text stringlengths 16 3.88k | source stringlengths 60 201 |
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< 1. The second conclusion
bounds the usual contamination breakdown point at a general law F0, e.g. a
normal law, assuming the functional T is defined and continuous at a related
(1 − γ)-degenerate law. Such an assumption seems not to hold for many
location and scatter functionals given in the literature for γ ≤ 1/2... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
linear subspace H := {x : x1 = 0} via M0. Suppose
15
that for some γ ∈ (0, 1) and law ρ on Rd, the law P := (1−γ)F˜0+γρ ∈ D and
if T = Σ, Σ(P ) is non-singular, or if T = µ, µ(P ) /∈ H. Then εR(T, P ) ≤ γ.
If in addition, T (·) is weakly continuous at P on D, and if for all a > 0,
∗
(1 − γ)F0 + γρ ◦ M −1 ∈ D, then... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
second conclusion follows. 2
a
For γ = 1/(d + 1), the hypothesis µ(P ) /∈ H of Theorem 7 holds by
Theorem 1(d) with n = d + 1 and P = Pd+1 an empirical measure if P ∈ D.
3 Univariate trimming and the shorth
Let J be a probability density function on [0, 1] such that J(y) ≡ J(1 − y)
for 0 ≤ y ≤ 1 and for some α > 0... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
J (Q)2
16
2 (Q) if desired by a constant
c >
exist and are finite. One may multiply σ
1
J
so that if Q is the standard normal distribution then cσ2 (Q) = 1.
It is
straightforward to verify that with or without such a multiplication, the two
functionals are defined for an arbitrary law Q on R and are affinely equivari... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
∗ to 2α,
for α < 1/4. Asymmetric trimming works well to prune asymmetric outlier
contamination from a symmetric true distribution [23], but apparently not
so well for an asymmetric true distribution.
C
−
There are various multivariate extensions of trimming, e.g. Donoho and
Gasko [10] and Liu, Parelius and Singh [... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
6] calls the “middle of the shortest half”
functional; Rousseeuw and Leroy [33, p. 169] call it the “least median of
:= µ and mSh,α(P )
17
squares” (LMS) functional, specializing a form of regression. Also, µSh,α(P )
is called the α-shorth of P and µSh(P ) := µSh,1/2(P ) is the shorth of P .
The LMS location func... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
weak convergence.
∗
ε
Proposition 8. (a) For any law P on R having a continuous density f and
any ε > 0, there is a law ζ ∈ N C (P ), also with a continuous density, for
which I1/2(ζ) contains more than one interval and so µSh(ζ) and mSh,1/2(ζ)
are not defined. Thus δC (mSh,1/2, P ) = 0, also if contamination neigh... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
Let
� 1 hδ (x)dx = 1/2.
g(x) = 1/2 for 0 ≤ x ≤ 1 and hδ := (1 − δ)f + δg. Then
For x > 0 let
0
gδ (x) :=
[f (x) − f (x + 1)] +
1 − δ
δ
1
.
2
18
� 1+x
x
x
� x+1 f (u)du = f (x + 1) − f (x) = 0 when x = 0, so f (0) =
We have (d/dx)
f (1). There is a γ > 0 such that γ < 1/2 and (1 − δ)[f (u + 1) − f (u)]... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
)
h
δ (u)du < 1/2.
(4)
x
�
x
Since
� 1+β
0
� 1+β
β
� 1+x hδ (u)du ≤
�
1+x g(u)du < 1/2 (for x ≤ 1 or x > 1), so (4) holds. To
For x > γ,
x
prove (4) for β < x ≤ γ it suffices to show
hδ (u)du, or
� 1+x
�
x −
]hδ (u) ≥ 0, or β (1 − δ)[f (u) − f (1 + u)] + δ du ≥ 0, which holds
[
1+β
β
by choice of γ.
... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
0 ≤ x ≤ β/2, so (a) is proved.
� 1+x hδ (u)du < 1
� 1+x uhδ (u)du/
hδ (u)du = 2
� ∞
x
x
x
x
2
1
For (b), if P exists, the conclusions follow from Theorem 1(c). To show
P exists, for 0 < α ≤ 1/2, let P = 2(U[0, 1] + U[3, 4]). For 1/2 < α < 1,
let P = (2α − 1)δ0 + (1 − α)(δ−1 + δ1). Then m = 0, σSh,α(P ) = 1, an... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
Pδ ([−ξ, ξ + δ]) > 1/2 for a shorter interval, proving (c) and the
proposition. 2
19
References
[1] Bassett, G. W. (1991). Equivariant, monotonic, 50% breakdown esti-
mators. American Statistician 45, 135-137.
[2] Bickel, P. J., and Lehmann, E. L. (1975). Descriptive statistics for
nonparametric models. I, Intro... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
). Desirable properties, breakdown and efficiency
in the linear regression model. Statistics and Probability Letters 19,
361-370.
[9] Davies, P. L. (1998). On locally uniformly linearizable high breakdown
location and scale functionals. Ann. Statist. 26, 1103-1125.
[10] Donoho, D. L., and Gasko, M. (1992). Breakdown... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
uw, P. J., and Stahel, W. A.
(1986). Robust Statistics: The Approach Based on Influence Functions.
Wiley, New York.
[18] Helmers, R. (1980). Edgeworth expansions for linear combinations of
order statistics with smooth weight functions. Ann. Statist. 8, 1361-
1374.
[19] Huber, P. J. (1967). The behavior of maximum l... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
Lopuha¨a, H. P., and Rousseeuw, P. J. (1991). Breakdown points of
affine equivariant 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... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/25202c93390aa2041da61d7a86594e7a_location_scatter.pdf |
, 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.
[35] Tatsuoka, K. S., and Tyler, D. E. (2000). On the uniq... | 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 |
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 = 2, x = 0.5
)
y
t
i
s
o
c
s
i
v
(
0
1
g
o
l
Viscosity isotherms
Y = 2
Y
8
... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
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 resistance
12
Various properties of lithium borate glass
More than two
structura... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
�� SiO2 : glass former
Al functions as a glass former in the form of AlO4 groups
(when x > y)
Each excess Al atom creates three NBOs (when x < y)
y = x
r
o
f
y
g
r
e
n
e
n
o
i
t
a
v
i
t
c
A
y
t
i
v
i
t
c
u
d
n
o
c
C
D
Al3+ in an oxygen octahedron
y / x
15
Chalcogenide glass (ChG)
Reduced mecha... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
cohalide, oxyhalide, etc.
Amorphous minerals
Opal, biominerals
and many others…
Amorphous calcium
carbonate in lobster
carapace
“The Formula for Lobster
Shell,” Max Planck Research
19
Representation of glass composition
In oxide glasses, the convention is to list the glass
network modifiers in increa... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/25224366c14974b897ee93b27def6225_MIT3_071F15_Lecture2.pdf |
No NBOs, corrugated layered
structure
Each ion creates 1 B4 group or
1 NBO; extremum in glass
properties (boron anomaly)
Each ion creates 2 B4 groups
or 2 NBOs; extremum in glass
properties (boron anomaly)
Alumina (Al2O3)
Each ion creates 3 NBOs; in the
presence of alkali ions Al3 → Al4
conversion occurs
B2... | 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 |
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 x is not near a zero of PΩx... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
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 pΩxædx
.
To see if this is true, we put
Ω7.9æ
çi... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
xæ 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-order zero at x0. Not all
zeroes of PΩxæ are of finite order, and an exa... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
— 2 —
The WKB Approximation
rapidly growing or rapidly decaying other than rapidly oscillatory, an example being the problems of
boundary layer which we will discuss in Chapter 9. Consider the equation
The WKB solutions are given by
yrr ? N2y : 0.
çyWKB Ωxæ :
çX NΩxædx
.
e
1
NΩxæ
These solutions are good a... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
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 the large paramater.
We also note that the integral X pdx is dimensionless and hence does not change with a change of
the scale of the independent variable. In the example of the Airy equation, we have
X RX1... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
.
Thus the WKB solutions are
both being oscillatory functions of x.
x?1/4eçi2x3/2/3,
(7.23)
C Higher-order WKB Approximation
We shall in this section find the higher-order terms of the WKB approximation.
For this purpose let us return to eq. (7.11). Since this equation has a small parameter K and is
linear, it ... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
1
PΩtæ
rr
dt.
(7.39)
(7.40)
#
Now we are ready to give a justification of the WKB method, which is approximating the solution of
(7.1) by truncating the series of (7.38). Strictly speaking, truncating a series is justified if we succeed
in proving that the sum of terms neglected is much less than the sum of term... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
higher-order terms of the solution from (7.39). This is done by gathering all the
terms in (7.39) proportional to Km and setting the sum to zero. We get
Thus
r + Pr
vm
2P
vm : ç i vm?1.
2P
rr
vmΩxæ : ç
i
PΩxæ
X dx
d2
PΩxæ dx2
2
vm?1Ωxæ.
(7.45)
(7.46)
From (7.46), we obtain the mth-order term of the per... | https://ocw.mit.edu/courses/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004/252a381c174fe21a39b59e15b9d173cd_nineten.pdf |
of the WKB solutions.
Homework due next Monday (Oct 18, 04) :
Chapter 7,
4a, 5b, 7b.
— 5 — | 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 |
portrait. We will use for this \standard" asymptotic analysis techniques.
The case n = 2 is of particular interest, because then the standard techniques fail, and some
extra tricks are needed to make things work.
Just so we know what we are dealing with, a computer made phase portrait for the system
2
(case n = 1) is s... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
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 (1.1) for n = 1. The qualitative
details of the portrait do not change in the range 0 < n
2. However, for n > 2 di(cid:11)erences arise.
(cid:20)
The last set of symmetries (A3) shows that we need only compute... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
. Along it the (cid:13)ow is
Simple invariant curves.
(cid:17)
in the direction of increasing y , with vanishing derivative at the origin only. This invariant
line is clearly seen in (cid:12)gure 1.1.
For n > 2, two further (simple) invariant lines are: y =
x.
pn
(cid:6)
n
2
p
(cid:0)
Whenever a one parameter family of... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
)
(cid:17)
This shows that the orbits are (strictly) concave in this quadrant. Note, however, that
the inequality breaks down for n > 2. Then the orbits are concave for (n
2)y
< nx
and
2
2
(cid:0)
convex for (n
2)y
> nx
.
2
2
(cid:0)
6
6
Tricky asymptotics (cid:12)xed point.
Notes: 18.385, MIT.
Rosales, Fall 200... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
it possible for such an orbit to reach in(cid:12)nity without ever leaving this
region? | in fact, this is precisely what happens when n > 2, when all the orbits in the region
p
p
n
2 y >
n x do this. Figure 1.1 seems to indicate that this is not the case, but: how can
(cid:0)
be sure that a very thin pencil of orbits ... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
to \extremes" (whatever this means). 100% mathematical rigor in
calculations like the ones that follow is possible in simple examples like the one we are doing | and
not even very hard | but quickly becomes prohibitive as the complexity of the problems increases.
But the type of techniques and way of thinking that we f... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
11)
(n
2)(cid:11)
= n ;
(3.2)
2
(cid:11)
(cid:0)
()
(cid:0)
(cid:18)
(cid:19)
which has no solution for 0 < n
2! It follows that an orbit approaching the critical point at
(cid:20)
a (cid:12)nite slope cannot occur | which is precisely what we wanted to show.
We now become more ambitious and ask the question: How exact... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
(cid:12)xed point.
Notes: 18.385, MIT.
Rosales, Fall 2000.
8
c.
For example:
Consistency with known facts.
c1. For 0 < n < 2, (3.5) is consistent with (3.3).
c2. For n > 2, (3.5) is not consistent with (3.3). However, our proof that the orbits approach
the critical point vertically (which is what (3.3) is based on) doe... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
1
and assume y
(cid:12) x
. Then use this to get an approximate equation for y
, solve it, and
1
1
n=2
(cid:28)
n=2
check that, indeed: y
(cid:12) x
.
1
(cid:28)
In the case 0 < n < 2 (the only one worth doing this for, since the other cases have already
failed the two prior tests) one can do not only this, but repeat ... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
(cid:0)
j
j
(cid:6)
j
j
p
p
n
2 y =
n x) and in(cid:12)nity at the other (ending paral lel to the y -axis there). In between their
(cid:0)
(cid:6)
slopes vary steadily (no in(cid:13)ection points) from one limit to the other.
Figure 3.1 shows a typical phase plane portrait for the n > 2 case. From the (cid:12)gure it s... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
7
works in (cid:12)xing some problems like the one we have.
7
In fact, there are some open research problems that have to do with failures of this type, albeit in contexts quite
a bit more complicated than this one.
Tricky asymptotics (cid:12)xed point.
Notes: 18.385, MIT.
Rosales, Fall 2000.
10
= y2 - x 2, n = 5.
... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
!
when we calculate the derivative
dy
d(cid:11)
= (cid:11) +
x ;
(4.2)
dx
dx
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 provide... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
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 basic check... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
, 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 we treat in this section. In
the end, though, each problem is its own thing and (at least with our present level of understanding)
the only way to learn how to do these things \well" is ... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
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 = 0 (see
what happens with the derivatives.) In the second case there is not even a limit.
The solutions for both cases, however, have the c... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
in the solution ((cid:11)) a slow function of x, with
13
14
Given by \standard" techniques.
These functions should also have properties (e.g.: large, small, ...
in some limit) that make the assumed form
consistent with the approximations that lead to the equations they solve.
Tricky asymptotics (cid:12)xed point.
Note... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
(cid:0)
(cid:20)
(cid:20)
2
n
n
!
(cid:0)
1
where R > 0 is a constant. For n = 1 these are circles of radius R, centered at (x; y ) = (R; 0).
Case
.
n
= 2
2
2
y
= (2 ln(x
)
2 ln(x)) x
;
for 0
x
x
;
(5.2)
0
0
(cid:0)
(cid:20)
(cid:20)
where x
> 0 is a constant.
0
For 2-D problems all sorts of theoretician luxuries are a... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/252b9c04191407857f471f018a6487f8_MIT18_385JF14_Tricky_Point.pdf |
.)
It aims at realistic scienti(cid:12)c problems, so it deals mostly with PDE’s (not ODE’s).
2. Bender, C. M., and Orszag, S. A. (1978). Advanced Mathematical Methods for Scientists and
Engineers, McGraw-Hill, New York.
This book has an extensive treatment of many of the ideas in asymptotic (and other) methods,
with m... | 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 |
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 drop
Gouy-Chapman layer : diffuse-double layer
exponential drop
Shear boundary : vz=0
Na... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/254962a15f6db8cf8544bcfdb6e6b187_electrokin_lec3.pdf |
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 field is distorted around the particle.
- Counterions are m... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/254962a15f6db8cf8544bcfdb6e6b187_electrokin_lec3.pdf |
. 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
direction
• Hydrodynamic interaction
screened
• Friction with solvents occurs at
every monomers
• ζ
friction
∼ Ν
• When driven by a ... | https://ocw.mit.edu/courses/20-330j-fields-forces-and-flows-in-biological-systems-spring-2007/254962a15f6db8cf8544bcfdb6e6b187_electrokin_lec3.pdf |
): see Figure 4-42.
• Isoelectric focusing (charge-based separation): see Figure 4-44.
• 2D protein separation: see Figure 4-45. | 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 |
the
intrinsic diffusivity. The intrinsic diffusivities and the self-diffusivities are related by KoM Eq. 3.13
and the relation involves a thermodynamic factor. Nonideality can either accelerate or retard interdif
fusion kinetics, relative to kinetics measured in the absence of a chemical concentration gradient.
• I... | 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 |
(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 such samples, these statistics are undefined if the scale parameter space ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
∗
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 define the breakdown point instead in terms of the smallest number
of r... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
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
0 for any Zj since for j = 1, if we let y1 → ∞ then Z ¯ → ∞ (for n fixed).
(ii) Let T = Z(1), the smallest numb... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
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) satisfies Z(j) ≤ X(n). So if
k = min(j − 1, n − j) and Z = k X, then X(1) ≤ Z(j) ≤ X(n) so Z(j) is bounded and
∗
ε (T, ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
If T has bounded
values, then it trivially has breakdown point 1 by our definition. Or, let T = minj |Zj|.
Then one can check that T has breakdown point 1 − 1 . n
1 − 1
n
2
For real-valued observations Z1, . . . , Zn, a real-valued statistic T = T (Z1, ..., Zn) will
be called equivariant for location if for all re... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/2575baf3b9f95fbcae3c35b4964a71e0_breakdown.pdf |
�. For such a Z, (3) and
�
(4) give a contradiction, proving Theorem 2.
REFERENCES
Frank R. Hampel, Peter J. Rousseeuw, Elvezio M. Ronchetti, and Werner A. Stahel
(1986). Robust Statistics: The Approach based on Influence Functions. Wiley, New York.
Peter J. Huber (1981) Robust Statistics. Wiley, New York.
3 | 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 |
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 program changes.
High correlation with program size.
No real intuitive reason for many of metrics.
Ignor... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
all possible inputs given it is correct for a specified
set of inputs.
Assumes outcome of test case given information about
behavior for other points close to test point.
Reliability Growth Models: Try to determine future
time between failures.
�
�
Reliability Growth Models
Software Reliability: The probabilit... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
707
1800
700
948
790
112
26
600
457
816
369
529
828
33
865
875
1082
6150
15
114
15
300
1351
748
379
1011
868
1435
245
22
3321
Execution time in seconds between successive failures.
(Read left to right in rows).
�
Using the Models
2400
2200
2000
1800
1600
1400
1200
1000
800 ... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
productive than if produce reliable and easy to maintain software
(measure only over software development phase).
More does not always mean better.
May ultimately involve increased system maintenance costs.
Common measures:
Lines of source code written per programmer month.
Object instructions produced per programme... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
X is variable
within scope of both p and q
Potential data binding:
- X declared in both, but does not check to see if accessed.
- Reflects possibility that p and q might communicate through the
shared variable.
Used data binding:
- A potential data binding where p and q use X.
- Harder to compute than potential d... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
Algorithmic Cost Modeling
Build model by analyzing the costs and attributes of completed projects.
Dozens of these around -- most well-known is COCOMO.
Assumes software requirements relatively stable and project will be
well managed.
Basic COCOMO uses estimated size of project (primarily in terms
of estimated... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/2595db843b206fcaa30fc563a3c1db94_cnotes7.pdf |
rapid buildup of staff correlates with project schedule slippages
- Throwing people at a late project will only make 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 ... | 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 |
csv = in.readLine();
St i
i
in.close();
dLi
()
Quoter is a state machine.
Draw it. What design
pattern does it use?
InputStreamReader(url.open..));
BufferedReader is also
BufferedReader is also
a state machine.
a state machine.
Draw it. What design
Draw it. What design
pattern does it use?
pattern does ... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
important measurement of modularity
decoupling achieved so far
¾ the website (Yahoo) and its format (CSV) have been decoupled from the
rest of the system
rest of the system
next step
¾ design the part of the system that generates the report
¾ report can be either HTML or RTF
© Robert Miller 2007
Design Option #2
... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
ize each design decision in exactly one place
¾ more crudely:“don’t repeat yourself”
why?why?
¾ ready for change: if decision needs to change, there’s only one place
¾ ease of understanding: don’t have to think about the details of that decision
when working on the rest of the system
¾ safety from bugs: fewer pla... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
Exception;
public void close ();
public void newLine ();
public void toggleBold ();
public void toggleItalic ();
public void write (String s);
}
public class RTFGenerator implements Generator {
public void open() throws FileNotFoundException { ... }
...}
public class HTMLGenerator implements Generator {
public void... | https://ocw.mit.edu/courses/6-005-elements-of-software-construction-fall-2008/259aaebfa525dbb0878464c24a9735f3_MIT6_005f08_lec08.pdf |
oter
HTMLGenerator
HTMLGenerator
RTFGenerator
RTFGenerator
obtains quotes
obtains quotes
formats text in HTML
formats text in HTML
formats text in RTF
formats text in RTF
© Robert Miller 2007
© Robert Miller 2007
Exercise
which modules would you need to modify to...
¾ handle new RTF syntax for italics?
¾ put t... | 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 |
) ∼= 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∗(L) ⊗ OP(E)(S))
= L ⊗ Sym sE
Comparing ranks, we see that... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
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∗
B )
The first object is Pic (B), the last is 0 since B is a curve, while the second... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
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 = −2 by the genus formula.
�
niCi: applying the genus formula and the above lemma,
If F is reducible, F =
2 = −1(−2 ≤ 2g(Ci)... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
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 minimal.
⇒ ·
π : X
Conversely, suppose X is minimal and φ : X ��� B × P1 is birational. Let
q : B ×P1 B be the projection, and consider q φ. There is a diagram factoring
this map through ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
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 the surjective morphism g∗u : g∗π∗E = f ∗E
L. Conversely, given a
line bundle L on Y and a surjective morphism v : f ∗E
L, we can define a
B-morphism g : Y → P(E) by associating to y ∈ Y the l... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
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
sufficiently large. Riemann-Roch for Dn gives h0(Dn) ≥ 2 Dn(Dn − K) = O(n).
Thus, |Dn| is nonempty for large enough n. Let E ∈ P1 . Since E · F = 0, every
componen... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008/25df5925f0a5221eb65308f166960a9a_lect6.pdf |
(b) = 0}. Σ is an irreducible variety of dimension k − 3 + 1 = k − 2, and
thus one cannot cover all of Pk−1 by projections of such correspondences. This
→ →
E
0 with E/sOX locally free,
gives us a sequence O → OX
�
implying the desired exact sequence.
E/sOX
→
Remark. The above generalizes to higher dimensions.... | 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 |
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 produce other functions)
}
→
(T J)(x) = min E
u∈U ... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
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 the optimal cost function of the N -
stage problem with terminal cost αN J
4 “SHORTHAND” THEORY – A SUMMA... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
= (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 later).
6 CONVERGENCE OF VALUE ITERATION
If J0
0,
≡
•
J ∗(x) = lim (T N J0)(x),
N→∞... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
•
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 CONTRACTION PROPERTY
Contraction property: For any bounded func-
•
tions J and J ′, and any µ,
max
x
(T J)(x)
−
(T J ′)(x... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
•
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),
k→∞
x
∀
Proof: Use
max
x
(T kJ)(x)
J ∗(x)
= max
x
−
(T kJ)(x)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
≤
(cid:12)
αk max
(cid:12)
x
J
(x)
−
( kJ ∗)(x)
T
−
J ∗(x)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
ry µ... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
.
Combining this with Bellman’s equation (J ∗ =
T J ∗), we obtain T J ∗ = TµJ ∗. Q.E.D.
11 COMPUTATIONAL METHODS - AN OVERVIEW
Typically must work with a finite-state system.
•
Possibly an approximation of the original system.
Value iteration and variants
Gauss-Seidel and asynchronous versions
−
Policy iteration and var... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/25f136d763929df0b7ce8db0fcbf4303_MIT6_231F15_Lec14.pdf |
(u)
j=1
X
(cid:18)
g(i, u, j) + α min Q∗(j, v)
v∈U (j)
(cid:19)
It has a unique solution.
•
13 USING Q-FACTORS II
We can equivalently write the VI method as
•
Jk+1(i) = min Qk+1(i, u),
u∈U (i)
i = 1, . . . , n,
where Qk+1 is generated for all i and u
U (i) by
∈
n
Qk+1(i, u) =
pij(u)
j=1
X
or Jk+1 = T Jk, Qk+1 = F Qk.
(... | 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 |
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
associated symmetric matrices.
51
�
3.1 The compani... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
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) =
[C
k
λi(A), and λi(Ak ) = λi(A)k
. Therefore, it follows
�
m
... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
ynomials, 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 A, denoted I(A), is the triple (n+, n0, n−),
where n+, n0, n− are the number of positive, zero, and negativ... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
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
f1
. . .
⎢
⎢
⎢
⎣
f
n−1
n
�
f T Hq(p)f =
=
⎤
T ⎡
�
n
i=1 q(xi)
�
n
i=1 q(xi)xi
.
.
.
n
n−1
i=1 q(xi)x
i
⎢
⎢
⎢
⎣
�... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
ic, 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
where x1, . . . ,... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/260460cc36cd5c2c78b0b04f9b3fe7bd_lecture_05.pdf |
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