text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
)
What is the controlling mechanism for
observed friction?
Is the friction due to adhesion?
(b)
(c) What is the role of wear particles in
determining the coefficient of friction?
(d) Why do different material combinations
give arise to different friction coefficient?
(e) What is the effect of environment?
S... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf |
5, and 1095 steel
Photos removed for copyright reasons.
See Figure 3.2 in [Suh 1986].
Coefficient of friction versus sliding distance
µ
µs
µi
∆µρ
µ
µs
µi
Distance slid
(a)
Distance slid
(b)
Effect of removing wear particles for an Armco
iron slider sliding against an Armco iron specimen
µ
µs
µI’
µi
Wear pa... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf |
0.0
5gf5gf
Al coated Si(5gf)
Flat surface
Undulated surface
t
n
e
i
c
i
f
f
e
o
C
n
o
i
t
c
i
r
F
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
Sliding distance(m)
Sliding distance(m)
Agglomeration of wear particles
Abrasive mark
Extrusion of Al layer
Adhesive mark
Plugged undulation
Delaminat... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf |
θ
O
Slip-line field solution for friction as a
function of the slope of asperities
1.0
m
0.5
q
a
=
20o
15o
10o
5o
0o
Figure 3.9
0
15
q
'
30
45
Figure by MIT OCW. After Suh, N. P., and H. C. Sin. "The Genesis of Friction." Wear 69 (1981): 91-114.
Effect of Boundary Lubrication
∼ µ ~ 0.1
• Cause?
– Plowing
• W... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf |
1.5
1
0.5
0
Voltage
Torque
50
0
-50
-100
-150
-200
-250
0
1000
2000
3000
4000
5000
Cycles
Friction at Polymeric Interfaces
• Thermoplastics
Highly linear semicrystalline polymers: HDPE, PTFE
Linear semicrystalline polymers
–
–
– Polymers with large pendant groups (amorphous polymers)
• Thermosetting pl... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf |
• Continuous vs. chopped fibers
• Example: Brake lining, carbon/carbon
composites, teflon/graphite fiber composites
Effect of Coatings on Friction
• Hard coatings on metals
–
TiN, DLC, TiC, Al
2O3-13TiO2, etc.
•
Soft coatings on metals (primarily to reduce wear)
–
Ni/Au/Steel, Cd/Steel, Au/steel, etc.
•
Po... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf |
particle agglomeration and plowing,
but still there is a metal-to-metal contact, which leads
to plowing and the observed coefficient of friction of
about 0.1.
Conclusions
7. Polymers are used extensively in diverse applications
because of their unique tribological properties. For
instance, highly linear polymer... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf |
Review on Geometrical Optics (02/26/14)
2.71/2.710 Introduction to Optics –Nick Fang
Reminder: Quiz 1 (closed book, Monday 3/3, in class)
Topics Covered:
(Pedrotti Chapter 2, 3, 18)
Reflection, Refraction, Fermat’s Principle,
Prisms, Lenses, Mirrors, Stops
Lens/Optical Systems
Analytical Ray Tracing... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/3a0560c6aa37dc11eae36f3f69a4ed7b_MIT2_71S14_lec7_notes.pdf |
𝑡
(𝑎 𝑖𝑠 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛; 𝑖𝑛 𝑎 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑓𝑖𝑒𝑙𝑑 𝑈( ), 𝑎 −
𝑑𝑈
𝑑𝑥
)
We may define Optical Lagrangian: ℒ 𝑛( , 𝑧)√ + 1
𝜕ℒ
𝜕
𝑑
𝑑𝑧
(
𝜕ℒ
𝜕
)
LHS: “Potential force”
RHS: “Acceleration”
Or
𝜕𝑛
𝜕
𝜕𝑛
𝜕
√ + 1
𝑑
𝑑𝑧
𝑛
√ + 1
1
√ + 1
𝑑
𝑑𝑧
𝑛
√ + 1
Example: Two Inte... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/3a0560c6aa37dc11eae36f3f69a4ed7b_MIT2_71S14_lec7_notes.pdf |
reduce NA (max)
o Effect of Aperture and field stops
3
(momentum)x(location)‘(momentum)X’(location)123123airglassRefractive index nS(x)xfF
Review on Geometrical Optics (02/26/14)
2.71/2.710 Introduction to Optics –Nick Fang
4
NAentrancepupilaperturestopexitpupilFoVentrancewindowexitwi... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/3a0560c6aa37dc11eae36f3f69a4ed7b_MIT2_71S14_lec7_notes.pdf |
.710 Introduction to Optics –Nick Fang
© Pearson Prentice Hall. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
Note: when prisms and mirrors are used in the optical train, consider “unfold” the optical
axis first!
7
... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/3a0560c6aa37dc11eae36f3f69a4ed7b_MIT2_71S14_lec7_notes.pdf |
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 11, NOVEMBER 2003
1949
Error Probability for Optimum Combining of � -ary
PSK Signals in the Presence of Interference and Noise
Marco Chiani, Senior Member, IEEE, Moe Z. Win, Senior Member, IEEE, and Alberto Zanella, Member, IEEE
Abstract—An exact expression for the... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
5]–[7]. The OC technique provides substan
tial improvement in performance over MRC when interference
is present. For OC, the receiver requires the knowledge of
the desired signal channel gain vector (as with MRC), and
the short-term covariance matrix of the overall disturbance
due to undesired interferers and ther... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
for Information and Decision Systems
(LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139 USA
Digital Object Identifier 10.1109/TCOMM.2003.819197
of OC is evident due to its much more efficient usage of the
radio spectrum. It should, however, be emphasized that the
analysis of systems with OC is more... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
the interferers. Unfortunately,
these bounds are generally not tight. Recently, moving from
the approach presented in [3], tighter bounds have been de
rived in [4], [16], and [17], in the context of multiple-input mul-
tiple-output (MIMO) systems [18]–[20]. It was shown in [3] and
[4] that the exact symbol-error pr... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
of the eigen
values of the short-term covariance matrix. The exact SEP is de
rived in terms of multiple integrals in Section III. In Section IV,
the efficient methods are developed to derive the SEP in terms
of a single integral with finite limits. Finally, in Section V, we
show some numerical results and in Secti... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
PSK, quadrature PSK, etc.
The interfering data samples are denoted by
for
, where the dependence of a (random) time delay
and
is written explicitly to emphasize the asynchronicity between
the desired signal and interfering users. They can be modeled
as uncorrelated zero-mean random variables, and without loss
a... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
8) into (5) as
[2]
(8)
(9)
where it is important to remark that
, varies at the fading rate.
the SINR
, and consequently also
A. Relation Between OC and MMSE
For what follows, it is worthwhile to recognize that (7),
with a proper choice of the scaling factor, also provides the
minimum mean-square error (MMSE)... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
)
was derived in [3] and [4] using the theory
of
of multivariate statistics, relating to complex Wishart matrices
[23]–[26]. It can be shown that the general expression for the
joint pdf of the first
values
, is
, valid for arbitrary
unordered eigen
and
of
eigenvalues of
are identically
The additional
eq... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
.i.d. elements
(17)
1Although this paper concerns the derivation of exact SEP, it is noted here
that the integrand of (21) is Schur monotonic [28], and this fact can be used to
obtain bounds on SEP.
1951
(18)
(20)
(21)
(22)
1952
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 11, NOVEMBER 2003
Using (19)... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
also be written as
.
(33)
For what follows, it is convenient to introduce the function2
(25)
can be obtained by a
The orthogonal system
, as shown
Gram–Schmidt procedure using the measure
in Appendix A. Hence, we have constructed an uncountable
number of orthogonal systems, each generated by the measure
3
in... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
53
by means of elementary row operations. Since the determinant
is invariant to such row operations [30]
TABLE I
PSEUDOCODE FOR EVALUATION OF � ���
(37)
We now let
function
permutes the integers
be written as
and let
be the set of all permutations of integers
denote the particular
which
. The determinant ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
efficiently and rapidly
evaluated using standard mathematical packages, even for a
large number of antennas and/or cochannel interferers, where
previous studies relied on highly time-consuming simulations.
; the former function
V. NUMERICAL RESULTS
In this section, the performance in terms of SEP of adap
tive ar... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
1000 error events per point and can be
considered quite reliable. The goodness of the exact analytical
results based on the Gaussian approximation can be appreciated
in the figure and justify the adoption of the Gaussian model for
the residual interference after combining.
Fig. 2 shows the SEP as a function of SNR... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
5Fig. 1 shows the SEP as low as ��
only to illustrate the asymptotic be
haviors of the SEP; these extremely low SEPs are not practical, especially for
wireless mobile communications. Similar comments apply to the extremely low
SEP ranges shown in Figs. 2 and 3.
Fig. 3. SEP as a function of SNR for � � �, 8-PSK, SI... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
the inner product (30) with
becomes
,
where
(44)
(45)
(46)
(47)
(48)
A closed-form expression for
from the integrals [31, eq. 3.353.5]
can be derived starting
(49)
is the exponential integral function [31, Sec. 8.2].
where
Using (49) in (48) and the relations between the exponential
integral function and... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
ers and/or antenna elements. This made pos
sible the exact SEP evaluation for wireless systems with many
users and antennas, where previous studies relied on highly
time-consuming simulations. Hence, performance evaluation of
wireless systems scenarios with optimum combining, that were
either extremely time consum... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
. Technol., vol. 49, pp.
1454–1463, July 2000.
[14] J. H. Winters and J. Salz, “Upper bounds on the bit-error rate of optimum
Comparing (45) and(51), we obtain the
th coefficient of the
[15]
th polynomial as
combining in wireless systems,” in Proc. 44th Annu. Int. Vehicular Tech
nology Conf., vol. 2, Stockholm,... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
.
[2] J. H. Winters, “Optimum combining in digital mobile radio with
cochannel interference,” IEEE J. Select. Areas Commun., vol. SAC-2,
pp. 528–539, July 1984.
[3] M. Chiani, M. Z. Win, A. Zanella, and J. H. Winters, “Exact symbol-
error probability for optimum combining in the presence of multiple
cochannel inte... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
5.
[7] M. Z. Win, G. Chrisikos, and J. H. Winters, “MRC performance for
� -ary modulation in arbitrarily correlated Nakagami fading channels,”
IEEE Commun. Lett., vol. 4, pp. 301–303, Oct. 2000.
[8] A. Shah, A. M. Haimovich, M. K. Simon, and M.-S. Alouini, “Exact
bit-error probability for optimum combining with a ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
]
, “A Laguerre polynomial-based bound on the symbol-error prob
ability for adaptive antennas with optimum combining,” IEEE Trans.
Wireless Commun., to be published.
[18] J. H. Winters, “On the capacity of radio communication systems with
diversity in Rayleigh fading environment,” IEEE J. Select. Areas
Commun., v... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
Wishart, “The generalized product moment distribution in samples
from a normal multivariate population,” Biometrika, vol. 20A, pp.
32–52, 1928.
[25] A. T. James, “Distributions of matrix variates and latent roots derived
from normal samples,” Annu. Math. Statist., vol. 35, pp. 475–501, 1964.
[26] A. Edelman, “Eige... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
Italy, on April 4, 1964. He received the Dr.lng. degree
(with honors) in electronic engineering and the Ph.D.
degree in electronic engineering and computer sci
ence from the University of Bologna, Bologna, Italy,
in 1989 and 1993, respectively.
From 1994 he has been with the Dipartimento di
Elettronica, Informati... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
01, he joined the Consiglio Nazionale delle
Ricerche-Centro di Studio per l’Informatica e i sis
temi di Telecomunicazioni (CNR-CSITE), now a sec
tion of CNR-Istituto di Elettronica e di Ingegneria
dell’Informazione e delle Telecomunicazioni (IEIIT), as a Researcher. His re
search interests include cellular and mob... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
1994 to 1997, he was a Research Assistant with
the Communication Sciences Institute at USC, where he played a key role in
the successful creation of the Ultra-Wideband Radio Laboratory. From 1998
to 2002, he was with the Wireless Systems Research Department, AT&T
Laboratories-Research, Middletown, NJ. Since 2002, h... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
the Secretary for the Radio Communi
cations Technical Committee, the current Editor for Equalization and Diversity
for the IEEE TRANSACTIONS ON COMMUNICATIONS, and was a Guest Editor
for the 2002 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, Spe
cial Issue on Ultra-Wideband Radio in Multiaccess Wireless Communi... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.334 Power Electronics
Spring 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
6.334: Power Electronics
By
David Perreault
Electrical Engineering and Computer Science Department
MIT
Cambridge, Massachusetts
Spring... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
1.1 Linear Regulator
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Considering Switching Power Convertor . . . . . . . . . . . . . . . . .
1.3 Add Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Simple Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
. . . . . . . . . . . . . . . . . .
1.13 Simple Rectifier with Free Wheeling Diode . . . . . . . . . . . . . . .
1.14 Rectifier with Free Wheeling Diode Waveform . . . . . . . . . . . . .
1.15 Linear Circuit with Sum of Fourier Sources . . . . . . . . . . . . . . .
1
2
3
3
4
4
5
5
5
6
7
7
8
... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
. . . . . . . . . . . . . . . . . . . . . .
15
2.8 Full-Bridge Rectifier
. . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.1 Resistor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.2
Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.3 Rect... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
. . . . . . . . . . . . .
28
4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.3 Diode Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.4 Thyristor Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.5 Output Voltage . . . . . .... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
.
36
5.4 Buck (down) Converter . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.5 Change the Location of Source and Load . . . . . . . . . . . . . . . .
39
5.6 Boost (up) Converter . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.7 Boost (up) Converter Drawn Left to Right . . . . . . . . .... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
. . . . . . . . . . . . . . . . . . . . .
42
5.15 Indirect DC/DC Converter . . . . . . . . . . . . . . . . . . . . . . . .
43
5.16 “Buck/Boost” or “up/down” converter . . . . . . . . . . . . . . . . .
44
5.17 Averaged Circuit Variables . . . . . . . . . . . . . . . . . . . . . . . .
45
5.18 Big L, C . . . .... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
5.25 Ripple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
5.26 Ripple Model with Inductor . . . . . . . . . . . . . . . . . . . . . . .
50
5.27 Ripple Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5.28 Boost Converter Waveforms . . . . . . . . . . . . . . . ... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
13.472J/1.128J/2.158J/16.940J
COMPUTATIONAL GEOMETRY
Lecture 13
N. M. Patrikalakis
Massachusetts Institute of Technology
Cambridge, MA 02139-4307, USA
Copyright c
(cid:13)
2003 Massachusetts Institute of Technology
Contents
13 O(cid:11)sets of Parametric Curves and Surfaces
13.1 Motivation . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
. . . . . . . . . . . . .
13.3.4 Self-intersections of o(cid:11)sets of explicit quadratic surfaces . . . . . . . . .
13.3.5 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography
Reading in the Textbook
Chapter 11, pp. 293 - 353
(cid:15)
2
2
5
5
6
9
10
10
11
13
14
21
22
1
Lecture ... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
)(cid:2)(cid:0)
(cid:0)(cid:2)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:2)(cid:0)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:2)(cid:0)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:2)(cid:0)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:2)(cid:0)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
:11)set curves
(cid:15)
(cid:15)
(cid:15)
A planar parametric curve r(t) is given by
where x and y are di(cid:11)erentiable functions of a parameter t.
r(t) = [x(t); y(t)] ;
t
[0; 1]
2
The unit normal vector of a plane curve, which is orthogonal to t, is given by
n = t
ez =
(cid:2)
( _y(t);
_x(t))
(cid:0)
_x2(t) + _y2(... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
ned
by
^r(t) = r(t) + dn(t)
(13.5)
where d > 0 corresponds to positive (\exterior") and d < 0 corresponds to negative
(\interior") o(cid:11)sets.
The unit tangent vector of the o(cid:11)set curve (see Figure 13.7 for illustration)
^t =
_^r
_^r
j
j
=
1 + (cid:20)d
1 + (cid:20)d
j
j
t
The unit normal vector of the o(cid:... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
equations (13.6) and (13.7) changes abruptly from -1 to 1
1 + (cid:20)d
j
when the parameter t passes through t = tc at an ordinary cusp, while at extraordinary
1 + (cid:20)d
points (1 + (cid:20)d)=
j
does not change its value, see Figure 13.7.
j
j
Equation (13.9) for r(t) =
can be reduced to
x(t); y(t)
g
f
d [(cid:127... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
(2t;
1)
(cid:0)
p1 + 4t2
The curvature and its derivative are given by
(cid:20)(t) =
(_r
(cid:127)r)
(cid:2)
3
_r
j
j
(cid:1)
ez
=
2
(1 + 4t2)
;
3
2
_(cid:20)(t) = (cid:0)
24t(1 + 4t2)
(1 + 4t2)3
1
2
Since _(cid:20)(0) = 0, (cid:20)(t) reaches an extremum at t = 0 and furthermore as (cid:127)(cid:20)(0) < 0, (cid:20)(0... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
Substitution of equation (13.2) in (13.11) yields the system [17]
x(s) +
y(s)
(cid:0)
_y(s)d
_x2(s) + _y2(s)
_x(s)d
_x2(s) + _y2(s)
p
= x(t) +
= y(t)
(cid:0)
_y(t)d
_x2(t) + _y2(t)
_x(t)d
_x2(t) + _y2(t)
p
(13.12)
If r(t) is a rational polynomial curve, this system can be converted to a nonlinear poly-
nomial system of... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
root involved in the expression of the unit normal vector. L(cid:127)u [13] for example
has shown that o(cid:11)set of a parabola is a rational curve and its singular point at in(cid:12)nity
was studied by Farouki and Sederberg [8]. However, this result has not been generalized
to higher order curves.
Farouki and Ne(ci... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
2. O(cid:11)set each leg of polygon by d.
3. Intersect consecutive legs of polygon to (cid:12)nd new vertices.
4. Check deviation of the approximate o(cid:11)set with the true o(cid:11)set using as weights (for
rational function) the weights of the progenitor curve.
5. If the deviation is larger than the given toleranc... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
ru
j
^ru
j
^rv
rv
(cid:2)
(cid:2)
j
K = (cid:20)max(cid:20)min; H =
(cid:20)max + (cid:20)min
2
equation (13.16) can be rewritten as follows:
^S ^n = S(1 + 2Hd + Kd2)n
If we take the norm of equation (13.16), we obtain
and substituting ^S into equation (13.16) yields
^S = S
(1 + d(cid:20)max)(1 + d(cid:20)min)
j
j
^n =... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
1 + d(cid:20)min)(cid:20)max
1 + d(cid:20)max
j
jj
(1 + d(cid:20)max)(cid:20)min
1 + d(cid:20)min
1 + d(cid:20)max
j
1 + d(cid:20)min
jj
(13.21)
(13.22)
(13.23)
(13.24)
j
j
Given an o(cid:11)set distance d, the critical curvature is de(cid:12)ned as (cid:20)crit =
categories arise [5]:
1=d then three
(cid:0)
(cid:20)ma... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
3.10: O(cid:11)set surface (left), region bounded by self-intersection curve (center) and
trimmed o(cid:11)set surface (right) of elliptic paraboloid z = 1
2 (1:75x2 + 2y2) with d = 0:6.
(cid:15)
In NC machining, the cutter radius must not exceed the smallest concave principal
radius of curvature of the surface to avoi... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
as a vector-valued mapping of two implicit curves in the uv-
1
parametric space to 3D space via the mapping (13.15), which satisfy (cid:20)max(u; v) =
d or
(cid:20)min(u; v) =
(cid:0)
1
d [9].
(cid:0)
Self-intersections:
Self-intersections of an o(cid:11)set surface are de(cid:12)ned by (cid:12)nding pairs of distinct ... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
(cid:12)x one of the four variables (s, t, u, v), the system of equations (13.26) to
(13.28) yields three equations with three unknowns.
We can replace S(s; t) and S(u; v) by auxiliary variables (cid:27) and ! such that (cid:27) 2 = S2(s; t)
and !2 = S2(u; v).
Consequently the system involving polynomials and square ro... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
; v)zu(u; v)
Nz(s; t) = xs(s; t)yt(s; t)
(cid:0)
Nz(u; v) = xu(u; v)yv(u; v)
yt(s; t)zs(s; t)
yv(u; v)zu(u; v)
(cid:0)
xs(s; t)zt(s; t)
xu(u; v)zv(u; v)
(cid:0)
xt(s; t)ys(s; t)
xv(u; v)yu(u; v):
(cid:0)
(13.29)
(13.30)
(13.31)
(13.32)
(13.33)
(13.34)
(13.35)
(13.36)
(13.37)
(13.38)
(13.39)
Since s = u, t = v are trivi... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
cid:28) ), t = t((cid:28) ), u = u((cid:28) ), v = v((cid:28) ), and by
di(cid:11)erentiating the equation for self-intersection curves of an o(cid:11)set with respect to (cid:28) yields
^rs
ds
d(cid:28)
+ ^rt
dt
d(cid:28)
= ^ru
du
d(cid:28)
+ ^rv
dv
d(cid:28)
(13.40)
13
If we denote ^r(s; t) = [^x(s; t); ^y(s; t); ^z... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
:16) =
(cid:6)
1
2 +
:
2
A2j
j
A1j
j
(13.45)
(13.46)
The points of the self-intersection curves are computed successively by integrating the initial
value problem for a system of nonlinear di(cid:11)erential equations (13.41) to (13.44) using the
variable step size and variable order Adams method [2]. The sign of (cid:... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
[x; y; h(x; y)]T , where h is a di(cid:11)erentiable
function with h(0; 0) = hx(0; 0) = hy(0; 0) = 0.
14
(a)
t (v)
s (u)
(b)
(c)
y
y
z
z
x
x
Figure 13.12: Self-intersection curves of the o(cid:11)set of bicubic patch when d=0.09. Figure (a)
shows the pre-images of the self-intersection curves in parameter domain. The ... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
; 0)x2y + 3hxyy(0; 0)xy2 + hyyy(0; 0)y3] + R (13.47)
where lim(x;y)!(0;0) R(x2 + y2)(cid:0)
3
2 = 0.
If we take into account that h(0; 0) = hx(0; 0) = hy(0; 0) = 0, we can consider
h(x; y) =
1
2
[hxx(0; 0)x2 + 2hxy(0; 0)xy + hyy(0; 0)y2]
(13.48)
as the second order approximation of h(x; y).
Let us denote E, F , G and L... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
0)min =
(cid:0)
N
G
N
G
(cid:0)
(cid:0)
=
=
hyy(0; 0)(13.49)
(cid:0)
hyy(0; 0)(13.50)
(cid:0)
If we set (cid:11) = hxx(0; 0) and (cid:12) = hyy(0; 0) (thus
(cid:12) = are principal curvatures)
and assuming that p is a nonplanar point, the surface can be written locally as a second
order approximation in the nonparametr... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
Self-intersection curves of o(cid:11)sets of the explicit quadratic surfaces r(x; y) =
[x; y; 1
2 ((cid:11)x2 + (cid:12)y2)]T and their corresponding curves in the xy-plane are as follows:
17
Signs of (cid:11) and (cid:12)
(cid:11)(cid:12) < 0
(cid:11)(cid:12) > 0 and (cid:11)
= (cid:12)
(cid:11) = (cid:12)
(cid:11) =... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
given by
z =
2((cid:12)
x2 +
(cid:11))
((cid:12)d)2 + 1
2(cid:12)
(cid:11)
(cid:12)
(cid:11)(cid:12)
(cid:0)
(cid:0)
(cid:11)
(cid:12)
(cid:0)
(cid:11)(cid:12)
(cid:18)(cid:0)
q
((cid:12)d)2
x
1
(cid:20)
(cid:20)
(cid:0)
(cid:11)(cid:12)
((cid:12)d)2
1
(cid:0)
(cid:19)
q
;
y = 0
(13.53)
and its corresponding curve in t... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
Figures 13.11, 13.15 (b)) given by equation (13.54). The self-intersection curve which
self-intersects in the x-direction is also a parabola given by
(cid:19)
z =
2((cid:11)
y2 +
(cid:12))
((cid:11)d)2 + 1
2(cid:11)
(cid:12)
(cid:11)
(cid:11)(cid:12)
(cid:0)
(cid:0)
(cid:12)
(cid:11)
(cid:0)
(cid:11)(cid:12)
(cid:18)(c... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
2 =
((cid:12)d)2
(cid:12)
1
(cid:0)
2
!
p
(13.57)
4. An o(cid:11)set of a parabolic cylinder ((cid:11) = 0 < (cid:12)) self-intersects only in the y-direction when
1
(cid:12) < d. The resulting self-intersection curve is a straight line in the xz-plane
((cid:12)d)2
2(cid:12)
; y = 0
z =
(cid:0)
1
(13.58)
and its corre... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
0
x
(c)
0.33
0.67
-0.67
-0.33
0.00
x
(d)
0.33
0.67
0.50
0.17
y
-0.17
-0.50
-0.33
0.00
x
(e)
0.33
0.67
-0.67
-0.33
0.00
x
(f)
0.33
0.67
Figure 13.15: Self-intersection and ridge curves of o(cid:11)sets of explicit quadratic surfaces. The
solid lines correspond to self-i... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
2) = 2, d = 0:6) (f) parabolic cylinder ((cid:11) = 0, (cid:12) = 2,
d = 0:6)
and
(cid:0)
(cid:0)
(cid:2)
(cid:0)
20
13.3.5 Approximations
Parametric O(cid:11)set Surface Approximation Algorithm [23]. (See Figure 13.16)
1. Input: NURBS surface patch.
2. O(cid:11)set each vertex of polygon by d ... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
Curves and Surfaces. Prentice-Hall, Inc.,
Englewood Cli(cid:11)s, NJ, 1976.
[4] R. T. Farouki. Exact o(cid:11)set procedures for simple solids. Computer Aided Geometric
Design, 2(4):257{279, 1985.
[5] R. T. Farouki. The approximation of non-degenerate o(cid:11)set surfaces. Computer Aided
Geometric Design, 3(1):15{43, ... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
7, June 1997, pages
faces.
230{237. IEEE Computer Society Press, 1997.
[12] T. Lozano-Perez and M. A. Wesley. An algorithm for planning collision-free paths amongst
polyhedral obstacles. Communications of the ACM, 25(9):560{570, October 1979.
[13] W. L(cid:127)u. Rational o(cid:11)sets by reparametrization. Technical r... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
Maekawa, F.-E. Wolter, and N. M. Patrikalakis. Umbilics and lines of curvature for
shape interrogation. Computer Aided Geometric Design, 13(2):133{161, March 1996.
[20] N. M. Patrikalakis and L. Bardis. O(cid:11)sets of curves on rational B-spline surfaces. Engi-
neering with Computers, 5:39{46, 1989.
[21] N. M. Patrik... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
quicha. O(cid:11)setting operations in solid modelling. Computer
Aided Geometric Design, 3(2):129{148, 1986.
[27] W. Tiller and E. G. Hanson. O(cid:11)sets of two-dimensional pro(cid:12)les. IEEE Computer Graphics
and Applications, 4(9):36{46, September 1984.
[28] T. J. Willmore. An Introduction to Di(cid:11)erential G... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
Example
Remark on complex numbers. As we saw in the session on Complex
Arithmetic and Exponentials in Unit I, the formula
D (c eat) = c a eat
(*)
remains true even when c and a are complex numbers. Therefore the rules
and arguments above remain valid even when the exponents and coeffi-
cients are complex. We illustrate ... | https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/3a93f22a9335f75ac746a02f761cb5b4_MIT18_03SCF11_s17_4text.pdf |
Admin
Arora talk.
No class Monday.
Review
Fingerprinting:
•
Universe of size u
• Map to random fingerprint in universe of size v ≤ u
• probability of collision 1/v
Freivald’s technique
• verify matrix multiplication AB = C
• check ABr = Cr for random r in {0, 1}n
• probability of success 1/2
• works to check a... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/3b0b8e79c6dd4381ed65ab37f7076165_n10.pdf |
prob. error at particular point most m/(t/ log t) from above
• so pick big t, union bound
• implement by add/sub as shift in bits
Fingerprints by Polynomials
Good for fingerprinting “composable” data objects.
• check if P (x)Q(x) = R(x)
• P and Q of degree n (means R of degree at most 2n)
• mult in O(n log n) usi... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/3b0b8e79c6dd4381ed65ab37f7076165_n10.pdf |
i Qi(x2, . . . , xn) with Qk () = 0 by choice of k
• By induction, prob Qk evals to 0 is at most (d − k)/ S|
|
i Q(r2, . . . , rn) is a nonzero univariate poly.
x1
• suppose it didn’t. Then q(x) =
�
• by base, prob. eval to 0 is k/|S|
• add: get d/|S|
• why can we add?
Pr[E1] = Pr[E1 ∩ E2] + Pr[E1 ∩ E2]
≤
Pr[E... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/3b0b8e79c6dd4381ed65ab37f7076165_n10.pdf |
)
• So compute mod p, where p is O((n log n + n log r)2)
• only need O(log n + log log r) bits
4
Treaps
Dictionaries for ordered sets
• New Operations.
– enumerate in order
– successor-of, predecessor-of (even if not in set)
– join(S, k, T ), split, paste(S, T )
Binary tree.
• child and parent pointers
• end... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/3b0b8e79c6dd4381ed65ab37f7076165_n10.pdf |
trace of a quicksort
• We proved O(log n) w.h.p.
• for x rank k, E[d(x)] = Hk + Hn−k+1 − 1
• S− = {y ∈ S | y ≤ x}
• Qx = ancestors of x
• Show E[Q−] = Hk .
x
6
•
to show: y ∈ Q
−
x iff inserted before all z, y < z ≤ x.
• deduce: item j away has prob 1/j. Add.
• Suppose y ∈ Q−
x .
– The inserted before x
– S... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/3b0b8e79c6dd4381ed65ab37f7076165_n10.pdf |
before x, but now assume
they arrive in random order in virtual priorities.
7 | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/3b0b8e79c6dd4381ed65ab37f7076165_n10.pdf |
Waterjet
first draft 9/23/04 from Prof. Carmichael notes.
9/17/06: modified to reflect w (V=>VA) and separate inlet
and outlet pressure loss (in addition to drag) to reflect paper
VA
w
Vs
Vj
velocity inlet
wake fraction
ship velocity
nozzle (outlet) velocity
VA := VsVs⋅(1 − w)
T = m_dot⋅(VJ − VA)
m_dot = ... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
absorbed by ideal pump is
therefore ...
Ppi = m_dot⋅
poj − pop
ρ
= m_dot⋅
⎡1 ⎛ 2
⎤
⋅⎝ Vj − VA ⎠ + g h
⎥
⎢
⎦
⎣
2
2⎞
ideal efficiency is then ...
and quasi propulsive coefficient is ...
ηi =
PTi
Ppi
ηD =
effective_power
power_delivered
=
PE
Ppi
=
R Vs
⋅
PPi
=
(1 − t ⋅
) T
⋅Vs
PPi
=
⋅
1 − t T VA ... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
1
⎜
VA
⎠
⎝
same as propeller (we
developed following in
actuator disk
2
Vj
VA
+ 1
from actuator_disk.mcd using new variables to avoid duplication
Δv := VVVVjj − VVA
VV := VVA +
ΔvΔv
2
ηIηI :=
T VVA
⋅
T V⋅
simplify → 2⋅
VVA
VVA + VVj
ηI =
1 +
2
VVj
VVA
what are implications of
VVj = VVA ?
h cannot... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
and outlet losses
assume internal losses are ... ~ 1/2*ρ*v^2
and the pump pressure rise is ...
Δploss = Δpin_loss + Δpout_loss
non-dimensional pressure loss coefficient is ....
and the real pump pressure rise is ...
poj − pop =
1
⋅ρ⋅⎝
⎛Vj
2
2
⎞ + ρ⋅g⋅h + Δploss =
2
− VA
⎠
1
⋅ρ⋅⎝
⎛Vj
2
Kin
=
Δpin_loss
1
2
... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
Ppi
Pp
Pp = actual_pump_power
Pp
=
PPi
ηp
=
m_dot
ηp
⋅
poj −
ρ
pop
=
1
⋅
2
2
m_dot VA
⋅
η
p
2
⎞
⎟
⎠
− 1
+ 2⋅
g h
⋅
2
V
+
Kin
⎛
+ Kout ⋅
⎜
⎝
Vj
⎞
⎟
VA
⎠
⎤
2
⎥
⎥
⎦
Vj
VA
⎡
⎛
⎢
⋅
⎜
⎢
⎝
⎣
⎞
− 1 −
⎟
⎠
⎡
⎛
2
m_dot VA
⋅
⋅
⎜
⎢
⎣
⎝
Vj
VA
CD⋅
⎤
1
⎥
2
⎦
ηreal
=
PT_net
Pp
=
2
m_dot VA
⋅
η... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
ηreal
=
and the quasi propulsive coefficient
is then ..
2 ηp
⋅
⋅
VA
⋅
Vj
2
⎛
⎡
1 −
⎜
⎢
⎣
⎝
VA
Vj
⎞
⎟
⎠
−
⋅CD
VA
⎤
1
⋅
⎥
2 Vj
⎦
2
+
+
⋅g h
2
⋅
2
Vj
⎛
Kin
⋅
⎜
⎝
VA
⎞
⎟
Vj
⎠
+
Kout
=
2
1 −
⎛
⎜
⎝
⎛
⋅
⎜
⎝
Vj
⎞
⎟
VA
⎠
VA
⎞
⎟
Vj
⎠
⋅2 η
⋅μ 1 − μ −
⋅
p
⋅CD
)
⎡(⎢
⎣
2
μ Kin − 1 + 1
⋅
(
)
+
Kout
+... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
− t
1 − w
⋅η
⋅
p
⎛
⎜
⎝
as from above ...
⎡(⎢
2⋅μ ⋅ 1 − μ −
⎣
)
CD
⋅
1
2
⎤
⋅μ
⎥
⎦
⋅
1 Kout
+
μ−
2
⋅(
1 −
)
Kin
2
⋅+
g h
⋅
2
Vj
net thrust power
=
PT_net Tnet ⋅V
=
2
m_dot VA
⋅
Vj
⎡
⎛
⋅
⎢
⎜
VA
⎣
⎝
⎞
− 1 −
⎟
⎠
⎤
1
CD
⋅
⎥
2
⎦
first some comments to relate to previous lecture/notes version and W... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
= inlet_loss_coefficient
Kin
=
at this point, assuming K, CD, and ηp are constant, could differentiate wrt Vj/VA (or
μ) and determine Vj/VA for max propulsive coefficient, but minimum weight usually
determines parameters.
pump background (Wislicenus)
example
Kout =
Δpout_loss
1
2
⋅ρ⋅Vj
2
pin_loss
1
2
⋅ρ⋅VA
Δ
2 | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
6.897: Selected Topics in Cryptography
27 February 2004
Lecture 8: The Dummy Adversary
Instructor: Ran Canetti
Scribed by: Jonathan Herzog
1 Previous Lecture
• Motivation for the Universally Composable (UC) framework
• Definition of an interactive Turing Machine (ITM) system with a variable number of
machines
•... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
adversary AD iff it realizes
F (according to the standard UC definition).
Proof. One direction is simple: if Π realizes F in the standard UC framework, then for
every adversary there exists a simulator S which is indistinguishable to every environment
Z. Hence, there must be such a simulator for the dummy adversary A... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
istinguishable from A to all environments in the real system?
We show the answer to be “yes.” For a given environment Z and adversary A, we construct
an environment ZD as in Figure 2 (ignoring the dashed box for the moment). That is, ZD
operates as follows:
• ZD simulates both the environment Z and the adversary A.... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
are passed and returned
as usual.
By definition, SD is indistinguishable from AD to all environments, so it is indistinguishable
to this one in particular. Now, look what happens when we give either AD or SD to the
dummy environment ZD as the real adversary (as indicated in the diagram above).
83
First, notice th... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
theorem crucially depends on the possibility of arbitrary communication between en
vironment and adversary, so that we can think of A as being both part of environment ZD
2
and outside of Z.
3 Hybrid execution
In this section, we define the hybrid execution model which will be used in the main UC
theorem. This exe... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
party playing the role pid
sends a message to the dummy party with ID = (sid , pid ). Likewise, a message
from the functionality is sent to the dummy party with ID = (sid , pid ) (for some
pid ).
• The adversary A can write a special “corrupt” message on the incoming message tape
of any party P . When this happens... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
and let P be a protocol that securely
realizes F. Then for every adversary A there exists another adversary H such that for every
environment Z:
Exec
F
Q,H Z,
≈
Exec
PQ
,A,Z
Proof. Taking advantage of our previous theorem, we prove this with respect to the dummy
adversary AD . That is, we will show that it is pos... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
is used, and routes the messages accordingly. Recall that since AD is the
dummy adversary, such messages actually originate with the environment Z.
• Likewise, messages from the simulated copies of SP are routed by H to the simulated
AD (and thus send by AD to the adversary).
• Since there is a bijection from copie... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
cryptography, and show this with the hybrid argument.
Suppose that there is an environment that distinguishes the two settings. That is, there
exists an environment Z so that
IdealF
Q,H,Z �≈ ExecQP
,AD ,Z
We use the hybrid argument to construct an environment Z� that distinguishes a single run
of P from SP . Let ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
,2
Q2
F2
S2
H1
Figure 5: Example of a hybrid system
• For the i + 1 execution of P , it starts up external real parties to execute it. Messages
from the simulated AD to parties running the i + 1st execution of P are routed to the
external dummy adversary. It also routes messages from the simulated parties to th... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.