text stringlengths 16 3.88k | source stringlengths 60 201 |
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We analyze S: when the verifier is corrupted, the views of Z in the hybrid and ideal interactions are identically
distributed. When the prover is corrupted, the only difference is that S may fail with probability 2−k;
conditioned on nonfailure, the views are identical. Therefore the views are statistically indistingui... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
. Receive (sid, T, R, v1, . . . , vm) from (sid, T ).
2. Receive (sid, R, T, i ∈ {1, . . . , m} from sid, R).
3. Output (sid, vi) to (sid, R).
4. Halt.
To realize F 2
ot, we can use the protocol from [EGL85]: let F be a family of trapdoor permutations and
let B be a hardcore predicate for F . Then the protocol i... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
(t0, t1). S works as follows:
obtain vi from F 2
ot, give the idealprocess input i to A, select (f, f −1) from F and give the pair to A,
receive (y0, y1) from A where yi = f (xi) and y1−i = x1−i for random x0, x1 due to semihonesty, and
send (t0, t1) to A where ti = vi ⊕ B(xi) and t1−i is random.
We now analyze ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
standard functionality;” i.e. it has a “shell” and a “core,” where the core does not know who
is corrupted. Our protocol evaluates the core only.
• F is written as twoinput ⊕ (addition mod 2) and ∧ (multiplication mod 2) gates.
• The circuit has 5 types of input lines: inputs of P0, inputs of P1, inputs of S, rando... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
gate, P0 and P1 use F 4 as follows: P0 chooses c0 at random, and acts as the
sender with input
ot
v00 = a0b0 + c0
v01 = a0(1 − b0) + c0
v10 = (1 − a0)b0 + c0
v11 = (1 − a0)(1 − b0) + c0
while P1 plays the receiver with input (a1, b1) and sets the output to be c1. It is easy to verify that
c0 ⊕ c1 = (a0 ⊕ a1)(b0 ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
P1 receives the corresponding information, plus random shares of all intermediate values from Fot.
This is also easy to simulate.
• Upon corruption, it’s easy to generate local state that is consistent with the protocol.
�
Some remarks: there is a protocol by Yao that works in a constant number of rounds, which can... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
zero
knowledge, statements about those values. Here is the formal definition of the F R
cp functionality for a given
polytime relation R:
1. Upon receiving (sid, C, V, commit, w) from (sid, C), add w to the list W of committed values, and output
(sid, C, V, receipt) to (sid, V ) and S.
2. Upon receiving (sid, C, ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
F Rp
zk ,
where
Rp = {((x, A), (W, R)) :
W = w1 · · ·
wn, A
= a1 · · ·
an, R = r1
· · ·
rn, R(x, W ) = 1 and ai = Com(wi, ri) for all i.}
4. Upon receiving (sid, C, V, (x, A), 1) from Fzk , V verifies that A agrees with its local list A, and if so,
Rp
outputs (sid, C, V, x).
Theorem 1 The above protocol realizes... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
) for random r. In the proof phase, S obtains
from Fcp a (sid, C, V, x) message, and simulates for A the message (sid, C, V, (x, A)) from Fzk , where
A is the list of commitments that S has generated so far.
Rp
Let’s analyze S: for a corrupted committer, the simulation is perfect. For a corrupted verifier, the only
... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
, commit, r) to Fcp.
Q0 receives r1 from Q1, and sets r = r0 ⊕ r1.
2. Commit to Q1’s randomness. Q0 receives (sid.1, Q1, Q0, receipt) from Fcp and sends a random value
s0 to Q1.
3. Receive the input x in the ith invocation. Q0 sends (sid.0, Q0, Q1, commit, x) to Fcp. Let M be
the list of messages seen so far. Q0 r... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
A there exists a semihonest adversary S such that for any envi
ronment Z we have:
EXECP,S,Z ≈ EXECQ,A,Z .
Fcp
Corollary 1 If protocol P securely realizes F for semihonest adversaries then Q = C(P ) securely realizes
F in the Fcphybrid model for malicious adversaries.
Proof Sketch: We will skip the details of th... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
composable two
party and multiparty secure computation. STOC 2002, pages 494–503, 2002.
[EGL85]
Shimon Even, Oded Goldreich, and Abraham Lempel. A randomized protocol for signing con
tracts. Communications of the ACM, 28(6):637–647, June 1985.
[GMW87] O. Goldreich, S. Micali, and A. Wigderson. How to play any men... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/40bda230e290076f112388b0d14497e7_l11.pdf |
6.241 Dynamic Systems and Control
Lecture 8: Solutions of State-space Models
Readings: DDV, Chapters 10, 11, 12 (skip the parts on transform methods)
Emilio Frazzoli
Aeronautics and Astronautics
Massachusetts Institute of Technology
February 28, 2011
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
F... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
0] = C [0]x0
y [1] = C [1] A[0] x[0]
x[0] = x0
x[1] = A[0] x[0]
x[2] = A[1] A[0] x[0] y [2] = C [2] A[1] A[0] x[0]
. . .
. . .
y [k] = C [k] Φ[k, 0] x[0]
x[k] = Φ[k, 0] x[0]
where we defined the state transition matrix Φ[k, �] as
�
Φ[k, �] =
A[k − 1] A[k − 2] . . . A[l], k > � ≥ 0
I ,
k = �
E. Frazzoli (M... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
�
. . . B[k − 1] ,
⎡
⎤
u[0]
u[1]
⎥
U = ⎢
⎥
⎢
⎣
. . .
⎦
u[k − 1]
.
The output is
y [k] = C [k]Γ[k, 0]U[k, 0].
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
Feb 28, 2011
4 / 19
Summary (DT)
In general, state/output trajectories of a DT state-space model can be
computed as:
x[k] = Φ[k, 0]x... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
t
Φ(t, τ ) = A(t)Φ(t, τ ),
Φ(t, t) = I .
The function Φ can in general be computed numerically, integrating a
differential equation in n unknown functions, with n initial conditions
(assuming x ∈ Rn).
Then, x(t) = Φ(t, t0)x0, and y (t) = C (t)Φ(t, t0)x0.
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Model... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
Φ(t, τ )B(τ )u(τ )]τ =t
= A(t)
Φ(t, τ )B(τ )u(τ ) dτ + B(t)u(t) = A(t)x(t) + B(t)u(t).
t0
Similarly for the output.
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
Feb 28, 2011
8 / 19
Further properties of the state transition function
Φ(t2, t0) = Φ(t2, t1)Φ(t1, t0).
Look up on the lecture notes... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
i.e., r = T −1x.
This is called a similarity transformation.
The standard state-space model can be written as
Tr + = ATr + Bu
y = CTr + Du
i.e.,
(T −1AT )r + (T −1B)u = ˆ
r + =
y = (CT )r + Du = ˆ
C r + ˆ
Du
Ar + ˆ
Bu
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
Feb 28, 2011
11 / 19
Modal Coor... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
shows that αi = wi
i-th mode.
�x0 is the contribution of the initial condition to the
i=1
E. Frazzoli (MIT)
Lecture 8: Solutions of State-space Models
Feb 28, 2011
13 / 19
Diagonalization of the system
If T = V = matrix of eigenvectors, then V −1AV = Λ (prove by AV = V Λ).
Decoupled system for each mode.
E. F... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/40ffa7a501e68278efd1715eae732170_MIT6_241JS11_lec08.pdf |
44
1.21. Bialgebras. Let C be a finite monoidal category, and (F, J) :
C → Vec be a fiber functor. Consider the algebra H := End(F ). This
algebra has two additional structures: the comultiplication Δ : H →
H ⊗ H and the counit ε : H
k. Namely, the comultiplication is
defined by the formula
→
Δ(a) = α−1 Δ(a)),
F,F
(... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
.1 is called
a bialgebra.
Thus, Theorem 1.21.1 claims that the algebra H = End(F ) has a
natural structure of a bialgebra.
Now let H be any bialgebra (not necessarily finite dimensional).
Then the category Rep(H) of representations (i.e., left modules) of
H and its subcategory Rep(H) of finite dimensional represent... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
Proof. Straightforward from the above.
�
Theorem 1.21.3 is called the reconstruction theorem for finite dimen
sional bialgebras (as it reconstructs the bialgebra H from the category
of its modules using a fiber functor).
Exercise 1.21.4. Show that the axioms of a bialgebra are self-dual
if H is a finite dimensional ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
G (not necessarily finite).
·
Exercise 1.21.6. Let H be a k-algebra, C = H −mod be the category
of H-modules, and F : C → Vec be the forgetful functor (we don’t
assume finite dimensionality). Assume that C is monoidal, and F is
given a monoidal structure J. Show that this endows H with the
structure of a bialgebra,... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
additional structure
on the bialgebra H = End(F ) from the previous subsection in the case
when the category C has right duals. In this case, one can define a
linear map S : H
H by the formula
→
S(a)X = a∗
X ∗ ,
where we use the natural identification of F (X)∗ with F (X ∗).
Proposition 1.22.1. (“the antipode axio... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
F (X)
� F (X) ⊗ F (X)∗ ⊗ F (X)
F (X)
Id
� �
F (X)
η1
� �
F (X)
JX,X∗
F (coevX )
� �
� F (X ⊗ X ∗) ⊗ F (X)
ηX⊗X∗
F (coevX )
� �
� F (X ⊗ X ∗) ⊗ F (X)
Id
� �
F (X) �
evF (X)
J −1
X,X∗
� �
F (X) ⊗ F (X)∗ ⊗ F (X),
for any η ∈ End(F ).
Namely, the commutativity of the upper and the lower square fol
lows from... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
sense: if H is a finite dimensional bialgebra with antipode
SH , then the bialgebra H ∗ also admits an antipode SH ∗ = S∗
H .
The following is a “linear algebra” analog of the fact that the right
dual, when it exists, is unique up to a unique isomorphism.
Proposition 1.22.4. An antipode on a bialgebra H is unique i... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
� b
b
3
j .
j
Then using the definition of the antipode, we have
�
�
S(ab) =
S(a
2
1
i b)ai S(ai ) =
3
S(a
11
3
i bj )ai bj S(bj )S(ai ) = S(b)S(a).
22
3
i
i,j
Thus S is an antihomomorphism of algebras (which is obviously unital).
The fact that it is an antihomomorphism of coalgebras then follows
using the... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
.
Proof. Part (i) follows from the antipode axiom and Proposition 1.22.5.
Part (ii) follows from part (i) and the fact that the operation of taking
�
the left dual is inverse to the operation of taking the right dual.
Remark 1.22.7. A similar statement holds for finite dimensional co
modules. Namely, if X is a fini... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
(Rep(H), Forget) are mutually inverse bijections between
1) finite tensor categories C with a fiber functor F , up to monoidal
equivalence and isomorphism of monoidal functors;
2) finite dimensional Hopf algebras over k up to isomorphism.
Proof. Straightforward from the above.
49
�
Exercise 1.22.12. The algebra of... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
ike elements is grouplike. In particular, grouplike elements
of any Hopf algebra H form a group, denoted G(H). Show that this
group can also be defined as the group of isomorphism classes of 1
dimensional H-comodules under tensor multiplication.
Proposition 1.22.15. If H is a finite dimensional bialgebra with an
ant... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
collapsing the first two factors using the antipode axiom, we get b =
a. Thus a = S�(S(a)) and thus a ∈ H1, so H = H1, a contradiction. �
Exercise 1.22.16. Let µop and Δop be obtained from µ, Δ by permu
tation of components.
(i) Show that if (H, µ, i, Δ, ε, S) is a Hopf algebra, then Hop :=
(H, µop, i, Δ, ε, S−1), ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
to generalize the reconstruction theory to the
situation when the category C is not assumed to be finite.
Let C be any essentially small k-linear abelian category, and F : C →
Vec an exact, faithful functor. In this case one can define the space
Coend(F ) as follows:
Coend(F ) := (⊕X∈CF (X)∗ ⊗ F (X))/E
where E is s... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
�CF (X)∗ ⊗ X. For
X, Y ∈ C and f ∈ Hom(X, Y ), let
jf : F (Y )∗ ⊗ X → F (X)∗ ⊗ X ⊕ F (Y )∗ ⊗ Y ⊂ Q
51
be the morphism defined by the formula
jf = Id ⊗ f − F (f )∗ ⊗ Id.
−→
Let I be the quotient of Q by the image of the direct sum of all jf . In
other words, I = lim (F (X)∗ ⊗ X).
The following statements are easy ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
since End(F )
may now be infinite dimensional, the algebra End(F ⊗ F ) is in general
isomorphic not to the usual tensor product End(F )⊗End(F ), but rather
to its completion End(F )⊗� End(F ) with respect to the inverse limit
topology. Thus the comultiplication of End(F ) is a continuous linear
map Δ : End(F ) → En... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
functors, and Hopf algebras
over k, up to isomorphism.
Remark 1.23.3. This theorem allows one to give a categorical proof
of Proposition 1.22.4, deducing it from the fact that the right dual,
when it exists, is unique up to a unique isomorphism.
Remark 1.23.4. Corollary 1.22.15 is not true, in general, in the infi... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
fine the
coproduct on H by setting Δ(x) = x ⊗ 1 + 1 ⊗ x for all x ∈ g. It is easy
to show that this extends to the whole H, and that H equipped with
this Δ is a Hopf algebra. Moreover, it is easy to see that the tensor
category Rep(H) is equivalent to the tensor category Rep(g).
This example motivates the following... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
Lie algebra over a field of characteristic zero. Show
that Prim(U (g)) = g.
Hint. Identify U (g) with Sg as coalgebras by using the symmetriza
tion map.
Example 1.24.5. (Taft algebras) Let q be a primitive n-th root of
unity. Let H be the algebra (of dimension n2) generated over k by g
and x satisfying the followi... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
of type h, g in a
Hopf algebra H.
(i) Show that ε(x) = 0, S(x) = −h−1xg−1 .
(ii) Show that if a, b ∈ H are grouplike elements, then axb is a skew-
primitive element of type (ahb, agb).
Example 1.24.9. (Nichols Hopf algebras) Let H = C[Z/2Z]�∧(x1, ..., xn),
where the generator g of Z/2Z acts on xi by gxig−1 = −xi. ... | https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/411b2cbdc7f4ccd26dc430d4c9ea9838_MIT18_769S09_lec05.pdf |
software
studio
an overview of Rails
Daniel Jackson
1 what is Rails?
an application framework
› full stack: web server, actions, database
a programming environment
› eg, rake (like make), unit testing
an open-source community
› many plugins
2 history of Rails
genesis in Basecamp
› project management tool... | https://ocw.mit.edu/courses/6-170-software-studio-spring-2013/41673afb1ac2f91f3724fd7d28c5da14_MIT6_170S13_10-rails-ovrvw.pdf |
?
› next slide...
11 Pull request by jeyb on GitHub.
12 in summary...
rich environment
many libraries
code generation
helpful community
friendly online guides
invisible magic
quirky conventions
no static checking
masking of failures
?
13 an easy life?
or a
deadly
cocktail?
14 actually, neither
every too... | https://ocw.mit.edu/courses/6-170-software-studio-spring-2013/41673afb1ac2f91f3724fd7d28c5da14_MIT6_170S13_10-rails-ovrvw.pdf |
MATH 18.152 COURSE NOTES - CLASS MEETING # 9
18.152 Introduction to PDEs, Fall 2011
Professor: Jared Speck
Class Meeting # 9: Poisson’s Formula, Harnack’s Inequality, and Liouville’s
Theorem
1. Representation Formula for Solutions to Poisson’s Equation
We now derive our main representation formula for solution’s to Poi... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
− )
ˆN σ Φ x σ dσ.
Recall also that
(1.0.4)
where
(1.0.5)
and
(1.0.6)
G(x, y) = Φ(x − y) − φ(x, y),
∆yφ(x, y
) = 0,
x ∈ Ω,
(
G x, σ
) =
0 when x Ω and σ ∂Ω.
∈
∈
The expression (1.0.3) is not very useful since don’t know the value of
fix this, we will use Green’s identit
and recalling that ∆yφ x, y
) =
(
y. Applying
Gree... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
P x, σ) = ∇ ˆN G(x, σ from (1.0.2)
a technique
works for special domains.
(
def
Warning 2.0.1. Brace yourself for a bunch of tedious computations that at the end of the day will
lead to a very nice expression.
(
)
The basic idea is to hope that φ x, y from the decomposition G x, y
(
)
φ x, y , where
the Newtonian
that
... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
which implies that
(2.0.9)
Simple algebra then leads to
1
4π∣x − y
∣
=
q
∗ − ∣
4π∣x
y
.
(2.0.10)
∗
∣
x
−
y 2
∣ =
q2 x y 2.
∣ − ∣
When ∣y∣ = R we
,
use
x∗ 2
∣
∣ −
(2.0.10) to compute
2x∗ ⋅ y + R2 = ∣x∗ − y
that
∣2 = q2∣ − y∣2 = q2
x
(∣
x∣2 − 2x ⋅ y + R ,
)
2
the
Euclidean dot product. Then performing simple
algeb
ra, it... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
2
x.
φ(x, y
) =
1
4π x R
∣
∣
R
2
− ∣
x∣2 x y
∣∣
,
(
φ 0, y
) =
1
4πR
,
where we took a limit as x
→ 0
Next, using (2.0.8), we have
in
(2.0.16)
to derive (2.0.17).
(2.0.18)
G(x, y) = −
(2.0.19)
(
G
0, y
)
= −
1
4π x − y∣
∣
+
1
4π ∣x∣∣
R
2
R
x 2 x − y
∣
∣
∣
,
1
∣
4π y
∣
+
1
4πR
.
For future use, we
also
compute
that
(2.0... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
∣
∣
x 2
R2
)
.
( −
1
=
= −
−
x σ
∣ − ∣3
σ
π
4 x
+
∣
∣
1 x 2
4π R2
R2
−
x 2 x σ
∣
∣
− σ∣3
∣x
Using (2.0.22) and the fact
that
(2.0.23)
∇ ( ) (
ˆN σ G x, σ
ˆ σ
( ) = 1
N
R σ, w
e deduce
def
) = ∇ (
σG x, σ N σ
) ⋅
( ) =
ˆ
R
2
−
4π
∣
∣
x 2
R
1
∣ − ∣
x σ 3
.
4
MATH 18.152 COURSE NOTES - CLASS MEETING # 9
Remark 2.0.2. If ... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
.
R2
x−p
R
−
(x
∣2
p
) − (y − p
)∣
,
≠
x p,
Furthermore, if x
∈ BR(p) and σ ∈ ∂BR(p), then
(2.0.25c)
∇
(
ˆN (σ)G x, σ
) =
∣
− 2
R2 − ∣x p
4πR
1
∣ − ∣
3
x σ
.
We can now easily derive a representation formula for solutions
to the Laplace equation on a ball.
Theorem 2.1 (Poisson’s formula). Let BR(p) ⊂
= (
)
(
x3
, x2,
x... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
g σ
∣
− σ
dσ,
n
Proof. The identity (2.0.27) follows immediately from Theorem 1.1 and Lemma 2.0.1.
(cid:3)
3. Harnack’s inequality
We will now use some of our tools to prove a famous inequality for Harmonic functions. The
theorem provides some estimates that place limitations on how slow/fast harmonic functions are
all... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
egativity of
(i.e. σ
we have that ∣x R x σ
∣
≤
e
g, w
− ∣ ≤
deduce
R
),
∈
∣ +
∣
x R.
that
(3.0.31)
)
∂BR(0
Now recall that by the mean value property, we have that
u(x) ≤
R
R2
+ ∣x
− ∣
x
∣
1
∣2 4πR
∫
( )
g σ dσ.
(3.0.32)
u( ) =
0
1
4πR2
Thus, combining (3.0.31) and (3.0.32), we have
( )
g σ dσ.
(0)
∫
∂BR
that
(3.0.33)
... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
�
∈
∣
def
=
(3.0.34)
Rn−2(R − ∣
∣)
x
−
( + ∣
∣)
R x n 1
v(0) ≤
v
)
(x
≤
−
( + ∣
n 2 R
R
∣)
∣
−
(
x
R
∣)
x
n−1
)
v(0 .
Allo
wing R
(and therefore u is to
→ ∞
o).
in (3.0.34), we conclude that v x
( ) = ( )
0
v
. Th
us, v is
a constant-valued function
6
MATH 18.152 COURSE NOTES - CLASS MEETING # 9
To handle the case u(x... | https://ocw.mit.edu/courses/18-152-introduction-to-partial-differential-equations-fall-2011/419f485442c5fb28d976ffa48ab91e3f_MIT18_152F11_lec_09.pdf |
18.413: ErrorCorrecting Codes Lab
February 24, 2004
Lecturer: Daniel A. Spielman
Lecture 6
6.1
Introduction
Begin by describing LDPC codes, and how they are described by many local constraints. Point out
that random graphs locally look like trees (from the birthday paradox), and so we will learn to do
belief p... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
|
Ppost [X1 = a1|Y1Y2 = b1b2] = cb1,b2Pprior [X1 = a1] Pext [X1 = a1 Y1Y2 = b1b2] ,
|
so it suffices to prove
Lemma 6.2.2.
Pext [X1 = a1|Y1Y2 = b1b2] = cb1,b2Pext [X1 = a1 Y1 = b1] Pext [X1 = a1 Y2 = b2] .
|
|
61
Lecture 6: February 24, 2004
62
Proof. We begin by examining the righthandsides. We have
Pext [X1 ... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
2 = b1b2 X1 = a1]
= cb1,b2
�
|
P [Y1Y2 = b1b2|X1X2 = a1a2] P [X2 = a2 X1 = a1]
|
= cb1,b2
a2:(a1,a2)∈C
�
a2:(a1,a2)∈C
= cb1,b2P [Y1 = b1|X1 = a1]
|
P [Y1 = b1|X1 = a1] P [Y2 = b2 X2 = a2] P [X2 = a2 X1 = a1]
|
�
a2:(a1,a2)∈C
P [Y2 = b2|X2 = a2] P [X2 = a2 X1 = a1] .
|
To conclude, we observe that this last te... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
1X3 = a1a3|X2 = a2] = P [X1 = a1 X2 = a2] P [X3 = a3 X2 = a2] .
|
|
In this case, we can say that all the information that X3 contains about X1 is transmitted through
X2. This fact can be used to simplify the belief computation.
Lemma 6.4.1.
Pext [X1 = a1|Y2Y3 = b2b3] =
�
a2:(a1,a2)∈C
P [X2 = a2|X1 = a1] Pext [X2... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
3 = b3]
|
|
6.5 Trees
A hypergraph is given by a collection of vertices x1, . . . , xn and a collection of edges e1, . . . , em,
where each ei ⊆ {x1, . . . , xn}. A path in a hypergraph is a sequence of vertices xi1, . . . , xik such
that each consecutive pair in the sequence lie in some edge. That is, for each 2 ≤ ... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/41ad43fa58543246660bb36a3b5bc28c_lect6.pdf |
6.763 Applied Superconductivity
Lecture 1
Terry P. Orlando
Dept. of Electrical Engineering
MIT
September 8, 2005
Outline
• What is a Superconductor?
• Discovery of Superconductivity
• Meissner Effect
• Type I Superconductors
• Type II Superconductors
• Theory of Superconductivity
• Tunneling and the Josephs... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
physics/laureates
5
Discovery of Superconductivity
“As has been said, the experiment left
no doubt that, as far as accuracy of
measurement went, the resistance
disappeared. At the same time,
however, something unexpected
occurred. The disappearance did not
take place gradually but (compare Fig.
Please see: Figu... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
the magnetic vortices
(blue cylinders feel a force Lorentz force that pushes the
(
vortices at right angles to the current flow. This movement
dissipates energy and produces resistance from D. J.
Bishop et al., Scientific American, 48 (Feb. 1993)].
[
)
i
Upper Critical Fields of Type II Superconductors
Image rem... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
Interaction
Image removed for copyright reasons.
The origin of superconductivity in conventional superconductors
http://www.physics.carleton.ca/courses/75.364/mp-2html/node16.html
13
14
•7
Cooper Pairs & Energy Gap
Images removed for copyright reasons.
Please see: Figure 5 and figure 6 from http://nobelprize.o... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
Tunneling between a normal metal and
another normal metal or a superconductor
Images removed for copyright reasons.
Please see: Figures 3 and 4 from http://nobelprize.org/physics/laureates/
1973/giaever-lecture.html
http://www.nobel.se/physics/laureates/1973/giaever-lecture.pdf
18
•9
Tunneling between two superco... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
important break-through in the
discovery of superconductivity in
ceramic materials"
Images removed for copyright reasons.
____________________
Please see: http://nobelprize.org/physics/
__________________
laureates/1987/index.html
Image removed for copyright reasons.
______________
Please see: Figure 1.5 from ht... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
the Earth’s magnetic field)
• Quantum Computing
Picture source: http://www.superconductors.org
Massachusetts Institute of Technology
25
26
•13
Large-Scale Applications
Technical Points
Application
Power cables
Current Limiters
Transformers
Motors/Generators
Energy Storage Magnets
NMR magnets (MRI)
Cavitie... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
, low cost of the raw materials, and absence of weak-link limitations that
allows the use of mature powder-in-tube technology to fabricate long wires. The
inclusion of MgB2 presentations in the symposium will bring together both communities
and will encourage the discussion of problems that are common to all superco... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
over existing computers
• Capable of reversible computation
• e.g. Can factorize a 250-digit number in seconds
while an ordinary computer will take 800 000
years!
Current Research in my group focuses on
Quantum Computation using Superconductors
Massachusetts Institute of Technology
34
•17
The “Magic” of Quantum... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/41b2882232b1b43823ec488969c21b6d_lecture1.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS
by A. Megretski
Lecture 3: Continuous Dependence On Parameters1
Arguments based on continuity of functions are common in dynamical system analysis.
They rarely apply to... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
) − a(¯
x2)| ∃ M |x1 − ¯
x2|
¯
x1, ¯
[t0, tf ] ∈� Rn of (3.1). The proof of both
for all ¯ x2 from a neigborhood of a solution x :
existence and uniqueness is so simple in this case that we will formulate the statement
for a much more general class of integral equations.
Theorem 3.1 Let X be a subset of Rn containin... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
. When a does not depend on the
third argument, we have the standard ODE case
x˙ (t) = a(x(t), t).
In general, Theorem 3.1 covers a variety of nonlinear systems with an infinite dimensional
state space, such as feedback interconnections of convolution operators and memoryless
nonlinear transformations. For example,... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
, t)|d�
∃ K|xk (� ) − xk−1(� )|d�
t0
∃ 0.5 max {|xk (t) − xk−1(t)|}.
t�[t0,tf ]
Therefore one can conclude that
max {|xk+1(t) − xk (t)|} ∃ 0.5 max {|xk (t) − xk−1(t)|}.
t�[t0,tf ]
t�[t0,tf ]
Hence xk (t) converges exponentially to a limit x(t) which, due to continuity of a with
respoect to the first argument, ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
≤ Br (¯
x1, ¯
x0), t0 ∃ � ∃ t ∃ t1,
and
x, �, t)| ∃ m(t) � ¯
where the functions K(·) and M (·) are integrable over [t0, t1].
x ≤ Br (¯
x0), t0 ∃ � ∃ t ∃ t1,
|a(¯
4
3.2 Continuous Dependence On Parameters
In this section our main objective is to establish sufficient conditions under which solutions
of ODE depend con... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
K ≤ R such that
|a(¯
x, �, t, q)| ∃ M � ¯
x ≤ X d, t0 ∃ � ∃ t ∃ tf , q ≤ (q0 − d, q0 + d);
(c) for every � > 0 there exists � > 0 such that
|x0(q1) − ¯
¯
x0(q2)| ∃ � � q1, q2 ≤ (q0 − d, q0 + d) : |q1 − q2| < �,
(3.6)
(3.7)
|a(¯
x, �, t, q1) − a(¯
x, �, t, q2)| ∃ � � q1, q2 ≤ (q0 − d, q0 + d) : |q1 − q2| < �, ¯
x ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
to derive qualitative statements about nonlinear
systems.
5
3.3.1 Differential flow
Consider a time-invariant autonomous ODE
where a : Rn ∈� Rm is satisfies the Lipschitz constraint
x˙ (t) = a(x(t)),
|a(¯
x1) − a(¯
x2)| ∃ M |x1 − ¯
x2|
¯
(3.8)
(3.9)
on every bounded subset of Rn . According to Theorem 3.1, this ... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
3.2, x : � ∈� Rn is a continuous function defined on an open
subset � ∀ R × Rn . With x considered a parameter, t ∈� x(t, ¯
x) defines a family of
smooth curves in Rn . When t is fixed, ¯
x) defines a continuous map form an open
subset of Rn and with values in Rn . Note that x(t1, x(t2, ¯
x)) = x(t1 + t2, ¯
x) whenever
... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
< �.
In other words, all solutions starting sufficiently close to an asymptotically stable
equilibrium ¯x0 converge to it as t � ⊂, and none of such solutions can escape far away
before finally converging to ¯x0.
6
Theorem 3.3 Let ¯x0 ≤ Rn be an asymptotically stable equilibrium of (3.8). The set
x0 as t � ⊂ is an o... | https://ocw.mit.edu/courses/6-243j-dynamics-of-nonlinear-systems-fall-2003/41d1588bb21baf8002a8b9b56ed1047c_lec3_6243_2003.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.969 Topics in Geometry: Mirror Symmetry
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
MIRROR SYMMETRY: LECTURE 4
DENIS AUROUX
1. Pseudoholomorphic Curves
For (X 2n, ω) symplectic, J a compatible a.c.s. ∈ J ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
if D∂ is onto.
Theorem 1. The set J reg(X, β) of J ∈ J (X, ω) s.t. every simple J-holomorphic
curve in class β is regular is a Baire subset. For J ∈ J reg(X, β), the subset of
simple maps M∗ (X, J, β) ⊂ Mg,k(X, J, β) is smooth and oriented of dimension
2d.
g,k
Let g(·, ) = ω( , J ) be the associated Riemannian met... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
uniformly to x �→ (x, 0). But if you reparameterize to ˜x = nx,
1 ) and away from x = ∞, it converges uniformly to ˜ → (0, 1
x ).
x˜ �→ ( n
˜
1 ˜ x
x, ˜
x
nx
The general idea is:
• Identify bubbling regions where sup |dun| → ∞.
• Away from those, ∃ convergent subsequences.
• Near them, we can rescale the doim... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
defined as follows: given α1, . . . , αk ∈ H ∗(X),
deg (αi) = 2d,
�
(8)
�α1, . . . , αk�g,β =
ev∗
1α1 ∧ · · · ∧ evk
∗αk ∈ Q
�
[M g,k(X,J,β)]
i
ev−1(Ci)) (or rather #(ev
X (choose Ci trans
Equivalently, if we represent P D(αi) by a cycle Ci
verse to the evaluation map), then the pairing is simply #([M g,k(X, ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
β
MIRROR SYMMETRY: LECTURE 4
3
is a universal curve C → M0,k (the fiber over a point is the corresponding curve),
and J is now given by a map C → J (X, ω). The holomorphic curve equation
becomes u : (Σ, j) X, du + J(u(z), z)du j = 0. We choose a superposition of
a finite number perturbations, which break the symme... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
xed j, we have a ∂-operator on sections of u∗T X, and
the cokernel of this operator is precisely H 1(Σ, u∗T X). Where du = 0, we have
u∗T X = T Σ ⊕ u∗N and H 1(Σ, T Σ) is simply the deformations of j. There is
also an obstruction bundle Obsu = H 1(Σ, u∗NΣ) if u is an immersion. We claim
that we can define an obstruc... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/41f318431154e73d835ed92bc5c40ab4_MIT18_969s09_lec04.pdf |
MIT 6.035
Semantic Analysis
Martin Rinard
Laboratory for Computer Science
Massachusetts Institute of Technology
Error Issue
• Have assumed no problems in building IR
• But are many static checks that need to be done
as part of translation
• Called Semantic Analysis
Goal of Semantic Analysis
• Ensure that program obe... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-sma-5502-fall-2005/41f7dfc1934579616240a609947159f0_7_semantic_check.pdf |
parameter descriptor for x
while (i < v.length)
v[i] = v[i]+x;
while
<
ldl
len
ldf
sta
ldf
ldl
+
lda
ldp
ldf
ldl
field descriptor for v local descriptor for i
parameter descriptor for x
while (i < v.length)
v[i] = v[i]+x;
while
<
ldl
len
ldf
sta
ldf
ldl
+
lda
ldp
ldf
ldl
field descriptor for v local descriptor for ... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-sma-5502-fall-2005/41f7dfc1934579616240a609947159f0_7_semantic_check.pdf |
and return
type declarations of superclass
Load Instruction
• What does compiler have? Variable name.
• What does it do? Look up variable name.
– If in local symbol table, reference local descriptor
– If in parameter symbol table, reference parameter
descriptor
– If in field symbol table, reference field descriptor
... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-sma-5502-fall-2005/41f7dfc1934579616240a609947159f0_7_semantic_check.pdf |
Int
• Class C compatible with Class D if
C inherits from D (but not vice-versa)
Store Instruction
• What does compiler have?
– Variable name
– Expression
• What does it do?
– Look up variable name.
• If in local symbol table, reference local descriptor
• If in parameter symbol table, error
• If in field symbol tabl... | https://ocw.mit.edu/courses/6-035-computer-language-engineering-sma-5502-fall-2005/41f7dfc1934579616240a609947159f0_7_semantic_check.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.969 Topics in Geometry: Mirror Symmetry
Spring 2009
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MIRROR SYMMETRY: LECTURE 2
DENIS AUROUX
Reference for today: M. Gross, D. Huybrechts, D. Joyce, “Calabi-Yau Mani
folds ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
· }
∂
q(X, E) = ker∂/im∂
H
∂
∂
∂
Deforming J to a “nearby” J � gives
(4)
Ω1,0 ⊆ T ∗C = Ω1,0 ⊕ Ω0,1
J
J �
J
is a graph of a linear map (−s) : Ω1
0 (acted
,
,1 . J � is determined by Ω1
,0 → Ω0
�
J
J
J
on by i) and Ω0,1 (acted on by i�). s is a section of (Ω1,0)∗ ⊗ Ω0,1 = T1,0 ⊗ Ω0,1
J
1,0X. If z1, . . . , z... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
, v�]
giving it the structure of a differential graded Lie algebra.
Proposition 1. J � is integrable
∂s + 2
Proof. We want to check that the bracket of two 0, 1 tangent vectors is still 0, 1,
i.e. that
1 [s, s] = 0.
⇔
(6)
[
∂ �
+
∂zk
�
s�k
∂ �
+
∂
,
∂z� ∂zk
s�k
∂
∂z�
�
1
0,
] ∈ T XJ
�
�
0,q⊗
q ΩX... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
(or, assuming that Aut(X, J) is discrete, we want that near J, ∃ a universal family
X → U ⊂ MCX (complex manifolds, holomorphic fibers ∼= X) s.t. any family of
integrable complex structures X � → S induces a map S → U s.t. X pulls back
to X �). We have an action of the diffeomorphisms of X: for φ ∈ Diff(X) close to
id... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
, this should satisfy
(12)
∂s(t) +
1
2
[s(t), s(t)] = 0
In particular, s1 = dt t=0 solves ∂s1 = 0. We obtain an infinitesimal action of
Diff(X): for (φt), φ0 = id , dt |t=0 = v a vector field,
dφ
|
ds
(13)
d
dt
|t=0(−(∂φt)−1 ◦ ∂φt) = −
d
dt
|t=0(∂φt) = −∂v
This implies that first-order deformations are given as
(... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
vij = −vji and vij + vjk = vik on Uijk.
Cech 1-cocycle
conditions in the sheaf of holomorphic tangent vector fields. Modding out by
holomorphic functions ψi : Ui → Ui (which act by φij �→ ψj φij ψ−1) is precisely
modding by the ˇ
Cech coboundaries. Thus, Def 1(X, J) = ˇ
This is precisely the ˇ
i
H 1(X, T X 1,0).
= φi... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
). If it is zero, then there is an s2 s.t. ∂s2 + 1
2 [s1, s1] = 0,
and the next obstructure is the class of [s1, s2] ∈ H 2(X, T X 1,0). We are basically
attempting to apply by brute force the implicit function theorem.
If it happens that H 2(X, T X) = 0, then the deformations are unobstructed
and the moduli space o... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
}
�
X
⇒
Jt
(17)
αt
=
i
�
I,J||I|=p,|J|=q
(
αIJ (t)dzi
1
t) ∧ · · · ∧ dzi
)
t
) ∧ · · · ∧ dz(
t
t) ∧ dz(
(
j
j
p
q
1
d
|
Taking dt t=0, the result follows from the product rule. We mostly get (p, q) terms
(t)
�
and a few (p + 1, q − 1), (p − 1, q + 1) forms (the latter from dt |t=0(dzik
).
d | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/42040c1f577d7178b66e4d6e3fbd12f4_MIT18_969s09_lec02.pdf |
Massachusetts Institute of Technology
18.413: ErrorCorrecting Codes Laboratory
Professor Daniel A. Spielman
Handout 0
February 3, 2004
Signing Up
If too many people sign up for the course, I will perform a lottery among those who have signed
up, and announce the results by email on Wednesday, Feb 4th.
First Re... | https://ocw.mit.edu/courses/18-413-error-correcting-codes-laboratory-spring-2004/4215f545fb81cbc74f2849d4596c802b_out0.pdf |
System Identification
6.435
SET 9
– Asymptotic distribution of PEM
Munther A. Dahleh
Lecture 9
6.435, System Identification
1
Prof. Munther A. Dahleh
Central Limit Theorem
(Generalization)
• Basic Theorem II:
Consider
are both ARMA processes, possibly correlated, with
underlying white noise (bounded 4th moment)
Lect... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/421a9e228140fc8e47002d404223b9bb_lec9_6_435.pdf |
leh
Examples
ARX:
Lecture 9
6.435, System Identification
11
Prof. Munther A. Dahleh
MA:
Lecture 9
6.435, System Identification
12
Prof. Munther A. Dahleh
ARMA:
Lecture 9
6.435, System Identification
13
Prof. Munther A. Dahleh
Comments: As , then . If , then
the model structure is over parametrized so
( has pol... | https://ocw.mit.edu/courses/6-435-system-identification-spring-2005/421a9e228140fc8e47002d404223b9bb_lec9_6_435.pdf |
20.110/5.60 Fall 2005
Lecture #1
page
1
Introduction to Thermodynamics
Thermodynamics:
→ Describes macroscopic properties of
equilibrium systems
→ Entirely Empirical
→ Built on 4 Laws and “simple” mathematics
0th Law ⇒ Defines Temperature (T)
1st Law ⇒ Defines Energy (U)
2nd Law ⇒ Defines Entropy (S)
3rd La... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
• Intensive: Independent of the size of the system
(T,p,
V =
V
n
,…)
The State of a System at Equilibrium:
• Defined by the collection of all macroscopic properties that
are described by State variables (p,n,T,V,…)
[INDEPENDENT of the HISTORY of the SYSTEM]
• For a one-component System, all that is required is “n... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
(g, 1 bar, 150 °C)
initial state
final state
20.110J / 2.772J / 5.601JThermodynamics of Biomolecular SystemsInstructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
20.110/5.60 Fall 2005
Lecture #1
page
5
• Path: Seque... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
are in thermal equilibrium.
Consequence of the zero’th law:
B acts like a thermometer, and , , and are all
C
A
B
at the same “temperature”.
20.110J / 2.772J / 5.601JThermodynamics of Biomolecular SystemsInstructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field
... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
(Kelvin)
)
=
T K t C
(
(
°
)
+ 273.15
T = 0K corresponds to absolute zero (
t
= −
273.15
)
C
°
Better reference points used for the Kelvin scale today are
T = 0K (absolute zero) and Ttp = 273.16K (triple point of H2O)
Ideal Gases
Boyle’s Law and the Kelvin scale
(
lim
→
0
p
pV
)
T
=
⎡
⎢
⎢
⎢
⎣
pV
)
(
lim
→
0
p
273... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
work to
surroundings (
the system. If the system does work on the
w
)0 then
, that
0 if
< .
0
∆ >
V
w
∆ <
V
0
>
convention:
• Heat:
“q”
That quantity flowing between the system and the
surroundings that can be used to change the temperature
of the system and/or the surroundings.
Sign convention:
If heat en... | https://ocw.mit.edu/courses/20-110j-thermodynamics-of-biomolecular-systems-fall-2005/425a9bc953f2930faf9816e9fcfe399e_l01.pdf |
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