text stringlengths 30 4k | source stringlengths 60 201 |
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, there is a dense set of laws for which
the MVE is not unique and “no affine equivariant choice can be made.” The
following fact, then, is to some degree known, but it gives strong forms of
denseness. Parts (a) and (c) of the following give contamination neighbor-
C (P ), which are included in total variation neighbo... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
is a P ∈ Nλ
for α = 1/2.
Proof. (a) Let P have a continuous density f . If I1/2(P ) contains more than
one interval we are done, so suppose I1/2(P ) contains just one interval [a, b],
� x+1 f (u)du < 1/2 for x (cid:11)= 0. Take any δ
which we can assume is [0, 1]. Thus
x
with 0 < δ < 1. Another continuous density ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
x < β/2 we have
�
1+β gδ (u − 1)du < γ/2. Define g(1 + x)
1
� 1+x
d
dx x
hδ (u)du = (1 − δ)[f (x + 1) − f (x)] + δ gδ (x) −
�
�
1
2
= 0,
hδ (u)du = 1/2 for 0 ≤ x ≤ β/2. For β/2 < x ≤ β we then have by
� 1+x
x
so
definition of g(1 + x)
h
δ (u)du < 1/2.
(4)
x
�
x
Since
� 1+β
0
� 1+β
β
� 1+x hδ (u)du ≤
... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
both probability
2 (1 − δ + δ). So (4) holds for all
densities. For any x < 0,
� 1+x hδ (u)du = 1/2 for 0 ≤ x ≤ β/2. Thus for ζ with
x /∈ [0, β/2] while
density hδ , we have σSh,1/2(ζ) = 1 and I1/2(ζ) = {[x, x + 1] : 0 ≤ x ≤ β/2}.
� 1+x uhδ (u)du is a strictly increasing
� 1+x
Now
x
function of x for 0 ≤ x ≤ β... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
F0. For any δ > 0 let Qδ be the law with density fδ
where fδ (−x) ≡ fδ (x), fδ (ξ + t) = t/δ2 for 0 ≤ t ≤ δ and fδ (x) = 0 for
all other x > 0. For fixed λ ∈ (0, 1) and Pδ
:= (1 − λ)F0 + λQδ , the
√
unique interval [−η, η] with Pδ ([−η, η]) = 1/2 has length 2ξ + 2δ + o(δ) as
δ ↓ 0. But Pδ ([−ξ, ξ + δ]) > 1/2 for a... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
B. R. (1983). Uniqueness and Fr´echet differentiability of func-
tional solutions to maximum likelihood type equations. Ann. Statist.
11, 1196-1205.
[5] Davies, [P.] L. (1992a). The asymptotics of Rousseeuw’s minimum vol-
ume ellipsoid estimator. Ann. Statist. 20, 1828-1843.
[6] Davies, [P.] L. (1992b). An efficient Fr... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
Probability, 2d ed. Cambridge
University Press.
20
[12] D¨umbgen, L. (1997). The asymptotic behavior of Tyler’s M-estimator
of scatter in high dimension. Preprint.
[13] D¨umbgen, L. (1998). On Tyler’s M-functional of scatter in high dimen-
sion. Ann. Inst. Statist. Math. 50, 471-491.
[14] D¨umbgen, L., and Tyler... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
P. J. (1967). The behavior of maximum likelihood estimates un-
der nonstandard conditions. Proc. Fifth Berkeley Symp. Math. Statist.
Probability 1, 221-233. University of California Press, Berkeley and Los
Angeles.
[20] Huber, P. J. (1981). Robust Statistics. Wiley, New York. Reprinted,
2004.
[21] Kent, J. T., and... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
ariant estimators of multivariate location and covariance
matrices. Ann. Statist. 19, 229-248.
[26] Maronna, R. A. (1976). Robust M -estimators of multivariate location
and scatter. Ann. Statist. 4, 51-67.
[27] Maronna, R. A., and Yohai, V. J. (1995). The behavior of the Stahel-
Donoho robust multivariate estimator... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
the median
absolute deviation. J. Amer. Statist. Assoc. 88, 1273-1283.
[33] Rousseeuw, P. J., and Leroy, A. Robust Regression and Outlier Detec-
tion. Wiley, New York.
[34] Stigler, S. M. (1974). Linear functions of order statistics with smooth
weight functions. Ann. Statist. 2, 676-693; corr. ibid. 7 (1979), 466. ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
MIT 3.071
Amorphous Materials
2: Classes of Amorphous Materials
Juejun (JJ) Hu
1
Network formers, modifiers and intermediates
Glass network formers
Form the interconnected backbone glass network
Glass network modifiers
Present as ions to alter the glass network
Compensated by non-bridging oxyge... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
of corners
shared per SiO4 tetrahedron
In a glass with molar composition:
x Na2O · (1-x) SiO2
Number of NBO per mole: 2x
Number of BO per mole: 2-3x
Number of corners shared per
mole: (2-3x) × 2 = 4-6x
Number of tetrahedra per
mole: 1-x
Y = (4-6x) / (1-x)
Onset of inverted glass structure:
Y =... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
O- Na+
O
B-
Na+
B3 → B4 conversion
O
O
11
The Boron anomaly
n
o
r
o
b
d
e
t
a
n
d
r
o
o
c
d
o
f
-
4
l
i
f
o
n
o
i
t
c
a
r
F
B3 → B4
conversion
NBO formation
Molar fraction of alkali (%)
Initial addition of
alkali ions increases
network connectivity,
reduces CTE and
enhances thermal &
chemical resista... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
Glassware images © Pyrex. All rights reserved. This content is excluded from
our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Space shuttle tile coating
14
(Alkali) aluminosilicate glass
Aluminosilicate glass: x M2O · y Al2O3 · (1 - x - y) SiO2
SiO2 : glass former ... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
difference in atomic size ratios (> 12%)
Negative enthalpy of mixing
18
Other glass groups and glass formers
Phosphate glass: P2O5
Heavy metal oxide (HMO) and transition metal oxide glass
TeO2, PbO, Bi2O3, V2O5, TiO2, etc.
Halide glass and alloys
e.g. ZBLAN: ZrF4-BaF2-LaF3-AlF3-NaF
Chalcohalide,... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
bonds
Tetrahedral glasses: passivation of dangling bonds
Amorphous metals: w/o grain boundaries
21
Summary of oxide glass chemistry
No modifier
Alkali oxide
Alkaline earth
oxide
SiO2 (silicate)
B2O3 (borate)
Structural unit: SiO4
No NBOs, low or negative CTE,
high softening point, low
diffusivity
Ea... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
The WKB Approximation
Lectures Nine and Ten The WKB
Approximation
The WKB method is a powerful tool to obtain solutions for many physical problems. It is
generally applicable to problems of wave propagation in which the frequency of the wave is very
high or, equivalently, the wavelength of the wave is very short. ... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
1.
p
dx
This condition is satisfied if pΩxæ is of the form
pΩxæ è RPΩxæ,
where R is a large constant, i.e.,
R ;; 1,
and PΩxæ is of the order of unity. Indeed, if pΩxæ is given by (7.5), the inequality (7.4) is
(7.6)
1 d
R dx PΩxæ
1
ò 1.
(7.7)
(7.4)
(7.5)
Clearly, (7.7) is satisfied if R ;; 1, provided that... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
The WKB Approximation
The magnitude of these solutions varies with x like 1/ pΩxæ . The Wronskian of yWKB Ωxæ is now
ç
exactly a constant. (See homework problem 1.) It is therefore tempting to surmise that, under the
condition (7.4) or equivalently, (7.7), yWKB Ωxæ are even better approximations than e
ç
çi X pΩxæd... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
hold, let PΩxæ near x0 be
approximately given by
Then (7.7) requires
PΩxæ u aΩx ? x0æn , x u x0.
|x ? x0| ô
1/Ωn+1æ
.
n
Ra
(7.14)
(7.15)
Eq. (7.15) tells us how far away from x0 it must be for the WKB approximate solutions to be valid.
If PΩxæ vanishes in the way given by (7.14), we say that PΩxæ has an nth... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
in which the Schrodinger equation is reduced to
the Hamilton-Jacobi equation satisfied by the classical action of Newtonian mechanics.
The WKB approximation can also be used to solve problems in which the functional behavior is
— 2 —
The WKB Approximation
rapidly growing or rapidly decaying other than rapidly osc... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
when x is very large. In this problem, x inherently
contains a large parameter. Indeed, let x be of the order of ?, with ? ô 1. We may put
x è ?X,
where X is of the order of unity. Then the Airy equation is
2
d
dX2
Comparing with (7.16), we have, if X is positive,
? ?3X y : 0.
NΩXæ : RX1 ,
/2
where R è ?3/2 is... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
|x|
?1/4eç2|x|3/2/3
.
(7.22)
We conclude immediately that, when x is large and negative, one of these solutions is an
exponentially increasing function of x and the other is an exponentially decreasing function of x.
When x is positive, we have, comparing with (7.1),
/2
p : x1 .
Thus the WKB solutions are
both b... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
3).
Setting to zero the sum of terms in (7.39) which are proportional to K, we get
v
r +
1
PrΩxæ
2PΩxæ
v1 : ç
i
2PΩxæ
rr
.
1
PΩxæ
Solving this first-order linear equation, we find that
v1Ωxæ : ç
i
PΩxæ
X
1
PΩtæ
2
1
PΩtæ
rr
dt.
(7.39)
(7.40)
#
Now we are ready to give a justification of the WKB m... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
x0æn , then v1Ωxæ blows up like
Ωx ? x0æ?1?3n/2 ,
while v0Ωxæ blows up like Ωx ? x0æ?n 2. Thus (7.42) requires
1
1+næ .
R1/Ω
|x ? x0| ô
/
(7.43)
(7.44)
Aside from a multiplicative constant, (7.44) is the same condition as (7.15).
We may find all higher-order terms of the solution from (7.39). This is done by ga... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
and is given by (7.14) near x0, the condition (7.47) is
satisfied provided that x is sufficiently far away from x0 so that (7.15) is satisfied. (See homework
problem 3.)
Similarly, to obtain successive approximations to the WKB solutions of (7.16), we put
Then we have
çX NΩxædx
v.
y è e
vr +
r
N
2N
v : @
1
N... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
Tricky Asymptotics Fixed Point Notes.
18.385, MIT.
Rodolfo R. Rosales
.
October 31, 2000.
(cid:3)
Contents
1 Introduction.
2
2 Qualitative analysis.
2
3 Quantitative analysis, and failure for n = 2.
6
4 Resolution of the diÆculty in the case n = 2.
9
5 Exact solution of the orbit equation.
14
6 Commented Bibliography.
... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
illustrate how one can analyze the behavior
of the orbits near this linearly degenerate critical point and arrive at a qualitatively
correct
1
(cid:20)
description of the phase portrait. We will use for this \standard" asymptotic analysis techniques.
The case n = 2 is of particular interest, because then the standard t... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
critical points; because such systems tend to have homogeneous simple structures.
Tricky asymptotics (cid:12)xed point.
Notes: 18.385, MIT.
Rosales, Fall 2000.
3
Dipole Fixed Point: x = 2xy/n, y = y2 - x 2, n = 1.
t
t
2
1
y 0
-1
-2
-2
-1
0
x
1
2
Figure 1.1: Phase plane portrait for the Dipole Fixed Point system (1.... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
1
Tricky asymptotics (cid:12)xed point.
Notes: 18.385, MIT.
Rosales, Fall 2000.
4
sets of orbits that are attracted by the critical point (we will show this later), the system is not
conservative. In fact, this is an example of a reversible, non-conservative system with a minimum
number of critical points.
B.
The y -a... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
2xy
2
x
y
(cid:0)
!
A simple computation then shows that:
2
d
y
n
1
x
dy
n
y
1
=
+
+
2
2
2
dx
2 x
y
dx
2 x
y
!
!
(cid:0)
n
2
2
2
2
=
(2
n)y
+ nx
y
+ x
< 0 :
(2.2)
2
3
(cid:0)
(cid:0)
4x
y
(cid:16)
(cid:17) (cid:16)
(cid:17)
This shows that the orbits are (strictly) concave in this quadrant. Note, however, that
the ineq... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
, and x will increase,
but in such a fashion that the orbit diverges to in(cid:12)nity, without ever making it to the x-axis?
The answer to this is very simple: this would require the orbit to have an in(cid:13)ection point,
which it cannot have.
5
II.
Considering the (cid:13)ow backwards in time, we
0
< x < y
y
_
>
0
... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
(D)
al l
n >
Tricky asymptotics (cid:12)xed point.
Notes: 18.385, MIT.
Rosales, Fall 2000.
6
III.
With this information, and using the symmetries in (A), we can draw a
Conclusion.
qualitatively correct phase plane portrait, which will look as the one shown in (cid:12)gure 1.1. It
should be clear from this (cid:12)gure... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
mono-
dy
! (cid:0)1
dx
tonically. Thus, it must have a well de(cid:12)ned limit (which may be
; in fact, the aim here is
1
to show that this limit is
.)
1
b. Suppose that there is an orbit that does not approach the critical point vertically. Then, the
result in item (a) shows that we should be able to write
y
(cid:11)... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
Using this, we should be able to replace equation (2.1) by the approximation
y
x for 0 < x
1 :
(3.3)
(cid:29)
(cid:28)
dy
n y
ny
2
dx
2xy
2x
(cid:25)
=
:
(3.4)
This yields
y
(cid:12) x
;
(3.5)
n=2
(cid:25)
where (cid:12) is a constant. This last step is not rigorous, by a long shot, and we must be a bit careful
before ... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
the invariant lines mentioned in (B) earlier.
So, there is no contradiction (see remark 3.1 below for a brief description of what the
situation is when n > 2.)
c3. For n = 2, (3.5) is not consistent with (3.3). Since our proof that the orbits approach the
critical point vertically (which implies (3.3)) does apply for n... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
of the form
n=2
y = (cid:12) x
+ y
+ y
+ y
+ : : : ;
(3.6)
1
2
3
where y
y
, can be systematically computed.
n+1
n
(cid:28)
Tricky asymptotics (cid:12)xed point.
Notes: 18.385, MIT.
Rosales, Fall 2000.
9
Remark 3.1
n >
2
What happens when
.
The same methods that work for 0 < n < 2 can be used to study this case (but a... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
cid:12)gure it should be
clear that we stil l
I
= 2
have for the index:
.
Remark 3.2
0
< n <
2
.)
Rate of approach to the critical point (
Substituting (3.5) into (1.1), we obtain (near the critical point, where both x and y are smal l)
dx
2(cid:12)
dy
(n+2)=
2
2
x
;
and
y
;
dt
n
(cid:25)
dt
(cid:25)
where (in the seco... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
, MIT.
Rosales, Fall 2000.
10
= y2 - x 2, n = 5.
= 2xy/n, y
Dipole Fixed Point: x
t
t
2
1
y
0
-1
-2
-2
-1
0
x
1
2
Figure 3.1: Phase plane portrait for the Dipole Fixed Point system (1.1) for n = 5. The qualitative
details of the portrait do not change in the range 2 < n, but di(cid:11)er from those that apply in ... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
we can neglect the second term. That is
d(cid:11)
(cid:11)
x ;
as x
0 :
(4.3)
(cid:29)
dx
!
We also expect that (cid:11)
as x
0;
since we know that the orbits must approach the critical
point vertically.
! 1
!
Notice that this proposal provides a very clean explanation of how it is that the step
from (3.3) to (3.5), vi... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
)
dx
(cid:11)
(cid:0)
)
(cid:0)
q
where c is a constant. It is easy to see that this is consistent with (4.3).
8
9
10
That is,
=
in (3.5).
(cid:11)
(cid:12)
That is to say: plug (4.1) into equation (2.1) and then expand, using the fact that
is large.
(cid:11)
Note that this answer must be sub ject to the same type of b... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
the \standard" methods. However, just as in the
standard methods one can give a vague | and rather short | list of things to do (e.g.: balance
terms and look for pairs that may dominate, therefore simplifying the problem
) we provide below a
11
list of hints as to what one can do when faced with problems like the one w... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
:12) is a constant):
(cid:0)
1
(cid:15)
n
n
c
y
(cid:12) x
;
where
(cid:15) =
;
when n < n
= 2
(cid:0)
c
(cid:25)
2
and
p
1
(cid:15)
n
n
c
y
x ;
where
(cid:15) =
;
when n > n
= 2 :
(cid:0)
(cid:0)
c
(cid:25)
p
(cid:15)
2
In the (cid:12)rst case the limit behavior is (cid:12) x, but it is a very non-uniform limit near x... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
failed solution in the (cid:12)rst
place. If you are lucky, and clever enough, this might (cid:12)x the problem.
In the example studied here, the failure occurs for n = 2, when equation (3.4) becomes
dy
y
= ; with solution y = (cid:11)x ((cid:11) a constant.)
dx
x
This solution is inconsistent with the assumption y
x u... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
:
2
2
dy
n
2
y
=
nx :
dx
x
(cid:0)
(cid:0)
(cid:0)
n
Now multiply the equation by x
, and integrate again, to obtain (assume x > 0):
(cid:0)
2
n
dy
x
(cid:0)
1
n
=
nx
:
dx
(cid:0)
From this the following solutions follow:
Case
.
0
2
< n <
(cid:15)
(cid:15)
15
2
n
2
2
n
n
2R(2
n)
y = 2Rx
x ;
for 0
x
;
(5.1)
(cid:0)
(cid... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
the orbits giving closed loops in (cid:12)gure 3.1), or
2
n
2
n
y
= C x
+
x
;
for 0
x ;
(5.4)
n
2
(cid:20)
(cid:0)
where C
0 is a constant (these are the orbits that diverge to in(cid:12)nity in the sectors around
(cid:21)
the y -axis in (cid:12)gure 3.1.)
6 Commented Bibliography.
Below I list a few books that I think... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
, without excuses or unnecessary jargon,
this is the place to go. Of course, it is a bit old, and a lot of the new theory is not here |
Tricky asymptotics (cid:12)xed point.
Notes: 18.385, MIT.
Rosales, Fall 2000.
16
but you cannot really appreciate (or understand) any proof in the newer theory without this
backgrou... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
Key Concepts for section IV (Electrokinetics and Forces)
1: Debye layer, Zeta potential, Electrokinetics
2: Electrophoresis, Electroosmosis
3: Dielectrophoresis
4: Inter-Debye layer force, Van-Der Waals forces
5: Coupled systems, Scaling, Dimensionless Number
Goals of Part IV:
(1) Understand electrokinetic phenomena... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/254962a15f6db8cf8544bcfdb6e6b187_electrokin_lec3.pdf |
is
Streaming
potential
(aqueous) medium,
Flow velocity (vm)
Navier-Stokes’ equation
Slip boundary, zeta potential
x
x
EEOv
zE
+
+
+
+
+
+
+
+
+
+
+
- +
- - - - - - - - - - - - - - - - - - -
+
+
+
+
+
Slip (shear) boundary
δ
~ κ−
1
zv
(0)Φ
ξ
Φ
Stern layer
Zeta potential
Stern layer : adsorbed ions, linear potential dr... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/254962a15f6db8cf8544bcfdb6e6b187_electrokin_lec3.pdf |
e
q
2R
Drag force
r
F
drag
=
R vπ μ
6
ep
r
F
net
=
r
F
el
−
r
F
drag
=
r
qE
−
π μ
6
R v
ep
=
0
∴
v
ep
E
z
=
u
ep
=
q
π μ
6
R
This is wrong!
Electrophoresis : real picture
r
u E
ep
r
v
ep
=
counterion motion
-
-
-
+
+
+
-
-
+
+
+
-
-
-
particle motion
r
E
μep is a complex, electromechanically coupled process.
- E fiel... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/254962a15f6db8cf8544bcfdb6e6b187_electrokin_lec3.pdf |
, S., Kashima, A.,
Mochizuki, S., Noda, M.,
Kobayashi, K. Protein Eng. 12
pp. 439 (1999)
Brown, T., Leonard, G. A., Booth,
E. D., Chambers, J, J Mol Biol
207 pp. 455 (1989)
Polyelectrolyte electrophoresis : Free-draining
• When driven by an electric field
• DNA and counterions are
dragged in the opposite
direct... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/254962a15f6db8cf8544bcfdb6e6b187_electrokin_lec3.pdf |
0,000
– 90% of total serum protein: albumin and globulin (~mg/ml level)
– biomarkers and cytokines : 10ng/ml or less (up to 109 dynamic range)
Electrophoresis is a complicated electrokinetic phenomena.
(determined by zeta potential, not the net charge of the molecule)
Three images removed due to copyright restrictio... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/254962a15f6db8cf8544bcfdb6e6b187_electrokin_lec3.pdf |
Massachusetts Institute of Technology
Department of Materials Science and Engineering
77 Massachusetts Avenue, Cambridge MA 02139-4307
3.21 Kinetics of Materials—Spring 2006
February 17, 2006
Lecture 3: Driving Forces and Fluxes for Diffusion. Self-Diffusion and Interdiffusion.
1. Balluffi, Allen, and Carter, Kine... | https://ocw.mit.edu/courses/3-21-kinetic-processes-in-materials-spring-2006/2558a1b5bd136913559d72f5d3441919_ls3.pdf |
gradients. The motion of each species in a labora
tory frame fixed to the crystal follows Fick’s first law, with a proportionality constant known as the
intrinsic diffusivity. The intrinsic diffusivities and the self-diffusivities are related by KoM Eq. 3.13
and the relation involves a thermodynamic factor. Nonidealit... | https://ocw.mit.edu/courses/3-21-kinetic-processes-in-materials-spring-2006/2558a1b5bd136913559d72f5d3441919_ls3.pdf |
18.465 notes, R. Dudley, March 8, 2005, revised May 2
INTRODUCTION TO ROBUSTNESS: BREAKDOWN POINTS
Let X = (X1, ..., Xn) and Z = (Z1, ..., Zn) be samples of real numbers. For j = 1, ..., n
let X = j Z mean that Xi = Zi except for at most j values of i. More specifically, for
y = (y1, ..., yj) let X = j,y Z mean that... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
such that 0 ≤ σ < ∞. Examples
of statistics with values in [0, ∞) are (i) the sample standard deviation and (ii) the median
of all |Xi − m| where m is the sample median. A variant of the scale parameter space is
the open half-line 0 < σ < ∞. Both examples (i) and (ii) can take the value 0 for some
samples, so on su... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
ε ∗ (T ; X1, ..., Xn) = max{j : {T (Z) : Z = j X} is compact}.
1
n
∗
In other words ε (T, X) = j/n for the largest j for which there is some compact set K ⊂ Θ
∗
such that T (Z) ∈ K whenever Z = j X. If ε (T, X) doesn’t depend on X, which is often
the case, then let ε (T ) := ε (T, X) for all X.
∗
∗
Some authors de... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
0, 1/n, 2/n, ..., 1. A statistic with a breakdown point of 0 is (by
definition) not robust. Larger values of the breakdown point indicate more robustness, up
to breakdown point = 1/2 which is the maximum attainable in some problems.
Examples. (i) For the sample mean T = Z ¯ = (Z1 + ... + Zn)/n, the breakdown point is... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
X} = +∞ (let y1, ..., yn−j+1 →
+∞). It follows that ε (T, X) ≤ 1 min(j − 1, n − j).
∗
If Z = j−1 X then the smallest possible value of Z(j) occurs when yi < Xk for all
i and k and for at least one r such that Xr = X(1), Xr is not replaced, so Z(j) ≥ X(1).
Similarly, if Z = n−j X the largest possible value of Z(j) sa... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
Theorem 1, no other order statistic has any larger
at least
breakdown point than the median, so ε (X(j)) < 1/2 for all j. This is typical behavior
for interesting estimators. But, larger breakdown points are possible. If T has bounded
values, then it trivially has breakdown point 1 by our definition. Or, let T = min... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
X + θ. Then
T (Z) = T (Y ) + θ. So
(4)
|T (Z) − θ| ≤ M whenever Z = j X + θ, and then 2M ≤ T (Z) ≤ 4M.
But if j ≥ n/2 there is a Z with Z = j X and also Z = j X + θ. For such a Z, (3) and
�
(4) give a contradiction, proving Theorem 2.
REFERENCES
Frank R. Hampel, Peter J. Rousseeuw, Elvezio M. Ronchetti, and Wer... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
Software Metrics
1.
2.
Lord Kelvin, a physicist
George Miller, a psychologist
�
Software Metrics
Product vs. process
Most metrics are indirect:
No way to measure property directly or
Final product does not yet exist
For predicting, need a model of relationship of predicted variable
with other measurable va... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
of a connected graph G is the
number of linearly independent paths in the graph or
number of regions in a planar graph.
R1
R2
R3
R5
R4
Claimed to be a measure of testing diffiiculty and
reliability of modules.
McCabe recommends maximum V(G) of 10.
Static Analysis of Code (Problems)
Doesn’t change as prog... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
omain Models: Estimate program reliability
using test cases sampled from input domain.
Partition input domain into equivalence classes,
each of which usually associated with a program path.
Estimate conditional probability that program correct
for all possible inputs given it is correct for a specified
set of inp... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
08
24
422
68
227
8
197
255
357
134
365 1222
300
290
1064 1783
843
1461
0 3110
446
10
1160 1864
386
100
9
88
180
65
193
193
543
529
860
12
1247
122
1071
4116
2
670
10
176
6
236
10
281
983
261
943
990
371
91
120
1146
58
79
31
16
160
707
1800
700
948
790
112
2... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
70 80 90 100 110 120 130
Different models can give varying results for the same
data; there is no way to know a priori which model
will provide the best results in a given situation.
‘‘The nature of the software engineering process is too
poorly understood to provide a basis for selecting a
particular model."
� ... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
project.
What is a source line of code? (declarations? comments? macros?)
How treat source lines containing more than a single statement?
More productive when use assembly language? (the more expressive
the language, the lower the apparent productivity)
All tasks subsumed under coding task although coding time rep... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
internal logic of module.
Actual data binding:
- Used data binding where p assigns value to x and q references it.
- Hardest to compute but indicates information flow from p to q.
Software Design Metrics (3)
Cohesion metric
Construct flow graph for module.
- Each vertex is an executable statement.
- For ea... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
size of project (primarily in terms
of estimated lines of code) and type of project (organic, semi-detached,
or embedded).
Effort = A * KDSI b
where A and b are constants that vary with type of project.
More advanced versions add a series of multipliers for other factors:
product attributes (reliability, database... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
it later.
Evaluation of Management Metrics (3)
Programmer ability swamps all other factors in factor analyses.
Accurate schedule and cost estimates are primarily influenced by
the experience level of those making them.
Warning about using any software metrics:
Be careful not to ignore the most important fa... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.005 Elements of Software Construction
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Today’s Topics
principles and concepts of system design
¾ modularity
¾ decoupling
¾¾ information hiding
information hiding
a... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
=noa");
}
public String getOpen () {return open;}
public String getAsk () {return ask;}
public int getChange () {return change;}
public void obtainQuote () throws IOException {
BufferedReader in = new BufferedReader(new
String csv = in.readLine();
St i
i
in.close();
dLi
()
Quoter is a state machine.
Draw it. What... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
choice?
choice?
}
out.close();
}
}
© Robert Miller 2007
Modularity and Decoupling
modularity is essential for managing complexity
¾ system is divided into parts (modules) that can be handled separately and
recombined in other combinations
coupling
coupling
¾ degree of dependence between parts of the system
¾ an ... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
a state machine!
¾ two versions, one RTF and one HTML
¾ but same interface
© Robert Miller 2007
Generator Machine
key design idea
(cid:190) develop generic interface for text formatting
OPEN
write,
newline
close
PLAIN
ROMAN
toggleBold
toggleBold
toggleItalic
toggleItalic
CLOSED
CLOSED
BOLD
BOLD
ITALIC
ITALIC
© Robe... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
oupled
¾ so we can plug in different generators
¾ without changing the formatter s code
¾ without changing the formatter’s code
solution
¾ formatter doesn’t refer to a particular generator class
¾ it refers to an interface instead
Interfaces, in Pictures
what we want
¾ two ways to configure formatter
how does form... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
Symbol ("MSFT");
formatter.generateOutput ();
plugin is selected here
plugin is selected here
Generator htmlg = new HTMLGenerator ("myQuotes.html");
formatter = new QuoteFormatter(htmlg);
formatter.addSymbol ("AAPL");
formatter.addSymbol ("INTC");
formatter.addSymbol ("JAVA");
formatter.addSymbol ("MSFT");
formatter.g... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
recording = new LinkedList<NoteEvent> ();
recording.add(...);
“marker” interfaces
¾ declare no methods
¾ declare no methods
¾ used to expose specification properties (e.g. java.util.RandomAccess)
¾ or as a hack to add functionality (e.g. java.io.Serializable)
Summary
system design principles
¾ modularity
¾ decoupl... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.727 Topics in Algebraic Geometry: Algebraic Surfaces
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
ALGEBRAIC SURFACES, LECTURE 6
LECTURES: ABHINAV KUMAR
Corollary 1. Assume that all the closed fibers of the ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
by the proposition below (about
Pic (P(E))), v∗OP(E�)(1) ∼= π∗(L) ⊗ OP(E)(S), where s ∈ Z, π : P(E) → B (resp.
π� : P(E�) → B) are the canonical projections. L is an invertible OB -module,
and
, so we get a locally free sheaf of rank 2.
(1)
E� ∼= π∗
� OP(E�)(1) ∼
= π
∼
= L ⊗ π
∗v∗OP(E�)(1) ∼
= π
∗OP(E)(S) ∼
∗(π∗(... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
sheaf of nonabelian groups
defined by G(U ) = Aut U (U × P1) = {morphisms of U into PGL 2(k)} Now, let
G = PGL 2(OB): we have
(2)
1 → O∗
B → GL 2(OB ) PGL 2(OB )
1
→
→
1
2
LECTURES: ABHINAV KUMAR
giving the associated long exact sequence
(3) H 1(B, O∗ → H 1(B, GL 2(OB )) →
B )
H 1(B, PGL 2(OB))
→ H 1(B, O∗ ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
−
i
least one being > 0 (because F is connected), so the sum is negative.
nj (Ci − Cj ). (Ci · Cj) ≥ 0, with at
�
Lemma 3. Let X be a minimal surface, B a smooth curve, π : X
B a→
morphism with generic fiber isomorphic to P1 . Then X is geometrically ruled by
π.
Proof. Let F be a fiber of π: then F 2 = 0 = ⇒ F ·K =... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
minimal
models of B × P1 are exactly the geometrically ruled surfaces over B, i.e. the
P1-bundles PB(E).
Proof. Let π : X
→
C be geometrically ruled. If E is an exceptional curve, then
E cannot be a fiber of π since E2 = −1. So π(E) = B, which is not possible
since E is rational and B has higher genus. Thus, X is m... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
∈ X, consider the corresponding line D ⊂ Eπ(X), and let NX = D. The bundle
OX (1) (the tautological bundle on X) is defined by
→
(4)
0 → N → π∗E → OX (1) → 0
→
→
B a morphism. If there is a morphism g : Y
Let Y be any variety, f : Y
→
P(E) s.t. degree commutes, then we can associate a line bundle L = g∗OX (1)
and t... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
·
Proof. For (a), let h be the class of OX (1). It is clear that h f = 1. Now, let
D ∈ Pic X, n = D f, D� = D − nh so D� f = 0. It is enough to show that
D� is the pullback under π∗ of a divisor on B. Let Dn = D� + nF for F a fiber,
D2 = D2 . Also, Dn ·K = D� ·K +nF ·K = D� ·K −2n, and
h0(K −Dn) = 0 for n
1
sufficie... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
for E ⊗ H, tensoring by H −1
gives the statement for E. So let s1, . . . , sk be global sections which generate E.
We claim that there is an element in the span of these sections s.t. sb ∈/ mbEb
for every b ∈ B. Consider the incidence correspondence Σ ⊂ B × Pk−1 =
{(b, s)|s(b) = 0}. Σ is an irreducible variety of d... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
6.231 DYNAMIC PROGRAMMING
LECTURE 14
LECTURE OUTLINE
We start a ten-lecture sequence on advanced
•
infinite horizon DP and approximation methods
We allow infinite state space, so the stochastic
•
shortest path framework cannot be used any more
Results are rigorous assuming a finite or count-
•
able disturbance space
−
−
T... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
−
−
If for two functions J and J ′ we have J(x) =
J ′(x) for all x, we write J = J ′
If for two functions J and J ′ we have J(x)
J ′(x) for all x, we write J
For a sequence
Jk
{
all x, we write Jk
J ′
≤
with Jk(x)
J; also J ∗ = minπ Jπ
J(x) for
→
≤
Shorthand notation for DP mappings (operate
•
on functions of state to ... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
cost function of π
Tµ0Tµ1 · · ·
for the N -stage problem with terminal cost
αN J
−
−
−
For any function J:
−
−
−
T J is the optimal cost function of the one-
stage problem with terminal cost function
αJ
T 2J (i.e., T applied to T J) is the optimal
cost function of the two-stage problem with
terminal cost α2J
T N J is t... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
r, and any µ
T (J + re)
(x) = (T J)(x) + αr,
x,
∀
(cid:0)
Tµ(J + re)
(cid:1)
(x) = (TµJ)(x) + αr,
x,
∀
1] (holds for
(cid:0)
where e is the unit function [e(x)
most DP models).
(cid:1)
≡
A third important property that holds for some
•
(but not all) DP models is that T and Tµ are con-
traction mappings (more on this la... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
)
J ∗(x) +
≤
αN M
α
1
−
,
where J0(x)
0 and M
g(x, u, w)
.
≥ |
Apply T to this relation and use Monotonicity
≡
|
•
and Constant Shift,
(T J ∗)(x)
αN +1M
−
1
−
α ≤
(
T N +1J0)(x)
(T J ∗)(x) +
≤
αN +1M
α
1
−
Take limit as N
•
→ ∞
and use the fact
lim (T N +1J0)(x) = J ∗(x)
N→∞
to obtain J ∗ = T J ∗. Q.E.D.
8 THE CONTRACT... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
)
≤
(T J)(x) + αc,
x
∀
Hence
(T J)(x)
−
(T J ′)(x)
αc,
≤
x.
∀
Similar for Tµ. Q.E.D.
(cid:12)
(cid:12)
(cid:12)
(cid:12)
9 IMPLICATIONS OF CONTRACTION PROPERTY
We can strengthen our earlier result:
•
Bellman’s equation J = T J has a unique solu-
•
tion, namely J ∗, and for any bounded J, we have
lim (T kJ)(x) = J ∗(x),... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
.,
T J ∗ = TµJ ∗.
Proof: If T J ∗ = TµJ ∗, then using Bellman’s equa-
tion (J ∗ = T J ∗), we have
J ∗ = T J ∗
µ
,
so by uniqueness of the fixed point of Tµ, we obtain
J ∗ = Jµ; i.e., µ is optimal.
Conversely, if the stationary policy µ is optimal,
•
we have J ∗ = Jµ, so
J ∗ = TµJ ∗.
Combining this with Bellman’s equatio... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
j) + αJ ∗(j)
j=1
X
(cid:0)
(cid:1)
for all (i, u)
Q∗(i, u) is called the optimal Q-factor of (i, u)
Q-factors have optimal cost interpretation in
•
an “augmented” problem whose states are i and
U (i) - the optimal cost vector is (J ∗, Q∗)
(i, u), u
∈
The Bellman Eq. is J ∗ = T J ∗, Q∗ = F Q∗ where
•
•
n
(F Q∗)(i, u) =
... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.231 Dynamic Programming and Stochastic Control
Fall 2015
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
MIT 6.972 Algebraic techniques and semidefinite optimization
February 23, 2006
Lecturer: Pablo A. Parrilo
Scribe: Noah Stein
Lecture 5
In this lecture we study univariate polynomials, particularly questions regarding the existence of
roots and nonnegativity conditions.
• When does a univariate polynomial have onl... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
roots
How many real roots does a polynomial have? There are many options, ranging from all roots being
real (e.g., (x − 1)(x − 2) . . . (x − n)), to all roots being complex (e.g., x2d + 1). We will give a couple of
different characterizations of the location of the roots of a polynomial, both of them in terms of some... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
do this by computing instead the eigenvalues of the associated (nonsymmetric)
companion matrix. In fact, that is exactly the way that MATLAB computes roots of polynomials; see
the source file roots.m.
For any A ∈ Cn×n, we always have TrA =
�
n
i=1
k = Tr p ]. As a consequence of linearity, we have that if q(x) =
... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
ring R[x]/�p(x)�. This will enable a very appealing generalization of companion
matrices to multivariate polynomials, in the case where the underlying system has only a finite number
of solutions (i.e., a “zero dimensional ideal”).
3.2
Inertia and signature
Definition 5. Consider a symmetric matrix A. The inertia of... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
fact that we will be using
symmetric matrices.
Let q(x) be a fixed auxiliary polynomial. Consider the following n × n symmetric Hankel matrix
Hq(p) with complex entries defined by
Like every symmetric matrix, Hq(p) defines an associated quadratic form via
i=1
[Hq(p)]jk =
n
�
j+k−2
q(xi)xi
.
(3)
⎥
⎥
⎥
⎦
⎡
f0
f... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
when p(x)
is monic, the entries of Hq(p) are actually polynomials in the coefficients of p(x). Notice that we have
used (2) in the derivation of the last step.
Define now the n × n Vandermonde matrix
V =
⎡
1 x1
1 x2
⎢
⎢
.
.
⎢
.
.
⎣
.
.
1 xn
⎤
⎥
⎥
⎥
⎦
n−1
n−1
. . . x1
. . . x2
.
.
.
.
.
.
. . . xn−1
n
... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
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