text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
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 Abrasive Slurries", Wear, 220, 9-21, 1998.
Design of seals for
rotating shaft
Functional Requirements of Seals
for Rotating Shaft
FR1 = Su... | 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 |
Sudan, Trevisan, and Vadhan based on error
correcting codes.
2 Nisan-Wigderson Generator
We begin by reviewing the notation of the previous lecture. Recall that a function f : {0, 1k → {0, 1}
is (cid:15)-easy for circuits of size m if there exists a circuit C of size at most m such that
Prx
←{0,1}k
[f (x) = C(x)]
1
≥ +... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
> (cid:15). By the
pigeonhole principle, there must exist an i such that αi − αi−1 > (cid:15)/n. We will show that this results
in a contradiction; specifically, we will construct a small circuit that approximately solves f , which
contradicts our hardness assumption.
←
In this lecture, we will show
In the previous lect... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
Ti), f2(z|T2 Ti), . . . , f
∩
∩
|
i−1(z|Ti−1 Ti)) = f (z|Ti)]
∩
≥
1
2
+
(cid:15)
n
Now, we wish to compute f (z(cid:48)) easily. To achieve this, we will choose our sets Ti so that all the
intersections Ti ∩ Tj are small. Then, since any function on (cid:96) variables can be simulated by a
circuit of size (cid:96)2(cid... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
, and hard problems like SAT need only be hard in the worst case.
In our discussion of pseudorandomness above, we needed problems that are hard not just in the
worst-case, but in the average-case. For example, our definition of a function being (cid:15)-easy to
compute was in terms of the probability over random k-bit s... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
Hardness Amplification and Error Correcting Codes
Recall that E is the complexity class given by DT IM E(2O(n)). The following theorem is due to
Sudan, Trevisan, and Vadhan.
3
Theorem 4. ([3]) If there exists an f : {0, 1}∗ → {0, 1} such that f ∈ E and f is not computable
˜
in size 2δn
, then there exists a function f ... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
log τ − (1 − τ ) log(1 − τ )
is the one-bit information entropy of τ (the Gilbert-Varshamov Bound further shows that such
codes exist for all τ < 1/2). Since H(1/2) = 1, this bound also shows that codes can only exist for
τ < 1/2. It is easy to see that in general, codes cannot exist for τ > 1/2; in particular, since t... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
the truth table of f as a message that is to be encoded using an error correcting code. The
encoded version then is the truth table of an average-case hard function f . To prove average-case
hardness, one shows that any efficient algorithm A that solves f on a 1/2 + (cid:15) fraction of inputs can
be treated as a “corrup... | https://ocw.mit.edu/courses/18-405j-advanced-complexity-theory-spring-2016/60827e3dcf88561447a094b8d05a1d84_MIT18_405JS16_P_vs_BPP2.pdf |
D will be able recover f exactly. This means that if f is hard, then
˜f should be hard to approximate, which is exactly the statement that we want.
˜
˜
References
[1] R. Lipton, New directions in testing, Distributed Computing and Cryptography, 2:191-202,
1991.
[2] N. Nisan, A. Wigderson, Hardness vs randomness, Journa... | 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 |
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
−∞
Z
i=1
Y
dφi exp
K
i... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
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 final expression can be represented in terms of the original coordi... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
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 form of eq.(II.60) is
exp(A)
h
i
= exp
A
ic +
h
(cid:20)
1
2
h
A2
,
ic
(cid:21)
(II.62)
where A =
i aiφi is any linear ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
)
(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 functional integral. There is a constant of
The notation
proportionality, (2π)N/2, left out of eq.(II.63). Although formally infin... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
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.69)
Comparing with eq.(II.44) implies K −1(x) =
by a less direct method.
φ(x)φ(0)
h
i
=
Id(x)/K, as obtained before
−
30
II.H Fluctuation Corrections ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
�V ,
−
·
and with corresponding eigenvalues K(q) = K(q2 + ξ−2). The resulting determinant of K
is a product of such eigenvalues, and hence
ln det K =
ln K(q) = V
q
X
dd q
(2π)d
Z
ln[K(q 2 + ξ−2)].
(II.71)
The free energy resulting from eq.(II.70) is then given by
f =
Z
ln
V
−
=
2
tm¯
2
+ um¯ 4 +
1
2... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
1
(II.74)
The integral has dimensions of (length)4−d, and changes behavior at d = 4. For d > 4
the integral diverges at large q, and is dominated by the upper cutoff Λ
1/a, where a is
≃
31
the lattice spacing. For d < 4, the integral is convergent in both limits. It can be made
dimensionless by rescaling q by ξ−1... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
looking at the fluctuation corrections to the heat capacity, we would have reached the same
−
≤
conclusion in examining the singular part of any other quantity, such as magnetization or
susceptibility. The contributions due to fluctuations always modify the leading singular
behavior, and hence the critical exponents,... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
−
ξ0|
≈
t
|
undergo ordering at the phase transition. For the liquid–gas transition, ξ0 can be estimated
as (vc)1/3, where vc is the critical atomic volume. In superfluids, ξ0 is approximately the
thermal wavelength λ(T ). Both these estimates are of the order of a few atomic spacings,
1–10˚A. On the other hand, the ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
inzburg reduced temperature tG. If so,
the apparent singularities at reduced temperatures t > tG may show saddle point behavior.
It is this apparent discontinuity that then appears in eq.(II.76), and may be used to self–
consistently estimate tG. Clearly, ΔCS.P. and ξ0 can both be measured in dimensionless
units; ξ... | https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/608f0e0b2ffa840d4e51ee0a592ef823_MIT8_334S14_Lec5.pdf |
��erent (by one or two orders of magnitude) for different quantities. So, it is in principle
possible to observe saddle point behavior in one quantity, while fluctuations are important
in another quantity measured at the same resolution. Of course, fluctuations will always
become important at sufficiently high resolution... | 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 |
) = 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 exclusively and instantaneously by the motion of the
boundaries and/or time-dependent forces as generated by the swimming objects.
14.2 Oseen’s solution
Cons... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
|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 is located at the origin, X(t) = 0, and
moves at velocity U (t). The corresponding solution of the Stokes equation with standard
bound... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
is the well-known Stokes drag coefficient for a sphere.
The O[(a/|x|)3]-part in (365a) corresponds to the finite-size correction, and defining the
Stokes tensor by
Sij = Gij +
(cid:18)
1
24πµ
a2
|x|3
δji − 3
(cid:19)
,
xjxi
2
|x|
we may rewrite (365a) as19
14.4 Dimensionality
ui(t, x) = SijFj.
(368)
(369)
We saw above that... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
|x|−n = ∂ (x x )−n/2 = −nx (x x )−(n+2)/2
j
j
j
i
i
i
i
19For arbitrary sphere positions X(t), replace x → x − X(t).
73
= −n
xj
|x|n+2 .
(372a)
(372b)
From this, we find
and
∂ip =
Fi
2π|x|2 − 2
Fjxjxi
2π|x|4 =
Fj
2π|x|2
(cid:18)
δij − 2
(cid:19)
xjxi
x|2
|
(373)
∂kJij =
=
=
1
4πµ
1
4πµ
1
4πµ
(cid:20)
∂k
−δij ln
(cid:18... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
(cid:18) xixjxk
|x|4
(cid:18) δik
|x|2 − 2
(cid:19)
−
xixk
|x|4
(cid:18) xi
|x|2
(cid:19)
xjxk
|x|4
xixjxkxk
|x|6
+ δjk
(cid:19)(cid:21)
(cid:19)
+
(cid:18) δij
|x|2 − 2
(cid:19)
xixj −
|x|4
(376)
(377)
The difference between 3D and 2D hydrodynamics has been confirmed experimentally
for Chlamydomonas algae.
74
14.5 Forc... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
x ∂kΓ (x) − x−∂ Γ (x) F +
ij
j
k k ij
ij
(cid:2) +
k
(cid:3)(cid:9)
= −2x+ [∂kΓij(x)] F +
j
k
(379)
2D case Using our above result for ∂kJ
|n| = 1, we find in 2D
ij, and writing x = (cid:96)n and F + = F n with
+
x+
ui(x) = − k
2πµ
F (cid:96)
2πµ
= −
(cid:18)
(cid:20)
−δij
(cid:18)
−
δik
xk
|x|2 +
x
knk
i |x|2 + ni
n
xj... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
=
Πik
|x|
,
∂nΠik = −
(xˆiΠnk + xˆkΠni) ,
and from this we find
∂kGij = − G
(Π
ij +
κ
x 2
|
xˆk
x
|
|
κ
(cid:0)−xˆkδij + xˆjδik + xˆiδjk − 3xˆkxˆixˆj
|x|2
ikxˆj + Πjkxˆi)
|
=
(383a)
(383b)
(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
|
|
(c... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
cylindrical
confinements.
14.6.1 Reminder: Cartesian vs. cylindrical corodinates
In a global orthornormal Cartesian frame {ex, ey, ez}, the po-
Cartesian coordinates
sition vector is given by x = xex + yey + zez, and accordingly the flow field u(x) can be
represented in the form
u(x) = ux(x, y, z) ex + uy(x, y, z) ey + uz... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
z ,
(cid:112)
r =
2
x + y2
and the flow field u(x) can be decomposed in the form
(387b)
(387c)
u(x) = ur(r, φ, z) er + uφ(r, φ, z) eφ + uz(r, φ, z) ez.
(387d)
The gradient vector takes 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... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
simply (u·∇)ur, but instead
er · [(u · ∇)u] = (u · ∇)u
1
2
r − u
r φ.
(388)
Physically, the term u2
urer + uφeφ + uzez 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 bounda... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
.
The flow speed is maximal at center of the pipe
u+
z =
P (0) − P (L)
4µL
R2
78
(392)
(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, ... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/60add870d6e8194319e9e22943d6298d_MIT18_354JS15_Ch14.pdf |
be achieved by inserting ansatz (395) into the Stokes equations and subsequently averaging
along the z-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 exp... | 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 |
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 hands with D and E
C, in addition, shook hands with F
B
E
AA
F
Ref [3]
Ref [3] ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
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, bc, ad, de, af, fg, fe}
7
Adjacency
Adjacency
•
•
•
If xy is an edge of G, the x and y are
adjacent vertices
Two adjacent ... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
two
vertices
So all graphs are multigraphs but not
vice versa
vice versa
•
• No loops are allowed in a multigraph – a
vertex cannot connect to itself
A
e1
e3
C
e2
e7
e6
e5
e4
D
B
Image by MIT OpenCourseWare.
10
Adjacency Matrix
•
In addition to set representation, we can
use matrices to represent multigraphs... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
⎥⎦
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 = 1 if vii is an endvertex of
edge ej, ot... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
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)
C
A
B
D
Image by MIT OpenCourseWare.
B
E
C
D
A
F
Degrees from A and B
• Given an adjacency matrix A
can we
Given an adj... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
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⎞
= ⎜ ⎟ =
⎝2 ⎠
n!
2!(n − 2)!
)
(
=
n n −1) n − 2)!
(
(
=... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
•
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 :
vertices and edges :
v0 e0 v1 e1 v2…vk‐1 ek... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
, 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
Ref [4]
e3
v1
e6
e
e7
v5
v4
e5
v3
e2
e4
e8
v2 v
e1
Ref [4]
e3
v1
v
Examples
Examples
... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
edges and is connected.
• A vertex v of a simple graph is called a
leaf if d(v) = 1.
( )
• Between every pair of distinct vertices
•
in T there is exactly one path.
Trees are useful in modeling
Trees are useful in modeling
applications such as hierarchy in a
business, directories in an operating
syystem, comp... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
or some other quantity of
interest.
Traveling Salesman Problem
A traveling salesman wants to make a
round trip through n cities, c1..ci..cn.
He starts in
c1, visits each remaining
exactly once, and ends in c1 where he
h
tl
started the trip.
d i
d
h
city ci
y
If he knows the distances between every
pair of cit... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
k‐regular if d(vi) = k for all
k for all
A graph g is called k regular if d(vi)
vi in G.
C5
•
•
•
The null graph is a 0‐regular graph.
The null graph is a 0 regular graph
The cycle Cn is a 2‐regular graph.
• A complete graph is an (n‐1) regular graph.
2-Regular
3-Regular
4-Regular
6-Regular
Image by MIT OpenCo... | https://ocw.mit.edu/courses/esd-00-introduction-to-engineering-systems-spring-2011/60d8431e128296df54df29c779477d86_MITESD_00S11_lec07.pdf |
shorten the distance between vertices.
that shorten the distance bet een ertices
Signal propagation speed is enhanced in such
systems; rumors can spread quickly the
systems; rumors can spread quickly, the
number of legs in an air or train journey is
small, infectious diseases spread more easily
in a population etc.... | 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 |
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 if it has positive volume
• The polyhe... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
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; P is empty.
∗
If xt
If xt
•
•
•
•
P stop; P is nonempty.
∈
/ P find a viola... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
∈/ 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 P is empty.
• We prove by induction on k that P ⊂ Ek for k = 0, 1, . . . , t ∗ . Note
that P ⊂ E0, by the assumptions of the algorithm, and this st... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
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
2.9 Binary Search
• P = x ∈ � | x ≥ 0, x ≥ 1, x ≤ 2, x ≤ 3
�
• E0 = [0, 5], centered at x0 = 2.5
�
4
Vol(Et∗ )
≤
≤
Slide 14... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
U )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 = (2n)n(nU )n
n
2
2.11 Full-dimensionality
�
�
Let P = {x ∈ �n | Ax ≥ b}. We assume that A and b have integer entries,
which are bounded in absolute val... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
entries with magni
tude bounded by some U and has full rank. If P is bounded and either
empty or full-dimensional, the ellipsoid method decides if P is empty in
O n log(V /v)
�
iterations
�
2
• v = n−n(nU )−n (n+1),
2
V = (2n)n(nU )n
•
•
�
• Number of iterations O n4 log(nU )
�
If P is arbitrary, we first f... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
first run the ellipsoid method to find a feasible solution x0 ∈ P =
x ∈ �n | Ax ≥ b
�
.
�
• We apply the ellipsoid method to decide whether the set
Slide 22
is empty.
P ∩ x ∈ �n | c � x < c � x0
�
�
• If it is empty, then x0 is optimal. If it is nonempty, we find a new solution
x1 in P with objective function v... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
the ellipsoid algorithm. Is it polynomial?
• In what?
4.2 The input
• Consider min c � x s.t. x ∈ P
• P belongs to a family of polyhedra of special structure
• A typical polyhedron is described by specifying the dimension n and an
integer vector h of primary data, of dimension O(nk), where k ≥ 1 is some
constant.
... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
solve linear optimization problems
If log U ≤ Cn� log� U0, then it is also
in time polynomial in n and log U .
polynomial in log U0
• Proof ?
• Converse is also true
• Separation and optimization are polynomially equivalent
6.2 MST
IZMST = min
s.t.
e∈E
�
cexe
xe ≥ 1
∀ S ⊆ V, S
�= ∅, V
Slide 29
Slide 30
e... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
the numbers pi and pij, which are between 0 and 1, are these beliefs
consistent?
6.4.1 Formulation
x(S) =
P
�
�
i∈S Ai
∩
∩
�
�
{S|i∈S}
�
{S|i,j∈S}
x(S)
x(S)
pi,
pij,
≤
≥
x(S) = 1,
∈S Ai
∩i /
,
�
�
�
i
N,
∈
i, j
∈
N, i < j,
�
S
x(S)
0,
≥
S.
∀
The previous LP is feasible if and only if the... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/60dd52896e183e2c60bfe40e076d83d3_MIT15_083JF09_lec10.pdf |
2
13 = −4, y ∗
14 = −4, y ∗
23 = −4, y ∗
24 =
−1, y ∗
34 = −7,
Slide 36
s
1,
2
1,
3
1,
4
3
4
4
2
4
4
4
1
7
1
2
3
4
9
6
4
2
t
• The minimum cut corresponds to S0 = {3, 4} with value c(S0) = 21.
• f (S0) =
y ∗
ij +
u ∗ = −7 + 4 + 2 = −1
i
�
i,j∈S0 ,i<j
�
i∈S0
• f (S) + z ∗ ≥ f (S0) + ... | 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 |
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 = Vo1
Vo2
A
1
o
V
=
2
i
V
δ
Vi1 = Vo2
Points A and
B are stable
(represent 0 & 1)
C
B
Vi 1 = Vo2
B
δ
Vi1 = V o2
L4: 6.111 Sprin... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
Q
G
clk
S
R
R and S
clock
(cid:131) A Positive D-Latch: Passes input D to output Q when CLK is
high and holds state when clock is low (i.e., ignores input D)
(cid:131) A Latch is level-sensitive: invert clock for a negative latch
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
7
Based Positive & Negativ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
k
Positive edge-triggered
register
7475
D
Q
C
Clk
Level-sensitive
latch
Bubble here
for negative
edge triggered
register
Timing Diagram:
D
Clk
Q
7474
Q
7475
Behavior the same unless input changes
while the clock is high
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
11
Important Timing Param... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
2006
Introductory Digital Systems Laboratory
14
JJ--K Master
Slave Register
K Master--Slave Register
Sample inputs while clock high
Sample inputs while clock low
J
K
S
R
P
P
Q
Q
S
R
Q
Q
CLK
Set
Reset
1's
Catch
T oggle
100
J
φ
K
Q
Q
J
K
Clk
P
\ P
Q
\ Q
Correct Toggle
Operation
Master
outputs
Slave
outpu... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
1
1
0
1
1
1
1
1
0
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
18
Realizing Different Types of Memory
Realizing Different Types of Memory
Elements
Elements
Characteristic Equations
D:
Q+ = D
J-K:
Q+ = J Q + K Q
T:
Q+ = T Q + T Q
E.g., J=K=0, then Q+ = Q
J=1, K=0, then Q+ = 1
J=0, K=1, then Q+ = 0
J=... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
K = D
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
20
Design Procedure (cont.)
Design Procedure (cont.)
Implementing J-K FF with a D FF:
1) K-Map of Q+ = F(J, K, Q)
2,3) Revised K-map using D's excitation table
its the same! that is why design procedure with D FF is simple!
JK
J
Q
0
1
00 01 11 1... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
Clk
Combinational
Logic
Th
In
D
Q
Clk
CLK
IN
FF1
CLout
Th
Tsu
Tcq,cd
Tl,cd
Tcq,cd + Tlogic,cd > Thold
L4: 6.111 Spring 2006
Introductory Digital Systems Laboratory
24
Register
ShiftShift--Register
(cid:132) Typical parameters for Positive edge-triggered D Register
D
CLK
Q
Tsu
20ns
Th
5ns
Tsu
20ns
Th
5ns
Tw 25ns
Tplh
2... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/60ded5de196ed8fa1ff1b65a29d27976_lec4.pdf |
System Architecture
Creativity
Thomas H. Speller, Jr.
Thomas H. Speller, Jr.
Engineering Systems Division
Engineering Systems Division
Doctoral Candidate
Doctoral Candidate
January 12, 2007
January 12, 2007
ESD.34
Professor Ed Crawley
1
Today’s Topics on Creativity
• Introduction
• Creativity
– Nature
– Design Rules a... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Solution Space
within the Creative Space
Definition
SA Sol’n Space:= SA’s satisfying the given
specification
DCI Variant
DCI Variant
DCI Variant
DCI Variant
DCI Variant
DCI Variant
DCI Variant
DCI Variant
DCI Variant
DCI Variant
DCI Variant
DCI Variant
DCI Variant
SA Sol’n Space
DCI Variant
DCI Variant
DCI Variant
DCI... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
.
Stephen Hawking, The
Universe in a Nutshell,
2001, pp. 156-168.
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
10
Accelerating rate of increase in the
Stocks of Knowledge and Technology
• The Development of Complexity since the
formation of the E... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Prahalad,
Competing for the Future, 1994)
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
14
Lack of Rigorous Algorithm to Select a
Best Architecture
• Given SA alternatives, how do you
compare them to determine which is better
than another?
• How do you de... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
enterprise sustainability.
1H. A. Simon, Quarterly Journal of Economics 69 (1955) 99.
2Clayton Christensen, The Innovator’s Dilemma, 1997
http://www.amazon.com/exec/obidos/tg/detail/-/0875845851/002-4197440-0554456?v=glance
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Te... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
.
Phylogeny of Archaea, Bacteria, and Eucarya.
Adapted from Baldauf (Science 2003) and Dawkins (2004)
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
23
Evolutionary Computation Approaches
characterized by as having in common:
1. a given and usually random popu... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
istic and combinatoric set of methods
that operate without regard to embedded physics,
discards possibly superior fits by not exploring the full
combinatoric space; are computationally bounded
arbitrarily – halting is user defined.
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Ins... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Genetic
Programming II (1994)
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
34
Courtesy of John Koza.
Used with permission.
Genetic and Evolutionary Conference (2005)
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute o... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Solution Space
Size:= 19,880
16103
3580
y
c
n
e
u
q
e
r
F
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
12 24
0
34
16
11
13
10 29
17
17
14
0
0
3
n
i
b
0
0
6
n
i
b
0
0
9
n
i
b
0
0
2
1
n
i
b
0
0
5
1
n
i
b
0
0
8
1
n
i
b
0
0
7
2
n
i
b
0
0
0
3
n
i
b
0
0
3
3
n
i
b
0
0
6
3
n
i
b
0
0
9
3
n
i
b
0
0
2
4
n
i
b
0
0
1
2
n
i
b... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
6000
4000
2000
0
12
24
0
34
16 11
13
10 29
17 17
14
0
0
3
n
i
b
0
0
6
n
i
b
0
0
9
n
i
b
0
0
2
1
n
i
b
0
0
5
1
n
i
b
0
0
8
1
n
i
b
0
0
1
2
n
i
b
0
0
4
2
n
i
b
0
0
7
2
n
i
b
0
0
0
3
n
i
b
0
0
3
3
n
i
b
0
0
6
3
n
i
b
0
0
9
3
n
i
b
0
0
2
4
n
i
b
Bins
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massac... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
ESD), Massachusetts Institute of Technology
43
SGCA to generate and two alternatives
to find the 20 unique solutions
• 20 unique solutions found by enumeration
– Time: ~40 hours at 2.66GHz processing speed and program
architecture
• 20 unique solutions found by evolutionary computation
using 100 initial pop., 50% se... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
: the creative process can be made into a
formal process
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
47
Today’s Topics on Creativity
• Introduction
• Creativity
– Nature
– Design Rules and Combinatorics
– Work of Vance and de Bono
• TRIZ theory
– TRIZ, Valu... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Serendipity
• Serendipity is defined by OED as: “The faculty of
making happy and unexpected discoveries by
accident. Also, the fact or an instance of such a
discovery.”
• Error and trial
– Mutation in nature
• Mistakes that are recognized as a solution,
ex.
– Telephone
– Microwave
– Post-it
© Thomas H. Speller, Jr. ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
, opportunity, re-phrase the
focus
Why is it done this way?
Why does it have to be done this way?
Are there other ways of doing it?
• Challenge --
– Continuity
– Validity of concept
– Dominating concept
– Assumptions
– Boundaries
– “Essential” factors
– “Avoidance” factors
– Either/Or propositions
© Thomas H. Speller,... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
PO -- shoplifters identify themselves
getting out of a pattern1
58
MOVEMENT TECHNIQUES
• Extract a principle: look for new media
PO -- bring back town crier
(cid:198) can’t turn town crier off
• Focus of the difference: postage stamps
... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
have explored all possible futures? You don’t know, but large fluctuations
in the “environment,” or disturbances, must be considered, not just the different normal,
possible paths out into the future. This is a ‘what if’ analysis. One must consider the
seemingly improbable disturbances that can, and as we have seen ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Web tool
• Radiant Thinking, Mind Mapping tool
• Appendix: Technological change: from its
creation to economic growth and societal
welfare
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
61
TRIZ: A Theory of Inventive Problem
Solving
Many thanks to Dan Frey, ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
move
toward ideality
66
Ideal Technological System
System
System
Function
Object
System
Function
Object
An ideal system does not exist as a physical entity,
but its function is fully performed
http://www.trizgroup.com/
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Tech... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Institute of Technology
72
Ideality: Practice
• A delivery company is shipping meat by
aircraft, but the refrigeration system is heavy
and is reducing the useful payload.
• A Swiss company manufactures confections.
Their process for cracking the shells
occasionally breaks the nut meats inside the
shell and they w... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
in MIT’s soldier nanotechnologies
– Nanomaterial that is normally flexible unless it is
impacted at which time it in microseconds changes
state to become hard and stiff
• Basic invention principle: separate the
requirements in time
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Inst... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Evolution of Subsystem
• Various sub-systems evolve along their own
S-curves at non-uniform rates.
• This causes development of System Conflicts,
which offer opportunities of innovation
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
85
Increasing Flexibility
... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Web tool
• Radiant Thinking, Mind Mapping tool
• Appendix: Technological change: from its
creation to economic growth and societal
welfare
© Thomas H. Speller, Jr. 2007, Engineering Systems Division (ESD), Massachusetts Institute of Technology
90
Holistic View of Creativity
91
Economic System Dynamics
Holistic View... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
Speller 2007
96
Technology
Creative
Environment
Social
Welfare
Capital
Investing
Maslow's
Hierarchy of
Needs
Unlimited
Wants
Delivering
Value to
Society's
Stakeholders
(its constituents)
Economic Growth
Human
Talent
Creating
Creativity
Inventing --
Technological
Change
Technology
© Thomas H. Speller, Jr. 2007, Enginee... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/60ee25fc760c67f2e8a05d57809ad073_creativity_wkshp.pdf |
ESD.33 -- Systems Engineering
Session #9
Critical Parameter Management &
Error Budgeting
Dan Frey
Plan for the Session
• Follow up on session #8
• Critical Parameter Management
• Probability Preliminaries
• Error Budgeting
– Tolerance
– Process Capability
– Building and using error budgets
• Next steps
S - Curves
Ati... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
JT9D-7R4H1 CFM56-5B
CF6-80A/A1 CFM56-5A1
RB211-535E4
2037
CF6-80C2
V2500
4460
RB211-524D4D
4056
4156
CFM56-5C2
4380
4168
4083
P&W
deHavilland
RR
GE
5
4
9
1
0
5
9
1
5
5
9
1
0
6
9
1
5
6
9
1
0
7
9
1
5
7
9
1
0
8
9
1
5
8
9
1
0
9
9
1
5
9
9
1
0
0
0
2
5
0
0
2
Certification date
Adapted from Koff, B. L. "Spanning the World Thro... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
the full sample space?
• What is an example of an “event”?
Probability Measure
• Axioms
– For any event A,
≥AP
)
(
0
– P(U)=1
– If A∩B=φ, then P(AUB)=P(A)+P(B)
For the case of rolling two dice:
A = rolling a 7 and
B = rolling a 1 on at least one die
Is it the case that P(A+B)=P(A)+P(B)?
Discrete Random Variables
• ... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
))
−
((
(
)
xExE
−
((
(
))(
yEy
−
(
)))
Sums of Random Variables
• Average of the sum is the sum of the
average (regardless of distribution and
independence)
y
)
=
xE
)(
+
yE
)(
xE
(
+
• Variance also sums iff independent
σ
• This is the origin of the RSS rule
x
)(
σ
+
=
+
y
x
)
(
2
σ
2
2
y
)(
– Beware of the indepe... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
subplot(3,1,3); hist(z,nbins); xlim([0 3]);
Probability Distribution of Sums
• If z is the sum of two random variables x and y
y
• Then the probability density function of z can be
x
+=
z
computed by convolution
)(
zp
z
=
z
∫
∞−
(
zx
−
d)()
y
ζζζ
Convolution
)(
zp
z
=
z
∫
∞−
(
zx
−
d)()
y
ζζζ
Convolution
)(
zp
z
=... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
• Or, knowledge of x provides no information
to update the distribution of y
Expectation Shift
S=E(y(x))- y(E(x))
S
Under utility theory (DBD),
S is a key difference
between probabilistic and
deterministic design
y(E(x))
E(y(x))
y(x)
fx(x)
x
fy(y(x))
E(x)
Plan for the Session
• Follow up on session #8
• Critical P... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
Process Capability Indices
p(q)
U L−
2
µ
L
U L+
2
U
q
• Process Capability Index
C
p ≡
/ 2
(
)
U L
−
3σ
• Bias factor
k
≡
−
µ
U L
+
2
) /
(
U L
−
2
• Performance Index
C
pk
C
p≡
1
−(
k
)
37
Concept Test
• Motorola’s “6 sigma” programs suggest
that we should strive for a Cp of 2.0. If this
is achieved but the mean i... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
⎞
⎟
⎟
⎠
Crankshafts
• What does a crankshaft do?
• How would you define the tolerances?
• How does variation affect performance?
Printed Wiring Boards
• What does the second level connection
do?
• How would you define the tolerances?
• How does variation affect performance?
Cp and k for the System
LL
UL
Cp = 0 82.
... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
Sources
• Kinematic errors
– Straightness
– Squareness
– Bearings
• Drive related errors
• Thermal errors
• Static loading
• Dynamics
Errors in a Linear Drive
)
m
µ
(
n
o
i
t
a
v
e
d
i
d
a
e
L
1 revolution
Once per revolution lead error (µm)
Cumulative lead error (µm/mm)
Nominal travel (mm)
Angular Errors
εy
y
x... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2004/6111ad8d6ab6cf23dd0a76e5d8c8eb7b_s9_err_bdgtng_v8.pdf |
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