text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
nd
a vector vi such that Eλ1 = ker(Bi) = span(vi). We form a basis B = (v1, v2, v3) and the
transition matrix P = (v1|v2 v3). Then we have
|
⎛
[A]B,B = ⎝ 0
0
λ1 0
λ2
0
0
0
λ2
⎛
⎞
⎠ , A = P ⎝ 0
0
λ1 0
λ2
0
0
0
λ2
⎞
⎠ P −1 .
(39)
We also have S = A and N = 0.
8
18.700 JORDAN NORMAL FORM NOTES ... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
three eigenvalues λ1 = −1, λ2 =
0, λ3 = 1. We define B1 = A − (−1)I3, B2 = A, B3 = A − I3. By GaussJordan elimination
we find
⎛⎛ ⎞⎞
3
⎛⎛ ⎞⎞
1
E−1 = ker(B1) = span ⎝⎝ 4 ⎠⎠ , E0 = ker(B2) = span ⎝⎝ 1
,
0
⎛⎛ ⎞⎞
2
E1 = ker(B3) = span ⎝⎝ 2 ⎠⎠ .
⎠⎠
2
We define
Then we have
1
⎞
⎠ .
⎛⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎞
1
⎠ , ⎝ 1
0 ... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
2 = ker(B2). There are two cases depending on
the dimension of Eλ1 .
The first case is when Eλ1 has dimension 2. Then we have a basis (v1, v2) for Eλ1 and a
basis v3 for Eλ2 . With respect to the basis B = (v1, v2, v3) and defining P = (v1 v2 v3), we
have
|
|
⎛
λ1 0
⎞
0
⎛
λ1 0
0
⎞
[A]B,B = ⎝ 0 λ1 0 ⎠ , A = P ⎝ ... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
λ1 0
⎞
0
⎛
λ1 0
0
[A]B,B = ⎝ 1 λ1 0 ⎠ , A = P ⎝ 1 λ1 0 ⎠ P −1 .
0
0 λ2
0
0 λ2
⎞
⎞
Also we have
⎛
λ1 0
0
λ1 0
0
⎞
⎛
[S]B,B = ⎝ 0 λ1 0 ⎠ , S = P ⎝ 0 λ1 0 ⎠ P −1 ,
and
0
⎛
0 λ2
0 0 0
0 0 0
⎞
⎛
0
0 λ2
⎞
0 0 0
0 0 0
[N ]B,B = ⎝ 1 0 0 ⎠ , A = P ⎝ 1 0 0 ⎠ P −1 .
Let’s see how this works in an example.... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
elimination we calculate that E2 = ker(B2) has a basis consisting of v3 =
(0, 1, 1)†. By GaussJordan elimination, we find that E3 = ker(B1) has a basis consisting of
(0, 1, 0)†. In particular it has dimension 1, so we have to keep going. We have
⎛
⎞
0 0 0
2 = ⎝ 1 0 1 ⎠ .
1 0 1
B1
By GaussJordan elimination (or ... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
(46)
(47)
(48)
(49)
(51)
(52)
(53)
10
18.700 JORDAN NORMAL FORM NOTES
We also have that
⎛
[S]B,B = ⎝ 0
0
3 0 0
0
3
2
0
⎛
⎞
⎠ , S = P ⎝ 0
0
3 0 0
0
3
2
0
⎛
⎞
⎠ P −1 = ⎝ −1
−1
3 0 0
−1
3
2
0
⎞
⎠ ,
⎛
0 0 0
0
0
0
0
⎛
⎞
⎠ , N = P ⎝ 1
0
0 0 0
0
0
0
0
⎛
⎞
⎠ P −1 = ⎝ 1
0... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
are nontrivial. We begin by finding a basis (w1, w2) for Eλ1 . Choose any vector v1 which
is not in Eλ1 and define v2 = B1v1. Then find a vector v3 in Eλ1 which is NOT in the span of
v2 . Define the basis B = (v1, v2, v3) and the transition matrix P = (v1 v2 v3). Then we have
|
|
⎛
λ1 0
0
⎞
⎛
λ1 0
0
⎞
[A]B,B = ⎝ 1 ... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
†, (0, 0, 1)†). Since this is 2dimensional, we are in the case above. So we choose any
vector not in E−2, say v1 = (1, 0, 0)†. We define v2 = B1v1 = (1, 1, 0)†. Finally, we choose a
vector in Eλ1 which is not in the span of v2, say v3 = (0, 0, 1)†. Then we define
⎛⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎞
1
⎞
0
B = ⎝⎝ 0 ⎠ , ⎝ 1 ⎠ , ⎝ 0 ⎠⎠ , P... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
. By GaussJordan we compute a basis for ker(B1
2v1.
vector v1 which is not contained in ker(B1
Then with respect to the basis B = (v1, v2, v3) and the transition matrix P = (v1 v2 v3), we
have
2). We define v2 = B1v1 and v3 = B1v2 = B1
|
|
⎛
λ1 0
⎞
0
⎛
λ1 0
0
⎞
[A]B,B = ⎝ 1 λ1 0 ⎠ , A = P ⎝ 1 λ1 0 ⎠ P −1 .
0
1... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
2 + 27X − 27. Also we see from the above that X = 3 is a root.
In fact it is easy to see that cA(X) = (X − 3)3 . So A has the single eigenvalue λ1 = 3.
We define B1 = A1 − 3I3, which is
⎛
⎞
B1 = ⎝ 1 −2 0 ⎠ .
2 −4 0
2 −3 0
(64)
By GaussJordan elimination we see that E3 = ker(B1) has basis (0, 0, 1)†. Thus we are... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
0 ⎠⎠ , P = ⎝ 0 1 0 ⎠ ,
1
0 2 1
1 2 0
⎛
2
0
we have
⎛
3 0 0
⎞
⎛
⎞
3 0 0
[A]B,B = ⎝ 1 3 0 ⎠ , A = P ⎝ 1 3 0 ⎠ P −1 .
0 1 3
0 1 3
We also have S = 3I3 and N = B1.
(66)
(67) | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.341: Discrete-Time Signal Processing
OpenCourseWare 2006
Lecture 8
DT Filter Design: IIR Filters
Reading: Section 7.1 in Oppenheim, Schafer & Buck (OSB).
In the last lecture we studied various forms of filter realizat... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/51e3beff8c8ce2289ba292fcdb0040f4_lec08.pdf |
)
| ≈ |
Hideal(ejω)| = H (ejω) .
|
|
Nonetheless, specifications can involve both magnitude and phase (or group delay). Such gen
eralized approximation is a harder problem, but may be desired in specific applications. In
particular, integer or fractional delays can only be achieved with FIR filters.
Most of our follow... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/51e3beff8c8ce2289ba292fcdb0040f4_lec08.pdf |
comparing different filters, one should pay attention to whether the #MAD
is measured per input sample, per output sample, or per unit time (clock cycle). It is
possible that an IIR of lower order actually requires more #MAD than an FIR of higher
order, because FIR filters may be implemented using polyphase structures.... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/51e3beff8c8ce2289ba292fcdb0040f4_lec08.pdf |
variable s to the discrete frequency variable z such that the imaginary axis in the s-plane
corresponds to one revolution of the unit circle in the z-plane:
s →
1 − z−1
1 + z−1
⇒
jΩ = j
2
T
tan
ω
2
,
Ω
Ωc
→
tan ω/2
tan ωc/2
π in the digital frequency domain corresponds to infinity in the analog frequenc... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/51e3beff8c8ce2289ba292fcdb0040f4_lec08.pdf |
−∞dB) as
frequency approaches π. This is because all zeros of the system are located at z = −1.
Since the gain diminishes quickly as frequency increases. It is possible that a lower order
filter exists such that it satisfies the given specifications, but does not exceed them as
greatly as the Butterworth design.
•
T... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/51e3beff8c8ce2289ba292fcdb0040f4_lec08.pdf |
)).
VN (x) = cos(N cos−1 x) is the N th-order chebyshev polynomial. Here we take the con
vention that cos−1 x becomes the inverse hyperbolic cosine and is imaginary if x > 1.
|
|
5
Some
examples of lower order Chebyshev polynomials are:
V0(x) = 1
V2(x) = cos(2 cos−1 x) = 2 cos2(cos−1 x) − 1 = 2x2 − 1
V1(x) = x
Se... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/51e3beff8c8ce2289ba292fcdb0040f4_lec08.pdf |
filters in
detail.
•
In both of the Chebyshev design methods, having a monotonic behavior in either the
passband or the stopband suggests a lower order system might exist such that it satisfies
the given set of specifications, but varies with equal ripple in both the passband and the
stopband.
Elliptic Filters
•
Fo... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/51e3beff8c8ce2289ba292fcdb0040f4_lec08.pdf |
I.C Phase Transitions
The most spectacular consequence of interactions among particles is the appearance
of new phases of matter whose collective behavior bears little resemblance to that of a few
particles. How do the particles then transform from one macroscopic state to a completely
different one. From a formal p... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
of densities ρg = 1/vg, and
ρl = 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 g... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
procedure may thus be appropriate to their description.
10
A related, but possibly less familiar, 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 discontinuit... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
hence can be used to distinguish between them. For a magnet, the
magnetization
m(T ) =
1
V h→0
lim M (h, T ),
serves 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 t... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
= 0)
−γ± ,
t
|
∝ |
(I.22)
where in principle two exponents γ+ and γ− are necessary to describe the divergences on
the two sides of the phase transition. 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 ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
− h
2
i
.
�
�
�
�
1
− Z2
�
tr [M exp (
β
−
H0 + βhM )]2
�
(I.24)
The overall magnetization is obtained by adding contributions from different parts of
the 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... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
in a factor of volume
V , and the susceptibility is given by
χ = βV
�
d3~r
h
m(~r )m(0)
ic .
(I.28)
The connected correlation function is a measure of how the local fluctuations in one
part of the system effect those of another part. Typically such influences occur over a
characteristic distance ξ, called the cor... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
the previous section that the singular behavior of thermodynamic func
tions at a critical point (the termination of a coexistence line) can be characterized by
a set of critical exponents
α, β, γ,
exponents are quite universal, i.e.
{
. Experimental observations indicate that these
· · ·}
independent of the materi... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
quantum
mechanical, involving such elements as itinerant electrons, their spin, and the exclusion
principle. Clearly a microscopic approach is rather complicated, and material dependent.
Such a theory is necessary to find out which elements are likely to produce ferromagnetism.
However, given that there is such beha... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
density. It is then useful to examine a generalized magnetization
for n-component spins, existing in d-dimensional space, i.e.
x
≡
(x1, x2, ..., xd)
d
∈ ℜ
(space)
~
, m
(m1, m2, ..., mn)
≡
n
∈ ℜ
(spin).
Some specific problems covered in this framework are:
n = 1 describes liquid–gas transitions, binary mixtures, ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
a rotation in the n-dimensional space.
= m(x)], where
m(x)]
[Rn ~
[ ~
H
H
Φ[ ~
m(x)]
are
m 2(x)
~m(x)
≡
n
n
≡
i=1
�
~m(x)
·
d
mi(x)mi(x)
, m 4(x)
≡
m 2(x)
2
, m 6(x)
,
,
· · ·
~m)2
(
∇
≡
i=1 α=1
� �
∂αmi∂αmi
,
2 ~m)2
, m 2(
(
∇
�
~m)2
∇
,
�
· · ·
.
Including a small magnetic field ~h, that... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
external parameters such as temperature and pressure.
{
· · ·}
It is essential to fully appreciate the latter point, which is usually the source of much con
fusion. The probability for a particular configuration of the field is given by the Boltzmann
β
{−
m(x)]
[ ~
This does not imply that all terms in the exponent ar... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
au–Ginzburg Hamiltonian in eq.(II.1).
Various thermodynamic functions (and their singular behavior) can now be obtained from
the associated partition function
Z =
D
�
m(x) exp
~
β
m(x)]
[ ~
.
}
H
(II.2)
{−
Since the degrees of freedom appearing in the Hamiltonian are functions of x, the symbol
m(x) refers to a f... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
) is replaced by
the maximum value of the integrand. The natural tendency of interactions in a magnet
is to keep the magnetizations vectors parallel, and hence we expect the parameter K in
eq.(II.1) to be positive. The configuration of ~m that maximizes the integrand is then
uniform, and
The uniform magnetization o... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
nite magnetization). The function Ψ(m) now has degenerate minima, at
a non–zero value of ¯m. There is thus a spontaneous magnetization, even at ~h =
0 indicating ferromagnetic behavior. The direction of the ~m is determined by the
systems preparation, and can be realigned by an external field ~h.
Thus a saddle point... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
-generic
situations which can presumably be removed by changing some other system parameter.
We now examine the singular behaviors predicted by eqs.(II.3) and (II.4).
Magnetization: In zero field, from ∂Ψ/∂m = tm¯ + 4um¯ 3 = m¯ (t + 4um¯ 2) = 0, and we
•
obtain
m¯ =
0
−t
4u
�
for t > 0,
for t < 0.
(II.6)
W... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/51eb0c6af0afd68616a47c6d070d358e_MIT8_334S14_Lec2.pdf |
h, we expect ~ = m(h)ˆ
m
¯
h, and from ∂Ψ/∂m 0, we
=
•
obtain tm¯ + 4um¯ 3 = h. Hence
χ−1 =
ℓ
∂h
∂m¯ h=0
�
�
�
�
= t + 12um¯ 2 =
t
2t
−
�
for t > 0,
for t < 0.
(II.9)
Thus the singularity in susceptibility is describable by χ±
−γ± , with γ+ = γ− =
1. Although the amplitudes A± are material dependen... | 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 |
LLC. All rights reserved. This
content is excluded from our Creative Commons license.
For more information, see https://ocw.mit.edu/fairuse.
5
• Machine Learning:
• Learn and improve from experience
without explicit programming;
• Environmental variables forecast.
© Desert Isle SQL.com. All... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/520e2bda9f4b8644c940fcf7acdad34e_MIT18_085Summer20_lec_JS.pdf |
10 MIT OpenCourseWare
https://ocw.mit.edu
18.085 Computational Science and Engineering I
Summer 2020
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms.
11 | 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 |
in probability due to incoming flux from x − y and the outgoing flux to 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 th... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
+
∂2
∂x2 [D(x)p(x, t)] .
v(x) ≡
dy yR(y, x) =
Z
h∆(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 t... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
corresponding to blue or
brown eye colors. The probability for a spontaneous mutation to occur that changes the
allele for eye color is extremely small, and effectively µ1 = µ2 = 0 in Eq. (1.23). Yet the
proportions of the two alleles in the population does change from generation to generation.
One reason is that some i... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
)
while
m2
hmi = N ×
n
N
= n ,
i.e
h(m − n)i = 0 ,
(m − n)2
= N ×
n
N
1 −
.
n
N
(cid:17)
(cid:16)
c =
(cid:11)
(cid:10)
We can construct 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), the... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
earlier for haploids, with A1 and A2 chosen with probabilities of x and 1 − x respectively.
Since, in a diploid population of N individuals, the number of alleles is 2N, the previous
result is simply modified to
Ddiploid(x) =
x(1 − x) .
(1.45)
1
4N
11
1.3.2 Chemical analog & Selection
Through the reactions in Eq. (1.25... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
1)p(n−1)−dn(N−n)p(n)−cn(N−n)p(n) ,
(1.49)
for 0 < n < N, and with boundary terms
dp(0, t)
dt
= d(N − 1)p(1),
and
dp(N, t)
dt
= c(N − 1)p(N − 1) .
(1.50)
When the number N is large, it is reasonable to take the continuum limit and construct
a Kolmogorov equation for the fraction x = n/N ∈ [0, 1]. The rates in Eq. (1.48)... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
the transition probabilities
after a whole generation (N steps of reproduction and removal). The selection process char-
acterized by Eq.(1.40) treats the two alleles as completely equivalent. In reality one allele
may provide some advantage to individuals carrying it. If so, there should be a selection pro-
cess by wh... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
It is not clear how
such a circumstance may arise in the context of population genetics, and we shall therefore
focus on circumstances where there is no probability current, such that
−v(x)p∗(x) +
∂
∂x
(D(x)p∗(x)) = 0.
We can easily rearrange this equation to
1
D(x)p∗
∂
∂x
(D(x)p∗(x)) =
ln (D(x)p∗(x)) =
v(x)
D(x)
.
∂
∂... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/5253e8ec9c66924411294ebb8e85e951_MIT8_592JS11_lec3.pdf |
1 − x′ +
Z
µ1 ln x + µ2 ln(1 − x) +
s
2
(cid:20)
(cid:21)
s
2
x
+ constant,
i
resulting in
p∗(x) ∝
h
1
x(1 − x)
· x4N µ1 · (1 − x)4N µ2 · e2N sx .
(1.63)
In the special case of no selection, s = 0 and (for convenience) µ1 = µ2 = µ, the steady-
state solution (1.63) simplifies to
p∗(x) ∝ [x(1 − x)]4N µ−1 .
(1.64)
If 4Nµ ... | 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 |
Slide 5
4 Goal
Develop an approach to address data uncertainty for optimization problems
that:
• 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, ... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
in its behavior, in that only a subset of the
coefficients will change in order to adversely affect the solution.
• 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 probabili... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
0
∈
= 0, j
Ji
∀
∀
∀
∀
∀
i = 1, . . . , 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: ... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
uncertain coefficients, and 2n + m + l constraints.
P
m
i=0 |Ji| 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) ... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
1}n .
4
Slide 15
Slide 16
Slide 17
Slide 18
Slide 19
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 ... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
x
X
n .
0, 1
}
⊂ {
∈
Robust Counterpart:
Z ∗ = minimize
′
c x +
max
{S| S⊆J,|S|=Γ}
subject to x
X,
∈
djxj
j∈S
X
WLOG d1
d2
≥
≥
. . .
≥
dn.
•
•
•
8.1 Remarks
Slide 23
• Examples: the shortest path, the minimum spanning tree, the minimum
assignment, the traveling salesman, the vehicle routing and m... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
j − θ, 0))
j
X
• Since X ⊂ {0, 1}n ,
max(dj xj − θ, 0) = max(dj − θ, 0) xj
d2
d1
≥
For dl
≥
θ
≥
•
•
•
•
Z ∗ = min
x∈X,θ≥0
θΓ +
(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
... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
intersection, are polynomially solvable.
9 Experimental Results
9.1 Robust Sorting
minimize
cixi
subject to
i∈N
X
xi = k
i∈N
X
x
n
0, 1
}
∈ {
7
Slide 28
.
Γ
0
10
20
30
40
50
60
70
80
100
¯
Z(Γ)
8822
8827
8923
9059
9627
10049
10146
10355
10619
10619
¯
% change in Z... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/52a0ea4ea461fd7ccf152704e726a0b3_MIT15_093J_F09_lec08.pdf |
X
i∈N
X
x
n
.
0, 1
}
∈ {
9.1.1 Data
• |N | = 200;
• k = 100;
• cj ∼ U [50, 200]; dj ∼ U [20, 200];
• For testing robustness, generate instances such that each cost component
independently deviates with probability ρ = 0.2 from the nominal value
cj to cj + dj.
9.1.2 Results
Slide 29
Slide 30
8
MIT OpenC... | 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 |
point (lfp)
Under the assumption of monotonicity and continuity least
fixed points are unique and computable
September 21, 2015
L02-4
Computing a Fixed Point
• Recursion requires repeated application of a function
• Self application allows us to recreate the original term
• Consider: W = (x. ... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/52b324114c211ad7e93fd14da38f6720_MIT6_820F15_L04.pdf |
ual Recursion
odd n = if n==0 then False else even (n-1)
even n = if n==0 then True else odd (n-1)
odd
even
where
= H1 even
= H2 odd
H1 = f.n.Cond(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... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/52b324114c211ad7e93fd14da38f6720_MIT6_820F15_L04.pdf |
• Taught around a formalization of all the
different correctness approaches with the Coq
proof assistant
• Will go into depth into different program
logics, different approaches to formalize
concurrency, behavioral refinement of
interacting modules, etc.
September 21, 2015
L02-11
Some useful references
• ... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/52b324114c211ad7e93fd14da38f6720_MIT6_820F15_L04.pdf |
)
September 21, 2015
L02-14
Tactics
• They instruct Coq on the steps to take to
prove a theorem
• reflexivity
– prove an equality goal that follows by normalizing
terms.
• induction x
– prove goal by induction on quantified variable [x]
– Structural Induction: X is any recursively defined
structure
–... | 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 |
0
0
1
0
0 0
0 1
0 1
1 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 bi... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
H = zH = 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 ... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
5 it has.
For H5∃G5 that has 31 columns, (31 − 5) = 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 ... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
H is just the bit
string for i. As a result, y H directly returns the location of the bit in which there is the error.
·
2.2 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(... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
number of bits in each
|
is the number of bits in each encoded word.
k
For binary and onebit errors,
|
K · Vol(t, n) ≤ Σ|
n
K(n + 1) ≤ 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... | https://ocw.mit.edu/courses/6-895-essential-coding-theory-fall-2004/52c97b8afa15b85a5feeddf2825b57ab_lect01.pdf |
Note that this is not necessarily the theory used
in modern CD players, cell phones, etc. which are based on average case.
Goals of course:
• Construct nice codes. “Optimize large |C vs. large min distance”
|
• Nice encoding algorithms (computationally efficient — polynomial, linear, or sublinear time)
• Decoding a... | 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 |
.
This is known as the multinomial coefficient, as it occurs in the expression
(p1 + p2 + . . . + 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 ... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
as much information.) As convenient measure, and taking clues from Statistical Mechanics,
we take the logarithm of Eq. (1.3), which gives
log N = log N ! −
(cid:88)
α
log Nα!
≈ N log N − N −
(cid:88)
(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... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
We gain a definite amount of knowledge by having advance insight about {pα}. Instead of
having to specify 2 bits per “letter” of DNA, we can get by with a smaller number. The
information gained per letter is given by
I({pα}) = 2 −
(cid:88)
α
pα log2
(cid:19)
.
(cid:18) 1
pα
(1.8)
If pα = 1/4, then Eq. (1.8) reduces to 0... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
stop the translation process.
The mutation is deleterious and the off-spring will not survive. However, as a result of the
3
redundancy in the genetic code, there are also mutations that are synonymous, in that they
do not change the amino acid which eventually results. Because these synonymous muta-
tions do not affect... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
say from α to β with a transition probability πβα. The q × q such elements
form the transition probability matrix ←→π . (Without the assumption that the sites evolve
←→
independently, we would have constructed a much larger (qN × qN ) matrix
Π . With the
assumption of independence, this larger matrix is a direct produc... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
∆t
(cid:88)
β(cid:54)=α
pβ(τ ) −
(cid:105)
pα(τ )
.
πβα
∆t
(1.12)
In the limit of small ∆t, [pα(τ + 1) − pα(τ )]/∆t ≈ dpα/dt, while
παβ
∆t
= Rαβ + O(∆t)
for α (cid:54)= β,
(1.13)
←→
are the off-diagonal elements of the matrix
R of transition probability rates. The diagonal
elements of the matrix describe the depletion r... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
∗ =
−→
p∗
,
and
←→
R
−→
p∗ = 0.
(1.17)
−→
p∗ represent the steady state probabilities for the process. These
The elements of the vector
probabilities no longer change with time. The other eigenvalues of the matrix determine how
an initial vector of probabilities approaches this steady state.
5
As a simple example, let... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
steady
state.
To make this explicit, let us start with a sequence that is purely A1, i.e. with p1 = 1 and
p2 = 0 at t = 0. The formal solution to the linear differential equation (1.18) is
(cid:19)
(cid:18) p1(t)
p2(t)
(cid:20)
= exp
t
(cid:18) −µ2
µ1
µ2 −µ1
(cid:19)(cid:21) (cid:18) p1(0)
p2(0)
(cid:19)
.
(1.20)
Decomp... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
sequence of length N can be recast and interpreted in
terms of the evolution of a population as follows. Let us assume that A1 and A2 denote two
forms of a particular allele. In each generation each individual is replaced by an offspring
that mostly retains its progenitor’s allele, but may mutate to the other form at so... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
(N − n + 1)p(n − 1) − µ2np(n) − µ1(N − n)p(n) , (1.23)
for 0 < n < N , and with boundary terms
dp(0, t)
dt
= µ2p(1) − µ1N p(0),
and
dp(N, t)
dt
= µ1p(N − 1) − µ2N p(N ) .
(1.24)
1.2.6 Enzymatic reaction
The appeal of the formalism introduced above is that the same concepts and mathematical
formulas apply to a host of d... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/52cb8a2b8e1236dc7077ad1bdb4891c0_MIT8_592JS11_lec1.pdf |
binary
7
elements independently distributed with probabilities p∗
B = a/(a + b).
Hence, the steady state solution to the complicated looking set of equations (1.23) is simply
A = b/(a + b) and p∗
p∗(n) =
(cid:18) N
n
(cid:19) bnaN −n
(a + b)N .
(1.27)
In fact, we can follow the full evolution of the probability to thi... | 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 |
state observer is given by
nx�
t
xıt+ 1
x A ıt
=
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 wil... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
and state estimation, however, are analogous;both based on the Prediction-Error-
Correction formula.
Luenberger’s state observer is strictly for deterministic systems. In actual systems,
sensor signals are to some extent corrupted with noise, and the state transition of the
actual process is to some extent disturbed... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
outputs with multiple sensors
F
irst-order density fXYZ(x
t,yt,zt)
Covariance: Ensemble mean
t X ) − mx (t)
(
⎡⎛
⎞
⎢⎜
⎟
(
(t Y
E
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 ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
and t2 , as shown in the above figure, the joint
probability density is given by: f
If mx= my= mz=0, the covariance is given by
( 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
)... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
nx1
xt+1
=
+x A t
t
u B t
t
+
w G t
t
(7)
ee F -4 ince the process noise is a random 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 i... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
[w ] = 0
t
rom eqF uation (6), the covariance of measurement noise vt ∈ R
� x 1
is given by
t ),
CV ( s
=
[
⋅E vt
Tv ∈
]
s
�
� x
R
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 ... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
t
=
t
x H
= t
+
t v +
t
w G
t
t
(8
)
(9)
nx1
where x ∈ R
that the process noise
, y ∈ R
t
t
, w ∈ R
t
, G
A
,
t
t
tw and the 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 covarianc... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
− xt ) ( xıt
−
x
t
)]
(20)
7
subject to the state eq
uation (9) with white, uncorrelated
process and measurement noises of zero mean and the covariant matrices given by
equations (16) - (18). (Necessary initial conditions are assumed.)
uation (8) and the output eq
Rudolf E. Kalman solved this problem around 196... | https://ocw.mit.edu/courses/2-160-identification-estimation-and-learning-spring-2006/52d79e15e4d3aa1add2641f941241070_lecture_5.pdf |
1
8
[
ı
x t
t − 1 = A E
1xıt
= At
−
1 xıt
1 + Gt − 1 w E
−
1wt
1 + Gt
1]
−
−
−
t − 1]
t
−
[
Ex
pected state based on xıt − 1
Estimated output
Form (9) and (10)
ıt =
y
ı
x H
t
t − 1
t
Note E[vt]=0
Correction of the state estimate
Assimilating the new measurement yt, we can update the state ... | 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 |
18.04 COURSE INTRODUCTION
2
0.4 Speed of the class
(Borrowed 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 b... | 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 |
� ⎤ dV = 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 ... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
2
∂ Φ
∂y
2 = 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
... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
kxe + D sin kxe
1
2
(
)
(
)
(
)
ky
-ky
ky
+ D 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,
(
... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
⎤
y ⎥
⎦
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
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 Z... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
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 Zah... | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2009/533990b05080585c0595ebfd764754f9_MIT6_641s09_lec10.pdf |
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