is too basis independent. Under this definition, we
have both
9-3
as well as
(cid:6)
1
0
0 −1
(cid... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
:5) LH .
Claim 6 Let G = (V, EG, wG) and H = (V, EH , wH ) be weighted graphs on the same vertex set such that
wG(i, j) ≥ wH (i, j) for all edges (i, j) ∈ E. Then, G (cid:5) H
2.2 Towards Spectral Sparsification
With this order relation on graphs, we can now restate the goal of spectral sparsification: Given a dense ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
probabilities are based on a linear algebra sense of importance, and have a nice interpretation in terms
of effective resistance of circuits. To proceed with our analysis, however, we need to develop the ideas of
pseudoinverses, calculating effective resistances, and a matrix version of the Chernoff Bound.
9-4
2.3 Ps... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
ent probability. It turns
out that the correct way to do this is to sample each edge with probability proportional to its “effective
resistance.”
The basic idea is to treat each edge as a resistor with resistance 1. If the edge had a capacity of c, we give
it a resistance of 1/c. After calculating these values, we s... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
ext = Lv, and v = L+iext
We define U (e, v) to be the adjacency matrix with ±1 values. Let ue be the eth row, and v = L+iext.
We have
and as a result,
T
Ref f (e) = ueL+ ue
Ref f (e) = (U L+U T )e,e
Thus, calculating the effective resistance of an edge is as simple as calculating the pseudoinverse of the
Laplacian... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
z
zT M T LGM z
for all vectors z, provided that x ⊥ (LG) ⇒ x ∈ range(M ). Choose M so that M T LGM is a projection.
Then, it suffices to show that
≤ 1 + (cid:2)
1 − (cid:2) ≤
(cid:10) M T LH M − M T LGM (cid:10)2≤ (cid:2)
From before, we have that LG = U T CU . Choose M = L+U T C 1/2 . Then, we have
G
Π = M T LGM =... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
)C 1/2)
(cid:12)
G
√
We then set τe =
Recall that
n−1
ceRef f (e) πe with (cid:10) τe (cid:10)= n − 1. Choose edges with probability pe =
c2Ref f (e)
n−1
.
(cid:3)
ceRef f (e) =
(cid:3)
Πe,e = n − 1
e
e
Then, we find that
E[τeτe
T ] =
(cid:3)
peτeτe
T =
(cid:3)
πeπe
T = Π
Sample q times with replacem... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
by solving logarithmically many linear systems. This
uses the Johnson-Lindenstrauss Lemma.
References
[1] “Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs, ” A. Benczur, D. Karger,
manuscript.
9-7
MIT OpenCourseWare
http://ocw.mit.edu
18.409 Topics in Theoretical Computer Science: An A... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/105bb5d8da4f17bc9ed3099824e0e59e_MIT18_409F09_scribe9.pdf |
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 2-1
Lecture 2 - Carrier Statistics in Equilibrium
February 8, 2007
Contents:
1. Conduction and valence bands, bandgap, holes
2. Intrinsic semiconductor
3. Extrinsic semiconductor
4. Conduction and valence band density of states
Reading ass... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
bands, bandgap, holes
y
g
r
e
n
e
Ec
Ev
conduction band
bandgap
↓
valence band
Ec
Ev
Eg
↓
space coordinate
Conduction and valence bands:
• bonding electrons occupy states in valence band
• ”free” electrons occupy states in conduction band
• holes: empty states in valence band
•
CB electrons and VB holes ... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
e
n
n
o
o
r
r
t
t
c
c
e
e
e
e
l
l
--
++
a)
υ > Eg
hhυ > Eg
--
++
hhυ >υ > EgEg
b)
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.72... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 2-6
Key dependencies of ni:
• Temperature:
• Bandgap:
T
↑⇒
ni
Eg ↑⇒
ni
What is detailed form of dependencies?
Use analogy of chemical reactions.
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 2-8
In general, relatively few bonds are broken. Hence:
and
Then:
Two important results:
• First,
[bonds] � no, po
[bonds] � constant
nopo ∼ exp(−
Eg
kT
)
ni ∼ exp... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
ni
Equilibrium np product in a semiconductor at a certain temper
ature is a constant specific to the semiconductor.
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
electronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 2-12
Representation of donor and acceptor states in energy band diagram:
Ed
Ec
Ev
Ea
ED
EA
E... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
electronic Devices - Spring 2007
Lecture 2-14
4. Conduction and valence band density of states
Image removed due to copyright restrictions.
Figure 1b) on p. 468 in Laux, S. E., M. V. Fischetti, and D. J. Frank.
"Monte Carlo Analysis of Semiconductor Devices: The DAMOCLES Program."
IBM Journal of Research and Develo... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
E
Common expressions for DOS:
gc(E) = 4π
⎛
⎝
2m∗
de
h2
⎞
⎠
3/2
√
E − Ec
E ≥ Ec
gv(E) = 4π
⎛
⎝
2m∗
dh
h2
⎞
⎠
3/2
√
Ev − E
E ≤ Ev
m∗
m∗
de ≡ density of states electron effective mass
dh ≡ density of states hole effective mass
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devi... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
in semiconductors
feature DOS ∼
E.
√
• Order of magnitude of key parameters for Si at 300 K:
– intrinsic carrier concentration: ni ∼ 1010 cm−3
– typical doping level range: ND, NA ∼ 1015 − 1020 cm−3
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCour... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/10707b9dced1bed9ea148baae9689b0b_lecture2.pdf |
Lecture 2
8.821/8.871 Holographic duality
Fall 2014
8.821/8.871 Holographic duality
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2014
Lecture 2
1.2: BLACK HOLE THERMODYNAMICS
1.2.1: IMPORTANT SCALES
Planck scale
We can construct physical units using fundamental constants l (reduced Planck constant), GN (... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/107c04c3759a2e244b8778e9bd616a72_MIT8_821S15_Lec2.pdf |
137
λG =
VG(rc) GN m 1
=
2
2
m
l/mc mc2 m2
p
=
=
l2
p
r2
c
mc2
Then λG « 1, for m « mp. For example, in the case of electron, me = 5 × 10−4GeV /c2, we have
λG ∼ 10−43
λEM
The gravity effect is quite weak in this case. But if the mass is at Planck mass scale mp, then λG ∼ O(1), which
means quantum gravity eff... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/107c04c3759a2e244b8778e9bd616a72_MIT8_821S15_Lec2.pdf |
important);
2. m « mp, rs « rc: rs is not relevant, gravity effect is weak and not important;
3. m ∼ mp, rc ∼ rs, quantum gravity effects are important.
If this were the whole story, life would be much simpler, but much less interesting. However, black holes can make
quantum gravity effects manifest at macroscopic lev... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/107c04c3759a2e244b8778e9bd616a72_MIT8_821S15_Lec2.pdf |
object of mass M. If we consider the object to be spherically symmetric, non-rotational, neutral, we have the
Schwarzschild metric solution:
ds2 = −f dt2 + dr2 + r 2(dθ2 + sin θ2dφ2),
1
f
f = 1 −
2GN M
r
= 1 −
rs
r
(1)
Note that from now on, we have adapted the convention to take c = 1.
The event horizon is de... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/107c04c3759a2e244b8778e9bd616a72_MIT8_821S15_Lec2.pdf |
· · = −dτ 2 + · · · . We have dτh = f 1/2(rh)dt, with τh to be the proper time for Oh. Then
2, t is the proper time for O∞. On the other hand, at r = rh:
h
dτh
dt
= (1 −
1
2
rs
)
rh
As rh → rs,
Consider some event of energy Eh happening at r = rh, to O∞ this event has energy
→ 0, i.e. compared to the time at... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/107c04c3759a2e244b8778e9bd616a72_MIT8_821S15_Lec2.pdf |
6.252 NONLINEAR PROGRAMMING
LECTURE 3: GRADIENT METHODS
LECTURE OUTLINE
• Quadratic Unconstrained Problems
• Existence of Optimal Solutions
• Iterative Computational Methods
• Gradient Methods - Motivation
• Principal Gradient Methods
• Gradient Methods - Choices of Direction
QUADRATIC UNCONSTRAINED PROBLEMS
... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/10cddea946dfe20a44ed08128ef2077d_6252_slides03.pdf |
): (cid:1)2 (cid:2)→ (cid:1) given by
f (x, y) = 1 αx2 + βy2 − x
(cid:1)
2
(cid:2)
for various values of α(cid:160)and β.
EXISTENCE OF OPTIMAL SOLUTIONS•
Consider
min f (x)
x∈X
Two possibilities:
(cid:3)
• The set f (x) | x ∈ X
and there is no optimal solution
• The set f (x) | x ∈ X is bounded below
(cid:... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/10cddea946dfe20a44ed08128ef2077d_6252_slides03.pdf |
angle with
∇f (x) that is greater than
90 degrees,
∇f(cid:160)(x)
�
d < (cid:160)0,(cid:160)
x + δd
d
there is an interval (0, δ)
of stepsizes such that f(cid:160)(x+
αd) < f (cid:160)(x) for all α(cid:160) ∈
(0, δ).
PRINCIPAL GRADIENT METHODS•
xk+1 = xk + αkdk ,
k = 0, 1, . . .
where, if ∇f (xk) (cid:9)= 0,... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/10cddea946dfe20a44ed08128ef2077d_6252_slides03.pdf |
) = 1.
Given xk(cid:160), the method ob
tains xk+1 as the minimum
of a quadratic approxima
tion of f(cid:160) based on a sec
ond order Taylor expansion
around xk(cid:160).
OTHER CHOICES OF DIRECTION
• Diagonally Scaled Steepest Descent
Dk = Diagonal approximation to
(cid:1)
∇2f (xk)
(cid:2)−1
• Modified New... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/10cddea946dfe20a44ed08128ef2077d_6252_slides03.pdf |
More Symple Types
Progress And Preservation
Armando Solar-Lezama
Computer Science and Artificial Intelligence Laboratory
M.I.T.
September 28, 2015
September 28, 2015
L06-1
Formalizing a Type System
Recap
September 28, 2015
Static Semantics
• Typing rules
– Typing rules tell us how to derive typing j... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/10f472f16fcff7bceab5d9b40aa8bfa3_MIT6_820F15_L06.pdf |
06-10
Preservation
• Using Big-Step semantics we can argue
global preservation
Γ ⊢ 𝑒1: 𝜏 ∧ 𝑒1 → 𝑒2 ⇒ Γ ⊢ 𝑒2: 𝜏
• Prove by induction on the structure of
derivation of 𝑒1 → 𝑒2
September 28, 2015
L06-11
Proof by induction on Structure
of Evaluation
• Base cases: trivial
• Inductive cas... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/10f472f16fcff7bceab5d9b40aa8bfa3_MIT6_820F15_L06.pdf |
September 28, 2015
L06-14
Small Step Example
• Contexts
H ::= o | H e1 | H + e | n + H |
if H then e1 else e2 |
H == e1 | n == H
• Local Reduction Rules
– n1 + n2 n (where n = plus n1 n2)
– n1 == n2 b (where b =(equals n1 n2))
– if true then e1 else e2 e1
– if false then e1 else e2 e2
– (𝜆x:... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/10f472f16fcff7bceab5d9b40aa8bfa3_MIT6_820F15_L06.pdf |
valid
context.
L06-18
MIT OpenCourseWare
http://ocw.mit.edu
6.820 Fundamentals of Program Analysis
Fall 2015
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/10f472f16fcff7bceab5d9b40aa8bfa3_MIT6_820F15_L06.pdf |
3.15
Photoconductors, Photovoltaics and Photodetectors
C.A. Ross, Department of Materials Science and Engineering
Reference: Pierret, chapter 9.2 and 9.3
Photoconductors – conductivity a function of light
Photovoltaics – generate power from light
Photodetectors – use a pn junction to detect light
Photoconducting materi... | https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/110ed969f7442b8ed4f4ff7a8e3d9026_lecture9.pdf |
orphous Si: uncertainly principle DxDp ≥ h
-the localization of carriers gives them an uncertain momentum, so direct
absorption of light can occur. Use PIN design because mobility is low.
Handout 5
†
1
Scanned article removed due to copyright restrictions.
Please See "This Month in Physics History, October 22, ... | https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/110ed969f7442b8ed4f4ff7a8e3d9026_lecture9.pdf |
Lecture 7
8.821/8.871 Holographic duality
Fall 2014
8.821/8.871 Holographic duality
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2014
Lecture 7
In fact, any orientable two dimensional surface is classified topologically by an integer h, called the genus. The
genus is equal to the number of “holes” that the surface ha... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/1119bf97378aff76fb5c02ae928d930d_MIT8_821S15_Lec7.pdf |
Theorem: Given a surface composed of polygons with F faces, E edges and V vertices, the Euler character satisfy
χ = F + V − E = 2 − 2h
Since each Feynman diagram can be considered as a partition of the surface separating it into polygons, then the
above theorem also works for our counting in N.
Thus in this limit, to t... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/1119bf97378aff76fb5c02ae928d930d_MIT8_821S15_Lec7.pdf |
of
Feynman diagrams.
General observables
Now we have introduced two theories:
(a)
L = −
1
g2 Tr
(cid:20)
1
g2
Y M
(cid:20) 1
2
1
4
−
(∂Φ)2 +
(cid:21)
Φ4
1
4
Tr FµνF µν
(cid:21)
− iΨ(D/ − m)Ψ
(b)
L =
(a) is invariant under the global U (N ) transformation: Φ→U ΦU † with U constant U (N ) matrix, i.e. the theory
has a gl... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/1119bf97378aff76fb5c02ae928d930d_MIT8_821S15_Lec7.pdf |
n)(cid:105)c
(1)
Note that it is enough to focus on single-trace operators since multiple-trace ones are products of them. Since we
are working in the t’ hooft limit, we want to know how correlation (Eq. 1) scales in the large N limit. There is a
trick, consider
ˆ
ˆ
Z [J1, · · · , Jn] =
DAµDΦ · · · exp (iSef f ) =
DA... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/1119bf97378aff76fb5c02ae928d930d_MIT8_821S15_Lec7.pdf |
∼ O(N ) + O(N −1) + · · ·
(cid:104)O1O2(cid:105)c ∼ O(N 0) + O(N −2) + · · ·
2O3(cid:105)c ∼ O(N − ) + O(N − ) + · · ·
1
3
(cid:104)O1O
All leading order contributions come from planar diagrams.
Physical implications:
1. In the large N limit, O(x)|0(cid:105) can be interpreted as creating a single-particle state (”glue... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/1119bf97378aff76fb5c02ae928d930d_MIT8_821S15_Lec7.pdf |
4)O(cid:105) =(cid:54)
(cid:104)O1O2(cid:105) = (cid:104)O1(cid:105)(cid:104)O2(cid:105) + (cid:104)O
the large N limit, it is more like a classical theory.
0 ∼ O(N ), the variance of (cid:104)O(cid:105) is (cid:104)O2(cid:105) − (cid:104)O(cid:105)2 = (cid:104)O2(cid:105)c ∼ O(1), i.e.
(cid:104)O2(cid:105)c N
(cid:104... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/1119bf97378aff76fb5c02ae928d930d_MIT8_821S15_Lec7.pdf |
Lecture Five: The Cacciopolli Inequality
1
The Cacciopolli Inequality
The Cacciopolli (or Reverse Poincare) Inequality bounds similar terms to the Poincare
inequalities studied last time, but the other way around. The statement is this.
Theorem 1.1 Let u : B2r → R satisfy u�u ≥ 0. Then
�
Br
2
|�u| ≤
4
2
r
� ... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2005/11294d6e66f18a2a5dc4230e13a6d8cc_lecture5.pdf |
||�u|
.
(5)
(6)
B2r
1
��
Recall the inequality f g ≤
of the CauchySchwarz inequality), and apply it above to get
f 2 �1/2 ��
�
g2 �1/2 for any functions f and g (this is one form
�
B2r
��
φ2|�u|2 ≤ 2
�1/2 ��
φ2|�u|2
�1/2
|u|2|�φ|2
.
B2r
B2r
Dividing and squaring then gives
�
B2r
�
φ2|�u|2 ≤ 4 ... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2005/11294d6e66f18a2a5dc4230e13a6d8cc_lecture5.pdf |
be a test function as before, and consider
�
B2r
|�(φu)|
2 =
=
�
B2r
�
B2r
2
|φ�u + u�φ|
φ2|�u|2 + u |�φ|2 + 2uφ�φ · �u.
2
Apply CauchySchwarz and lemma 1.2 to get
2
�
B2r
|�(φu)| ≤
2
�
φ2
�
|�u|2 +
��
u |�φ|2 + 2
2
�1/2 ��
2
|�u|
φ2
�1/2
|�φ|2
2
u
B2r
�
≤ 2
B2r
�
φ2|�u|2 + 2
2
|�φ|2
.
u
... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2005/11294d6e66f18a2a5dc4230e13a6d8cc_lecture5.pdf |
Br
�
(1 + k(n))
B2r \Br
�
2
u ≤
2
u .
Br
B2r
and
This completes the proof.
(11)
(12)
(13)
(14)
(15)
2.2 Bounding the growth of the energy of a harmonic function
We will now prove a similar inequality for the Dirichlet energy of a harmonic function.
Proposition 2.2 There are dimensional constants c(n) suc... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2005/11294d6e66f18a2a5dc4230e13a6d8cc_lecture5.pdf |
Lecture 2
Acoustics of Speech and Hearing
6.551J / HST.714J
LECTURE 2: One-Dimensional ‘Traveling Waves’
Main Points
- Exponential and sine-wave solutions to the one-dimensional wave equation.
- The distributed compressibility and mass in acoustic plane waves are analogous with the
distributed capacitance and in... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/1149b68a00a45f7c67d4a5c24ed08b51_lec_2_2004.pdf |
(the sound pressure) at any position (x) and time (t) in a one
dimensional system like Figure 2.1.
Newtons 2nd Law:
∂p(x, t)
∂x
= −ρ0
∂vx (x,t)
∂t
Conservation of Mass-Compressibility Relationship:
∂v x (x,t)
∂x
= −
1
BA
∂p(x,t)
∂t
.
The Wave-Equation for Sound Pressure in a Plane Wave:
14-Sept-2004
(1.2.7)
(... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/1149b68a00a45f7c67d4a5c24ed08b51_lec_2_2004.pdf |
C
C
C
C
C
C
C
A
A
A
A
A
A
A
A
D
D
D
D
D
D
D
E
D
D
E
E
E
E
E
E
E
E
14-Sept-2004
page 2
Lecture 2
Acoustics of Speech and Hearing
6.551J / HST.714J
In the top row the balls and springs are at rest
In the second row, ball A is displaced to the left, stretching the AB spring
In the third row, ball B h... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/1149b68a00a45f7c67d4a5c24ed08b51_lec_2_2004.pdf |
pressure).
The inductance/length:
The compliance per length:
∂e(x,t)
∂x
= −Lx ∂i(x,t)
∂t
∂i(x,t)
∂x
= −C x ∂e(x,t)
∂t
.
(2.2)
(2.3)
The variations in voltage and current in time and space can be described by wave equations:
∂2e(x,t)
∂x 2
=
1
c 2
∂2e(x,t)
∂t 2
∂2i(x,t)
∂x 2
=
1
c 2
∂2i(x,t)
∂t 2
, and
c =
1
LxC... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/1149b68a00a45f7c67d4a5c24ed08b51_lec_2_2004.pdf |
) determined by the absolute time t
and the time needed to travel to x, i.e. x/c.
You should note:
–vx and p in each wave are related by the characteristic impedance of the medium z0.
– The two scalar pressure terms add.
– Because of a difference in direction, the two velocity terms subtract.
– Because of the dif... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/1149b68a00a45f7c67d4a5c24ed08b51_lec_2_2004.pdf |
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