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6.854J / 18.415J Advanced Algorithms
Fall 2008
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18.415/6.854 Advanced Algorithms
October 1, 2008
Lecturer: Michel X. Goemans
Lecture 8
Previously, we introduced the dynamic... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
(we omit the arrows here) from top to bottom.
1.2 Virtual Trees
The union of disjoint paths described above can be used to represent virtual trees. In a virtual tree,
each solid path is represented by a splay tree such that the following conditions hold:
•
A successor node in a splay tree is an ancestor in the roo... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
and that the splay tree representing the path to
which v belongs is rooted at v. We can describe this operation in three steps. In our example, we
run Expose on node 15.
2.1 Step 1
Step 1 consists of walking from v to the root of the virtual tree. Whenever the walk enters a splay
tree (solid edges) at some node w,... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
in the splay tree corresponds to the part of the solid path that is below w in the rooted tree;
this is because w is the root of its splay tree. Exchanging that solid subpath with the solid path
corresponding to the splay tree rooted at v still leaves the rooted tree decomposed into a node-disjoint
union of paths.
... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
these operations require keeping track of different
costs in the tree, so we first consider an efficient way of doing this.
3.1 Maintaining Cost Information
When performing operations on the dynamic tree, we need to keep track of cost(x) for each node x,
and we need to be able to find the minimum cost along paths to the... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
a
splice, which will allow us to maintain cost(x) and mincost(x) in O(1) time.
We first consider a rotation, see Figure 6 for the labelling of the nodes. Let Δ cost(x) and
Δ cost�(x) correspond to before and after the rotation, respectively. Similarly define Δ min(x) and
Δ min�(x). Observe that during a rotation, onl... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
, and so
only those Δ min values change. Using (1), one can see that
Δ min�(w) = max(0, Δ min(b) − Δ cost�(b), Δ min(c) − Δ cost(c))
Δ min�(v) = max(0, Δ min(a) − Δ cost(a), Δ min�(w) − Δ cost�(w)).
Notice that Δ min�(v) depends on Δ min�(w) that was just computed.
Similar when we perform the splicing step given i... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
min(right(v))}.
If Δ min(v) = 0, then splay(v) and then return v, as the minimum is achieved at v. Else,
if −Δ cost(left(v)) + Δ min(left(v)) > −Δ cost(right(v)) + Δ min(right(v)), then the minimum
is contained in the left subtree and we walk down it recursively. Otherwise, the minimum is
contained in the right sub... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
fine
s(x) =
�
y∈Tx
w(y),
where Tx is the entire virtual tree subtree attached at x. Then, consider r(x) = log2 s(x) and take
our final potential function to be
�
φ(T ) = 3
r(x).
This differs from the potential function for splay trees in 2 ways. First Tx is defined over the entire
virtual tree and secondly we ha... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
We are using the splay operation on the k nodes v, p(t(v)), . . . , (p
meaning that we get a total amortized runtime of
◦
t)k−1(v) in this step,
�
9 r(t((p ◦ t)i(v))) − r((p ◦ t)i(v)) + 1 ≤ 9[r(root) − r(v)] + k,
�
−1
k
�
i=0
since we have that r(t(p ◦ t)i−1(v)) ≤ r((p ◦ t)i(v)), so the sum telescopes. The amort... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
O(log n), for
a total of O(log n).
4.3 Runtimes of all Operations
We can now briefly summarize the runtimes of all other operations in terms of Expose.
• find-cost, find-root, find-min, add-cost
Each of these operations requires at most one use of Expose, at most one run of splay, and
at most one search of the tre... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
cost is at most
O((m + n) log n). We now have the following theorem.
Theorem 1 Any m operations on a dynamic tree with n nodes run in O((m + n) log n) time.
8-8 | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/5cccdb7dca0aa81810781ba75d0a1d46_lec8.pdf |
Lab 1 - Revisited
• Display signals on scope
• Measure the time, frequency, voltage
visually and with the scope
• Voltage measurement*
• Build simple circuits on a protoboard.*
• Oscilloscope demo
6.091 IAP Lecture 2
1
RMS Voltage
• 0-5v square wave (50%) duty cycle has a
rms value of 5 / = 3.54v
2
• 5v peak-peak sq... | https://ocw.mit.edu/courses/6-091-hands-on-introduction-to-electrical-engineering-lab-skills-january-iap-2008/5cf021d5f88ec7ce343a8f31bff527f4_lec2.pdf |
controlled device
~25 V max operating
6.091 IAP Lecture 2
D
S
OFF
ON
Vgs < Vt
Vgs ≥ Vt
7
Bipolar Junction Transistors
collector
ic = β ib
+
Vce
-
β ~50-300
emitter
ie = β ib +ib
NPN
base
ib
PNP
• BJT can operate in a linear
mode (amplifier) or can
operate as a digital switch.
• Current controlled device
• Two famil... | https://ocw.mit.edu/courses/6-091-hands-on-introduction-to-electrical-engineering-lab-skills-january-iap-2008/5cf021d5f88ec7ce343a8f31bff527f4_lec2.pdf |
3
TLP732
1: Anode
2: Cathode
3: NC
4: Emitter
5: Collector
6: Base
1
2
3
6
5
4
6
5
4
• Optical Isolators are
used to transmit
information optically
without physical
contact.
• Single package with
LED and photosensor
(BJT, thyristor, etc.)
• Isolation up to 4000
Vrms
6.091 IAP Lecture 2
12
Figure by MIT OpenCourse... | https://ocw.mit.edu/courses/6-091-hands-on-introduction-to-electrical-engineering-lab-skills-january-iap-2008/5cf021d5f88ec7ce343a8f31bff527f4_lec2.pdf |
)
dB
=
20
log
⎛
⎜⎜
⎝
V
o
V
i
⎞
⎟⎟
⎠
dB
=
10
log
⎛
⎜⎜
⎝
P
o
P
i
⎞
⎟⎟
⎠
log10(2)=.301
3 dB point = ?
6.091 IAP Lecture 2
19
741 Open Loop Frequency Gain
6.091 IAP Lecture 2
20
741 vs 356 Comparison
Input device
741
BJT
356
JFET
Input bias current
0.5uA
0.0001uA
Input resistance
0.3 MΩ
106 MΩ
Slew rate*
0.5 v/us
7.5 v/u... | https://ocw.mit.edu/courses/6-091-hands-on-introduction-to-electrical-engineering-lab-skills-january-iap-2008/5cf021d5f88ec7ce343a8f31bff527f4_lec2.pdf |
91 IAP Lecture 2
23
Lab Exercise - Oscillator
• Wire up a comparator on the
proto-board using 741 op-amp.
Be sure to supply power
and ground.
• R1=10k, R2=4.7k, R3=10K,
C=.33uf
• Display V- and Vout on the
scope. Describe what is
happening. Set R3=4.7k.
Predict what happens to the
frequency.
6.091 IAP Lecture ... | https://ocw.mit.edu/courses/6-091-hands-on-introduction-to-electrical-engineering-lab-skills-january-iap-2008/5cf021d5f88ec7ce343a8f31bff527f4_lec2.pdf |
“on” time.
6.091 IAP Lecture 2
27
Lab Exercise - Schmitt Trigger
Vin
V-
V+
Vo
R2
R1
• Schmitt trigger have
different triggers
points for rising edge
and falling edge.
• Can be used to
reduce false triggering
• This is NOT a
negative feedback
circuit.
6.091 IAP Lecture 2
28
Current Source
+15V
• Household applic... | https://ocw.mit.edu/courses/6-091-hands-on-introduction-to-electrical-engineering-lab-skills-january-iap-2008/5cf021d5f88ec7ce343a8f31bff527f4_lec2.pdf |
RE=100, βF=100, Vin=5
•
iE ~iC
• R1=10k, R2=4.7k,
R3=10K, C=.33uf
• R1, R2 can be replaced
with a pot.
6.091 IAP Lecture 2
* Fall 2006 6.101 lab exercise
30
Lecture 2 Summary
• BJT, MOSFET
• Op Amp circuits
– Comparator
– Oscillator
– Schmitt trigger
– Current Source
6.091 IAP Lecture 2
31 | https://ocw.mit.edu/courses/6-091-hands-on-introduction-to-electrical-engineering-lab-skills-january-iap-2008/5cf021d5f88ec7ce343a8f31bff527f4_lec2.pdf |
3.15
Transistors
C.A. Ross, Department of Materials Science and Engineering
Reference: Pierret, chapter 10, 15.1-2, 16.1-2 and 17.1.
Transistors are three-terminal devices that use a small voltage (or current)
applied to one contact to modulate (i.e. control) a large voltage (or current)
between the other two contacts.... | https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/5d3ca8e27b90844252ee7d300b5c9c89_lecture7.pdf |
)
+ eRB (recombination in base -small)
The gain β = IC /IB ~ IE /IB = NA,E / ND,B is determined by doping.
‘Common base’ circuit: by setting VEB and IE we control VCB and IC
‘Common emitter’ circuit: by setting VEB and IB we control VEC and IC
Digital logic: make the transistor act like a switch by running between
... | https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/5d3ca8e27b90844252ee7d300b5c9c89_lecture7.pdf |
type.
MOSFET
G
oxide
S
n+
D
n+
n-channel
p-type substrate
depletion region
Source and substrate are grounded. Applying a large positive VG creates an
n-channel which conducts current between S and D.
If VD = 0, channel has uniform width.
If VD > 0, channel is thinner towards the D and may pinch-off; also the... | https://ocw.mit.edu/courses/3-15-electrical-optical-magnetic-materials-and-devices-fall-2006/5d3ca8e27b90844252ee7d300b5c9c89_lecture7.pdf |
Lecture 4
Acoustics of Speech and Hearing
Lecture 4:The Interaction of Sound and Objects: Scattering and Diffraction
6.551J/HST714J
P(r, θ)
A. Definitions:
1. Scattering
Scattering refers to the alteration in
sound path produced by the interaction of a
sound ‘ray’ and an object. Scattering implies
that the inc... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
4.2 Scattering and Diffraction by a large and small object (From Olsen). Which of these
situations is better explained by ray theory and which by wave theory and diffraction?
Image removed due to copyright considerations.
Source: Olsen, H. F. Music Physics and Engineering. Dover Press, 1967 .
Figure 4.3 (a) Scatteri... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
an analytic solution is the
diffraction and scattering of plane waves by a rigid sphere. This solution based on a simple
geometric view of spherical symmetry ( only a and θ are significant) is mathematically complicated
(Rayleigh, 1986; Morse & Ingard 1968).
Figure 4.5A
A geometry to describe the sound pressure at... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
-490.
A) The sound pressure measured at the
surface of a sphere, relative to the free-field
pressure in the stimulating plane wave, as a
function of ka (the x-scale at the top) or
frequency assuming a sphere of the size of the
human head (the x-scale at the bottom). The
parameter is θ, the angle between the soun... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
ener FM (1947). J. Acoust. Soc. Am. 19: 444-451.
Acoustics of Speech and Hearing
6.551J/HST714J
21-Sept-2004
Courtesy of Acoustical Society of America.
Used with permission.
page 6
B
Helix (pf)
Fossa of Helix
Antihelix (pf)
Cymba (concha)
Cavum (concha)
Antitragus
A
Lobule (pf)
Crus helias
Tragus
A'... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
74).
Image removed due to copyright considerations.
Source: Shaw, E. A. G. "The External Ear." In Handbook of Sensory
Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff.
New York: Springer-Verlag, 1974, pp. 455-490.
21-Sept-2004
page 9
Lecture 4
Acoustics of Speech and Hearing
6... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
1974, pp. 455-490.
Measurements and theoretical values for azimuthal dependence (HRTF normalized by HRTF(0)left )
and interaural intensity differences (right) of the magnitude of sound pressure at the ear drum
normalized by the stimulus pressure in the plane wave. (SHAW 1974)
21-Sept-2004
page 11
Lecture 4 ... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
ITD in this model is
independent of Frequency..
In the sinusoidal steady state, the
interaural phase difference is:
IPD = ω ITD =
2π
period
ITD
Adapted from Shaw, E. A. G. "The External Ear." In Handbook of Sensory
Physiology: Vol V/1: Auditory System. Edited by W. D. Keidel, and W. D. Neff.
New York: Springer-Ver... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
aural time is
consistent with models of diffraction about a rigid sphere.
From Kuhn 1977.
Courtesy of Acoustical Society of America. Used with permission.
While the measured and calculated ITDs at frequencies above 1000 Hz in man ka > 2 are consistent
with the spherical model of Figure 4.13: the Normalized ITD,:
... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
On the minimum
audible angle. J. Acoust. Soc. Am. 30: 237-
246.)
Courtesy of Acoustical Society of America. Used with permission.
21-Sept-2004
page 15
Lecture 4
Acoustics of Speech and Hearing
6.551J/HST714J
Measurements of MAA
From Mills AW (1958). On the
minimum audible angle. J.
Acoust. Soc... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
(kHz)
8
Courtesy of Acoustical Society of America. Used with permission.
While the difference between elevation and
azimuth is irrelevant when discussing the
symmetric sphere, asymmetries in the
placement of the external ear and in the
structures of the external ear, generate
differences in the frequency dependen... | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
Wiener, FM, Sound diffraction by rigid spheres and circular cylinders. J. Acoust. Soc. Am. 19
(1947) pp. 444-451.
21-Sept-2004
page 18 | https://ocw.mit.edu/courses/6-551j-acoustics-of-speech-and-hearing-fall-2004/5d477f7380c5b24a6c6f5e739036f9a2_lec_4_2004.pdf |
BETA DRAFT- For Review Only
Space Systems, Policy, and Architecture Research Consortium
(SSPARC)
“Beta plus” Revised Draft 5/26/04
“SSPARC BOOK” MATERIAL
for Lecture 4
Prepared by:
Hugh McManus
Metis Design
Adam Ross
MIT
----
Draft for evaluation only. Do not distribute.
© 2004 Massachusetts Institute of Tech... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
© 2004 Massachusetts Institute of Technology
2/4/04
2
BETA DRAFT- For Review Only
DEDICATION AND NOTE ON SOURCES
This document is an excerpt of a future book or hyper-book on the MATE-CON method. It is
provided for class use as a draft. Suggestions for improvement are welcome, as are warnings of
errors or omissi... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
other students, staff, and undergraduate
researchers also contributed. Bill Borer, Kevin Ray, and John Ballenthin of the Air Force
Research Laboratory, Steve Wall of NASA JPL, and Pete Hendrickson of the Department of
Defense aided with the development of the method and the development of the case studies.
SSPARC r... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
: Preferences and Value Tradeoffs.5 Chapters 18-21 of de Neufville are strongly
suggested reading, covering the same ground as this section in considerably more detail, and
with explicit procedures for collecting user preferences. Keeney and Raiffa state the formal
theory behind the method in great detail. It is imp... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
containing the prespecified alternatives) is not correctly
scoped and assessed. Also, the output of the utility theory will not be the final word on the issue.
A stakeholder presented with the results of a tradespace study may well alter both his or her view
of what alternatives might be interesting (changing the bo... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
BETA DRAFT- For Review Only
If the stakeholder really does have a firm need for specific values of a proposed attribute, it is not
a good attribute. It should instead be treated as a constraint on the tradespace. Typically,
however, the stakeholder needs are not as absolute as Figure 4-1 might suggest. Figure 4-2
s... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
Ui = Ui( xi)
(4-1)
We also define the utility to be zero at the lowest (or least desirable) acceptable level of xi, xi*,
and one at the highest (most desirable) level of xi, xi*.
€
Ui(xi*) = 0
Ui(xi *) = 1
(4-2)
Zero represents the lowest possible level of user satisfaction that the user might still consider; it... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
. normalized cost
per billable minute of a broadband telecom system), in others (e.g. usefulness, to scientists, of
scientific data) it can only be used as a relative metric. In the latter case, interpretation of these
metrics is somewhat dangerous—a higher metric is better than a lower one, but a utility of 0.5
ma... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
Attribute value
b) diminishing returns
y
t
i
l
i
t
U
y
t
i
l
i
t
U
1
0
1
0
Attribute value
c) threshold
Attribute value
d) non-monotonic
Figure 4-4 Typical forms of utility functions
© 2004 Massachusetts Institute of Technology
2/4/04
8
BETA DRAFT- For Review Only
Requirements for Single Attribu... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
40°C; and this statement would be equally true in the Fahrenheit
system although the numbers would be different. In utility terms, zero utility is defined
arbitrarily above (as the lowest utility of interest). The users preference for an attribute with a
utility of 0.2 over one with a utility of 0.1 should be just as s... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
ies”) to tease out user utilities in a way that is as free from biases as humanly possible. It
is NOT a technique for dealing with true technical uncertainties in the results. These are dealt
with later using separate techniques. The probabilities are simply an artifice for extracting the
basic utility curves for the a... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
able of maintaining the detachment from his or her own biases and/or technical problems
necessary to go through the interview process. In practice, this requirement has eliminated about
half of potential interview subjects for either psychological reasons (e.g. failure to suspend
disbelief) or knowledge bias (e.g. too ... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
A Note on Risk Aversion
Most texts on utility theory, including de Neufville, include a discussion of risk aversion or risk-
seeking. These discussions often lead to needless confusion, due mostly to the historical basis of
the terminology. True risk aversion, which can be either a psychological effect or a rational
re... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
4-7 SAU versus performance with confusing risk terminology included
For the purposes of the method, as used here, the above assumption should not be made. The
utilities are intended to express the users preference for various design choices that are assumed
to be available in the tradespace. The SAU interviews are expr... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
real and
psychological reasons to be risk averse or risk taking when answering these questions; see de
Neufville, Chapter 18.
Below are scenarios for three of the six X-TOS attributes. The user is asked to choose between
the “new” and “old” technology in each case. The questions were asked repeatedly, with
different pr... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
data. The
boat-based sensor offers a ## chance of getting 180 degrees of diversity of your
data or a 1-## chance of getting 0 degrees of diversity of your data
In this case, the “traditional method” is a satellite system, for which we are extracting the
utilities. Various XX and ## values were substituted, and the user... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
alternative was 0 deg. diversity. The first quest ion was
easy – a 45% chance of excellent performance vs. a 50% chance of only 30 deg. of diversity.
The second was less so, pairing a 10% chance of excellence vs. a 50% chance of 30 deg.; the
user chose the latter. The fourth question, with 20% chance of excellence, has... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
i
l
i
t
U
0
0
50
100
150
Diversity in Latitude (degrees)
Figure 4-8 Single attribute utility function for Diversity of Latitude
A similar process was used to obtain utility points for all the attributes. In Figure 4-8 we see a
very typical case. The utility shows a diminishing return on the diversity of latitude in the... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
Equatorial Region
© 2004 Massachusetts Institute of Technology
2/4/04
17
BETA DRAFT- For Review Only
Sample Altitude
The question was:
A commercial television provider has offered to place a sensor on its geo-
synchronous satellite with a lookdown capability to extract pertinent data at 150
kilometers. However, ... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
available to interview.
They utilities were explored as one of the parameters in the tradespace exploration.
The delta-V utility is shown in Figure 4-11. Delta-V is a continuous attribute calculated for each
system considered. Utility is assumed to increase linearly with delta-V, with diminishing returns
above the leve... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
(
i
v
V
y
t
i
l
i
t
U
V
-
a
t
l
e
D
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Leo-Geo RT
Leo-Geo
)
s
s
e
l
n
o
i
s
n
e
m
d
(
i
v
V
y
t
i
l
i
t
U
V
-
a
t
l
e
D
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Leo-Geo RT
Leo-Geo
0
2000
4000
6000
8000
10000
12000
0
5000
10000
15000
20000
25000
30000
350... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
(the weighted sum)
The simplest multi-attribute utility function is the weighted sum:
U =
n
∑
i=1
kiUi
(4-4)
where ki is a scalar weight for utility i. It can be found by asking the user the combined utility of
a design with attribute i set to its best single-attribute-utility value xi(max), and all the other
attribute... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
- For Review Only
utility. This function represents a demanding user who wants all attributes of the system to excel
and/or a demanding system function that requires high performance of all of its parts.
Simple Inverse-Multiplicative Utility Function
A final simple function is
or
1− U =
€
U = 1−
n
∏
i=1
n
∏
i=1
(
1− U... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
Only
n
∑ > 1
ki
i=1
(4-12)
the user or system is satisfied with partial solutions. In this case, Eq. 4-11 yields K values less
than one. In the limit of even some of the ki’s approaching 1, K approaches –1 and the Keeney
and Raiffa function reduces to
€
1− U =
n
∏
i=1
(
1− kiUi
)
(4-13)
which is a modified form of the... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
first criteria, referred to as “preference independence,” requires not only that a user’s
preference order in one attribute be independent of the values of the other attributes, but that a
choice that trades two attributes against each other be independent of the value of third. To
understand the later point, imagine t... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
shows the additive utility function, with total utility rising in proportion to the constituent
utilities. The lowest line shows the multiplicative utility function. Total utility rises slowly at
first, reaching a value of only .25 when the two component utilities have reached 0.5. It rises
steeply as both constituent ... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
Interestingly, when the ki’s
were changed quite drastically late in the study, K changed to 0.28, implying the MUA was still
very close to a weighted sum.
© 2004 Massachusetts Institute of Technology
2/4/04
24
BETA DRAFT- For Review Only
k1=k2->1 K=-1
k1=k2=0.9 K=-.99
k1=k2=0.7 K=-.82
k1=k2=0.5 K=0
k1=k2=0.3 K=4.4
k1=... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
U functions for two attributes moving in opposite directions (U2=1-U1)
© 2004 Massachusetts Institute of Technology
2/4/04
25
BETA DRAFT- For Review Only
In contrast to these examples, the B-TOS study7 used a Keeney-Raiffa function which had a K of
–0.998. The MAU was clearly a modified inverse multiplicative func... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
Cost Per Function (CPF). To have an ascending measure of value, one could simply
invert CPF and measure Function Per Cost.
A single, quantifiable metric such as CPF makes comparisons between a broad range of different
systems easy. This approach is limited, however, to situations where the function desired can be
expre... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
users.
If the user has preferences, but they are not independent, this does not invalidate the tradespace
exploration concept, but it does require some creativity for determining user preferences across
many possibilities without requiring the user to rank preferences exhaustively across thousands
of choices. A very in... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
advantage. In either case, tradespace
understanding is a powerful tool for moving the human process of multi-stakeholder decision-
making forward.
© 2004 Massachusetts Institute of Technology
2/4/04
27
BETA DRAFT- For Review Only
4.5. X-TOS Multi-attribute Utilities
The X-TOS study used a Keeney-Raiffa MUA function. T... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
ights
Old
New
Lifespan Utility Function
OLD
NEW
)
1
o
t
0
(
y
t
i
l
i
t
U
1
0.8
0.6
0.4
0.2
0
Latency
Latitude
Equator Time Lifespan
Altitude
0
2
4
6
8
10
Data Lifespan (years)
0.5
0.4
0.3
0.2
0.1
0
Figure 4-17: New weights and Lifespan utility function
© 2004 Massachusetts Institute of Technology
2/4/04
28
BE... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
ly formed in any case. The point is
to gain insight into, and quantify (even if imperfectly), the users preferences, in order to proceed
with the exploration of the tradespace. No choices are fixed, correctly or incorrectly, at this
stage. We will close with two quotes. From Otto and Antonsson,19 a defense against crit... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
design functions, by identifying
the best tradeoffs—particularly under uncertainty!—and by disaggregating design
team decision problems into subproblems on which consensus can be reached.
So, while decision theory by itself does not constitute a theory of design,
integrating it throughout several design phases, includi... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
designs using:
a) Lexographic ordering
b) A weighted sum, “normalized” (by multiplying by a constant factor) so that the
multiattribute utility scales from 0 to 1
c) A multiplicative function
d) A modified inverse multiplicative function
e) a Keeney-Raiffa function
Comment. What ranking is most likely closest to the us... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
001.
8 Seshasai, Satwiksai, “Knowledge Based Approach to Facilitate Engineering Design,”
Masters Thesis in Electrical Engineering, Massachusetts Institute of Technology, May 2002.
9 Spaulding, Timothy J., “Tools for Evolutionary Acquisition: A Study of Multi-Attributes
Tradespace Exploartion (MATE) Applied to the Space... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
berg, E., Yukish, M. A., and Simpson T. W., “Attribute-Based
Multidisciplinary Optimization of Undersea Vehicles,” AIAA Paper 2000-4865, 8th
AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization,
Long Beach, CA.
16 Arrow, K. J., Social Choice and Individual Values, 1st ed., J. Wiley, New York NY... | https://ocw.mit.edu/courses/16-892j-space-system-architecture-and-design-fall-2004/5d55ba3f4c86851ee274c8914b58d6c4_04010lec4frame13.pdf |
OpenCourseWare
2.5 Pitch Accents with more than one tone: Bi-tonal accents L+H* and L*+H
9 August 2006
All pitch accents described so far have been produced with a single tonal element, either a High
(H*) or a Low (L*) aligned with the accented syllable. There are other choices open to speakers
however, to convey ... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/5d68a5b367b536661becde2f4202e4dd_chap2_5.pdf |
accent on the same syllable (the –
mel- of Amelia), followed by a Low phrase accent / High boundary tone sequence (L-H%), the
same break labels, and the same words (Amelia knew him), but in one case the prominence is
associated with the H* and in the other case with the L*.
1
Figure 2.5.1 L+H*
<ameli... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/5d68a5b367b536661becde2f4202e4dd_chap2_5.pdf |
tone with the prominent syllable corresponds to the perception that pitch accent is one of a Low-
tone prominence—the “starred” tone in this bitonal pitch accent is the L (in an L*+H). We’ll see
later that f0 peaks and valleys do not invariably occur on the accented syllable; here we simply
illustrate that the L*+H /... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/5d68a5b367b536661becde2f4202e4dd_chap2_5.pdf |
(the -mel- of
Amelia) is followed by a rise to a peak, corresponding to the +H portion of the pitch accent.
(The subsequent Low and the final rise of the contour are due to the Low phrase accent and High
boundary tone, L-H%). The example with the single-tone L*, <amelia3>, also shows a low f0 on
the pitch-accented ... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/5d68a5b367b536661becde2f4202e4dd_chap2_5.pdf |
onational phrase of the file, with the tone labels L+H* L-L%,
notice the sharp rise in pitch at the beginning of the vowel of the prominent syllable, -mel-,
towards the peak of the High tone. The f0 at the beginning of the word, while starting at a mid
range, falls through the word-initial vowel A- and the following... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/5d68a5b367b536661becde2f4202e4dd_chap2_5.pdf |
syllable, then a sharp rise into the
High tone of the prominent syllable. In <marmalade2>, with the H* pitch accent, the rise to the
peak of the H tone on the pitch-accented syllable is more gradual, starting from the beginning of
the intonational phrase and across the first two syllables of the word. (The slightly ... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/5d68a5b367b536661becde2f4202e4dd_chap2_5.pdf |
variety of
contexts, and a specific context will not necessarily lead all speakers to select the
same intonation contour.
2.5.2.3 Constraints on the use of the L+H* label
The use of the label L+H* is constrained to places where a low f0 cannot be accounted for by
some other tonal event. For example a rise from a L... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/5d68a5b367b536661becde2f4202e4dd_chap2_5.pdf |
is a common occurrence when
speakers produce a pitch that is at the lower end of their pitch range, such as with L-L%. In these
regions, pitch periods come at irregular intervals, so there is no regular period for the f0 tracker
to track. This irregularity in the pitch periods produces irregularity in the pitch trac... | https://ocw.mit.edu/courses/6-911-transcribing-prosodic-structure-of-spoken-utterances-with-tobi-january-iap-2006/5d68a5b367b536661becde2f4202e4dd_chap2_5.pdf |
OUTLIERS: Feb. 4, 2005, R. Dudley, 18.465 notes
A rough definition of an outlier is that it’s an observation far away from the bulk of
the data. There may be multiple outliers in a given data set, especially if it’s large. For
example, Bill Gates’s wealth would be an outlier among those of all individuals.
One of th... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/5d70b3b85c48f657003af200517c2930_outliers.pdf |
don’t change
at all. The same is true if X(1) → −∞, when it keeps the same rank 1.
But, what is an outlier? It turns out to be even harder to give a precise definition
than for a sample quantile. Some books give examples of outliers and a few try to give
specific rules for identifying them.
An example, given in a bo... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/5d70b3b85c48f657003af200517c2930_outliers.pdf |
distribution, it must be from some
other distribution, sometimes called a contaminating distribution. It might have resulted
from some error, or a wrong normality assumption.
√
There’s a problem though with defining outliers in terms of standard deviations if the
standard deviation is estimated from the sample, bec... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/5d70b3b85c48f657003af200517c2930_outliers.pdf |
samples, quantiles have slightly varying definitions). The interquartile
range IQR is defined as q3 − q1. That’s a scale statistic that’s robust, not sensitive to
if we move data in the lower quarter or upper quarter of the order statistics
outliers:
outward, it won’t change the IQR. An attempted definition of outlier... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/5d70b3b85c48f657003af200517c2930_outliers.pdf |
we’d be calling amounts of rain “outliers” if they were larger than 4q3 which
might still not be that large.
By the way, the median rainfall per day in either case would be 0, which is very
uninformative about rainfall.
It seems that we might only want to call a daily amount of rain or snow an outlier if
we compar... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/5d70b3b85c48f657003af200517c2930_outliers.pdf |
18.417 Introduction to Computational Molecular Biology
Lecture 6: September 28, 2004
Lecturer: Ross Lippert
Scribe: Lele Yu
Editor: Sam Kaufman
Dynamic Programming: Gene Discovery
Introduction
This lecture discusses some techniques for gene discovery using dynamic programming
algorithms. We begin with some nece... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/5d9b3d4a5a1557de90cbcc60b2c718f5_lecture_06.pdf |
2004
If there are too many ORFs of length 40 aa, analysis using statistics of codons is no
longer useful because its much easier to randomly get a region of 40aa than 300aa.
6.3
Intron/exon structure
In eukaryotes, splicing (the presence of introns, i.e. noncoding regions interspersed in
exons) can foul this up. ... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/5d9b3d4a5a1557de90cbcc60b2c718f5_lecture_06.pdf |
can be done in O(n) time using dynamic programming.
6.5 Similarity based gene searching/structuring
Another way to find genes in DNA is to match to their known products, mRNA and
proteins.
Lecture 6: September 28, 2004
6-3
6.5.1 Matching known (spliced) mRNA
Suppose we have known (spliced) mRNA, corresponding ... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/5d9b3d4a5a1557de90cbcc60b2c718f5_lecture_06.pdf |
hypothetical intermediate DNA sequence
(”iDNA”) against the input DNA sequence, and then align the translation of the
iDNA to the given amino acid sequence, and optimize the sum of the scores of the
two alignments.
A semi-principled method limits the inserts in DNA relative to iDNA.
6.6 Spliced alignment problem
... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/5d9b3d4a5a1557de90cbcc60b2c718f5_lecture_06.pdf |
CMOS Technology and Logic Gates
Only 15,432,758 more
mosfets to do...
meta
l
pdiff
ndiff
poly
6.884 – Spring 2005
2/07/2005
L03 – CMOS Technology 1
Quality of Design
Quality of a hardware design primarily judged by:
– Price
– Performance
– Power and/or Energy
Other important metrics can include:
– Operatin... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
ors
– ~10,000nm gates originally, now down to 90nm in production
– scaling will stop somewhere below 30nm (over 100 billion
trans./chip)
• Future:
– 3D CMOS (10 trillion transistors/system?)
– Carbon Nanotubes?
– Molecular Electronics?
CMOS VLSI is the digital implementation technology of choice for
the foreseea... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
in course)
6.884 – Spring 2005
2/07/2005
L03 – CMOS Technology 7
Basic CMOS Fabrication Steps
Growing silicon dioxide to serve as an
insulator between layers deposited on the
surface of the silicon wafer.
Doping the silicon substrate with acceptor
and donor atoms to create p- and n-type
diffusions that form isol... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
Source
diffusion
Eh
gate
Ev
bulk
inversion
happens here
Drain
diffusion
INVERSION:
A sufficiently strong vertical field
will attract enough electrons to the If a channel exists, a horizontal
surface to create a conducting n-
type channel between the source
and drain.
field will cause a drift current
from th... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
.18µm has
0.18µm transistor
length)
2
3
2
2
2
3
3
2x2
1
2
3
3
1
1
diffusion (active)
poly
metal1
contact
6.884 – Spring 2005
2/07/2005
L03 – CMOS Technology 13
Static CMOS Gates
F = (A+B).(C+D)
6.884 – Spring 2005
Figure by MIT OCW.
2/07/2005
L03 – CMOS Technology 14
Simplified FET Model
Binar... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
NAND Gate
A
B
(A.B)
B
(A.B)
A
When both A and B are high, output is low
When either A or B is low, output is high
6.884 – Spring 2005
2/07/2005
L03 – CMOS Technology 17
NOR Gate
A
B
(A+B)
A
B
(A+B)
When both A and B are low, output is high
When either A or B is high, output is low
6.884 – Spring 2005 ... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
, f=(A.B)
Pulldown f = A.B
Pullup p = f = A.B
= A+B
(De Morgan’s Laws)
For NOR gate, f=(A+B)
Pulldown f = A+B
Pullup p = f = A+B
= A.B
parallel
switches
form OR
(A.B)
series
switches
form
AND
A
B
series
switches
form
AND
(A+B)
parallel
switches
form OR
B
A
6.884 – Spring 2005
2/07/2005
L03 – CMOS Te... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
drives another, all capacitance on the node must be
charged or discharged to change voltage to new state. Delay is
proportional to driving resistance and connected capacitance.
CgateP
CgateP
R
onP
CdrainP
R
onP
CdrainP
0⇒1
R
onN
CdrainN
R
onN
CdrainN
C
gateN
C
gateN
Delay ~ RonN(CdrainN+CdrainP+CgateP+CgateN)... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
.884 – Spring 2005
2/07/2005
L03 – CMOS Technology 27
Method of Logical Effort
(Sutherland and Sproul)
Figure by MIT OCW.
• Easy way to estimate delays in CMOS process.
• Indicates correct number of logic stages and transistor sizes.
• Based on simple RC approximations.
• Useful for back-of-the-envelope circuit... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
4
4
1
1
Inverter
NAND
NOR
Input Cap = 3 units
Input Cap = 4 units
Input Cap = 5 units
L.E.=1 (definition)
L.E.=4/3
L.E.=5/3
6.884 – Spring 2005
2/07/2005
L03 – CMOS Technology 31
Electrical Effort
Cin
Logic Gate
Cout
Ratio of output load capacitance over input capacitance:
Electrical Effort = Cout/... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
, F = GH
Minimum delay when each of N stages has equal effort
(gi = L.E. stage i)
(hi = E.E. stage i)
(pi = P.D. stage i)
Min. D = NF1/N + P
i.e. gi hi = F1/N
6.884 – Spring 2005
2/07/2005
L03 – CMOS Technology 35
Optimal Number of Stages
Cin
Cout
Minimum delay when:
stage effort = logical effort x electr... | https://ocw.mit.edu/courses/6-884-complex-digital-systems-spring-2005/5db68ba6abd100ecc232604ccfeb9f46_l03_cmos_gates.pdf |
6.252 NONLINEAR PROGRAMMING
LECTURE 7: ADDITIONAL METHODS
LECTURE OUTLINE
• Least-Squares Problems and Incremental Gra-
dient Methods
• Conjugate Direction Methods
• The Conjugate Gradient Method
• Quasi-Newton Methods
• Coordinate Descent Methods
• Recall the least-squares problem:
minimize f (x) = 1
2
(cid:1)g... | https://ocw.mit.edu/courses/6-252j-nonlinear-programming-spring-2003/5db87e8d64f415730c362e7abd4042a6_6252_slides07.pdf |
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