text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
• Uncertainty
• Emergence
• Various definitions have been proposed
What are the relationships, especially trade-offs,
between forms, functions, ilities, performance and
these characteristics?
Adv Sys Arch intro
8/24/2006
© Daniel E Whitney
26
Other Words That We Will Use and
Need to Understand
• Element, module, co... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/31440ed9b876651c3287b430ba75f77e_lec1.pdf |
2.997 Decision-Making in Large-Scale Systems
MIT, Spring 2004
March 8
Handout #13
Lecture Note 10
1 Value Function Approximation
DP problems are centered around the cost-to-go function J ⁄ or the Q-factor Q⁄. In certain problems, such as
linear-quadratic-Gaussian systems, J ⁄ exhibits some structure which allows for it... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
position (i; j) of the board is fllled, and b(i; j) = 0 otherwise.
If there are p difierent types of pieces, and the board has dimension n £ m, the number of states is on the
order of p £ 2n£m, which grows exponentially with n and m.
Since exact solution of large-scale of MDPs is intractable, we consider approximate solu... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
)
(cid:4)(cid:1)(cid:4)(cid:1)(cid:4)
(cid:6)(cid:1)(cid:6)(cid:1)(cid:6)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)
(cid:3)(cid:1)(cid:3)(cid:1)(cid:3)(cid:1)(cid:3)
(cid:5)(cid:1)(cid:5)(cid:1)(cid:5)(cid:1)(cid:5)
(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)
(cid:4)(cid:1)(cid:4)(cid:1)(cid:4)
(cid:6)(ci... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
Cost-to-go Function Approximation
Another approach to approximating the dynamic programming solution is to approximate the cost-to-go
function. The underlying idea for cost-to-go function approximation is that J ⁄ has some structure that
allows for approximate compact representation
J ⁄(x) … ~J(x; r);
for some paramete... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
, such that yj =
i=1 rij „xi. The vector y is then used as the input to a sigmoidal layer, which
outputs a vector z 2 <m with the property that zi = f (yi), and f (:) is a sigmoidal function. A sigmoidal
function is any function with the following properties:
P
1. monotonically increasing
2. difierentiable
3. bounded
Fi... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
all set of weights, but flnding the global optimum may be a di–cult
problem.
2.2 State Space Partitioning
Another common choice for approximation architecture is based on partitioning of the state space. The
underlying idea is that \similar" states may be grouped together. For instance, in an MDP involving
continuous st... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
130
RICHARD B. MELROSE
18. Solutions to (some of) the problems
Solution 18.1 (To Problem 10). (by Matjaˇz Konvalinka).
Since the topology on N, inherited from R, is discrete, a set is com
C)→
pact if and only if it is finite. If a sequence {xn} (i.e. a function N
is in C0(N) if and only if for any � > 0 there exist... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
0, the mapping
�
|
∞
=1n
n=1
xnyn| ≤ ∞
=1n
�
|xn||yn|
≤ �x�0 �y�1) by
Φ : l1
�−→ c∗
0
defined by
�
�→ y
x
�→
�
∞
�
xnyn
n=1
is a (linear) welldefined mapping with norm at most 1. In fact, Φ is
an isometry because if |xj| = �x�0 then Φ(x)(ej)| = 1 where ej is
the jth unit vector. We claim that Φ is also su... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
ϕ(0)) = −iδ(ϕ),
we get DxH = Cδ for C = −i.
0
LECTURE NOTES FOR 18.155, FALL 2004
131
Solution 18.3 (To Problem 40). (Matjaˇz Konvalinka) Let us prove this
in the case where n = 1. Define (for b = 0)
U (x) = u(b) − u(x) − (b − x)u�(x) − . . . −
(b − x)k−1
(k − 1)!
u
(k−1)(x);
then
U �(x) =
(b − x)k−1
(k − 1)... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
2
x + . . . +
u(k−1)(0)
(k − 1)!
x k−1 +
u(k)(0)
k!
k
x ,
v(x) = u(x) − p(x) =
u(k)(ζ) − u(k)(0)
k!
k
x
for ζ between 0 and x, and since u(k) is continuous, (u(x) − p(x))/xk
tends to 0 as x tends to 0.
The proof for general n is not much more difficult. Define the func
R by wx(t) = u(tx). Then wx is ktime... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
wx (0)
k!|x|k
u(x) − p(x)
|k
|
x
=
=
,
(k)
(k)
�
�
�
�
∂x l1
∂lu
1 ∂x l2
2 · · ·
li
∂x i
(ζxx) −
∂lu
1 ∂x l2
2 · · ·
∂x l1
li
∂x i
(0)
�
�
�
�
with l1 + . . . + li = k and 0 < ζx < 1. This tends to zero as x →
because the derivative is continuous.
0
Solution 18.4 (Solution to Problem 41). (Matjˇz Kon... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
for
|
|
x < �/M.
|
|
Solution 18.5. (partly Matjaˇz Konvalinka)
For any ϕ ∈ S(R)
∞
�
|
−∞
ϕ(x)dx| ≤
|
ϕ(x) dx ≤ sup((1+x
|
we have
�
∞
−∞
�
�
∞
2
|
|
) ϕ(x) )
|
(1+
x|
|
2)−1dx
−∞
2
|
≤ C sup((1 + x ) ϕ(x) ).
|
|
Thus S(R) � ϕ
�−→ R ϕdx is continous.
�
Now, choose φ ∈ C∞(R) with R φ(x)dx = 1. Then, for ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
φ(t) ≤ Ckt−k−1 in t ≥ 1 it follows from (18.2) that
x kAψ(x) ≤ Cx k
|
|
∞
t−k−1dt ≤ C �, k > 1, in x > 1.
�
x
A similar estimate as x → −∞ follows from (18.1). Now, A is clearly
linear, and it follows from the estimates above, including that on the
integral, that for any k there exists C and j such that
sup x α... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
)m|u|
�
2 dξ =
But that is true since
�
Rn
�
=
Rn
�ξ�2m�
⎛
⎝
�
|α|≤m
Cαξ2α
�
Rn
⎞
⎠ |�|2 dξ =
u
�
|α|≤m
Cα
��
Rn
�
�ξ�2m� ξ2α u
|�|2 dξ
u = ξ�m� Dαu is in L2(Rn) (note that u ∈ H m(Rn)
�
� (Rn), α ≤ m). The converse is also true since
and since �ξ�m� ξα
�
follows from Dαu ∈ H m
Cα in the formula ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
n + 1/|xj
�1/2 ≤ (2n)1/2 ,
2
|
134
RICHARD B. MELROSE
n xjwj for wj ∈ L2(Rn).
j=1
= xjwj for wj ∈ L2(Rn). But that means that �x�v = w0 +
� x�vj
so �
If u is in L2(Rn) then �u ∈ L2(Rn), and so there exist w0, . . . , wn ∈
L2(Rn) so that
�
∞
0
ϕ�(x) dx = i(0−ϕ(0)) =
−
iδ(ϕ),
ξ�
�
u� = w0 +
n
�
ξjwj,
j=1
in ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
ξ�
2m has a finite integral over
this is equivalent to finding m such that �
Rn . One option is to write �ξ� = (1 + r2)1/2 in spherical coordinates,
and to recall that the Jacobian of spherical coordinates in n dimensions
has the form rn−1Ψ(ϕ1, . . . , ϕn−1), and so �ξ�2m is integrable if and only
if
rn−1
(1 + r2)m... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
1}
{ | |
|
|
i
LECTURE NOTES FOR 18.155, FALL 2004
135
is in H m for any m < 1 + n/2 so, by the Sobolev embedding theorem,
each vi ∈ C0 (Rn) and
0
(18.4)
1 = vˆ0
n
�
n+1
ξi
i=1
v�i =
⇒
δ = v0 +
�
n+1
Di
vi.
i
How to see that this cannot be done with n or less derivatives? For
the moment I do not have a... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.014 Calculus with Theory
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-014-calculus-with-theory-fall-2010/316f0420829f20de483bbc975159dea1_MIT18_014F10_ChBnotes.pdf |
The Challenges of Delivering
The Challenges of Delivering
Content on the Internet
Content on the Internet
Tom Leighton
Tom Leighton
Chief Scientist
Chief Scientist
Akamai Technologies
Akamai Technologies
Outline
Outline
How the Web Works
How the Web Works
Services
Akamai’s Services
Akamai’s
Technology Overview
Techn... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
bones and long distances
Unreliable performance
•• Unreliable performance
Content may be blocked by congestion or backbone
-- Content may be blocked by congestion or backbone
peering problems
peering problems
Not scalable
•• Not scalable
Usage limited by bandwidth available at master site
-- Usage limited by bandwidt... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
N
7
1
y
a
M
n
o
o
N
8
1
y
a
M
n
o
o
N
9
1
y
a
M
n
o
o
N
0
2
y
a
M
n
o
o
N
1
2
y
a
M
n
o
o
N
2
2
y
a
M
n
o
o
N
3
2
y
a
M
n
o
o
N
4
2
y
a
M
n
o
o
N
5
2
y
a
M
n
o
o
N
6
2
y
a
M
n
o
o
N
7
2
y
a
M
n
o
o
N
Web object delivered without Akamai
Web object delivered by Akamai
... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
Live or On
Streaming Video
Streaming Video
Speaker Support
Speaker Support
e.g. PowerPoint
e.g. PowerPoint
Other Features:
Other Features:
Ask a Question
•• Ask a Question
Live Audience
•• Live Audience
Phone--inin
Phone
Viewer
•• Viewer
Registration
Registration
•• EE--mail mail
promotion
promotion
Download
•• D... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
Suite
personalized content at Akamai’s
personalized content at
servers
servers
edge
Akamai’s edge
Outline
Outline
How the Web Works
How the Web Works
Akamai’ Services
Akamai’ Services
Technology Overview
Technology Overview
Technological Challenges
Technological Challenges
The Future
The Future
Downloading www.xy... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
Akamai
•• Browser requests HTML
Browser requests HTML
•• Akamai server assembles
Akamai server assembles
page, contacting customer
page, contacting customer
Web server if necessary
Web server if necessary
Customer
Web server
44
HTML
Akamaized HTML
•• Optimal Akamai
Optimal Akamai server
server
returns Akamaized ... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
Every few seconds
-- Every few seconds
for LLDNS
for LLDNS
Time To Live
Root
HLDNS
LLDNS
1 day
30 min.
30 sec.
TTL of DNS responses gets shorter
further down the hierarchy
Page Assembly
Page Assembly
Container Page
[TTL=5d]
Site owners create
container pages that
can be populated
with varying content
[TTL=8h]
[X... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
Edge
to the Source
to the Source
End User
X
Source Server
•• Maintain path performance data so that the
Maintain path performance data so that the
optimal path can be used to reach optimal
optimal path can be used to reach optimal
Akarouting))
customer location (Akarouting
customer location (
Connecting from the ... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
widespread and
unpredictable
unpredictable
--
•• Must load balance widely varying kinds of traffic, optimize
Must load balance widely varying kinds of traffic, optimize
multiple kinds of resources, and minimize various costs
multiple kinds of resources, and minimize various costs
Must tolerate large numbers of compo... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
on connections to
improve performance
improve performance
Support for interactive and personalized
•• Support for interactive and personalized
messaging; e.g., Q&A
messaging; e.g., Q&A
time data aggregation for polling, etc.
•• RealReal--time data aggregation for polling, etc.
Synchronized delivery of audio, video, ... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
L1: 6.111 Course Overview
L1: 6.111 Course Overview
Acknowledgements:
Materials in this lecture are courtesy of the following sources and are used with
permission.
Rex Min
J. Rabaey, A. Chandrakasan, B. Nikolic. Digital Integrated Circuits: A Design Perspective.
Prentice Hall/Pearson, 2003.
L1: 6.111 Spring 2006
Intro... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
:134) Must have basic background in circuit theory
(cid:134) Some basic material might be a review for those who have taken 6.004
L1: 6.111 Spring 2006
Introductory Digital Systems Laboratory
3
Overview of Labs
Overview of Labs
(cid:132) Lab 1: Basics of Digital Logic (Discrete Devices)
(cid:134) Learn about lab equip... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
(cid:134) Approximate breakdown:
(cid:122) Quiz
(cid:122) 3 Problem Sets
(cid:122) 4 Lab exercises
(cid:129) Lab 1
(cid:129) Lab 2
(cid:129) Lab 3
(cid:129) Lab 4
(cid:122) Writing (Lab 2 revision- part of CIM requirement)
(cid:122) Participation (lecture, recitation, project presentations)
(cid:122) Final Project ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
132) 1854: George Boole shows that logic is math, not just
philosophy!
(cid:132) Boolean algebra: the mathematics of binary values
L1: 6.111 Spring 2006
Introductory Digital Systems Laboratory
8
Key Link Between Logic and Circuits
Key Link Between Logic and Circuits
0
1
0
1
0
+
1
The
Vacuum
Tube
Lee de Forest, 1906... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
132) Programmable Logic
L1: 6.111 Spring 2006
Introductory Digital Systems Laboratory
11
Building Digital Systems with HDLsHDLs
Building Digital Systems with
(cid:132) Logic synthesis using a Hardware Description Language (HDL)
automates the most tedious and error-prone aspects of design
Problem Statement
(cid:132) ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
flow Level
(cid:134) The flow of data through components is specified based on the idea of how
data is processed
(cid:132) Gate Level
(cid:134) Specified as wiring between logic gates
(cid:134) Not practical for large examples
(cid:132) Switch Level
(cid:134) Description in terms of switching (modeling a transistor)
... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
buses, lights)
(cid:132) Digital processing systems consist of a datapath, memory, and control.
Early machines for arithmetic had insufficient memory, and often
depended on users for control
(cid:132) Today’s digital systems are increasingly embedded into everyday places
and things
(cid:132) Richer interaction with t... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
VOL = f (VOH)
VM = f (VM)
V OL
V
OH
V(x)
Nominal Voltage Levels
L1: 6.111 Spring 2006
Introductory Digital Systems Laboratory
20
Example Noise Sources in Digital Circuits
Example Noise Sources in Digital Circuits
v(t)
VDD
Capacitive coupling
Power and ground
noise
(cid:132) Noise sources: coupling, cross talk, supply ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
to 11:45 PM
(cid:122) Friday – 9:00 AM to 5:15 PM
(cid:122) Saturday – CLOSED
(cid:122) Sunday – noon to 11:45 PM
(cid:132) Please do not move or reconfigure computers and other lab equipment
(logic analyzers, scopes, power supplies, etc.). Please turn off the
power switch for the labkit when you are done for the day... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
Determinants 2. Area and Volume
Area and volume interpretation of the determinant:
(1)
±
�
� a
1 a
2
�
b2
b1
�
�
�
�
�
= area of parallelogram with edges A = (a1, a2), B = (b1, b2).
B
θ
A
(2)
�
�
� a1 a2 a3 �
�
�
b3 � = volume of parallelepiped with edges row-vectors A, B, C.
± � b1
b2
�
�
c3 �... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/317eb93dbd9e833fbe51dfc6b2d3e034_MIT18_02SC_MNotes_d2.pdf |
1
′ ,
by the above observations
b
1
B
b
2
by the geometric definition of dot product
by the formula for B ′
This proves the area interpretation (1) if A and B have the position shown. If their positions
are reversed, then the area is the same, but the sign of the determinant is changed, so the
formula has to ... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/317eb93dbd9e833fbe51dfc6b2d3e034_MIT18_02SC_MNotes_d2.pdf |
D-Lab
Spring 2010
1 Today in class:
• Review of the Design Process
• Design for Manufacture
– No Spare Parts
• Books Assignment
• Readings
2
D-Lab
Design for Manufacture
3 DfM Definition:
Adapting a design to make it more
easily manufactured and to reduce
its manufacturing costs.
4 DfM Definition:
To g... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/31bc1da7c60661b5848601aa2f9eba07_MITEC_720JS10_lec12.pdf |
your detailed design
19 Design for Assembly
"a process for improving product
design for easy and low-cost
assembly, focusing on functionality
and on assemblability
concurrently.”
--Vincent Chan & Filippo A. Salustri
20 Design for Assembly
• Reduce cost of assembly
• Improve quality and reliability
• Reduce ... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/31bc1da7c60661b5848601aa2f9eba07_MITEC_720JS10_lec12.pdf |
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/31bc1da7c60661b5848601aa2f9eba07_MITEC_720JS10_lec12.pdf |
III. The Scaling Hypothesis
III.A The Homogeneity Assumption
In the previous chapters, the singular behavior in the vicinity of a continuous transi
tion was characterized by a set of critical exponents {α, β, γ, δ, ν, η, · · ·}. The saddle–point
estimates of these exponents were found to be unreliable due to the imp... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
The singularities in the free energy can in fact be described by a single homogeneous
function† in t and h, as
f (t, h) = |t|2 gf
h/|t|Δ
.
(III.2)
The function gf only depends on the combination x ≡ h/|t|Δ, where Δ is known as the gap
exponent. The asymptotic behavior of gf is easily obtained by comparing eqs.(II... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
other thermodynamic quantity) retains
the homogeneous form
fsing.(t, h) = |t|2−α gf
h/|t|Δ
.
(III.4)
The actual exponents α and Δ depend on the critical point being considered. The depen
(cid:0)
(cid:1)
dence on t is chosen to reproduce the heat capacity singularity at h = 0. The singular part
of the energy is... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
t, h) = |t|−α±g±
h/|t|Δ±
,
(III.7)
with different functions and exponents for t > 0 and t < 0, that match at t = 0. However,
(cid:0)
(cid:1)
this is ruled out by the condition that the free energy is analytic everywhere except on the
coexistence line for h = 0 and t < 0, as proven as follows: Consider a point at... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
leads to
C±
t ≪ hΔ
= A±h−α±/Δ± + B±h−(1+α±)/Δ± |t| + · · · .
(III.10)
(cid:0)
(cid:1)
Continuity at t = 0 now forces α+/Δ+ = α−/Δ−, and (1 + α+)/Δ+ = (1 + α−)/Δ−,
which in turn implies
α+ = α− ≡ α
(
Δ+ = Δ− ≡ Δ
.
(III.11)
Despite using |t| in the postulated scaling form, we can still ensure the analyticity... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
� , =⇒
β = 2 − α − Δ.
(III.13)
On the other hand, if x → ∞, gm(x) ∼ xp, and
m(t = 0, h) ∼ |t|2−α−Δ
h
|Δ
|t
(cid:18)
p
.
(cid:19)
(III.14)
Since this limit is independent of t, we must have pΔ = 2 − α − Δ. Hence
m(t, h = 0) ∼ h(2−α−Δ)/Δ , =⇒
δ = Δ/(2 − α − Δ) = Δ/β.
(III.15)
• Similarly, the susceptibilit... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
of exponent identities. For example,
eqs.(III.13), (III.15), and (III.16) imply
α + 2β + γ = α + 2(2 − α − Δ) + (2Δ − 2 + α) = 2
γ
β
2Δ − 2 + α
2 − α − Δ
Δ
2 − α − Δ
δ − 1 =
− 1 =
=
(Rushbrooke ′ s Identity),
(Widom ′ s Identity).
(III.17)
These identities can be checked against the following table of crit... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
from, diverging response functions. In order to obtain an identity involving
the exponent ν for the divergence of the correlation length, we replace the homogeneity
assumption for the free energy, with the following two conditions:
(1) The correlation length ξ has a homogeneous form,
ξ(t, h) ∼ |t|−ν g
h/|t|Δ
.
(I... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
units
of the size of the correlation length. Each unit is then regarded as an independent random
variable, contributing a constant factor to the critical free energy. The number of units
grows as (L/ξ)d, leading to eq.(III.19).
The consequences of the above assumptions are:
(1) The homogeneity of fsing.(t, h) emer... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
Thus all correlations decay as a power of the separation. As
discussed in the previous chapter, the magnetization correlations fall off as
Gc
m,m
(x) ≡ hm(x)m(0)i − hmi2 ∼ 1/|x|d−2+η .
(III.22)
Similarly, we can define an exponent η ′ for the decay of energy–energy correlations as
E,E(x) = hH(x)H(0)i − hHi2 ∼ 1/|x|... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
additional dilation symmetry. Under a change of scale, the critical correlation functions
behave as
Gcritical(λx) = λpGcritical(x).
(III.26)
This implies a scale invariance or self–similarity:
if a snapshot of the critical system
is blown up by a factor of λ, apart from a change of contrast (multiplication by λp)... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/31f360cf7db5b66068eacc5240c17aeb_MIT8_334S14_Lec6.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
2.161 Signal Processing: Continuous and Discrete
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Massachusetts Institute of Technology
Department of Mechanical Engineering
2.161 Signal Processing - Continuous and Di... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
Low Frequency Behavior: There are a pair of zeros at the origin so that
lim |H(jΩ)| = 0
Ω→0
and the low frequency asymptotic slope is +40dB/decade.
Mid Frequency Behavior: The response in the region Ω ≈ a0 is determined by
the systems damping ratio ζ, and will exhibit a resonant peak if ζ < 0.707.
√
(cid:2) (cid... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
(jΩ)| = 1.
Ω→∞
Low Frequency Behavior: There are no zeros at the origin and
lim |H(jΩ)| = 1
Ω→0
Mid Frequency Behavior: There are a pair of imaginary zeros at s = ±j a0 forcing
√
√
the response magnitude to zero at a frequency Ω = a0.
√
|H(j a0)| = 0
which defines the band rejection (notch) center frequency.
(cid... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
)
(cid:5)
(cid:3)
(cid:11)
(cid:21)
(cid:22)
(cid:14)
(cid:17)
(cid:5)
(cid:8)
(cid:2)
(cid:11)
(cid:10)
(cid:11)
(cid:3)
(cid:8)
(cid:2)
(cid:18)
(cid:3)
(cid:11)
(cid:13)
(cid:6)
(cid:14)
(cid:17)
(cid:5)
(cid:8)
(cid:2)
(cid:10)
(cid:11)
(cid:11)
(cid:10)
(cid:3)
(cid:2)
(cid:10)
(cid:3)
(cid:11)
(cid:13)
(cid:6)
(c... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
low-
pass filter. For practical filters, however, the “skirts” of the pass-bands will be a warped
representation of the low-pass prototype filter. This does not usually cause problems.
Example 1
Transform the first-order low-pass filter
Hlp(s) =
Ωc
s + Ωc
to a high-pass filter Hhp(s).
Using the transformation g(s) = Ω... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
2 + ΔΩs + Ω2
s
o
Example 4
Design a second-order band-stop filter with center frequency Ωo and notch-width
ΔΩ.
Step 1: Design a first-order prototype low-pass filter with cut-off frequency
ΔΩ:
Hlp(s) =
ΔΩ
s + ΔΩ
Step 2: Transform the prototype using
g(s) =
sΔ2
Ω
s2 + Ω2
o
so that
H(s) = �
ΔΩ
�
sΔ2
Ω + ΔΩ
2... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
(cid:14)
(cid:6)
(cid:3)
The design of each low-order block can be handled independently.
The state-variable filter design method is based on the block diagram representation used
in the so-called phase-variable description of linear systems that uses the outputs of a chain
of cascaded integrators as state variabl... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
(cid:2)
(cid:5)
(cid:2)
(cid:18)
The first equation may be rewritten explicitly in terms of the highest derivative
d2x
dt2
= −a1 − a0x + u.
dx
dt
Consider a pair of cascaded analog integrators with the output defined as x(t) so that the
derivatives of x(t) appear as inputs to the integrators:
(cid:12)
(cid:24) ... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
(cid:26)
(cid:12)
(cid:12)
(cid:24) (cid:17) (cid:17) (cid:22)
(cid:12)
(cid:24) (cid:23)
(cid:16) (cid:8) (cid:23) (cid:9)
(cid:18)
(cid:14)
(cid:27) (cid:8) (cid:23) (cid:9)
(cid:18)
(cid:18)
(cid:26)
(cid:5)
(cid:5)
(cid:24) (cid:22)
(cid:24) (cid:23)
(cid:7)
(cid:5)
(cid:18)
(cid:26)
(cid:22) (cid... | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
)sX(s)s X(s)2aa10U(s)+--a00++Y (s) (low -pass)Y (s) (band-pass)Y (s) (high-pass)Y (s) (band-stop)1234a1a | https://ocw.mit.edu/courses/2-161-signal-processing-continuous-and-discrete-fall-2008/32157b1c3d282eb849fbda43d513c5e3_lecture_08.pdf |
Introduction to Engineering
Introduction to Engineering
Systems, ESD.00
Lecture 1
Lecturers:
Professor Joseph Sussman
Dr Afreen Siddiqi
Dr. Afreen Siddiqi
TA: Regina Clewlow
Motivation I
Motivation I
Society faces many large-scale problems-- we
call them critical contemporary issues (CCIs)
They aren’t simply ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/321cd9d9f5c380b76f1939b678e8b9c9_MITESD_00S11_lec01.pdf |
picking up a
tangled skein of wool; all the threads are
interwoven - recreation and pollution and
interwoven recreation and pollution and
mental health. and the crime rate, and rapid
transit and the war on poverty, and
parks....everything leads to something else."
-
Lady Bird Johnson
Lady Bird Johnson s 's Obi... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/321cd9d9f5c380b76f1939b678e8b9c9_MITESD_00S11_lec01.pdf |
I
Systems Parallel to our CSS I
Our CSS has various
parallel systems. These
parallel systems These
might include:
the air quality in
Boston/Cambridge,
MIT itself, whose behavior
will affect the
Boston/Cambridge
transportation system:
transportation system:
when MIT is on spring
break, the loads on th... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/321cd9d9f5c380b76f1939b678e8b9c9_MITESD_00S11_lec01.pdf |
make a choice of what
mode to use: driving, taking public
transportation of one sort or another,
traveling intermodally (bike to the station
and take the train from there)
The Micro Macro Question
The Micro-Macro Question
In our CSS,, do we need to know how an
internal combustion engine works?
Probably not. D... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/321cd9d9f5c380b76f1939b678e8b9c9_MITESD_00S11_lec01.pdf |
Partial derivatives
Partial derivatives
Let w = f (x, y) be a function of two variables. Its graph is a surface in xyz-space, as
pictured.
Fix a value y = y0 and just let x vary. You get a function of one variable,
(1)
w = f (x, y0),
the partial function for y = y0.
Its graph is a curve in the vertical plane y ... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/3299372c0b77500ada4ea558fea4db80_MIT18_02SC_MNotes_ta1.pdf |
is common in science
and engineering, where you are just dealing with relations between variables and don’t
mention the function explicitly; the third and fourth indicate the point by just using a
single subscript.
Analogously, fixing x = x0 and letting y vary, we get the partial function w = f (x0, y),
whose graph... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/3299372c0b77500ada4ea558fea4db80_MIT18_02SC_MNotes_ta1.pdf |
respect to x, for instance, hold all the other variables constant and take the
ordinary derivative with respect to x; the notations are the same as above:
d
dx
f (x, y0, z0, . . . ) = f x(x0, y0, z0, . . . ),
∂f
∂x
�
0
,
�
�
1
∂w
∂x
.
0
�
MIT... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/3299372c0b77500ada4ea558fea4db80_MIT18_02SC_MNotes_ta1.pdf |
6.172
Performance
Engineering
of Software
Systems
LECTURE 14
Caching and Cache-
Efficient Algorithms
Julian Shun
© 2008-2018 by the MIT 6.172 Lecturers
1
!"##$*
%&'&(!"#)*+)$#)*+,*-./01*
!"##$*
%&'&(!"#)*+)$#)*+,*-./01*
CACHE HARDWARE
© 2008-2018 by the MIT 6.172 Lecturers
2
Multicore Cache Hierarchy
DRAM... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
block must be evicted to make room for a new block.
The replacement policy determines which block to evict.
© 2008-2018 by the MIT 6.172 Lecturers
4
Direct-Mapped Cache
w-bit
address
space
0x0000
0x0004
0x0008
0x000C
0x0010
0x0014
0x0018
0x... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
address
tag
w – lg(M/k)
bits
set
lg(M/kB)
offset
lg B
© 2008-2018 by the MIT 6.172 Lecturers
6
To find a block in the
cache, only the k
locations of its set
must be searched.
Taxonomy of Cache Misses
Cold miss
∙ The first time the cache block is accessed.
Capacity miss
∙ The previous cached copy would h... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
*
IDEAL-CACHE MODEL
© 2008-2018 by the MIT 6.172 Lecturers
9
Ideal-Cache Model
Parameters
! Two-level hierarchy.
! Cache size of M
bytes.
! Cache-line length
of B bytes.
! Fully associative.
! Optimal, omniscient
replacement.
memory
cache
P
M/B
cache lines
B
Performance Measures
! work W (ordinary running ti... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
29) (cid:3564)(cid:16)
(cid:30)
(cid:16)
(cid:85)(cid:75)
(cid:75)
(cid:19)
si
B
B
B
(cid:3074)
(cid:30)
B
(cid:16)(cid:3553)
B
© 2008-2018 by the MIT 6.172 Lecturers
(cid:20)
(cid:48)
(cid:48)
(cid:16)(cid:3553)
(cid:48)
(cid:16)(cid:3553)
(cid:21)(cid:48)
(cid:16)(cid:3553)
(cid:16)(cid:3553) (cid:15)
(cid:30)
(cid:... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
to read all
A’s elements is at most 3n2/B.
Proof. We have N = n2, n = r = si, B ≤ n = N/r, and N
< M /3. Thus, the Cache-Miss Lemma applies. ∎
© 2008-2018 by the MIT 6.172 Lecturers
15
!"##$*
%&'&(!"#)*+)$#)*+,*-./01*
CACHE ANALYSIS OF
MATRIX MULTIPLICATIO... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
< n; i++)
for (int64_t j=0; j < n; j++)
for (int64_t k=0; k < n; k++)
C[i*n+j] += A[i*n+k] * B[k*n+j];
}
Assume row major and tall cache
A
© 2008-2018 by the MIT 6.172 Lecturers
B
19
Case 2
c’M1/2< n < cM/B.
Analyze matrix B.
Assume LRU.
Q(n) = n·!(n2/B) =
!(n3/B), since
matrix B can exploit
spatial local... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
exploit
spatial locality.
!"##$*
%&'&(!"#)*+)$#)*+,*-./01*
TILING
© 2008-2018 by the MIT 6.172 Lecturers
22
Tiled Matrix Multiplication
void Tiled_Mult(double *C, double *A, double *B, int64_t n) {
for (int64_t i1=0; i1<n/s; i1+=s)
for (int64_t j1=0; j1<n/s; j1+=s)
for (int64_t k1=0; k1<n/s; k1+=s)
for (int64... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
2).
! Submatrix Caching Lemma implies
"(s2/B) misses per submatrix.
"((n/s)3(s2/B))
Q(n) =
= "(n3/(BM1/2)).
!
Remember
this!
© 2008-2018 by the MIT 6.172 Lecturers
n
! Optimal [HK81].
24
Tiled Matrix Multiplication
void Tiled_Mult(double *C, double *A, double *B, int64_t n) {
for (int64_t i1=0; i1<n; i1+=s)... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
8-2018 by the MIT 6.172 Lecturers
26
Two-Level Cache
n
s
t
t
s
void Tiled_Mult2(double *C, double *A, double *B, int64_t n) {
for (int64_t i2=0; i2<n; i2+=s)
for (int64_t j2=0; j2<n; j2+=s)
for (int64_t k2=0; k2<n; k2+=s)
for (int64_t i1=i2; i1<i2+s && i1<n; i1+=t)
for (int64_t j1=j2; ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
172 Lecturers
29
Recursive Matrix Multiplication
Divide-and-conquer on n × n matrices.
C11
C12
A11
A12
B11
B12
C21
C22
=
A21
A22
×
B21
B22
A11B11
A11B12
A12B21
A12B22
=
A21B11
A21B12
+
A22B21
A22B22
8 multiply-adds of (n/2) × (n/2) matrices.
© 2008-2018 by the MIT 6.172 Lecturers
30
Recursive ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
2008-2018 by the MIT 6.172 Lecturers
31
Recursive Code
// Assume that n is an exact power of 2.
void Rec_Mult(double *C, double *A, double *B,
int64_t n, int64_t rowsize) {
if (n == 1)
C[0] += A[0] * B[0];
else {
int64_t d11 = 0;
int64_t d12 = n/2;
int64_t d21 = (n/2) * rowsize;
int64_t d22 = (n/2) * (rowsi... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
64_t d22 = (n/2) * (rowsize+1);
Rec_Mult(C+d11, A+d11, B+d11, n/2, rowsize);
Rec_Mult(C+d11, A+d12, B+d21, n/2, rowsize);
Rec_Mult(C+d12, A+d11, B+d12, n/2, rowsize);
Rec_Mult(C+d12, A+d12, B+d22, n/2, rowsize);
Rec_Mult(C+d21, A+d21, B+d11, n/2, rowsize);
Rec_Mult(C+d21, A+d22, B+d21, n/2, rowsize);
Rec_Mult(C+... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
Analysis of Work
W(n) = 8W(n/2) + #(1)
recursion tree
1
8
W(n/2)
1
W(n/2)
1
8
!
W(n/2)
1
lg n
W(n/4) W(n/4)
1
1
!
W(n/4)
1
#leaves = 8lg n = nlg 8 = n3
"
#(1)
Note: Same work as looping versions.
© 2008-2018 by the MIT 6.172 Lecturers
G
e
o
m
e
t
r
i
c
1
8
64
$
#(n3)
#
W(n) = #(n3)
37
... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
, A+d22, B+d22, n/2, rowsize);
} }
Submatrix
Caching
Lemma
Q(n) =
!(n2/B) if n2<cM for suff. small const c"1,
8Q(n/2) + !(1) otherwise.
© 2008-2018 by the MIT 6.172 Lecturers
38
Analysis of Cache Misses
Q(n) =
!(n2/B) if n2<cM for suff. small const c"1,
8Q(n/2) + !(1) otherwise.
recursion tree
Q(n)
© 2008... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
/M3/2).
G
e
o
m
e
t
r
i
c
1
8
64
"
#(cM/B)
Same cache misses as with tiling!
© 2008-2018 by the MIT 6.172 Lecturers
#(n3/BM1/2)
#(
Q(n) = #(n3/BM1/2)
42
Efficient Cache-Oblivious Algorithms
• No voodoo tuning parameters.
• No explicit knowledge of caches.
• Passively autotune.
• Handle multileve... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
cilk_spawn Rec_Mult(C+d21, A+d21, B+d11, n/2, rowsize);
cilk_spawn Rec_Mult(C+d12, A+d12, B+d22, n/2, rowsize);
Rec_Mult(C+d22, A+d21, B+d12, n/2, rowsize);
cilk_sync;
} }
© 2008-2018 by the MIT 6.172 Lecturers
44
Cilk and Caching
Theorem. Let QP be the number of cache misses in a
deterministic Cilk computatio... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
1)
C[0] += A[0] * B[0];
else {
int64_t d11 = 0;
int64_t d12 = n/2;
int64_t d21 = (n/2) * rowsize;
int64_t d22 = (n/2) * (rowsize+1);
cilk_spawn Rec_Mult(C+d11, A+d11, B+d11, n/2, rowsize);
cilk_spawn Rec_Mult(C+d21, A+d22, B+d21, n/2, rowsize);
cilk_spawn Rec_Mult(C+d12, A+d11, B+d12, n/2, rowsize);
Rec_Mult(... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/329bfc6e1808c375afa517feb3c4c273_MIT6_172F18_lec14.pdf |
LECTURE 13
Homotopy Coinvariants, Abelianization, and Tate
Cohomology
Recall that last time we explicitly constructed the homotopy invariants X hG of a
qis
−−→ Z,
G is a canonical complex of free G-modules in non-positive degrees. Then
complex X of G-modules. To do this, we constructed the bar resolution P can
where P ... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
m − m for all
g ∈ G. The upshot is that
H 1(G, M ) := H 1(M hG) = {1-cocycles}/{1-coboundaries}.
As a corollary, if G acts trivially on M , then H 1(G, M ) = HomGroup(G, M ), since
the 1-coboundaries are all trivial, and the 1-cocycles are just ordinary group homo-
morphisms. This also shows that zeroth cohomology is j... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
��G (cid:16) IG via 1 (cid:55)→ g − 1 on the gth coordinate.
Z[G]⊕G → Z[G] → Z → 0
Remark 13.1. The correct algorithm for computing tensor products is as fol-
lows: recall that tensor products are right-exact, that is, they preserve surjections,
and tensoring with the algebra gives the original module. To tensor with a... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
preserves
injections.
We now ask if PG is flat. In fact:
Claim 13.4. Any projective complex is flat.
An easier claim is the following:
Claim 13.5. Any complex F that is bounded above with F i flat for all i is flat.
13. HOMOTOPY COINVARIANTS, ABELIANIZATION, AND TATE COHOMOLOGY
59
To prove this claim, we will use the fact... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
A F is as well by the long exact sequence on cohomology. A similar (inductive)
argument gives the case where F is bounded.
Case 3. In the general case, form the diagram
F0
F1
F2
...
· · ·
· · ·
· · ·
0
0
0
...
0
0
0
F 1
id
F 2
d
F 1
id
...
id
...
d
d
F 0
id
F 0
id
F 0
id
...
0
0
0
...
· · ·
· · ·
· · ·
Clearly all squa... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
).
60
13. HOMOTOPY COINVARIANTS, ABELIANIZATION, AND TATE COHOMOLOGY
Definition 13.9. Hi(G, X) := H −i(XhG) (where we note that the subscript
notation is preferred as XhG is generally a complex in non-positive degrees only).
We now perform some basic calculations.
Claim 13.10. If X is bounded from above by 0, then H0(... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
.
Z[G]/I 2
G → Gab × Z,
g (cid:55)→ (¯g, 1)
This would imply that IG/I 2
G = Ker((cid:15))/I 2
G = Gab, as desired.
Proof. First note that the map above is a homomorphism. Indeed, letting
[g] ∈ Z[G] denote the class of g, we have
[g] + [h] (cid:55)→ (¯g¯h, 2)
[g] (cid:55)→ (¯g, 1)
[h] (cid:55)→ (¯h, 1)
for any g, h ∈ G... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
maps (using tensor-hom adjunction).
We then set
X tG := hCoker(XhG
N−→ X hG),
which we claim generalizes what we had previously for cyclic groups up to quasi-
isomorphism, so that we may define
Soon we will prove:
ˆH i(G, X) := H i(X tG).
Claim 13.13 (lcft). For a finite group G and extension L/K of local fields,
is an is... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/32b9450c555c522ecc6bab733675b350_MIT18_786S16_lec13.pdf |
18.600: Lecture 33
Entropy
Scott Sheffield
MIT
1 Outline
Entropy
Noiseless coding theory
Conditional entropy
2 Outline
Entropy
Noiseless coding theory
Conditional entropy
3 I Familiar on some level to everyone who has studied chemistry
or statistical physics.
I Kind of means amount of randomness or disorder.
... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
it’s quite common to use log to mean
log2 instead of loge. We follow that convention in this lecture.
In particular, this means that
log P{X = x} = −k
I Since there are 2k values in S, it takes k “bits” to describe an
for each x ∈ S.
element x ∈ S.
I Intuitively, could say that when we learn that X = x, we have
learned... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
the random sequence (so X is a random variable), then
for each x ∈ S we have P{X = x} = 2−k .
10 I Since there are 2k values in S, it takes k “bits” to describe an
element x ∈ S.
I Intuitively, could say that when we learn that X = x, we have
learned k = − log P{X = x} “bits of information”.
Information
I Suppose we... | https://ocw.mit.edu/courses/18-600-probability-and-random-variables-fall-2019/32dce72d547d449d4cfe2012a8297ba2_MIT18_600F19_lec33.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.