text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
= 1/vl, at temperatures T < Tc.
≡
(3) Due to the termination of the coexistence line, it is possible to go from the gas phase to
the liquid phase continuously (without a phase transition) by going around the critical
point. Thus there are no fundamental differences between liquid and gas phases.
From a mathematical ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
phase transition occurs between paramagnetic
and ferromagnetic phases of certain substances such as iron or nickel. These materials
become spontaneously magnetized below a Curie temperature Tc. There is a discontinuity
in magnetization of the substance as the magnetic field h, goes through zero for T < Tc.
The phase... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
as the order parameter. In zero field, m vanishes for a paramagnet and is non–zero
in a ferromagnetic, i.e.
m(T, h = 0)
0
t
|
|
β
∝
�
for
for
T > Tc,
T < Tc,
(I.20)
T )/Tc is the reduced temperature. The singular behavior of the order
where t = (Tc −
parameter along the coexistence line is therefore indicate... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
. Actually in almost all cases, the same singularity
governs both sides and γ+ = γ− = γ. The heat capacity is the thermal response function,
and its singularities at zero field are described by the exponent α, i.e.
C±(T, h = 0)
−α± .
t
|
∝ |
(I.23)
Long–range Correlations: Since the response functions are related ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
system, i.e.
(For the time being we treat the magnetization as a scalar quantity.) Substituting the
�
M =
d3 ~r m(~r ).
(I.25)
above into eq.(I.24) gives
kBT χ =
�
d3~r d3~r ′ (
h
m(~r )m(~r ′ )
m(~r )
m(~r ′ )
) .
i
i h
i − h
(I.26)
Translational symmetry of a homogeneous system implies that
= m is a const... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
Typically such influences occur over a
characteristic distance ξ, called the correlation length. (It can be shown rigorously that
this function must decay to zero at large separations; in many cases Gc(~r )
decays as exp(
ic
> ξ.) Let g denote a typical value of the correla
/ξ) at separations
m(~r )m(0)
≡ h
~r
|
|... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
the nature of the phase transition. For example, the vanishing of the
coexistence boundary in the condensation of CO2 has the same singular behavior as that
of the phase separation of protein solutions into dilute and dense components. This univer
sality of behavior needs to be explained. We also noted that the dive... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
of the collection of interacting electrons is excessively
complicated. The important degrees of freedom close to the Curie point, whose statistical
mechanics is responsible for the phase transition, are long wavelength collective excitations
of spins (much like the long wavelength phonons that dominate the heat capa... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
superconductivity, and planar magnets;
n = 3 corresponds to classical magnets.
While most physical situations occur in three–dimensional space (d = 3), there are also
important phenomena on surfaces (d = 2), and in wires (d = 1). Relativistic field theory
is described by a similar structure, but in d = 4.
As in the case... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
)2
∇
,
�
· · ·
.
Including a small magnetic field ~h, that breaks the rotational symmetry, the lowest order
terms in the expansion of Φ lead to,
=
β
H
�
dd x
t
2
�
m 2(x) + um 4(x) +
K
2
(
∇
m)2 +
~h
·
· · · −
m(x)
~
,
(II.1)
�
which is known as the Landau–Ginzburg Hamiltonian. (The magnetic field also ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
integrating over (coarse graining) the microscopic degrees of freedom, while constraining
their average to ~m(x). It is precisely because of the difficulty of carrying out such a first
principles program that we postulate the form of the resulting effective free energy on the
basis of symmetries alone. The price paid is... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
N
m(x)
F
�
D
�
m(x),
∂m
∂x
,
N
· · · ≡
�
lim
→∞
N
i=1
�
dmiF
mi,
�
mi+1 −
a
mi
,
.
· · ·
�
(There are some mathematical concerns regarding the existence of functional integrals.
The problems are associated with having too many degrees of freedom at short distances,
allowing rather badly behaved functio... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
vicinity of the critical point, m is a small quantity, and it is justified to keep
only the lowest powers in the expansion of Ψ(m). (We can later check self consistently that
the terms left out are indeed small corrections.) The behavior of Ψ(m) depends strongly
on the sign of the parameter t.
(1) For
t > 0, we can... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
· ·
−
Tc) +
) =u + u1(T
· · ·
−
(T
O
Tc) +
Tc)2 ,
(T
−
−
O
Tc)2 ,
(II.5)
where a and u are unknown positive constants, dependent upon material properties. The
basic idea is that the phenomenological parameters are functions of temperature that can
be expanded in a Taylor series in T
−
Tc. The minimal conditions ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
.
Since t = a(T
Tc) + ; ∂/∂T
· · ·
−
∝
∂/∂t, and
C =
∂2
∂2f
∂T 2 ∝ − ∂t2
T
−
βF
V
�
=
� �
0
1
8u
for t > 0,
for t < 0.
(II.7)
(II.8)
We observe a discontinuity, rather than a divergence, in the heat capacity. If we insist
upon describing the singularity by a power law, we have to choose the exponent... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
There is also a transverse susceptibility that is always infinite
below Tc.)
Equation of State: On the critical isotherm t = 0, the magnetization behaves as
•
m¯ = (h/4u)1/3, i.e.
m¯ (t = 0, h)
∼
h1/δ, with
δ = 3.
(II.10)
18
MIT OpenCourseWare
http://ocw.mit.edu
8.334 Statistical Mechanics II: Statistical Physi... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
Wave Energy Generation
Jorge Manuel Marques Silva
1
• Bachelor + Master in Electrical
Engineering (Energy) in 2015;
• Superconductors in Electrical Machines;
• Started MIT Portugal PhD Program –
Sustainable Energy Systems in 2017:
• Renewable Energy (Ocean Waves);
• Machine Learning Forecasting;
• Predic... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/520e2bda9f4b8644c940fcf7acdad34e_MIT18_085Summer20_lec_JS.pdf |
w.mit.edu/fairuse.
5
• Machine Learning:
• Learn and improve from experience
without explicit programming;
• Environmental variables forecast.
© Desert Isle SQL.com. All rights reserved. This content
is excluded from our Creative Commons license. For
more information, see https://ocw.mit.ed... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/520e2bda9f4b8644c940fcf7acdad34e_MIT18_085Summer20_lec_JS.pdf |
1.3 Forward Kolmogorov equation
Let us again start with the Master equation, for a system where the states can be ordered
along a line, such as the previous examples with population size n = 0, 1, 2 · · · , N. We start
again with a general Master equation
dpn
dt
= −
Rmnpn +
Rnmpm .
(1.28)
m6=n
X
m6=n
X
In many relevant... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
x + y, leading to
∂
∂t
∗
p(x, t) =
dy [R(y, x − y)p(x − y) − R(y, x)p(x)] .
(1.31)
Z
We now make a Taylor expansion the first term in the square bracket, but only with respect
to the location of the incoming flux, treating the argument pertaining to the separation of
the two points as fixed, i.e.
R(y, x − y)p(x − y) = R(y... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
∆(x)i
∆t
,
D(x) ≡
1
2
dy y2R(y, x) =
1
2
h∆(x)2i
∆t
.
Z
(1.35)
(1.36)
(1.37)
Equation (1.35) is a prototypical description of drift and diffusion which appears in many
contexts. The drift term v(x) expresses the rate (velocity) with which the position changes
from x due to the transition rates. Given the probabilistic n... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
1.23). Yet the
proportions of the two alleles in the population does change from generation to generation.
One reason is that some individuals do not reproduce and leave no descendants, while others
reproduce many times and have multiple descendants. This is itself a stochastic process and
the major source of rapid cha... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
a continuum evolution equation by setting x = n/N ∈ [0, 1], and replacing
p(n, t + 1) − p(n, t) ≈ dp(x)/dt, where t is measured in number of generations. Clearly, from
Eq. (1.41), there is no drift
(cid:10)
(cid:11)
while the diffusion coefficient is given by
v(x) = h(m − n)i = 0 ,
Dhaploid(x) =
1
2N 2
(m − n)2
=
1
2N
x(1... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
chemical reaction that mimicks a
mutating population. Consider a system where a reaction between molecules A and B can
lead to two outcomes:2
A + B ⇀c A + A or A + B ⇁d B + B ,
(1.46)
at rates c and d. In a “mean-field” approximation the number of A molecules changes as
dNA
dt
= (c − d)NANB = (c − d)NA(N − NA) .
(1.47)
... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
by ±1, and hence
v(x) =
h∆ni
N
Rn+1,n − Rn−1,n
N
= N(c − d)x(1 − x) ,
=
=
1
N
[cn(N − n) − dn(N − n)]
while
D(x) =
=
Rn+1,n + Rn−1,n
2N 2
=
1
2N 2 [cn(N − n) + dn(N − n)]
h∆n2i
2N 2 =
c + d
2
x(1 − x) .
(1.51)
(1.52)
2In a sense these reactions mimic the mating process in which the offspring of a heterozygote (a diploid... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
quantified by
a parameter s, which is related to c and d by
c =
1
4N
(1 + s)
and d =
1
4N
(1 − s) .
In the following, we shall employ the nomenclature of population genetics, such that
v(x) =
s
2
x(1 − x) ,
and D(x) =
1
4N
x(1 − x) .
(1.53)
(1.54)
1.3.3 Steady states
While it is usually hard to solve the Kolmogorov equa... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
(x′)
Z
+ constant,
p∗(x) ∝
1
D(x)
exp
x v(x′)
D(x′)
,
(cid:21)
(cid:20)Z
with the proportionality constant set by boundary conditions.
Let us examine the case of the dynamics of a fixed population, including mutations, and
reproduction with selection. Adding the contributions in Eqs. (1.38), (1.39) and (1.54), we
have
v... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
15.093 Optimization Methods
Lecture 8: Robust Optimization
1 Papers
• B. and Sim, The Price of Robustness, Operations Research, 2003.
• B. and Sim, Robust Discrete optimization, Mathematical Programming,
2003.
2 Structure
Motivation
Data Uncertainty
Robust Mixed Integer Optimization
Robust 0-1 Optimization
• ... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
• It allows to control the degree of conservatism of the solution;
• It is computationally tractable both practically and theoretically.
5 Data Uncertainty
minimize c x
subject to Ax ≤ b
′
l ≤ x ≤ u
xi ∈ Z,
i = 1, . . . , k,
Slide 6
Slide 7
WLOG data uncertainty affects only A and c, but not the vector b.
Slid... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
.
• We will guarantee that if nature behaves like this then the robust solution
will be feasible deterministically. Even if more than Γi change, then the
robust solution will be feasible with very high probability.
Slide 9
Slide 10
2
6.1 Problem
minimize
′
c x +
max
{S0| S0⊆J0,|S0|≤Γ0}
dj xj
|
|
)
(
j∈S0
X
... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
. . , k.
∈
6.3 Proof
Given a vector x ∗ , we define:
∗
βi(x ) =
max
{Si| Si⊆Ji,|Si|=Γi}
∗
aˆij xj
|
.
|
)
(
j∈Si
X
Slide 11
Slide 12
Slide 13
This equals to:
∗
βi(x ) = max
s.t.
j∈Ji
X
j∈Ji
X
0
≤
∗
aˆij xj zij
|
|
zij
zij
Γi
1
≤
≤
i, j
∀
∈
Ji.
Slide 14
Dual:
βi(x ∗ ) = min
pij + Γizi
s.t.
j∈... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
is the
6.5 Probabilistic Guarantee
6.5.1 Theorem 2
∗
Let x be an optimal solution of robust MIP.
(a) If A is subject to the model of data uncertainty U:
Pr
∗
a˜ijxj > bi
j
X
1
≤ 2n
!
µ)
(1
−
n
n
l
l=⌊ν⌋
X
(cid:18)
(cid:19)
n =
Ji
|
(b) As n
, ν = Γi+n and µ = ν
|
2
ν
− ⌊
; bound is tight.
⌋
→ ... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
Approx bound
Bound 2
0
10
−1
10
−2
10
−3
10
−4
10
0
1
2
3
4
5
Γ
i
6
7
8
9
10
Γ
0
2.8
36.8
82.0
200
Violation Probability
0.5
4.49 × 10−1
5.71 × 10−3
5.04 × 10−9
0
Optimal Value
5592
5585
5506
5408
5283
Reduction
0%
0.13%
1.54%
3.29%
5.50%
• w˜i are independently distributed and... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
•
•
•
8.1 Remarks
Slide 23
• Examples: the shortest path, the minimum spanning tree, the minimum
assignment, the traveling salesman, the vehicle routing and matroid inter
section problems.
• Other approaches to robustness are hard. Scenario based uncertainty:
minimize max(c1x, c2x)
subject to x ∈ X.
′
′
is NP... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
(cj + max(dj − θ, 0)) xj
j
X
Slide 26
dn+1 = 0.
dn
≥
dl+1,
. . .
≥
≥
min
x∈X,dl≥θ≥dl+1
θΓ +
n
l
cj xj +
(dj
j=1
X
j=1
X
θ)xj =
−
dlΓ + min
x∈X
n
l
cj xj +
(dj
j=1
X
j=1
X
n
dl)xj = Zl
−
l
∗ Z = min
l=1,...,n+1
dlΓ + min
x∈X
cjxj +
(dj
j=1
X
j=1
X
dl)xj.
−
8.4 Theorem 3
Slide 27
• Algori... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
7
8923
9059
9627
10049
10146
10355
10619
10619
¯
% change in Z(Γ)
0 %
0.056 %
1.145 %
2.686 %
9.125 %
13.91 %
15.00 %
17.38 %
20.37 %
20.37 %
σ(Γ)
501.0
493.1
471.9
454.3
396.3
371.6
365.7
352.9
342.5
340.1
% change in σ(Γ)
0.0 %
-1.6 %
-5.8 %
-9.3 %
-20.9 %
-25.8 %
-27.0 %
-29.6 %
-31.6 % ... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
Recursion and Intro to Coq
Armando Solar Lezama
Computer Science and Artificial Intelligence Laboratory
M.I.T.
With content from Arvind and Adam Chlipala. Used with permission.
September 21, 2015
September 21, 2015
L02-1
Recursion and Fixed Point Equations
Recursive functions can be thought of as
solutions o... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/52b324114c211ad7e93fd14da38f6720_MIT6_820F15_L04.pdf |
Computing a Fixed Point
• Recursion requires repeated application of a function
• Self application allows us to recreate the original term
• Consider: W = (x. x x) (x. x x)
• Notice b-reduction of W leaves W : W W
• Now to get F (F (F (F ...))) we insert F in W:
WF = (x.F (x x)) (x.F (x x))
which b-reduce... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/52b324114c211ad7e93fd14da38f6720_MIT6_820F15_L04.pdf |
(n=0, False, f(n-1))
H2 = f.n.Cond(n=0, True, f(n-1))
Can we express
odd using Y ?
substituting “H2 odd” for even
odd
odd
= H1 (H2 odd)
= H odd where H =
= Y H
f. H1 (H2 f)
September 21, 2015
L02-7
Self-application and Paradoxes
Self application, i.e., (x x) is dangerous.
Suppose:
u ... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/52b324114c211ad7e93fd14da38f6720_MIT6_820F15_L04.pdf |
endent Types
– A draft is available online (http://adam.chlipala.net/cpdt/)
– most of what it covers goes beyond the scope of 6.820.
• Another popular book: Bertot & Casteran,
Interactive Theorem Proving and Program
Development (Coq'Art)
– https://www.labri.fr/perso/casteran/CoqArt/
• A popular online book that us... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/52b324114c211ad7e93fd14da38f6720_MIT6_820F15_L04.pdf |
� reduction
• rewrite H
– use (potentially quantified) equality [H] to rewrite
in the conclusion.
• intros
– move quantified variables and/or hypotheses
"above the double line.
• apply thm
– apply a named theorem, reducing the goal into one
new subgoal for each of the theorem's hypotheses,
if any.
September ... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/52b324114c211ad7e93fd14da38f6720_MIT6_820F15_L04.pdf |
6.895 Essential Coding Theory
September 8, 2004
Lecturer: Madhu Sudan
Scribe: Piotr Mitros
Lecture 1
1 Administrative
Madhu Sudan
To do:
• Sign up for scribing – everyone must scribe, even listeners.
• Get added to mailing list
• Look at problem set 1. Part 1 due in 1 week.
2 Overview of Class
Historical ov... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
1
1
0
1
1
⎞
⎟
⎟
⎠
1
1
0
1
(b1, b2, b3, b4) −→ (b1, b2, b3, b4) G·
Here, the multiplication is over F2.
Claim: If a, b ∈ { 0, 1} , a = b then a · G and b · G differ in ≥ 3 coordinates.
This implies that we can correct any one bit error, since with a one bit error, we will be one bit away
from the correct stri... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
0.
We can weaken it by asking how few coordinates can x = y − z be nonzero on, given that xH = 0?
0.
We’ll make subclaim 1: If x has only 1 nonzero entry, then x H =�
We’ll make subclaim 2: If x has only 2 nonzero entries, then x H =�
Subclaim 1 is easy to verify. If we have exactly one nonzero value, x H is just... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
) = 26 rows, and:
�
xG5|
x ∈ {0, 1}
26
�
=
{y|
yH5 =
0}
G5 also has full column rank, so it maps bit strings uniquely.
Note that our efficiency is now 31 , so we’re asymptotically approaching 1. In general, we can encode
26
n − log2(n + 1) bits to n bits, correcting 1 bit errors.
2.1 Decoding
We can figure out if ... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
2 Theoretical Bounds
2.2.1 Definitions
Define the Hamming Distance as Δ(x, y) =
i xi � yi, or the number of bits by which x and y differ.
Define a ball around string x of radius (integer) t as B(x, t) = {y ∈ Σn Δ(x, y) ≤ t}, or the set of strings
that differ by at most t bits.
We can describe a code in terms of the max... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
) ≤ 2n
Taking log of both sides,
k ≤ n − log2(n + 1)
Notice that the 26 bit Hamming code is as good as possible:
31 − 5 = 26
3 Themes
Taking strings and writing them so they differ in many coordinates. This is called an error correcting
code. In general, you have a finite alphabet Σ. Popular examples: {0, 1}, ASCI... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
algorithms (computationally efficient — polynomial, linear, or sublinear time)
• Decoding algorithms
• Theoretical bounds/limitations
• Applications
Hamming’s paper did all of the above — it described a code, an efficient encoding algorithm (poly
time. Challenge: Construct lineartime encoding algorithm for Hamming co... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
1 Sequence
1.1 Probability & Information
We are used to dealing with information presented as a sequence of letters. For example,
each word in English languate is composed of m = 26 letters, the text itself includes also
spaces and punctuation marks. Similarly in biology the blueprint for any organism is the
string of ... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
. . + pm)N =
(cid:48)
(cid:88)
{Nα}
pN1
1 pN2
2
· · · pNm
m ×
N !
α=1 Nα!
(cid:81)m
,
(1.3)
(1.4)
where the sum is restricted so that (cid:80)m
α=1 Nα = N . Note that because of normalization, both
sides of the above equation are equal 1. The terms within the sum on the right-hand side
are known the multinomial probabi... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
)
(Nα log Nα − Nα)
= −N ·
(cid:18) Nα
N
(cid:88)
α
α
(cid:19)
log
(cid:18) Nα
N
(cid:19)
.
(Stirling’s approximation for N ! is used for all Nα (cid:29) 1.) The above formula is closely related
to the entropy of mixing in thermodynamics, and quite generally for any set of probabilities
{pα}, we can define a mixing entro... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
4, then Eq. (1.8) reduces to 0, which is consistent—we gain no information. On
the other hand, if pA = pT = 0 and pC = pG = 1
2 , then
I = 2 −
(cid:88)
G,C
1
2
log2 2 = 1 bit per base.
1.2 Evolving Probabilities
As organisms reproduce the underlying genetic information is passed on to subsequent gen-
eration. The copyi... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
shed it by comparing such single nucleotide
polymorphisms (SNPs). Non-synonymous mutations are not necessarily deleterious and may
lead to viable off-spring.
1.2.2 Classical Genetics
The study of heredity began long before the molecular structure of DNA was understood.
Several thousand years of experience breeding anima... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
+ 1) =
m
(cid:88)
β=1
παβpβ(τ ),
or in matrix form (cid:126)p(τ + 1) = ←→π (cid:126)p(τ ) = ←→π τ (cid:126)p(1),
(1.9)
where the last identity is obtained by recursion, assuming that the transition probability
matrix remains the same.
Probabilities must be normalized to unity, and thus the transition probabilities are ... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
(cid:88)
α
Rαβ = 0,
or Rββ = −
(cid:88)
α(cid:54)=β
Rαβ.
dpα(t)
dt
=
(cid:88)
β(cid:54)=α
(Rαβpβ(t) − Rβαpα(t))
,
(1.14)
(1.15)
which is known as the Master equation.
1.2.4 Steady state
Because of the conservation of probability in Eqs. (1.10) and (1.14), the transition probability
←−
matrix ←→π , and by extension the ... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
µ1
µ2 −µ1
(cid:19) (cid:18) p1
p2
(cid:19)
.
(1.18)
The above 2 × 2 transition rate matrix has the following two eigenvectors
(cid:18) −µ2
µ1
µ2 −µ1
(cid:19) (cid:18) µ1
µ1+µ2
µ2
µ1+µ2
(cid:19)
= 0,
and
(cid:18) −µ2
µ1
µ2 −µ1
(cid:19) (cid:18) 1
−1
(cid:19)
= −(µ1 + µ2)
(cid:18) 1
−1
(cid:19)
.
(1.19)
−→
p∗ with eigenv... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
µ1+µ2
(cid:33)
µ1
µ2 −µ1
+ e−(µ1+µ2)t µ2
− e−(µ1+µ2)t µ2
µ1+µ2
µ1+µ2
=
(cid:32) µ1
µ1+µ2
µ2
µ1+µ2
(cid:19)
+
µ2
µ1 + µ2
(cid:18) 1
−1
(cid:19)(cid:21)
.
(1.21)
At long times the probabilities to find state A1 or A2 are in the ratios µ1 to µ2 as dictated
by the steady state eigenvector. The rate at which the probabilitie... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
stays the same, or changes
by unity. Thus the transition rate matrix only has non-zero terms along or adjoining to the
diagonal. For example
Rn,n+1 = µ2(n + 1),
and Rn,n−1 = µ1(N − n + 1),
(1.22)
where the former indicates that a population of n + 1 A1s can decrease by one if any one of
them mutates to A2, while the po... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
mean numbers of constituents evolve to the steady state with N ∗
A/N ∗
B = b/a.
However, in a system where the number of particles is small, for example for a variety
of proteins within a cell, the mean number may not be representative, and the entire dis-
tribution is relevant. The probability to find a state with NA =... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
2.160 System Identification, Estimation, and Learning
Lecture Notes No. 5
February 22, 2006
4. Kalman Filtering
4.1 State Estimation Using Observers
In discrete-time form a linear time-varying, deterministic, dynamical system is
represented by
xt +1
= t
x A t
+
u B
t
t
(1
)
nx1
is a n-dimensional state vect... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
yı = H t xıt
t
t
+
u B
t
t L ( y
+
t
t
−
yı )
t
(3)
To differentiate the estimated state from the actual state of the physical system, the
estimated state residing in the real-time simulator is denoted xıt . With this feedbackthe
state of the simulator will follow the actual state of the real system, and ther... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
with noise, and the state transition of the
actual process is to some extent disturbed by noise. If stochastic properties of these noise
sources are available, state estimation may be performed more effectively than simply
using sensor signals as noise-free signals and estimating the sate based on noise-free state
... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
m
t
( )
t X
(
) −
⎟
⎜
⎢
Y
⎟
⎜
⎢ ⎣
(t Z ) − m (t)
⎠
⎝
( )
t
C
XYZ
=
Z
) −
( )
m t
x
If mx=my=mz=0
t Y
(
) −
( )
mY t
t Z
(
) −
)(tm
Z
⎤
⎥
) ⎥
⎥ ⎦
(4)
3
[
2
⎡
X E
⎢
[
C XYZ (t ) = ⎢
t X E
⎣ [
⎢
t X E
(t )]
)]
( ) (
t Y
)]
( ) (
t Z
[
)]
( ) (
t X E
t Y
[ 2
(t )]
Y E
[ ( ) (
t Y E
t Z ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
xt , y , z , x , y , z )
t 2
t 1
Z Y X Z Y X
1
2
2 2
t 2
t 1
t 2
1
1
1
C XYZ (
t
t
1, 2
⎡
) = ⎢
⎢
⎢
⎣
[
t X E
(
[ (
t Y E
[ (
t Z E
1)
1)
1)
t X
(
2
t X
(
2
t X
(
2
)]
)]
)]
[
t X E
( )
1
[ ( )
t Y E
1
[ ( )
t Z E
1
2 )]
t Y
(
2 )]
2 )]
t Y
(
t Y
(
[
t X E
( )
1
[ ( )
t Y E
1
[ ( )
t ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
process, the state xt driven by wt is a
S igure 4 . S
, is a deterministic term. In
random process. The second term on the right hand side, u B
t
the following stochastic state estimation, this deterministic part of inputs is not important,
since its influence upon the state xt is completely predictable and hence ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
If the noise signals at any two time slices are uncorrelated,
,
t ) = E [v ⋅
CV ( s
t
T ]
v
s
= ,0
t∀
≠
s
(10)
(11)
(12)
the noise is called “White”. (We will discuss why this is called white later in the
following chapter.) Note that, if t= s , the above covariance is that of the first order density,
i.e. a... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
measurement noise
, v ∈ R
t
nx1
� x1
� x1
n
n
∈ R × , and H ∈ R
tv have zero mean values,
� xn
t
. Assume
E[wt]=0, E[vt]=0.
and that they have the following covariance matrices:
t CV
),(
s
=
[
vv E
⋅
t
s
T
]
=
t CW
),(
s
t[
= w E
⋅ w T
s ]
=
CWV ( s
,
t ) = w E
[
vt
⋅
T ]
s
0
⎧
⎨
R... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
uation (8) and the output eq
Rudolf E. Kalman solved this problem around 1960.
Kalman Filter: two major points of his seminal workin 1960.
I)
II)
If we assume that the optimal filter is linear, then the Kalman filter is the state
estimator having the smallest unconditioned error covariance among all linear
filter... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
t
t − 1
t
Note E[vt]=0
Correction of the state estimate
Assimilating the new measurement yt, we can update the state estimate in
proportion to the output estimation error.
xıt = xı
t − 1
t
+ Kt ( y −
t
x H ı
t
)
t − 1
t
(21)
(22)
(23)
Equation (23) provides a structure of linear filter in recursive form... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
Topic 0 Notes
Jeremy Orloff
0 18.04 course introduction
This class is an adaptation of a class originally taught by Andre Nachbin. He deserves most of
the credit for the course design. The topic notes were written by me with many corrections and
improvements contributed by Jörn Dunkel. Of course, any responsibility f... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/532ad5a9f97c5222ee8098c374ec3df7_MIT18_04S18_topic0.pdf |
ed from R. Rosales 18.04 OCW 1999)
Do not be fooled by the fact things start slow. This is the kind of course where things keep on building
up continuously, with new things appearing rather often. Nothing is really very hard, but the total
integration can be staggering - and it will sneak up on you if you do not watc... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/532ad5a9f97c5222ee8098c374ec3df7_MIT18_04S18_topic0.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.641 Electromagnetic Fields, Forces, and Motion
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 10: Solutions to Laplace’s Equation In C... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
V = Φ ∇ Φ
d
d
⎣ d
d ⎦
(cid:118)∫
S
i da = ∇ Φ d dV = 0
2
∫
V
on S, Φ d = 0 or ∇Φ d i da = 0
Φ d = 0 ⇒ Φ a = Φ b on S
∇Φ d i da = 0 ⇒
∂Φ a =
∂n
∂Φ b on S ⇒ E
∂n
na = E
nb on S
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 10
Page 1 of 8
A problem is uniquely posed when ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
= 0
)
1. Try product solution: Φ (x, y ) = Χ x Y
(
(y )
)
(
Y y
)
(
2
d Χ x
Y
dx 2
+
(
X
)x
2
d
)
(
y
dy 2
= 0
Multiply through by 1 :
XY
2
1 d Χ
X dx 2
= −
2
1 d Y
Y dy 2
= − k
2
k=separation constant
only a
function
of x
only a
function
of y
2
d Χ
dx 2
= − k Χ
2
;
2
d Y
dy 2
2
= k Y
2. Zer... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
cos kxe + D cos kxe
-ky
3
4
= E 1 sin kx sinh ky + E2 sin kx cosh ky + E3 cos kx sinh ky + E 4 cos kx cosh ky
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 10
Page 4 of 8
4. Parallel Plate Electrodes
Neglecting end effects,
(
Φ x .
) Boundary conditions are:
Φ (x = 0) = Φ 0 , Φ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
Lecture 10
Page 5 of 8
Electric field lines:
dy
dx
=
E y =
E x
x
y
ydy = xdx
2
y
2
=
2
x
2
+ C
2
y = x + y 0 − x 0 (field line passes through (x0, y0))
2
2
2
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 10
Page 6 of 8
6. Spatially Periodic Potential Sheet
Φ (x, y )... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
a sin ay
Electric Field Lines:
dy
dx
=
E y
E
x
⎧−cot ay
⎪
= ⎨
⎪
⎩+cot ay
x > 0
x < 0
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 10
Page 7 of 8
x > 0
cos ay e -ax = constant
x < 0
cos ay e +ax = constant
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Z... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
6.895 Theory of Parallel Systems
Lecture 9
Analysis of Cilk Scheduler
Lecturer: Michael A. Bender
Scribe: Alexandru Caraca¸s, C. Scott Ananian
Lecture Summary
1. The Cilk Scheduler
We review the Cilk scheduler.
2. Location of Shallowest Thread
We define the depth of a thread and the shallowest thread. Next, We pr... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
(which is not in a deque).
(c) If the deque is empty and the processor is unable to execute α’s parent (because the parent is
busy), then the processor work steals.
3. Procedure α Syncs. If there are no outstanding children, then continue: we’re properly synced. Oth-
erwise (when there are no outstanding children), ... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
Structural lemma) Consider any processor p at time t. Let u0 be the current root thread (of
procedure α0) executing on processor p. Let u1, u2, . . . , uk be the threads (of procedures α1, α2, . . . , αk ) in p’s
deque ordered from bottom to top. Let d(ui) be the depth of thread ui in DAG G. Then,
d(u0) ≥ d(u1) > · ... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
Change in the deque of a processor when the processor returns from a procedure.
• Case 1: Steal. A steal removes the top entry from the deque (see Figure 2). The processor performing
the steal begins executing uk with an empty deque, trivially satisfying the inequality as in the base
case. The processor from whom th... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
to ua through u0 which has
length d(u0) + 1. Therefore,
d(ua) ≥ d(u0) + 1,
9-3
uk
uk−1
.
.
.
u2
u1
u0
=⇒
uk
uk−1
.
.
.
u2
u1
ua
Figure 4: Change in the deque of a processor when the processor reaches a sync point or continues a procedure.
· · ·
u0
ua
· · ·
α0
Figure 5: A piece of a computatio... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
a shallowest thread
may be the currently-executing thread instead of the thread at the top of the deque.
Since the base case satisfies the property, and every action maintains the property, then at any time t in the
execution for all deques
d(u0) ≥ d(u1) > · · · > d(uk−1) > d(uk ).
=⇒
uk
uk−1
.
.
.
u2
u1
u0... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
2) by making a (back) edge to every
spawn thread from the parent’s continuation thread. Note that if a sync is immediately followed the spawn,
we need to insert an extra continuation thread before the sync to prevent cycles in the DAG. Figure 8 shows
the execution graph of Figure 9 augmented with these edges.
We sh... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
ors left in the DAG, but it is neither being executed nor at the
top of the deque.
Observation 8 (Depth of Augmented DAG G(cid:2)) The critical path of the augmented DAS G(cid:2) is 2T∞.
We can now get to a spawned thread via our back edge from the continuation edge, adding a distance of one
to the longest path to ... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
argument based on two buckets to help in proving the bound.
9-6
Figure 10: The work and steal buckets used in the accounting argument.
Theorem 10 (Cilk Scheduler execution time bound) Consider the execution of any fully strict mul
tithreaded computation with work T1 and critical path T∞ by Cilk’s work-stealing alg... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
the number of dollars in the steal bucket is O(P T∞),
with high probability.
We give an exact definition of high probability during the course of the proof. In the rest of the lecture,
we prove the previous lemma. We introduce several concepts to prove the bound.
Observation 13 (Dollars) Per time-step P dollars ente... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
1 Rounds
As long as no processor is stealing, the computation is progressing efficiently. What divide the computation
into rounds, which (more or less) contain the same number of steals (see Figure 11).
Definition 16 (Round) A round of work stealing attempts is a set of at least P and at most 2P − 1 steal
attempts. We... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
done at random and there are
Proof
P places that a processor could steal from. Hence, the probability that a critical thread would be stolen
during a round is 1/P , and the probability that the critical thread will not be stolen is 1 − 1/P . Hence, for
C rounds we obtain the probability that a critical thread is no... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
Base case: During last time-step t, last thread in G(cid:1) is uL and it is critical and executed (see Figure 12).
Induction step: Suppose during time-step t, thread uj is critical. During time-step t − 1, either uj is
also critical, or if not it is because it has un-executed predecessors in G(cid:2). Some threads ui... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
A delay sequence occurs if ∀i at least πi rounds occur while ui is critical and un-executed.
A delay sequence is a combinatorial object. A delay sequence occurs if there is a directed path in G(cid:2)
in which one of the threads in the path is critical at each time-step.
Lemma 21 If at least 2P (2T∞ + R) steal attem... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
∞ + R)
steal attempts occur
(cid:5)
(cid:4)
≤ Pr
(cid:5)
exists delay sequence
(U, R, Π) that occurs
(cid:6)
≤
≤
Pr [(U, R, Π) occurs]
delay sequences
(U, R, Π)
(cid:1)
Number of
Delay Sequences
9-10
⎛
⎝
(cid:2) Maximum Probability
that any delay
sequence occurs
⎞
⎠
.
Lemma 23 (Probability of Del... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
than partitioning R into 2T∞ pieces.
We simply count up all the possible ways of partitioning the path. Thus, by multiplying the number of paths
with the number of ways in which a path can be partitioned we obtain the number of delay sequences, which
is at most
(cid:1)
22T∞
2T∞ + R
2T∞
(cid:2)
.
Our goal is to... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
considerably large, while the second term is exponentially
small in R. We want to choose a value for R such that the product of the two terms is small. We let
R = 2CT∞ where C is a constant. Replacing in the previous equation we have:
Pr [any (U, R, Π) occurs] ≤
(cid:1)
[2e(C + 1)]1/C
e
(cid:2)R
.
(4)
Definiti... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
the
possible events we have shown that with high probability there are few steal attempts. With probability
at least 1 − ε the number of steal attempts is O(P T∞ + lg ε
We have started our analysis using an accounting argument, based on a work bucket and a steal
bucket.We have shown that at the end of the computati... | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
y
x
(cid:12) (cid:13)
ey x
x
.
≤
Figure 13: Death-bed formulae.
13 | https://ocw.mit.edu/courses/6-895-theory-of-parallel-systems-sma-5509-fall-2003/533da8dd57712f1c0414ef6c84baf90a_lecture9.pdf |
16.920J/SMA 5212
Numerical Methods for PDEs
Lecture 5
Finite Differences: Parabolic Problems
B. C. Khoo
Thanks to Franklin Tan
SMA-HPC ©2002 NUS
Outline
• Governing Equation
• Stability Analysis
• 3 Examples
• Relationship between σand λh
• Implicit Time-Marching Scheme
• Summary
SMA-HPC ©2002 NUS
2
G... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/5350b98a660c073d845f9128a3851a8b_lec5.pdf |
u
u
2
+ −
j
1
2
x
∆
u
−+
j
1
+
(
O x
∆
2
)
which is second-order accurate.
• Schemes of other orders of accuracy may be
constructed.
SMA-HPC ©2002 NUS
5
Stability Analysis
Discretization
We obtain at
x
1
x
2
:
:
du
1
dt
du
2
dt
=
=
υ
2
x
∆
υ
2
x
∆
( u −
o
2
u u
+
1
2
)
(... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/5350b98a660c073d845f9128a3851a8b_lec5.pdf |
��
u
υ
N
∆
2
x
0
1
−
0
A
SMA-HPC ©2002 NUS
7
Stability Analysis
PDE to Coupled ODEs
Or in compact form
(cid:71)
du
dt
(cid:71) (cid:71)
Au b
+
=
where
(cid:71)
u
(cid:71)
b
[
u
=
1
o u
υ
=
2
x
∆
u
2
0
0
0
T
]
u −
1
N
Nu
υ
2
x
∆
T
We hav... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/5350b98a660c073d845f9128a3851a8b_lec5.pdf |
−
diagonalizes the matrix by
E
A
N
1) matrix formed by the (
N
−
1) columns
1E AE−
= Λ
where
Λ =
λ
1
λ
2
0
0
1
λ −
N
SMA-HPC ©2002 NUS
10
Stability Analysis
Coupled ODEs to
Uncoupled ODEs
Starting from
(cid:71) (cid:71)
Au b
+
=
Premultiplicati... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/5350b98a660c073d845f9128a3851a8b_lec5.pdf |
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