text stringlengths 21 3.88k | source stringlengths 82 201 |
|---|---|
K 2 = 1
·
so |−K = ∅. Take a hyperplane section H of X. Then there is an n ≥ 0
| �
nK| =� ∅ but |H + (n + 1)K| = ∅. Since −K ∼ an effective nonzero
s.t. |H +
·
divisor, H K < 0 and H (H + nK) is eventually negative and H + nK is not
effective. Let D ∈ |H + nK|: then |D + K| = ∅ and K · D = K(H + nK) =
K H < 0 sinc... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/01d543f81d0b743f06deed68d9eec77d_lect9.pdf |
Again, let H be a hyperplane section: if K H < 0,
we can take E = H; if K H = 0, we can take K + nH for n >> 0; so assume
K H > 0. Let γ = −K· > 0 so that (H + γK) K = 0. Also,
·
·
·
H
·
K2
(3)
(H + γK)2 > H 2 + 2γ(H K) + γ2K = H 2 +
·
So take β rational and slightly larger than γ to get
(K · H)2
(−K 2)
> 0... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/01d543f81d0b743f06deed68d9eec77d_lect9.pdf |
as in reduction 2, i.e.
K · D ≥ 0 for every effective divisor D s.t. |K + D| = ∅. We will obtain a
contradiction.
Lemma 1. If X is a minimal surface with p2 = q = 0, K 2 > 0 and K D ≥ 0
for every effective divisor D on X s.t. |K + D| = ∅, then
·
(1) Pic (X) is generated by ωX = OX (K), and the anticanonical bundle
... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/01d543f81d0b743f06deed68d9eec77d_lect9.pdf |
irreducible,
and similarly D is not a multiple. Furthermore, pa(D) = 2 D(D + K) + 1 = 1,
showing (2).
Next, we claim that the only effective divisor s.t. D + K = ∅ is the zero
|
divisor. Assume not, i.e. ∃D > 0 s.t. |K + D| = ∅.
∈ D: then since
h0(−K) ≥ 1 + K 2 ≥ 2, there is a C ∈ |−K| passing through x. C is an in... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/01d543f81d0b743f06deed68d9eec77d_lect9.pdf |
of
positive self-intersection, so by the useful lemma D +nK is not effective for n >>
0. Now, let Δ be an arbitrary effective divisor. Then ∃n ≥ 0 s.t. Δ + nK = 0
|
but |Δ + (n + 1)K| = ∅. Take D ∈ |Δ + nK| effective. |D + K| =
=
0 from above. Since any divisor is a difference of effective divisors, Pic (X) is
generate... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/01d543f81d0b743f06deed68d9eec77d_lect9.pdf |
theorem,
(8)
0 → OX (−K) ⊗ Ix ⊗ Iy → OX (−K) → k6 → 0
2 have dimension 3 over k. Taking the long exact sequence,
since OX,x/m2 , OX,y/my
we find that h0(OX (−K) ⊗ Ix ⊗ Iy) = 0, and get a nonzero section of that sheaf.
x
�
�
4
LECTURES: ABHINAV KUMAR
It is a divisor of zero passing through x and y with multiplicit... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/01d543f81d0b743f06deed68d9eec77d_lect9.pdf |
X
=
X
Since q
=
an
X )∗) ∼
= H 1(O∗
X ) = Pic X ∼
(10)
This implies that b2 = rank H 2(X, Z) = rank Pic X = 1 contradicting b2 ≥ 5.
For positive characteristic, we will sketch a proof: the first proof was given by
Zariski, and the second using ´etale cohomology by Artin and by Kurke. Our
proof will be by reductio... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/01d543f81d0b743f06deed68d9eec77d_lect9.pdf |
Bluespec Tutorial: Rule
Scheduling and Synthesis
Michael Pellauer
Computer Science & Artificial Intelligence Lab
Massachusetts Institute of Technology
Based on material prepared by Bluespec Inc,
January 2005
March 4, 2005
BST-1
Improving performance via
scheduling
Latency and bandwidth can be improved by
performing... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/01e06522cd4ab035dbbbd38279045351_t03_bluespec.pdf |
Rule: fetch
Predicate: the_bf.i_notFull_ && the_started.get
Blocking rules: imem_put, start
more urgent rules which can
block the execution of this rule
(more on urgency later)
March 4, 2005
BST-6
3
Static execution order
When multiple rules execute in a single
clock cycle, they must appear to
execute in sequence
Th... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/01e06522cd4ab035dbbbd38279045351_t03_bluespec.pdf |
2 decrements register y
0
+1
-1
1
+1
-1
2
x
y
rule proc0 (cond0);
x <= x + 1;
endrule
rule proc1 (cond1);
y <= y + 1;
x <= x – 1;
endrule
rule proc2 (cond2);
y <= y – 1;
endrule
(* descending_urgency = “proc2, proc1, proc0” *)
show what happens under different urgency annotations
March 4, 2005
BST-11
Example2.bsv Demo
... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/01e06522cd4ab035dbbbd38279045351_t03_bluespec.pdf |
a
e
r
t
r
n
d i t i o
e l y
t i v
March 4, 2005
BST-13
Demo rule splitting:
Example 3
(* descending_urgency = "r1, r2" *)
// Moving packets from input FIFO i1
rule r1;
Tin x = i1.first();
if (dest(x)== 1) o1.enq(x);
else o2.enq(x);
i1.deq();
if (interesting(x)) c <= c + 1;
endrule
// Moving packets from input FIFO i2
r... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/01e06522cd4ab035dbbbd38279045351_t03_bluespec.pdf |
rule urgency
Sometimes, an urgency warning or a conflict can be due
to a mistake or oversight by the designer
(cid:132) A rule may accidentally include an action which shouldn’t
be there
(cid:132) A rule may accidentally write to the wrong state element
(cid:132) A rule predicate might be missing an expression which ... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/01e06522cd4ab035dbbbd38279045351_t03_bluespec.pdf |
synthesized
The attributes apply to all methods in
the interface
March 4, 2005
BST-21
always_ready
This attribute has two effects:
Asserts that the ready signal for all
methods is True
(cid:132) It is an error if the tool cannot prove this
Removes the associated port in the
generated RTL module
(cid:132) Any users o... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/01e06522cd4ab035dbbbd38279045351_t03_bluespec.pdf |
1;
endrule: cycle
method Action start (Tin d_init, Tin r_init) if (r == 0);
d <= zeroExtend(d_init);
r <= r_init; product <= 0;
endmethod
method Tout result () if (r == 0);
return product;
endmethod
endmodule: mkMult1
March 4, 2005
BST-26
13
Test bench for Example 1
module mkTest (Empty);
// arrays a, b contain the nu... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/01e06522cd4ab035dbbbd38279045351_t03_bluespec.pdf |
Lecture 23: FaultTolerant Quantum Computation
Scribed by: Jonathan Hodges
Department of Nuclear Engineering, MIT
December 4, 2003
1
Introduction
Before von Neumann proposed classical faulttolerance in the 1940’s, it was assumed that a compu
tational device comprised of more than 106 components could not perfor... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
the strings received are 1110101 on a and 0111111
on b. If one performs a bitwise AND on each successive bit of the bit strings a and b, the result is
0110101. Taking “triples” of bits of this resulting addition, one performs a majority vote. Thus, if
one bit has an error probablity of p , two bits in a triple bein... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
a
1
P. Shor – 18.435/2.111 Quantum Computation – Lecture 23
2
specific point, checking the result, then starting the computation again. The probablistic nature of
quantum mechanics and the nocloning theorem make this technique useless for QC. Classically, one
might make many copies of the computation to perform a... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
(KitaevSolovay Theorem). Given a set of gates on SU(2) ( or SU(k) generally)
that generates a dense set in SU(2), then any gate U ∈ SU(2) can be approximated to � using
O (logc 1
� ) gates where 1 ≤ c ≤ 2. See Appendix 3 of Nielsen and Chuang for more details.
3.1 Fault Tolerance of σx
In order to show that a σx ... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
one of the qubits, where the errors on each qubit are
uncorrelated. Then this code will be able to correct for these errors, resulting in a quantum error
correcting code and operation that performs σx with faulttolerance.
|
|
3.2 Fault Tolerance of σz
By using the Steane code above, the equivalent of σz on an unen... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
fault tolerance of the Hadamard gate under this CSS encoding can be seen under the additional
⊥. If the function E(x) represents the act of encoding the bit, the action of a
constraint C1 = C2
Hadamard on an encoding qubit must follow the transformations:
1
E(|0�) −→ √
2
1
E(|1�) −→ √
2
(E( 0�) + E( 1�))
|
|
(E( ... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
), thus appropriately adding a phase to the states in the code that are E( 1�).
|
|
|
|
3.4 Fault Tolerance of CNOT Gate
The σx, σz, and H gates can all be performed on a single encoded qubit with faulttolerance because
these gates are always applied to single qubits. Likewise, given two singlequbit encoded states,... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
added to all states y�, which is just the encoding E( 1�). If
an error occurs in any of the twoqubit CNOT operations, this will result in va or vb not being all
0’s or all 1’s, and the CSS code will correct the the appropriate state.
|
|
4 Error Correction With FaultTolerant Precision
Both classical and quantum e... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
in time due to the backaction of the CNOT. The stringent requirement of each error not
affecting more than a single qubit (or pair in the FT CNOT construction) is not fulfilled.
Using the idea of single failure points between qubits, as seen in the FT CNOT construction,
we start our parity check register on k qubits ... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
superposition of the states |0�
and 1�. The Hadamard transform of this state:
|
|
H ⊗k(α 0� + β 1�) = (
|
|
α + β
√
2
)
1
2k−1
�
s∈even
|s� + (
α − β
)√
2
1
2k−1
�
s∈odd
|s�
(10)
Thus if α = β, the state is all zeros and no backaction will occur. The all ones state simply
adds the ones vector to the qubits. ... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/01e4b2ffc6380061eb0e47822d7813de_qc_lec23.pdf |
18.445 Introduction to Stochastic Processes
Lecture 13: Countable state space chains 2
Hao Wu
MIT
1 April 2015
Hao Wu (MIT)
18.445
1 April 2015
1 / 5
Recall Suppose that P is irreducible.
The Markov chain is recurrent if and only if
Px [τ +
x < ∞] = 1,
for some x.
The Markov chain is positive recurrent if and only if
... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/01eb8f31f3e72b4532887f64419f2267_MIT18_445S15_lecture13.pdf |
.
Theorem
Suppose that the Markov chain is irreducible and aperiodic. If the chain
is positive recurrent, then
lim ||Pn(x, ·) − π||TV = 0.
n
Hao Wu (MIT)
18.445
1 April 2015
5 / 5
MIT OpenCourseWare
http://ocw.mit.edu
18.445 Introduction to Stochastic Processes
Spring 2015
For information about citing these materials ... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/01eb8f31f3e72b4532887f64419f2267_MIT18_445S15_lecture13.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.01 Single Variable Calculus
Fall 2006
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Lecture 32: Exam 4 Review
18.01 Fall 2006
Exam 4 Review
1. Trig substitution and trig integrals.
2. Partial fractions.
3. Integrati... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/0202fa3893049a6a502c4f7079eca657_exam4_review.pdf |
find C,
C =
(−2)2 − 2 + 1
(−2 − 1)2
=
1
3
(x → −2)
To find A, one method is to plug in the easiest value of x other than the ones we already used
(x = 1, −2). Usually, we use x = 0.
1
A
(−1)2(2) −1
=
+
1
(−1)2
+
1/3
2
and then solve to find A.
The Review Sheet handed out during lecture follows on the next p... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/0202fa3893049a6a502c4f7079eca657_exam4_review.pdf |
required to cover every case systematically, namely, completing the square1 and long division.2
3. Integration by parts:
� b
a
b
�
�
uv�dx = uv
�
�
�
a
� b
−
a
u�vdx
This is used when u�v is simpler than uv�. (This is often the case if u� is simpler than u.)
4. Arclength: ds = dx2 + dy2. Depending on wheth... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/0202fa3893049a6a502c4f7079eca657_exam4_review.pdf |
3.
2Long division is used when the degree of P is greater than or equal to the degree of Q. It expresses P (x)/Q(x) =
P1(x) + R(x)/Q(x) with P1 a quotient polynomial (easy to integrate) and R a remainder. The key point is that the
remainder R has degree less than Q, so R/Q can be split into partial fractions.
3
� ... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/0202fa3893049a6a502c4f7079eca657_exam4_review.pdf |
x + 1
(To carry out this long division, do not factor the denominator Q(x) = x2 − 2x + 1, just leave it
alone.) The quotient x + 2 is a polynomial and is easy to integrate. The remainder term
3x − 2
(x − 1)2
has a numerator 3x − 2 of degree 1 which is less than the degree 2 of the denominator (x − 1)2 .
Therefore... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/0202fa3893049a6a502c4f7079eca657_exam4_review.pdf |
out using the trigonometric substitution u = k tan θ du = k sec2 θdθ. This then leads to
sec-tan integrals, and the actual computation for large values of n are long.
There are also other cases that we will not cover systematically. Examples are below:
1. If Q(x) = (x − a)m(x − b)n, then the expression is
A1
x − a... | https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/0202fa3893049a6a502c4f7079eca657_exam4_review.pdf |
2.160 Identification, Estimation, and Learning
Lecture Notes No. 3
February 15, 2006
2.3 Physical Meaning of Matrix P
The Recursive Least Squares (RLS) algorithm updates the parameter vector
t y ) in such a way that the overall squared error may
(
ˆ
θ(t − 1) based on new data ϕT (t ),
be minimal. This is done by... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
has all real
eigenvalues. The eigen vectors associated with the individual eigenvalues are also real.
Therefore, the matrix ΦΦT can be reduced to a diagonal matrix using a coordinate
transformation, i.e. using the eigen vectors as the bases.
λ 0 L 0
1
0
M
∈ R mxm
ΦΦT ⇒ D =
M
0 L
λm ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
input data: ϕ(i)L. This direction has been
well explored, well excited. Although new data are obtained, the correction to the
parameter vector θ(t − 1) is small, if the new input data ϕ(t) is in the same direction as
that of λ . See the second figure above.
ˆ
max
The above observations are summarized as follows:
1) ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
of θ (0) and Po , the (best) estimation thereafter will be
different.
ˆ
Question: How do the initial conditions influence the estimate? The following theorem
shows exactly how the RLS algorithm works, given initial conditions.
Theorem
The Recursive Least Squares (RLS) algorithm minimizes the following cost functio... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
t − 1
t y )ϕ (t ) + ∑ i y ) ϕ ϕ
(i ) T
= Pt
(
i = 1
(
)− 1
(i ) + P θ (0)
0
(23)
(24)
3
− 1 ˆ − 1)
Pt
θ(t
− 1
− 1 = ϕ ϕ
T
(t )
− 1
(t ) + P t − 1
Recall Pt
)
t [ − 1 )
1)
θ (t ) = P P θ (t
+
−
P ϕ )[ (
t
(
t y
1)
+
−
Postmultiplying ϕ (t ) to both sides of (14)
)
=θ (t
... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
) −ϕ (t )θ (t − 1))
(
ˆ
T
(25)
(26)
(18)
Q.E.D.
Discussion on the Theorem of RLS
)
θ (t ) =
arg
min
θ
t
1
∑ (
i y ) − ϕ (i )θ)
(
2 i = 1
1 4 4 42
4 4 43
Squared estimation error
A
T
2
+
1
2
1
Weighted squared
)
(0)) P 0 ( θ θ
( θ θ
)
−
−
4 4 4 4
4 4 4 4
from
(0))
3 )
− ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
RLS theorem from the batch processing case of Pt
− 1 = ∑ ϕ ϕ
(i )
T
t
i = 1
− 1
P t
= ∑ ϕ ϕ
(i )
T
− 1
(i ) + P
o
t
i = 1
(i ) to:
(28)
4
Other important properties of RLS include:
• Convergence of θ (t ) . It can be shown that
lim ˆ(t )
θ
ˆ
θ (t 1)
−
−
t →∞
=
0
See Goodwin and Sin’s book, Ch.3, f... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
) T
1
t
)
( P
Pt − 1 ϕ ϕ
t − 1
α Pt − 1 − ( ϕ α
t ϕ (t ))
+ T
) t − 1
( P
Exercise: Obtain (34) and (35) from (32) and (33).
t ϕ (t ))(
Pt =
t y ) −ϕ (t )θ (t − 1))
(
ˆ
T
(32)
(33)
(34)
(35)
A drawback of the forgetting factor approach
When the system under consideration enters “steady stat... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
ning the whole m -dim space
Set P0 =I (the m xm identity matrix) and θ(0) arbitrary
Compute
ˆ
ˆ
θ (t ) = θ (t
ˆ
−
1)
+
Pt −
1ϕ (t )
t ϕ (t )
ϕ ( P
)
t − 1
T
t y ) −ϕ (t )θ (t − 1))
(
(
ˆ
T
(38)
(39)
where matrix Pt-1 is updated with the same recursive formula as RLS
Note that +1 involved in the denomin... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
e 1
r
te ) = M =
(
e
Consider that each squared error is weighted differently, or
Weighted Multi-Output Squared Error:
) r(
r T
J t (θ ) = ∑ e (
ieWi
ˆ
θ (t ) = min
r
) =∑ (
T
(
iy ) Ψ− θ ) W (
r(
iy ) Ψ− θ )
ˆ
θ (t )
i = 1
T
T
t
t
Β Ρ =
t
t
J (θ )
t
i = 1
arg
θ
=Ρ
t
T
t... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/0223ccdb1348e84e3a5f04fc0af46edf_lecture_3.pdf |
2.160 Identification, Estimation, and Learning
Lecture Notes No. 1
February 8, 2006
Mathematical models of real-world systems are often
too difficult to build based on first principles
alone.
Figure by MIT OCW.
System Ident cation;
“Let the data speak about the system”.
ifi
Figure by MIT OCW.
Image removed fo... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/024038d962b1e7d63130fc746e03ef71_lecture_1.pdf |
structure
3. Not available until an actual
system has been built
2
Introduction: System Identification in a Nutshell
b 3
b 2
b 1
u(t )
y(t )
FIR
Finite Impulse Response Model
t y ) =
(
(
t u b
1
− 1) +
(
t u b
2
− 2)
⋅ ⋅ ⋅ +
t u − m )
⋅ ⋅ ⋅ + b
m
(
Define
θ := [b , b ,
2
1
]T
m
⋅ ⋅ ⋅ , b ∈ R
m
ϕ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/024038d962b1e7d63130fc746e03ef71_lecture_1.pdf |
)θ )(−ϕ ) = 0
(
T
∑ t y )ϕ (t ) = ∑ (ϕ (t )θ )ϕ (t )
(
T
N
t = 1
3
N
∑ (ϕ(t )ϕ (t )
t =1
T
N
θ = ∑ t y )ϕ(t )
(
t =1
=
RN
∴ θN = RN ∑ t y )ϕ(t )
−1
(
ˆ
N
t =1
Question1 What will happen if we repeat the experiment and obtain θˆ again?
N
Consider the expectation of θN when the experiment is r... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/024038d962b1e7d63130fc746e03ef71_lecture_1.pdf |
∑ϕ(
−1
ˆ
) (
t e
)
t
N
t =1
Taking expectation
E [θ −θ ] = E RN ∑ϕ(
−1
0
ˆ
N
N
t =1
)
t e
t
) (
−1
= RN ∑ϕ(t ) ⋅ E [
N
t =1
(
t e )] = 0
Question2 Since the true parameter θ is unknown, how do we know how close
0
ˆ
N will be toθ ? How many data points, N , do we need to reduce the errorθ −... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/024038d962b1e7d63130fc746e03ef71_lecture_1.pdf |
t e
( ) (
s e
t
)
t = 1 s = 1
T
)]ϕ (s )
R − 1
N
Assume that {e(t)} is stochastically independent
[ ( ) (
t e E
s e
)] =
)] = 0
s e
t e
( ) (
2
t ≠ s
E [
E [e (t )] = λ t = s
Then PN = RN ∑ϕ (t )λϕ (t )RN = λRN
− 1
T
− 1
N
− 1
t = 1
As N increases, RN tends to blow out, but... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/024038d962b1e7d63130fc746e03ef71_lecture_1.pdf |
. The convergence of
ˆ
θN
to
θ0 may be accelerated if we design inputs such that R is
large.
IV. The covariance does not depend on the average of the input signal. Only the
second moment
What will be addressed in 2.160?
A) How to best estimate the parameters
What type of input is maximally informative?
•
Inform... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/024038d962b1e7d63130fc746e03ef71_lecture_1.pdf |
Introduction to C++
Massachusetts Institute of Technology
January 12, 2011
6.096
Lecture 5 Notes: Pointers
1 Background
1.1 Variables and Memory
When you declare a variable, the computer associates the variable name with a particular
location in memory and stores a value there.
When you refer to the variable b... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/0240aeefb6d5fb9c0a20587ed98fa7ca_MIT6_096IAP11_lec05.pdf |
ffer
ent memory locations
• Use polymorphism – calling functions on data without knowing exactly what kind of
data it is (more on this in Lectures 7-8)
2 Pointers and their Behavior
2.1 The Nature of Pointers
Pointers are just variables storing integers – but those integers happen to be memory ad
dresses, usuall... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/0240aeefb6d5fb9c0a20587ed98fa7ca_MIT6_096IAP11_lec05.pdf |
a variable of type data type * – a “pointer to a data type value.”
2.2.2 Using Pointer Values
Once a pointer is declared, we can dereference it with the * operator to access its value:
cout << * ptr ; // Prints the value pointed to by ptr ,
// which in the above example would be x ’s value
We can use deferenced poin... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/0240aeefb6d5fb9c0a20587ed98fa7ca_MIT6_096IAP11_lec05.pdf |
keyword can be placed within a pointer variable declaration.
This is because there are two different variables whose values you might want to forbid
changing: the pointer itself and the value it points to.
const int * ptr ;
declares a changeable pointer to a constant integer. The integer value cannot be changed
thro... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/0240aeefb6d5fb9c0a20587ed98fa7ca_MIT6_096IAP11_lec05.pdf |
pointer the function returns is invalid.
As with any other variable, the value of a pointer is undefined until it is initialized, so it
may be invalid.
3 References
When we write void f(int &x) {...} and call f(y), the reference variable x becomes
another name – an alias – for the value of y in memory. We can decla... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/0240aeefb6d5fb9c0a20587ed98fa7ca_MIT6_096IAP11_lec05.pdf |
3.1 The Many Faces of * and &
The usage of the * and & operators with pointers/references can be confusing. The * operator
is used in two different ways:
1. When declaring a pointer, * is placed before the variable name to indicate that the
variable being declared is a pointer – say, a pointer to an int or char, not... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/0240aeefb6d5fb9c0a20587ed98fa7ca_MIT6_096IAP11_lec05.pdf |
of using subtraction and addition of pointers to move around
between locations in memory, typically between array elements. Adding an integer n to a
pointer produces a new pointer pointing to n positions further down in memory.
4.1.1 Pointer Step Size
Take the following code snippet:
1
2
3
long arr [] = {6 ,
l... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/0240aeefb6d5fb9c0a20587ed98fa7ca_MIT6_096IAP11_lec05.pdf |
notation, in which you explicitly add your offset to the pointer
and dereference the resulting address. For instance, an alternate and functionally identical
way to express myArray[3] is *(myArray + 3).
4.2 char * Strings
You should now be able to see why the type of a string value is char *: a string is actually
a... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/0240aeefb6d5fb9c0a20587ed98fa7ca_MIT6_096IAP11_lec05.pdf |
2.092/2.093 — Finite Element Analysis of Solids & Fluids I
Fall ‘09
Lecture 5 - The Finite Element Formulation
Prof. K. J. Bathe
MIT OpenCourseWare
In this system, (X, Y, Z) is the global coordinate system, and (x, y, z) is the local coordinate system for the
element i.
We want to satisfy the following equations... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/0252bdf2e6c34bca6819c5be82f3bcb0_MIT2_092F09_lec05.pdf |
uˆ =
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u1
v1
w1
. . .
uN
vN
wN
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
N is the number of nodes (3N = n) and H is the displacement interpolation matrix. For the moment, let’s
assume Su = 0. We use
�
uˆT = u1 u2 u3
. . . un
�
Then, we obtain
We also assume
ε (m) = B (m) uˆ
n×1
6×n
6×1
= H (m)u ¯ˆ
(m)u¯
... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/0252bdf2e6c34bca6819c5be82f3bcb0_MIT2_092F09_lec05.pdf |
V (m)+ Σ Σ
�
H Si(m)T f Si(m)
f
f
dSi(m)
f
�
m i Si(m)
f
uˆ is the unknown to be found. When evaluated on Sf
i(m) ,
i(m)
u¯Sf
= H Sf
i(m)
u ¯ˆ
With the transformed equation above, we can insert the following identity matrices:
H Si(m)
f
= H (m)
�
�
Sf
i(m)
�
Let u ¯ˆ T =
Then u ¯ˆ T =
Then u¯ˆ =
T
... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/0252bdf2e6c34bca6819c5be82f3bcb0_MIT2_092F09_lec05.pdf |
T = [ u1 u2 u3 ]. We want to find:
u(1)(x) = H (1) ⎢
⎡
⎡
⎤
⎤
u1
u1
⎦ ; u(2)(x) = H (2) ⎢
⎣ u2 ⎥
⎣ u2 ⎥
⎦
u3
u3
3
MIT OpenCourseWare
http://ocw.mit.edu
2.092 / 2.093 Finite Element Analysis of Solids and Fluids I
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mi... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/0252bdf2e6c34bca6819c5be82f3bcb0_MIT2_092F09_lec05.pdf |
Lecture 1
8.821/8.871 Holographic duality
Fall 2014
8.821/8.871 Holographic duality
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2014
Lecture 1
1: HINTS FOR HOLOGRAPHY
In this chapter, we will get a favor of the holographic duality. We first study gravity system and derive black hole
thermodynamics where holography p... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/0252c3933c4e097296c1f43cfbc98e37_MIT8_821S15_Lec1.pdf |
in a fixed spacetime
(2)
The two sides should be be considered as different descriptions of the same quantum system. This duality provides
a ”unification”, which has far-reaching implications for both sides of the equation. Maldacena’s original paper has
been cited by more than 10,000 times in SLAC database. But the subje... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/0252c3933c4e097296c1f43cfbc98e37_MIT8_821S15_Lec1.pdf |
itten [1].
2
Theorem 1 : A theory that allows the construction of a Lorentz-covariant conserved 4-vector current J µ cannot
´
contain massless particles of spin > 1 with non-vanishing values of the conceived charge
2
J 0d3x.
Theorem 2 : A theory that allows a conserved Lorentz-covariant stress tensor T µν cannot contai... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/0252c3933c4e097296c1f43cfbc98e37_MIT8_821S15_Lec1.pdf |
j(helicity)
ˆR(θ, k)|k, σ(cid:105) = eiσθ|k, σ(cid:105)
ˆ
ˆ
ˆ
where R(θ, k) is the rotational operator by an angle θ around k = k . More about representations of Poincare
group can be found in Ref. [2]. The conserved, Lorentz-covariant current is J µ, with the conserved charge
ˆQ =
J 0d3x; the conserved, Lorentz-covari... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/0252c3933c4e097296c1f43cfbc98e37_MIT8_821S15_Lec1.pdf |
when looking at 0-component of Eq. (7), we have (cid:104)k, σ|J 0|k(cid:48), σ(cid:105) −k−→−k→ q
.
(cid:48)
(2π)3
For massless particles, k2 = k(cid:48)2 = 0. This implies that kµkµ
such that k + k(cid:48) = 0 and kµ = (E, 0, 0, E), k(cid:48)µ = (E, 0, 0, −E). In this frame, a rotation by θ around the z-axis has
the e... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/0252c3933c4e097296c1f43cfbc98e37_MIT8_821S15_Lec1.pdf |
= Λµ
λ(θ)(cid:104)k(cid:48), j|T ρλ|k, j(cid:105)
ρ (θ)Λν
ˆ
ν only has eigenvalues e±iθ, 1, thus (cid:104)k(cid:48), j|R−1(θ)J µR(θ)|k, j(cid:105) can only be nonzero if j (cid:54) 1 . Otherwise,
2
, j|R−1(θ)T µν ˆR(θ)|k, j(cid:105) can only be nonzero if j (cid:54) 1. Otherwise, Eq. (8) is
(13)
ˆ
Note that Λµ
Eq. (7) ... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/0252c3933c4e097296c1f43cfbc98e37_MIT8_821S15_Lec1.pdf |
Consequence of Electrons as Waves on
Free Electron Model
• Boundary conditions will produce quantized energies for
all free electrons in the material
• Two electrons with same spin can not occupy same
electron energy (Pauli exclusion principle)
Imagine 1-D crystal for now
Traveling wave picture
Standing wave pic... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/028c4235617140f9b44c51728098cc4f_lecture_4.pdf |
dk dE L π h
=
2m
=
2k πh
dE
dk
1
−
2
E
2mE
h
g(E)=density of states=number of electrons per energy per length
n =
2k
N
=
L π
F =
2 2mEF
hπ
or kF =
nπ
2
•n=is the number of electrons per unit length, and is
determined by the crystal structure and valence
•The electron density, n, determines the energy ... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/028c4235617140f9b44c51728098cc4f_lecture_4.pdf |
Rb
Cs
Cu
Ag
Au
Be
Mg
Ca
Sr
Ba
Nb
Fe
Mn
Zn
Cd
Hg
3.25
3.93
4.86
5.20
5.62
2.67
3.02
3.01
1.87
2.66
3.27
3.57
3.71
3.07
2.12
2.14
2.30
2.59
2.65
eF
4.74 eV
3.24
2.12
1.85
1.59
7.00
5.49
5.53
14.3
7.08
4.69
3.93
3.64
5.32
11.1
10.9
9.47
7.47
7.13
TF
5.51 x 104 K
3.77
2.46
2.15
1.84
8.16
6.38
6.42
16.6
8.23
5.44
4.57
4.23
... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/028c4235617140f9b44c51728098cc4f_lecture_4.pdf |
2.07
2.19
2.41
2.48
2.22
2.30
2.25
2.14
13.6
12.1
10.0
9.46
11.8
11.0
11.5
12.7
Fermi energies, fermi temperatures, fermi waves vectors, and fermi velocities for representative metals*
* The table entries are calculated from the values of rs / a0 given in Table 1.1 using m = 9.11 x 10-28 grams.
1.75
1.66
1.51
1.46
1.64... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/028c4235617140f9b44c51728098cc4f_lecture_4.pdf |
e- in a metal):
cv = ⎜
⎛ ∂U
⎞
⎟
⎝ ∂T ⎠v
U ~ ΔE ⋅ ΔN ~ kbT ⋅[g(EF )⋅ kbT ] ~ g(EF )⋅(kbT )2
U=total energy of
electrons in system
⎛
cv = ⎜
⎝
U
∂
T
∂
⎞
⎟
⎠v
©1999 E.A. Fitzgerald
= 2 ⋅ g(EF ) ⋅ kb
2T
Right dependence, very close to exact derivation
9
Heat Capacity (cv) of electrons in Metal
• Rough deri... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/028c4235617140f9b44c51728098cc4f_lecture_4.pdf |
6.087 Lecture 8 – January 21, 2010
Review
Pointers
Void pointers
Function pointers
Hash table
1
Review:Pointers
• pointers: int x; int∗ p=&x;
• pointers to pointer: int x; int∗ p=&x;int∗∗ pp=&p;
• Array of pointers: char∗ names[]={"abba","u2"};
• Multidimensional arrays: int x [20][20];
1
Review: Stacks
• ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/02aef29b821a0258e53ba95a648207f9_MIT6_087IAP10_lec08.pdf |
("%d",∗p); /∗ invalid ∗/
void∗ p; int ∗px=(int∗)p; printf ("%d",∗px); /∗valid ∗/
5
Function pointers
• In some programming languages, functions are first class
variables (can be passed to functions, returned from
functions etc.).
• In C, function itself is not a variable. But it is possible to
declare pointer to... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/02aef29b821a0258e53ba95a648207f9_MIT6_087IAP10_lec08.pdf |
void ∗ pa , void ∗ pb )
{
r e t u r n ( ∗ ( i n t ∗ ) pb − ∗ ( i n t ∗ ) pa ) ;
i n ascending o r d e r ∗ /
}
/ ∗ s o r t
q s o r t ( a r r , s i z e o f ( a r r ) / s i z e o f ( i n t ) , s i z e o f ( i n t ) , asc ) ;
/ ∗ s o r t
q s o r t ( a r r , s i z e o f ( a r r ) / s i z e o f ( i n t ) , s i z e o ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/02aef29b821a0258e53ba95a648207f9_MIT6_087IAP10_lec08.pdf |
void ∗ arg )
{
s t r u c t node∗ np =( s t r u c t node ∗ ) p ;
p r i n t f ( "%d " , np−>data ) ;
}
a p p l y ( phead , p r i n t , NULL ) ;
10
Callback (cont.)
Counting nodes:
void d o t o t a l ( void ∗ p , void ∗ arg )
{
s t r u c t node∗ np =( s t r u c t node ∗ ) p ;
i n t ∗ p t o t a l
∗ p t o t a ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/02aef29b821a0258e53ba95a648207f9_MIT6_087IAP10_lec08.pdf |
drawfn f p [ 4 ] =
{& draw_square ,& draw_rec ,& d r a w _ c i r c l e ,& draw_poly } ;
void draw ( s t r u c t shape∗ ps )
{
( ∗ f p [ ps−>t y p e ] ) ( ps ) ;
/ ∗ c a l l
t h e c o r r e c t
f u n c t i o n ∗ /
}
13
6.087 Lecture 8 – January 21, 2010
Review
Pointers
Void pointers
Function pointers
Hash tab... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/02aef29b821a0258e53ba95a648207f9_MIT6_087IAP10_lec08.pdf |
MULTIPLIER 31
s t r u c t wordrec
{
char ∗ word ;
unsigned long count ;
s t r u c t wordrec ∗ n e x t ;
} ;
/ ∗ hash bucket ∗ /
s t r u c t wordrec ∗ t a b l e [ MAX_LEN ] ;
17
Hash table: example
unsigned long h a s h s t r i n g ( const char ∗ s t r )
{
unsigned long hash =0;
while ( ∗ s t r )
{
}
hash=... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/02aef29b821a0258e53ba95a648207f9_MIT6_087IAP10_lec08.pdf |
7. Rational Cherednik algebras and Hecke algebras for varieties with
group actions
7.1. Twisted differential operators. Let us recall the theory of twisted differential oper
ators (see [BB], section 2).
Let X be a smooth affine algebraic variety over C. Given a closed 2-form ω on X, the
algebra Dω(X) of differential op... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
2,cl, given by the De Rham differential acting from 1-forms to closed
plex of sheaves Ω1
X
2-forms (sitting in degrees 1 and 2, respectively). If X is projective then this space is
isomorphic to H2,0(X, C) ⊕ H1,1(X, C). We refer the reader to [BB], Section 2, for details.
ΩX
→
X
X
Remark 7.1. One can show that Dω... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
OX (Z) ⊗OX
→
We have an exact sequence of OX -modules:
0 → OX → OX (Z) → i∗N
φ
−
→
0
Thus we have a natural surjective map of OX -modules ξZ : T X → OX (Z)/OX .
7.3. The Cherednik algebra of a variety with a finite group action. We will now
generalize the definition of Ht,c(G, h) to the global case. Let X be an affi... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
fficients.
Definition 7.2. A Dunkl-Opdam operator for (X, G) is an element of Dω/t(X)r[c] given by
the formula
(7.1)
D := tLv −
· fY (x) · (1 − g),
� 2c(Y, g)
1 − λY,g
(Y,g)∈S
where λY,g is the eigenvalue of g on the conormal bundle to Y , v ∈ Γ(X, T X) is a vector
field on X, and fY ∈ OX (Z) is an element of the ... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
and G is a subgroup in GL(h). Let v be a constant
vector field, and let fY (x) = (αY , v)/αY (x), where αY ∈ h∗ is a nonzero functional vanishing
on Y . Then the operator D is just the usual Dunkl-Opdam operator Dv in the complex
reflection case (see Section 2.5). This implies that all the Dunkl-Opdam operators in the... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
(the principal symbol homomorphism).
The homomorphism ψ is clearly surjective, and our job is to show that it is injective (this
is the nontrivial part of the proof). In each degree, ψ is a morphism of finitely generated
OG -modules. Therefore, to check its injectivity, it suffices to check the injectivity on the
form... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
�
has been accomplished already. We are done.
Remark 7.9. The following remark is meant to clarify the proof of Theorem 7.7. In the case
X = h, the proof of Theorem 7.7 is based, essentially, on the (fairly nontrivial) fact that the
usual Dunkl-Opdam operators Dv commute with each other. It is therefore very import... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
by affine G-invariant open sets. Then the quotient variety X/G
exists.
For any affine open set U in X/G, let U � be the preimage of U in X. Then we can
define the algebra Ht,c,0(G, U �) as above. If U ⊂ V , we have an obvious restriction map
Ht,c,0(G, V �) Ht,c,0(G, U �). The gluing axiom is clearly satisfied. Thus the c... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
relatively prime to the characteristic.
51
�
(2) The construction and main properties of the (sheaves of) Cherednik algebras of alge
braic varieties can be extended without significant changes to the case when X is a
complex analytic manifold, and G is not necessarily finite but acts properly discon
tinuously. In t... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
1 = z, z2, . . . , zd near y. A
Dunkl-Opdam operator near y for the vector field ∂z can be written in the form
�
1 n−1 2c(Y, gm) m
(
z
m=1
Conjugating this operator by the formal expression zη(Y ) := (zm)η(Y )/m, we get
1 − λm (g − 1) + η(Y )).
∂
∂z
D =
+
Y,g
∂
z η(Y ) ◦ D ◦ z−η(Y ) =
∂
∂z
+
n−1
1 � 2c(Y,... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
such Y will be called a reflection hypersurface). For (Y, g) ∈ S, let GY be the subgroup of
G whose elements act trivially on Y . This group is obviously cyclic; let nY = |GY |. Let CY
be the conjugacy class in BG corresponding to a small circle going counterclockwise around
the image of Y in X/G, and TY be a represe... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
we will sometimes drop x0 form the notation.
The main result of this section is the following theorem.
Theorem 7.15. Assume that H2(X, C) = 0. Then A = Hτ (G, X) is a flat formal defor
mation of A0, which means A = A0[[τ ]] as a module over C[[τ ]].
Example 7.16. Let h be a finite dimensional vector space, and G be a... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
C+ be a complex number
with a positive imaginary part, and G�� = G � (Q∨ ⊕ ηQ∨) be the double affine Weyl group.
Then Hτ (h, G��) is (one of the versions of) the double affine Hecke algebra of Cherednik ([Ch]),
and it is flat by Theorem 7.15. The fact that this algebra is flat was proved by Cherednik,
Sahi, Noumi, Stokma... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
, k = 1, . . . , nj . Define the Hecke algebra Hτ (Σ)
of Σ to be generated over C[[τ ]] by the same generators al, bl, cj with defining relations
l
nj
�
(cj − e 2πji/nj
e τkj ) = 0,
c1c2 · · ·
cm =
�
albla−1b−1 .
l
l
k=1
l
Thus Hτ (Σ) is a deformation of C[Γ].
This deformation is flat if H is a Euclidean plane ... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
see that Hτ (Σ) fails to be flat in the following “forbidden” cases:
g = 0, m = 2, (n1, n2) = (n, n);
m = 3, (n1, n2, n3) = (2, 2, n), (2, 3, 3), (2, 3, 4), (2, 3, 5).
Indeed, the orbifold Euler characteristic of a closed surface Σ of genus g with m special
points x1, . . . , xm whose orders are n1, . . . , nm is
χ... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
1
→
π1(X, x)
→
orb (X/G, x) G
→ →
π1
1.
54
�
�
7.8. Hecke algebras of wallpaper groups and del Pezzo surfaces. The case when H
is the Euclidean plane (i.e., Γ is a wallpaper group) deserves special attention. If there are
elliptic elements, this reduces to the following configurations: g = 0 and
m = 3, (n1, n2... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
Γ, H) be the projector to an eigenspace of c.
Consider the “spherical” subalgebra Bτ (Γ, H) := eHτ (Γ, H)e.
�
·
Theorem 7.19 (Etingof, Oblomkov, Rains, [EOR]).
(i) If � = 0 then the algebra Bτ (Γ, H)
is commutative, and its spectrum is an affine del Pezzo surface. More precisely, in
the case (2, 2, 2, 2), it is a ... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
H N with the action of ΓN = SN �ΓN .
If H is a Euclidean or Lobachevsky plane, then by Theorem 7.15 Hτ (ΓN , X N ) is a flat
deformation of the group algebra C[ΓN ]. If N > 1, this algebra has one more essential
parameter than for N = 1 (corresponding to reflections in SN ). In the Euclidean case, one
expects that an... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
let M be a module over H1,c,η,0,G,X which is a locally free coherent sheaf when
restricted to X �/G. Then the restriction of M to X �/G is a G-equivariant D-module on
X � which is coherent and locally free as an O-module. Thus, M corresponds to a locally
constant sheaf (local system) on X �/G, which gives rise to a ... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
Y
− η(Y ))/nY .
Proposition 7.22. The functor KZ maps the category Cc,η to the category of representations
of the algebra Hτ (c,η)(G, X).
Proof. The result follows from the corresponding result in the linear case (which we have
already proved) by restricting M to the union of G-translates of a neighborhood of a ge... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
o
sition 7.22 and it deforms flatly the module regG. This implies Hτ (c,η)(G, X) is flat over
C[[τ ]].
Remark 7.23. When X is not simply connected, the theorem is still true under the as
sumption π2(X) ⊗ C = 0 (i.e. H2( �
X� is the universal cover of X), and the
proof is contained in [E1].
X, C) = 0, where
56
7.... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
1 double affine Hecke algebra. If we set three of the four Ti’s
2 = 1, we get the double affine Hecke algebra of type
satisfying the undeformed relation Ti
A1. More precisely, this algebra is generated by T1, . . . , T4 with relations
T2
2 = T3
2 = T4
2 = 1,
(T1 − t)(T1 + t−1) = 0, T1T2T3T4 = q.
Another presentation ... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
2 = 1.
The Hecke algebra of the partial braid group is then defined to be the group algebra of B
plus an extra relation: (T − q1)(T + q2) = 0.
A common way to present this Hecke algebra is to renormalize the generators so that one
has the following relations:
T XT = X −1, T −1Y T −1 = Y −1, Y −1X −1Y XT 2 = q, (T −... | https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/02ec876b48928aad62c32c782ce96699_MIT18_735F09_ch07.pdf |
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