text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
E[X 2
∞] = (cid:80)∞
E[ξ2
j ].
(cid:80)
j=1
j= ξ1 j a.s. and in L2.
Hao Wu (MIT)
18.445
15 April 2015
9 / 10
Example
Let (ξj )j≥1 be non-negative independent random variables with mean
one. Set
X0 = 1, X
n
n = Πj=1ξj .
1
(Xn)n≥0 is a non-negative martingale.
2 Xn converges a.s. to some limit X∞ ∈ L1.
Question :
1 Do w... | https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/5fe9145062f4c67f4f675b6e18bc2065_MIT18_445S15_lecture17.pdf |
MODULE ConcurrentTransactions [
V, % Value
S, % State of database
T % Transaction ID
] EXPORT Begin, Do, Commit =
CONST s0:S := init()
% initial state
A = S -> [v,s]
E = [a: A, v: V]
H = SEQ E
Y = T -> H
% Action
% Event
% History
% histories of transactions
TS = SET T
XC = (T, T)-> Bool % eXternal Cons... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/5ffe1e8aa354be6f61ceb68a116f8392_20slides.pdf |
-> Bool =
RET (EXISTS to: TO |
t.set=ts /\ Consistent(to, xc) /\ Valid(y,to))
FUNC Invariant(com: TS, act: TS, xc, y) -> Bool =
Serializable(com, xc, y)
APROC Begin() -> T = <<
VAR t: T | ~ t IN (active \/ committed) =>
y(t) := {};
active := active \/ {t};
xc(t,t) := true;
DO VAR t’ :IN committed | ~xc.closu... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/5ffe1e8aa354be6f61ceb68a116f8392_20slides.pdf |
AC, CC, EO, OD, OC1, OC2, NC
AC: (ALL ts: TS | (com <= ts <= current) ==>
Serializable(ts, xc0, y0))
CC:
Serializable(current, xc0, y0)
EO: (ALL t :IN act | EXISTS ts | com <= ts <= current /\
Serializable(ts + {t}, xc0, y0))
OD: (ALL t :IN act | EXISTS ts | AtBegin(t) <= ts <= current /\
Serializable(ts + {t},... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/5ffe1e8aa354be6f61ceb68a116f8392_20slides.pdf |
/\ h2 <<= h1
/\ IsInterleaving(h3, {h2, h.reml})
/\ h.last.a(Apply(+ : (to * y0) + h2 + h.reml, s0) = h.last.v))
NC: true
FUNC Prefixes(h: T) -> SET H = RET {h’ | h’ M= h /\ h’ # {}}
FUNC AtBegin(t: T) -> TS = RET {t’ | xc.closure(t’,t)}
FUNC IsInterleaving(h: H, s: SET H) -> Bool =
... sequence h is interleavin... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/5ffe1e8aa354be6f61ceb68a116f8392_20slides.pdf |
y(t) | protect(e.a) <= locks(t)
% I3: ALL t1 :IN active, t2 :IN active | t1 # t2 ==>
%
%
ALL lk1 :IN locks(t1), lk2 :IN locks(t2) |
~conflict(lk1,lk2) | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/5ffe1e8aa354be6f61ceb68a116f8392_20slides.pdf |
New Bedford Steel Coking Coal Supply Problem 1
New Bedford Steel (NBS) is a small steel manufacturing company. Coking coal
is a necessary raw material in the production of steel, and NBS procures 1.0 - 1.5 million
tons of coking coal per year. It is now time to plan for the 1997 production, and Stephen
Coggins, coa... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
(volatility is the percent of volatile or burnable
matter in the coal). Also, as a hedge against adverse labor relations, NBS has decided to
procure at least 50% of its coking coal from union mines (United Mine Workers).
Finally, Steve Coggins needs to keep in mind that capacity for bringing in coal by rail is
limi... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
575
0
600
0
0
0
0
0
1
-4
1
1
1
-3
1
1
1
-1
-1
1
1
1
1
1
1
2
-1
1
1
3
1
1
1
4
-1
1
Hopt
0
490
1
6
-1
1
COST
TOTAL
0
49.50
50.00
61.00
63.50
66.50
71.00
72.50
80.00
15.066J
3
Summer 2003
... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
80
1
-4
1
1
1
-3
1
1
1
-1
-1
1
1
1
1
1
1
2
-1
1
1
3
1
1
1
4
-1
1
Hopt
0
490
1
6
-1
1
49.50
50.00
61.00
63.50
66.50
71.00
72.50
80.00
15.066J
4
Summer 2003
... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
$8
$K$8
$D$8
$E$8
$F$8
$G$8
$H$8
$I$8
$J$8
$K$8
Amount from Ashley
Amount from Bedford
Amount from Consol
Amount from Dunby
Amount from Earlam
Amount from Florence
Amount from Gaston
Amount from Hopt
Amount from Ashley
Amount from Bedford
Amount from Consol
Amount from Dunby
Amount from E... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
5
475
680
0
490
15.066J
5
Summer 2003
Sensitivity Report from Solver for NBS Problem
Microsoft Excel 4.0 Sensitivity Report
Worksheet: New Bedford
Changing Cells
Cell
Name
$D$8
$E$8
$F$8
$G$8
$H$8
$I$8
$J$8
$K$8
Amount from Ashley
Amount from Bedfo... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
5
20
20
1E+30
125
25
100
3.33
145
1E+30
15.066J
6
Summer 2003
Sensitivity Report : Notes for NBS Case Discussion
Shadow Price for a constraint is how much the objective function for the optimal
solution will increase if we increase the RHS (right-hand-side) of ... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
assure that you are making the right interpretation. Also, different solvers
may refer to dual prices, rather than shadow prices: they are the same.
15.066J
7
Summer 2003
Now, let's consider the volatility constraint from the NBS case. Suppose that we re-
write the constraint as follows:
0.15A + 0.... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
optimal solution, the additional cost to supply 1
mton of volatile matter is $300,000 (or $300 for 1 ton of volatile matter).
We can reinterpret this in terms of the volatility content of coal. Each percentage point
of volatility will supply 0.01 tons of volatile matter per ton of coal. Since the cost to
supply 1 t... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
he be willing to
go?
If Florence is uneconomical, how much must NBS insist that they lower
their price before they would take them on as a supplier?
If NBS contracts with Earlham, and their actual volatility rating goes
down, how much of a penalty should NBS insist on in their contract?
15.066J
9
Summer 2003
... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
the net volatility cost (or premium) for a coal source is the difference between its
volatility and the target of 19%, multiplied by $3/ton.
Observations:
•
In the optimal solution, coal sources A, D and E are used but are not at their upper
bounds. They all have an imputed cost of $61.50, the shadow price for the... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/60367b40dc46191eb2ff5a1d44b8b66e_ses1_freund.pdf |
Lecture # 22
Date: 4/30/09
8.592J–HST.452J: Statistical Physics in Biology
1 Kinetics of protein–DNA interaction
1.1 Reaction Kinetics
1 The rate of change with time of the concentration of a protein–DNA complex is the sum
of two terms. A positive contribution due to complex formation between a previously free
specific ... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/6039e032dce5df28031d2100fe2fe33f_MIT8_592JS11_lec19.pdf |
×
d
dt
[P
|
DNA]
≈
(ka[DNA])[P ] = [P ]/τ,
(2)
DNA]) as the characteristic time scale for a free
where we have identified τ = 1/(ka[P
repressor to locate the operator sequence. For the measured value of ka, this search time is
of order τ
0.1s.
|
∼
1.2 Debye-Smoluchowski theory
In this section we will compute the on–rate... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/6039e032dce5df28031d2100fe2fe33f_MIT8_592JS11_lec19.pdf |
molecules
diffusing from the outer to the inner sphere. To obtain this current, we must solve Laplaces
law:
with the boundary conditions C(R) = CR and C(b) = 0 (because diffusing particles disappear
at r = b). Spherical symmetric solution is
2C = 0
∇
(4)
C(r) = C0
1
(cid:20)
b
r
,
(cid:21)
−
(5)
where C0 = CR/(1
the radi... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/6039e032dce5df28031d2100fe2fe33f_MIT8_592JS11_lec19.pdf |
mechanism to find specific
site quickly.
×
≈
3
2
1.3 Berg – von Hippel theory
In 1980s Berg and von Hippel proposed that proteins use combination of 1D (sliding) and
3D (jumps) diffusion to quickly find the target site on the DNA (Figure 1). Proteins are able
TF
Figure 1: Schematics of 1D/3D search for target site on the ... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/6039e032dce5df28031d2100fe2fe33f_MIT8_592JS11_lec19.pdf |
in first
where the 1
1 rounds and factor q reflects the probability that protein has found the target in the
NR
last round. The average number of rounds needed for protein to find the target is:
−
−
−
∞
NR =
NRq(1
NR=1
X
−
3
q)NR−1 =
1
q
=
M
n
(8)
The average search time for protein to find the target site is:
ts = NR(τ1 ... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/6039e032dce5df28031d2100fe2fe33f_MIT8_592JS11_lec19.pdf |
]NR−1 =
1
q
h
i
=
Mb
2√D1τ1
,
(11)
∞
0 n(τ1)ρ(τ1)dτ1 = 2
D1τ1/b2. Calculating average search time is a
where we used
=
bit more complicated:
n
i
h
ts
h
i
ts
h
i
ts
ts
h
h
i
i
=
=
=
=
∞
NR
NR=1
X
NR
NR
NR
h
h
h
i
i
i
τ3 +
τ3 +
h
(cid:0)
R
NR
[τ1,i + τ3]
!
*
i=1
X
∞
p
NR−1
q(τ1,NR)
[1
q(τ1,i)]
+
−
i=1
Y
)NR−2 [
h
i
(NR
... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/6039e032dce5df28031d2100fe2fe33f_MIT8_592JS11_lec19.pdf |
1
a lot and spends too much time with 3D diffusion. Protein dissociation rate from the DNA
(opt), protein spends too much time sliding. In the other case τ1 < τ1
4
k(ns)
d = 1/τ1 strongly depends on the binding strength Ens (homework), which depends on
the salt concentration in the cytoplasm. Increasing the salt concen... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/6039e032dce5df28031d2100fe2fe33f_MIT8_592JS11_lec19.pdf |
)
ts
np−1
ρ1(t)dt
=
(cid:19)
1
(cid:18)
−
0
Z
np
ts
h
i
exp(
tsnp/
−
ts
)
i
h
(15)
The mean search time of the fastest of the np proteins to find the target location goes as
ts
100 copies of proteins searching for target at
h
the same time, which greatly speeds up the search time of proteins for the target site.
/np. Th... | https://ocw.mit.edu/courses/8-592j-statistical-physics-in-biology-spring-2011/6039e032dce5df28031d2100fe2fe33f_MIT8_592JS11_lec19.pdf |
Chapter 11B
Design of Seals
Design of Face Seals
Wear of Face Seals
(From Ayala, et al. 1998)
Diagram removed for copyright reasons.
See Ayala, H.M., Hart, D.P., Yeh, O.C., Boyce, M.C. "Wear of Elastomeric
Seals in Abrasive Slurries", Wear, 220, 9-21, 1998.
Percentage of lip worn as a function of the
number... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/605d69102d68f79501ee511dae851cf7_ch11b_seal_des.pdf |
Wear of Elastomeric
Seals in Abrasive Slurries", Wear, 220, 9-21, 1998.
Percentage of lip worn as a function of the
number of cycles for textured lip surfaces
(From Ayala, et al., 1998)
Graph removed for copyright reasons.
See Ayala, H.M., Hart, D.P., Yeh, O.C., Boyce, M.C. "Wear of Elastomeric
Seals in Abrasiv... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/605d69102d68f79501ee511dae851cf7_ch11b_seal_des.pdf |
, L
DP7 = Channel for lubricant circulation
DP8 = Seal material
Design Equation and Design
Matrix
Seals for Rotating Shaft
FR1
⎧
⎪
FR2
⎪
FR3
⎪⎪
FR4
⎨
FR5
⎪
FR6
⎪
FR7
⎪
⎪
FR8
⎩
⎫
⎪
⎪
⎪⎪
=
⎬
⎪
⎪
⎪
⎪
⎭
X 0 000 00 X
⎡
0 X 000 00 x
⎢
⎢
00 X 00 00 x
⎢
x 00 X 00 00
⎢
00 00 X 00 0
... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/605d69102d68f79501ee511dae851cf7_ch11b_seal_des.pdf |
(cid:20)(cid:27)(cid:17)(cid:23)(cid:19)(cid:24)(cid:45)(cid:18)(cid:25)(cid:17)(cid:27)(cid:23)(cid:20)(cid:45)(cid:29)
Advanced Complexity Theory
Spring 2016
Prof. Dana Moshkovitz
Lecture 21: P vs BPP 2
(cid:54)(cid:70)(cid:85)(cid:76)(cid:69)(cid:72)(cid:29)(cid:3)(cid:36)(cid:81)(cid:82)(cid:81)(cid:92)(cid:80)(cid... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
derson pseudorandom generator which (if the Nisan-Wigderson assumption
holds) will imply that BP P = P .
In this lecture we will complete the proof that the Nisan-Wigderson assumption implies that BP P =
P . We will then discuss the relation between worst case hardness and average case hardness,
and present a hardness ... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
pseudorandom generator, there is a
distinguisher D of size less than size 22δk such that
1
Prz
←{
0,1 s[D(G(z)) = 1] − Prn
y
}
0,1[D(y) = 1] > (cid:15)
∈{
If we then let αi = Prw Hi[D(w) = 1], then this condition is simply that αn − α0 > (cid:15). By the
pigeonhole principle, there must exist an i such that αi − αi−1 ... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
−1)) = f (z|Ti) only depends on z
|Ti.
Therefore, letting Ti be the complement of Ti in {1, . . . , s}, by the averaging principle, there must
be a setting for z| ¯Ti such that the claim still holds. Fixing z| ¯T to this setting, we can write
fj(z|Ti Tj ) = f (z
|Tj ) for 1 ≤ j < i. Our claim then becomes:
¯
∩
i
Prz Ti... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
1, ..., Tn ⊂ {1, . . . , s} such that |Ti| = k and |Ti ∩ Tj| ≤ δk. Moreover, we can find such a set in
polynomial time.
2
Such collections of subsets are known as combinatorial designs. There exists a simple greedy
algorithm for finding such sets; for more details, interested readers should consult Section 20.2 of
the cl... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
be computed in polynomial time (via Gaussian elimination, for example),
the permanent is very hard to compute. It is known (due to a theorem of Valiant) that computing
the permanent is #P complete [4].
It turns out (due to an argument of Lipton) that computing the permanent is also hard on average.
In particular, we ha... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
k
D(E(x) ⊕ η) = x ∀η ∈ {0, 1n s.t.wt(η) ≤ τ n
and
that satisfy the relation
.
Here ⊕ is the XOR operation, and wt(s) for a binary string s is equal to the number of bits in s
set to 1.
As suggested by the name, error-correcting codes are primarily used to store information with
some redundancy so the information is res... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
τ, L)-error-correcting code is a pair of functions E : {0, 1}k → {0, 1}n and
D : {0, 1}n → {{0, 1}k}L such that
4
.
x ∈ D(E(x) ⊕ η) ∀η ∈ {0, 1}n s.t.wt(η) ≤ τ n
In particular, our local decoder D now generates L possible messages, and we only require that
the actual message belongs to this set. For these error-correct... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
Let (E, D) be a 1/2 − (cid:15) local list-decoding code (for some (cid:15) to be determined
0, 1 N , with K = 2k. Then a
later) whose decoder runs in sublinear time, where E : 0, 1 K
function f : {0, 1}k
→ {0, 1} can be viewed as a K-bit string and thus and element of the domain
of E. E(f ) is then an N -bit string, wh... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
CourseWare
https://ocw.mit.edu
18.405J / 6.841J Advanced Complexity Theory
Spring 2016
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
II.G Gaussian Integrals
In the previous section, the energy cost of fluctuations was calculated at quadratic
order. These fluctuations also modify the saddle point free energy. Before calculating
this modification, we take a short (but necessary) mathematical diversion on performing
Gaussian integrals.
The simplest G... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
/K, and
φ2
ic =
h
h
i − h
φ
i
φ
ic =
h
φ
i
h
cumulants of the Gaussian distribution are zero since
e −ikφ
(cid:10)
∞
"
ℓ=1
X
exp
≡
(cid:11)
(
−
ik)ℓ
ℓ!
φℓ
c
(cid:10)
(cid:11)
#
= exp
ikh
−
−
(cid:20)
k2
2K
.
(cid:21)
Now consider the following Gaussian integral involving N variables,
I N =
∞ N ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
=
form a
qˆ
}
{
q
φi}
{
n o
28
P
the integration variables from
transformation is unity, and
φi}
{
to φ˜q
n o
. The Jacobian associated with this unitary
I N =
N ∞
−∞
Z
q=1
Y
dφ˜q exp
− 2
(cid:20)
Kq φ˜
2 + h˜qφ˜q =
q
N
s
q=1
Y
2π
Kq
exp
"
h˜q
K −1
q h˜q
2
.
#
(II.58)
(cid:21)
The fi... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
i,j
X
Moments of the distribution are obtained from derivatives of the characteristic function
i,j
X
(cid:28) P
(cid:29)
with respect to ki, and cumulants from derivatives of its logarithm. Hence, eq.(II.60)
implies
φiic =
h
φiφjic =
h
K
−1
i,j hj
j
X
−1
K
i,j
.
(II.61)
Another useful f... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
0)
(det K)−1/2 exp
Z
∝
−
Z
(cid:20)Z
dd xdd x ′ K(x, x ′ )
2
dd xdd x ′ K −1(x, x ′ )
2
φ(x)φ(x ′ ) +
Z
h(x)h(x ′ )
(cid:21)
,
dd xh(x)φ(x)
(cid:21)
(II.63)
where the inverse kernel K −1(x, x ′ ) satisfies
dd x ′ K(x, x ′ )K −1(x ′ , x ′′ ) = δd(x
−
x ′′ ).
Z
(II.64)
φ(x) is used to denote the function... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
′ φ(x ′ )δd(x
−
x ′ )(
2 + ξ−2)φ(x),
−∇
(II.66)
K(x, x ′ ) = Kδd(x
−
x ′ )(
2 + ξ−2).
−∇
(II.67)
Following eq.(II.64), the inverse kernel satisfies
K dd x ′′ δd(x
x ′′ )(
2 + ξ−2)K −1(x ′′
−∇
x ′ ) = δd(x ′
x),
−
−
(II.68)
−
Z
which implies the differential equation
2 + ξ−2)K −1(x) = δd(x).
K(
−∇
(II.6... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
:20)
dd x (
(cid:20)
φℓ)2 +
∇
φt)2 +
∇
2
φ
ℓ
2
ξℓ
2
φ
t
2
ξt
(cid:21)(cid:27)
.
(cid:21)(cid:27)
(II.70)
Each of the Gaussian kernels is diagonalized by the Fourier transforms
φ˜(q) =
dd x exp (
Z
iq x) φ(x)/√V ,
−
·
and with corresponding eigenvalues K(q) = K(q2 + ξ−2). The resulting determinant of ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
��
1
8u
The correction terms are proportional to
0 +
n
2
Z
+ 2
Z
1
dd q
(2π)d (Kq2 + t)2
dd
(2π)d (Kq2
2t)2
q
1
−
for t > 0
for t < 0
.
(II.73)
CF =
1
K 2
Z
dd q
(2π)d (q2 + ξ−2)2 .
1
(II.74)
The integral has dimensions of (length)4−d, and changes behavior at d = 4. For d > 4
the i... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
correction term from eq.(II.75) which is more important than
singularity, a discontinuity in C, is not changed. For d < 4, the divergence of ξ
∝
the original discontinuity. Indeed, the correction term corresponds to an exponent α =
(4
d)/2. However, this is only the first correction to the saddle point result. The di... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
superconductors than in other phase transitions?
Eq.(II.75) indicates that fluctuation corrections become important due to the diver
gence of the correlation length. Within the saddle point approximation, the correlation
T )/Tc is the reduced temperature, and
length diverges as ξ
√K is a microscopic length scale. I... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
�
ξ0
∝
in
1/u, and the correction term CF .
−dt−(4−d)/2 . Thus fluctuations are
0
−dt− 4−d
ξ0
2
≫
ΔCS.P., =
⇒ |
t
| ≪
1
tG ≃ (ξdΔCS
0
.P.)
.
2
4−d
(II.76)
The above requirement is known as the Ginzburg criterion. Naturally in d < 4, it is
satisfied sufficiently close to the critical point. However, the resolu... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
G < 10−18 is necessary to see any fluctuation effects. This
ξ0 ≈
is much beyond the ability of current apparatus. The newer ceramic high temperature
superconductors have a much smaller coherence length of ξ0 ≈
10a, and they indeed show
some effects of fluctuations.
10−1
−
ξ
0
Again, it is worth emphasizing that a sim... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
/2,
For d less than a lower critical dimensions (dℓ = 2 for continuous symmetry, and dℓ = 1
•
for discrete symmetry) fluctuations are strong enough to destroy the ordered phase.
d
du, fluctuations are strong enough to change
•
the saddle point results, but not sufficiently important to completely destroy order. Un
... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
18.354J Nonlinear Dynamics II: Continuum Systems
Lecture 14
Spring 2015
14 Low-Reynolds number limit
In this section, we look at the limit of Re → 0 which is relevant to the construction of
microfluidic devices and also governs the world of swimming microbes.
70
Bacteria and eukaryotic cells achieve locomotion in a ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
identify such solutions, these equations must still be endowed with appropriate initial and
boundary conditions, such as for example
(cid:40)
u(t, x) = 0,
p(t, x) = p ,∞
as
|x| → ∞.
(360)
Note that, by neglecting the explicit time-dependent inertial terms in NSEs, the time-
dependence of the flow is determined exclusive... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
|x|2
2
jk
(cid:19)
,
(362a)
(362b)
(363)
as can be seen from
GijG−1
jk
(cid:19) (cid:18)
(cid:18)
xixj
ij +
=
δ
|x|2
= δik − ixk
x
2|x|2 +
xixk
2|x|2 +
= δik −
δjk −
−
xixk
|x|2
xixk
2|x|2
xjxk
2|x|2
xixj
|x|2
(cid:19)
xjxk
2|x|2
= δik.
(364)
14.3 Stokes’s solution (1851)
Consider a sphere of radius a, which at time t ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
θ, φ)) = U ,
µ
a2 Uj aj(θ, φ) + p∞,
p(t, a(θ, φ)) =
3
2
(366a)
(366b)
17Proof by insertion.
18Proof by insertion.
72
corresponding to a no-slip boundary condition on the sphere’s surface. The O(a/|x|)-
contribution in (365a) coincides with the Oseen result (362), if we identify
F = 6π µa U .
(367)
The prefactor γ = 6π... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
4πµ
(cid:20)
−δij ln
(cid:18) |x|
a
(cid:19)
+
xixj
x|2
|
(cid:21)
(371b)
with a being an arbitrary constant fixed by some intermediate flow normalization condi-
tion. Note that (371) decays much more slowly than (370), implying that hydrodynamic
interactions in 2D freestanding films are much stronger than in 3D bulk solu... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
2
(cid:18) xixj
|x|2
(cid:19)(cid:21)
(cid:18)
δik
xj
|x|2 + δjk
xi
|x|2 − 2
xixjxk
x|4
|
(cid:19)(cid:21)
.
(374)
To check the incompressibility condition, note that
∂iJij =
=
1
4πµ
1
4πµ
= 0,
(cid:20)
−δij
xi
|x|2 +
x
− j
|x|2 + 2
(cid:18)
(cid:18)
δii
xj
|x|2 + δji
xj
|x|2 − 2
xi
|x|2 −
(cid:19)
xj
|x|2
xj
|x|2 +
(c... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
k
|x|4
xiδjkxk
|x|4 +
1 (cid:19) (cid:18) δij
+
|x|2
δ
kk
|x|2
(cid:18) δikxjxk
|x|4
2
|x|2 − 2
xixj
|x|4 + 2
(cid:19)
|x|4 +
(cid:18) xjxi
xixj
|x|4 − 4
xixjδkk
|x|4 − 4
xjxi
|x|4
(cid:19)(cid:21)
|x|2 − 2
xixj
|x|4
2
2
(cid:20)
1
4πµ
δ
− ij
(cid:18)
1
2πµ
ij
δ
|x|2 − 2
xjxi
x|4
|
Hence, by comparing with (373), we se... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
In the absence of external forces, microswimmers must satisfy the force-free constraint. This
simplest realization is a force-dipole flow, which provides a very good approximation for the
mean flow field generated by an individual bacterium but not so much for a biflagellate
alga.
To construct a force dipole, consider two ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
(cid:20)
−δij
(cid:18)
−
δik
xk
|x|2 +
x
knk
i |x|2 + ni
n
xj
|x|2 + δjk
xjnj
|x|2
+ nknk
F +
j
xi
|x|2 − 2
xi
|x|2 − 2
xixjxk
|x|4
nkxixjxknj
|x|4
(cid:19)
(cid:19)(cid:21)
and, hence,
where xˆ = x/|x|.
u(x) =
F (cid:96)
πµ|x|
2
(cid:2)2(n · xˆ)2 − 1(cid:3) xˆ
(380)
3D case To compute the dipole flow field in 3D, we nee... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
3b)
(383c)
(cid:1).
(384)
Inserting this expression into (379), we obtain the far-field dipole flow in 3D
u(x) =
F (cid:96)
4πµ x 2
|
|
(cid:2)
3(n · xˆ)2 − 1 xˆ.
(cid:3)
(385)
Experiments show that Eq. (385) agrees well with the mean flow-field of a bacterium.
Upon comparing Eqs. (380) and (385), it becomes evident that h... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
(386a)
The gradient vector is given by
∇ = ex∂x + ey∂y + ez∂z,
(386b)
76
and, using the orthonormality ej · ek = δjk, the Laplacian is obtained as
∆ = ∇ · ∇ = ∂2 + ∂2
y + ∂2
z .
x
(386c)
One therefore finds for the vector-field divergence
∇ · u = ∂iui = ∂xux + ∂yuy + ∂zuz
(386d)
and the vector-Laplacian
∂2
2
x + ∂y ux ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
the form
∇ = er∂r + eφ ∂φ + ez∂z,
1
r
yielding the divergence
∇ · u =
1
r
∂r(rur) + ∂φuφ + ∂zuz.
1
r
The Laplacian of a scalar function f (r, φ, z) is given by
∇2
f =
1
r
∂r(r∂rf ) + ∂2 +
1
2 φf
r
∂2
z f
and the Laplacian of a vector field u(r, φ, z) by
∇2u = Lrer + Lφeφ + Lzez
77
(387e)
(387f)
(387g)
(387h)
where
Lr =... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
and some of the unit vectors change with φ (e.g., ∂φeφ = −er).
φ/r corresponds to the centrifugal force, and it arises because u =
14.6.2 Hagen-Poiseuille flow
To illustrate the effects of no-slip boundaries on the fluid motion, let us consider pressure
driven flow along a cylindrical pipe of radius R pointing along the z-... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
393)
and the average transport velocity is
uz =
1
πR2
(cid:90) R
0
uz(r) 2πrdr = 0.5u+.
z
(394)
Note that, for fixed pressure difference and channel length, the transport velocity uz de-
creases quadratically with the channel radius, signaling that the presence of boundaries can
substantially suppress hydrodynamic flows.... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
-direction20, yielding
0 = ∇ · U ,
0 = −∇P + µ∇2U − κU
(397)
where κ = 12µ/H 2 and ∇ is now the 2D gradient operator. Note that compared with
unconfined 2D flow in a free film, the appearance of the κ-term leads to an exponential
damping of hydrodynamic excitations. This is analogous to the exponential damping in the
Yuka... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
Introduction to Engineering
Systems, ESD.00
Networks
Lecture 7
Lecture 7
Lecturers:
Professor Joseph Sussman
Dr. Afreen Siddiqi
TA: Reggina Clewlow
The Bridges of Königsberg
The Bridges of Königsberg
•
•
The town of Konigsberg in 18th century
The town of Konigsberg in 18
century
Prussia included two islands ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
Discrete+Mathematics/Graph+Theory/konisberg
-multigraph-bridge.aspx
Modeling ample
Modeling Example
Ex
‐ II
There were six people: A, B, C, D, E, and F in a party and
following handshakes among them took place:
following handshakes among them took place:
A shook hands with B, C, D, E and F
B, in addition, shook... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
E, we write G=(V,E)
Each edge {x,y} of G is usually denoted by
Each edge {x y} of G is usually denoted by
xy, or yx.
•• What is the vertex set V(G) and edge set
What is the vertex set V(G) and edge set
E(G) of the graph G shown?
G
b
Ref [3]
a
f
c
g
d
e
V((G))={{a,, b,
, c, d, e, f,
,
,
,
, g}
g}
E(G)={ab, ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
edge between a pair of vertices
• What if there are more edges?
• Consider Euler’s graph for the
Köninggsberg problem
g p
The graph K is a multigraph
•
• A multigraph has finite number of edges
(including zero) between any two
(including zero) between any two
vertices
So all graphs are multigraphs but not
vic... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
⎢
A
⎥
⎢
⎥
⎢
⎢2 2 1 0 1⎥
⎣⎢3 2 1 1 0 ⎥⎦
Whhy are thhe ellements off thhe ddiagonall allways zero?
What is the order (n) of G?
What is the size (m) of G?
from A?
How can you determine mm from A?
How can you determine
12
Incidence Matrix
•
•
•
The Incidence Matrix, B is a binary, n x m
matrix,, where bijij = ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
multigraph, the sum of the degrees of
its vertices is twice its size (number of
edges).
•
n
∑
i=1
d vi( ) = 2m
This is also known as the Hand Shaking Theorem
This is also known as the ‘Hand-Shaking Theorem’
-
• A vertex with the highest degree is called a
hub in a graph (or network).
hub in a graph (or network) ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
if
we pick k objects at a time?
This is given by the Binomial coefficient:
•
•
•
K1: 0
K2: 1
K3: 3
K4: 6
K5: 10
K6: 15
K7: 21
K8: 28
Image by MIT OpenCourseWare.
For a complete graph with order n, Kn, the
size m is:
(
⎛ ⎞⎞ n n −1)L ( n − k +1)
⎛n
⎜ ⎟ =
⎝k⎠
k(k −1)L 1
k ≤ n
=
!
n
k!(n − k)!
m
⎛n⎞
= ⎜ ⎟... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
P1
PP3
P4
C3
C4
C5
Walks
Walks
•
In a graph, we may wish to know if a route exists
from one vertex to another‐ two vertices may not
be adjacent, but maybe connected throughh a
b
b dj
sequence of edges.
t d th
t b t
C
e3
• A walk in a graph G is an alternating sequence of
A
B
vertices and edges :
vertice... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
uits
i
i
cycles
Ref [4]
Examples
Examples
•
The walk t is a trail of length 5:
t = (v1, e2, v3, e3, v1, e1, v2, e8, v5, e7, v4)
)
t
t is not a path since v1 appears twice
(
• The walk p is a path of length 4:
)
p = ((v1, e2, v3, e4, v2, e8 , v5, e7, v4)
e6
e7
e7
v5
v4
e5
v3
e2
e4
e8
e8
v2
e1 ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
Image by MIT OpenCourseWare.
Trees
Trees
• A tree, T, is a connected graph that has
no cycle as a subgraph
• A tree is a simple graph on n vertices‐
a tree cannot have any loops or
multiple edges between two vertices.
•
T has n‐1 edges and is connected.
• A vertex v of a simple graph is called a
leaf if d(v) = ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
s
Weighted Graphs
• A connected graph G is called a
weighted graph if each edge e in G is
weighted graph if each edge e in G is
assigned a number w(e), called the
weight of e.
• Depending on the application, the
weight of an edge may be a measure of
physical distance, time consumed, cost,
physical distance time... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
t t t l ti
th
The critical path determines project completion
time.
i th
iti
l
•
•
•
•
A
BB
C
DD
E
F
G
H
AA
A
B,CB,C
B,C
E
D,F
G
0
1010
20
3030
20
40
20
0
26
regular graphs
kk‐regular graphs
• A graph g is called k‐regular if d(vi) = k for all
k for all
A graph g is called k reg... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
model our world as a collection of
people – each pperson is a vertex (node)) and
p p
two people (vertices) are connected if they
are acquainted. What will be the diameter of
this graph (or social network)?
(
V
u
Image by MIT OpenCourseWare.
Small World Networks
Small World Networks
•
Small world networks are ‘highl... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
00 Introduction to Engineering Systems
Spring 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
15.083J/6.859J Integer Optimization
Lecture 10: Solving Relaxations
Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
1 Outline
• The key geometric result behind the ellipsoid method
• The ellipsoid method for the feasibility problem
• The ellipsoid method for optimization
• Problems with exponentially many constrain... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
n + 1 a�Da
.
�
D
�
The matrix D is symmetric and positive definite and thus E� = E(z, D) is an
ellipsoid
E�
H
E
Vol(E�) < e−1/(2(n+1)) Vol(E)
⊂
∩
•
•
•
•
•
1
Et+1
Et
0011
xt
P
0011
xt+1
a�x ≥ b
a�x ≥ a�xt
x2
E
E'
x1
2
2.3
Illustration
2.4 Assumptions
• A polyhedron P is full-dimensional... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
(x0, r2I) with volume at most V , such that P ⊂ E0
Output: A feasible point x ∗ ∈ P if P is nonempty, or a statement that P is
empty
2.6 The algorithm
1. (Initialization)
Let t ∗ = 2(n + 1) log(V /v) ; E0 = E(x0, r 2I); D0 = r 2I; t = 0.
2. (Main iteration)
�
�
Slide 6
Slide 7
Slide 8
Slide 9
If t = t stop... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
for which the prior information x0, r, v, V is available. Then, the ellipsoid
method decides correctly whether P is nonempty or not, i.e., if xt∗−1 ∈/ P , then
P is empty
2.8 Proof
• If xt ∈ P
nonempty
for t < t ∗, then the algorithm correctly decides that P
is
• Suppose x0, . . . , xt∗−1 ∈/ P . We will show that... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
⊂
Ek+1 and the induction is
Vol(Et+1)
Vol(Et)
−1/(2(n+1))
< e
Vol(Et∗ )
Vol(E0)
∗
−t
< e
/(2(n+1))
Slide 11
Slide 12
Slide 13
Vol(Et∗ ) < V e−�2(n+1) log
V �/(2(n+1))
v
V e− log
V
v
= v
≤
∗
If the ellipsoid method has not terminated after t iterations, then Vol(P )
v. This implies that P is empty
2... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
the polyhedron P = {x ∈ �n | Ax ≥ b} satisfies
−(nU )n ≤ xj ≤ (nU )n ,
j = 1, . . . , n
• All extreme points of P are contained in
PB = x ∈ P
�
|xj| ≤ (nU )n, j = 1, . . . , n
�
�
�
• Since PB ⊆ E 0, n(nU )2nI , we can start the ellipsoid method with E0 =
E 0, n(nU )2nI
�
�
�
•
�
V ol(E0) ≤ V = 2n(nU )n ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
b are integer and have absolute value bounded by U . Then,
�
�
Slide 15
Slide 16
Slide 17
Slide 18
Vol(P ) > v = n −n(nU )−n (n+1)
2
5
Slide 19
Slide 20
2.12 Complexity
• P = {x ∈ �n | Ax ≥ b}, where A, b have integer entries with magni
tude bounded by some U and has full rank. If P is bounded and either
... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
�π
max
s.t. A�π = c
Slide 21
0.
π
≥
By strong duality, both problems have optimal solutions if and only if the following
system of linear inequalities is feasible:
�
b p = c x,
�
Ax
b,
≥
�
A p = c,
0.
p
≥
LO with integer data can be solved in polynomial time.
3.1 Sliding objective
• We first run the ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
slow convergence, close to the worst case
• Contrast with simplex method
• The ellipsoid method is a tool for classifying the complexity of linear
programming problems
4 Problems
4.1 Example
min
cixi
�
i
aixi ≥ |S|,
for all subsets S of {1, . . . , n}
�
i∈S
• There are 2n constraints, but are described conc... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
P ⊂ �n and a vector x ∈ �n, the separation problem is
to:
Slide 28
• Either decide that x ∈ P , or
• Find a vector d such that d� x < d� y for all y ∈ P
What is the separation problem for
aixi ≥ |S|,
for all subsets S of {1, . . . , n}?
�
i∈S
6 Polynomial solvability
6.1 Theorem
If we can solve the separati... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
δ(S)
�
xe = 2,
e∈δ(i)
�
xe ∈ {0, 1}
i ∈ V
How can you solve the LP relaxation?
6.4 Probability Theory
•
•
•
•
Events A1, A2
P (A1) = 0.5, P (A2) = 0.7, P (A1
Are these beliefs consistent?
A2)
0.1
≤
∩
General problem: Given n events Ai i
N =
1, . . . , n
, beliefs
∈
pi,
P(Ai)
P(Ai
Aj)
∩
≥
≤
pij,
}
N, ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
0,
∀ S
• Given solution (y ∗ , u ∗ , z ∗) is feasible
7 Conclusions
• Ellipsoid algorithm can characterize the complexity of solving LOPs with
an exponential number of constraints
• For practical purposes use dual simplex
• Ellipsoid method is an important theoretical development, not a practical
one
Slide 37
Sli... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
L4: Sequential Building Blocks
L4: Sequential Building Blocks
flops, Latches and Registers)
(Flip(Flip--flops, Latches and Registers)
Acknowledgements:
.,
Materials in this lecture are courtesy of the following people and used with permission.
- Randy H. Katz (University of California, Berkeley, Department of Electric... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
N inputs (N-1 Adders)
in0
in1
in2
inN-1
Using a sequential (serial) approach
in
reset
D Q
clk
Current_Sum
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
4
stability
Implementing State: Bi--stability
Implementing State: Bi
Vo1 = Vi2
Vo2 = Vi1
Vo1
Vi2
1
o
V
=
2
i
V
Point C is
Metastable
C
Vi1
A
Vi 2 = Vo... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
are also
called flip-flops) – this circuit is not clocked and outputs change
“asynchronously” with the inputs
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
6
Making a Clocked Memory Element:
Making a Clocked Memory Element:
Latch
Positive D--Latch
Positive D
Q
hold
sample
hold
sample
hold
D
CLK
D Q
G... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
2) Master-Slave Register
(cid:134) Use negative clock phase to latch inputs into first latch
(cid:134) Use positive clock to change outputs with second latch
(cid:132) View pair as one basic unit
(cid:134) master-slave flip-flop twice as much logic
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
10
Trigg... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
2006
Introductory Digital Systems Laboratory
12
Triggered Register)
74HC74 (Positive Edge--Triggered Register)
74HC74 (Positive Edge
Images removed due to copyright restrictions
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
13
The J--K Flip
The J
K Flip--FlopFlop
S
R
Q
Q
100
J
0
0
1
1
K
Q+
Q+
0
1
0
1... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
2006
Introductory Digital Systems Laboratory
16
Triggered Registers
Pulse--Triggered Registers
Pulse
Ways to design an edge-triggered sequential cell:
Master-Slave Latches
Pulse-Based Register
L1
D
Q
L2
D
Q
Clk
Clk
Data
Clk
Latch
D
Q
Clk
Data
Clk
Short pulse around clock edge
(cid:131) Pulse registers are widely used ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
D
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
19
Design Procedure
Design Procedure
Excitation Tables: What are the necessary inputs to cause a particular kind of
change in state?
Q
0
0
1
1
Q +
0
1
0
1
J
0
1
X
X
K
X
X
1
0
T
0
1
1
0
D
0
1
0
1
Implementing D FF with a J-K ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
FF. Of course it is identical
to the characteristic equation for a J-K FF.
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
21
System Timing Parameters
System Timing Parameters
In
D
Q
Clk
Combinational
Logic
D
Q
Clk
Register Timing Parameters
Logic Timing Parameters
Tcq : worst case rising edge
clock to ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
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