text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
over the surface of a
nylon copolymer as a function of temperature (load 1050 g).
(b) Low -frequency vicoelastic loss data for the same polymer
as a function temperature.
Graphs removed for copyright reasons.
See Figure 6.14 in [Suh 1986].
Frictional Behavior of Composites
Fiber orientation
•
• Continuous vs.... | https://ocw.mit.edu/courses/2-800-tribology-fall-2004/39bee0a6dd75b805338d29c25b8a0810_ch3_friction.pdf |
wear particles by the use of undulated
surface reduces the coefficient of friction to a level of
boundary lubricated cases with boundary lubricants.
6. Boundary lubricants lower the friction coefficient by
preventing wear particle agglomeration and plowing,
but still there is a metal-to-metal contact, which leads ... | 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 |
𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛; 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 Interpretation of R... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/3a0560c6aa37dc11eae36f3f69a4ed7b_MIT2_71S14_lec7_notes.pdf |
more information, see http://ocw.mit.edu/fairuse.
Problem: spherical aberration (𝑐𝑜𝑠 ≈ 1 −
𝜃2
+
𝜃2
4!
+ ⋯);
Mitigation: use proper apertures to reduce NA (max)
o Effect of Aperture and field stops
3
(momentum)x(location)‘(momentum)X’(location)123123airglassRefractive index nS(x)xfF
... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/3a0560c6aa37dc11eae36f3f69a4ed7b_MIT2_71S14_lec7_notes.pdf |
2ndPPEFLEFLIC’COsosihiho
Review on Geometrical Optics (02/26/14)
2.71/2.710 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 ... | 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 |
. 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 thermal noise. In modern
communication systems, especially in light of the on-going
development of digital signal p... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
radio spectrum. It should, however, be emphasized that the
analysis of systems with OC is more difficult than those with
MRC and that the performance evaluation of the former is even
more complicated if fading is taken into account for interfering
and the desired signals.
Closed-form expressions for the bit-error ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
values distribution of Wishart complex
matrices. However, numerical evaluation of SEP requires the
evaluation of multiple integrals, with the number of integrals
depending on the minimum of the number of antennas and in
terferers.
To alleviate this problem, we develop in this paper an efficient
-ary PSK
method t... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
(4)
II. PRELIMINARIES
We now derive the weight vector
that maximizes the output
We consider OC of multiple received signals in flat fading
environment with coherent demodulation, where the fading rate
is assumed to be much slower than the symbol rate. Throughout
stands for
is the transposition operator and
the... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
a white Gaussian random
vector with independent and identically distributed (i.i.d.) ele
ments with
, where
is the two-sided thermal noise power spectral density per
denotes the short-term co-
antenna element. In the following,
and
SINR defined by
(5)
Note that the mean is here taken over the “fast” processes,... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
found
with
(11)
which is in the form (7). Moreover, if
explicit MMSE solution is readily obtained as
is nonsingular, the
where the constant
is the minimum residual MSE given by
(12)
(13)
Thus, we have shown that MMSE and OC receivers are equiva
lent and will be used interchangeably throughout the paper.
B. ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
first perform
channel ensemble of the desired signal to obtain the conditional
. We
SEP, conditioned on the random vector
to average out the channel ensemble of the
then perform
interfering signals.
, denoted by
In many cases of interest, the contribution from the cochannel
interference and noise at the output... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
We first note that the term
in
(16) can also be seen as the determinant of the Vandermonde
given by
matrix
equipped with the inner product and norm defined, respectively,
by
(30)
(31)
If the nonnegative function
increasing in at least
is
, then the elements
are linearly
independent. This implies that there... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
system generated by
.
Proof:
(35)
For any given
, the Vandermonde matrix
canbetransformed,usingtheorthogonalsystemgenerated
by
, into
defined by
. . .
. . .
. . .
. . .
2The dependence of the parameters �
� �
, and
�
is suppressed to simplify
the notation.
3Other choices of measure lead to other orth... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
to a simple single integral over
integration limits. The integrand is a product of two functions
involves trigonometric
can
functions and is given by (22), and the latter function
be evaluated easily as described by the pseudocode provided
in Table I, based on the approach illustrated in Appendix A.
Finally, the... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
pared to time-consuming simulation results obtained for asyn
chronous cochannel interferers.5 We remark that here the simu
lation is at the signal level, thus without any adjoint hypothesis.
The simulation, which took several days of computation time,
was obtained with at least 1000 error events per point and can b... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
when the number of an
4This was always assumed, explicitly or implicitly, in all previous literature
on OC [2], [8]–[15].
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 c... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
can be expressed as
Using 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 exponen... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
studies relied on highly
time-consuming simulations. Hence, performance evaluation of
wireless systems scenarios with optimum combining, that were
either extremely time consuming or impossible by simulation
with current computing power, becomes feasible.
APPENDIX
Derivation of the Orthogonal System
As pointed ou... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
wireless systems,” in Proc. 44th Annu. Int. Vehicular Tech
nology Conf., vol. 2, Stockholm, Sweden, June 1994, pp. 942–946.
, “Upper bounds on the bit-error rate of optimum combining in
wireless systems,” IEEE Trans. Commun., vol. 46, pp. 1619–1624, Dec.
1998.
(52)
with
.
Exploiting the previous relations (47) ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
2, San Antonio, TX, Nov. 2001, pp. 1182–1186.
[4] M. Chiani, M. Z. Win, A. Zanella, R. K. Mallik, and J. H. Winters,
“Bounds and approximations for optimum combining of signals in the
presence of multiple cochannel interferers and thermal noise,” IEEE
Trans. Commun., vol. 51, pp. 296–307, Feb. 2003.
[5] M. Z. Win ... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
antenna array systems with
optimum combining in a Rayleigh fading environment,” IEEE Commun.
Lett., vol. 4, pp. 387–389, Dec. 2000.
[10] T. D. Pham and K. G. Balmain, “Multipath performance of adaptive
antennas with multiple interferers and correlated fadings,” IEEE Trans.
Veh. Technol., vol. 48, pp. 342–352, Mar.... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
of linearly modulated cochannel in
terferers,” IEEE Trans. Commun., vol. 45, pp. 73–79, Jan. 1997.
[22] M. Chiani and A. Giorgetti, “Statistical analysis of asynchronous QPSK
cochannel interference,” in Proc. IEEE Global Telecommunications
Conf., vol. 2, Taipei, Taiwan, Nov. 2002, pp. 1855–1859.
[23] R. A. Fisher,... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
[30] P. D. Lax, Linear Algebra, 1st ed. New York: Wiley, 1996.
[31] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Prod
ucts, 5th ed. San Diego, CA: Academic, 1994.
Marco Chiani (M’94–SM’02) was born in Rimini,
Italy, on April 4, 1964. He received the Dr.lng. degree
(with honors) in electronic... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
Ferrara, Italy, in 1996, and
the Ph.D. degree in electronic and computer science
from the University of Bologna, Bologna, Italy, in
2000.
In 2001, 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 E... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
space
exploration missions. From 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-Resea... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/3a0d5a7fe6f51ba428022500c7d1756a_oc_othopoly_tc.pdf |
CTIONS 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 Communications.
He received the IEEE Communications Society Best Student Paper Award at
the Fourth Annual IEEE NetWorld+Interop’97 Conference... | 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 |
. .
1.2 Considering Switching Power Convertor . . . . . . . . . . . . . . . . .
1.3 Add Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Simple Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Diode
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
. . . . . . . .
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
8
9
2.1 Simple Half-wave Rectifier . . . . . . . . . . . . . . . . . . . . . . . .
10
iii
... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.2
Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.3 Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
Thyristor Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.5 Output Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.6 Power Factor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.7 AC-Side Reactance . . . . . . . . . . . . . . . . . . . . . . .... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
Converter Drawn Left to Right . . . . . . . . . . . . . . .
40
5.8 Direct Canonical Cell . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.9 MOSFET
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
5.10 BJT
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
. . . . . . . . . . . . . . . . .
45
5.19 Direct Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
5.20 Indirect Converters (neglecting ripple)
. . . . . . . . . . . . . . . . .
47
5.21 Direct Converters (neglecting ripple)
. . . . . . . . . . . . . . . . . .
47
v
5.22 Boost Conve... | https://ocw.mit.edu/courses/6-334-power-electronics-spring-2007/3a361ea36eebd72f804991dcb713c90d_content.pdf |
. . . . . . . . . . .
56
5.31 DCM Operation Model . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.32 Parasitic Ringing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
5.33 Design in DCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
vi
List of Tables
5.1 Effect of Al... | 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 |
. . . . . . . . . . . . . . . . . . . .
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 13
O(cid:11)sets of Parametric Curves and
Surfaces
13.1 Motivation
O(cid:11)sets are de(cid:12)ned as the locus of points at a signed distance d along the normal of... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
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)
Figure 13.3: Access space representations in robotics.
3
Curved plate (shell) representation in solid modeling [23]. (See Figure 13.4)
(cid:15)
Figure 13.4: Plate repr... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
)
dn
ds
= (cid:20)t
where (cid:20) is the signed curvature of the curve given by
(cid:20) =
(_r
ez
(cid:2)
(cid:127)r)
v3
(cid:1)
=
_x(cid:127)y
_y(cid:127)x
(cid:0)
( _x2 + _y2)
3
2
(13.3)
(13.4)
_r(t)
j
where v =
is the parametric speed. The curvature (cid:20) of a curve at point P is positive
when the direction of n... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
There are two kinds of singularities on the o(cid:11)set curves, irregular points and self-intersections.
Irregular points
(cid:15)
Isolated points: This point occurs when the progenitor curve with radius R is a circle and
the o(cid:11)set is d =
R.
(cid:0)
Cusps: This point occurs at a point t where the tangent vector... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
0)
By setting (cid:28) 2 = _x2 + _y2 and if r(t) is a rational polynomial curve, the computation of
cusps can be reduced to system of two nonlinear polynomial equations that can be solved
using the methods of Chapter 10.
Examples (see Figures 13.7 and 13.8)
Given a parabola r = (t; t2), the unit tangent and principal n... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
exist two symmetric values t1, t2.
(cid:0)
(cid:0)
(cid:0)
Self-intersections
(cid:15)
7
Self-intersections of an o(cid:11)set curve (see also Figures 13.7 and 13.8) can be obtained by
seeking pairs of distinct parameter values s
= t such that
r(s) + dn(s) = r(t) + dn(t):
(13.11)
Substitution of equation (13.2) in (13... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
8 and cutter path
(cid:0)
8
6
13.2.3 Approximations
(cid:15)
In general, an o(cid:11)set curve is functionally more complex than its progenitor curve because
of the square 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 cur... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
a NURBS curve.
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... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
)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 =
(1 + d(cid:20)max)(1 + d(cid:20)min)
(1 + d(cid:20)max)(1 + d(cid:20)min)
... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
(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)max > (cid:20)min > (cid:20)crit: The normal vector of the progenitor and its o(cid:11)set are directed in the
same direction. Also the sign of Gaussian and principal curvatures of the ... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
ure is a set of points of self-intersection curve in the xy-plane mapped
onto the progenitor surface. A pair of thin solid straight lines emanating from two distinct
points on the surface r(s; t), r(u; v) and intersecting along the parabola are the pairs of vectors
dn(s; t) and dn(u; v).
(cid:15)
(cid:15)
(cid:15)
Crit... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
zs(s; t)
(cid:0)
S(s; t)
d = x(u; v) +
12
yu(u; v)zv(u; v)
yv(u; v)zu(u; v)
(cid:0)
S(u; v)
d(13.26)
6
y(s; t) +
z(s; t) +
xt(s; t)zs(s; t)
xs(s; t)zt(s; t)
(cid:0)
S(s; t)
xs(s; t)yt(s; t)
xt(s; t)ys(s; t)
(cid:0)
S(s; t)
d = y(u; v) +
d = z(u; v) +
xv(u; v)zu(u; v)
xu(u; v)zv(u; v)
(cid:0)
S(u; v)
xu(u; v)yv(u; v)
x... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
)] = 0
(cid:27)![z(s; t)
N 2
(cid:27)2
N 2
!2
(cid:0)
x(s; t)
x(u; v)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
N 2
y (s; t)
(cid:0)
N 2
y (u; v)
(cid:0)
N 2
z (s; t) = 0
N 2
z (u; v) = 0
(cid:0)
where
Nx(s; t) = ys(s; t)zt(s; t)
(cid:0)
Nx(u; v) = yu(u; v)zv(u; v)
Ny(s; t) = xt(s; t)zs(s; t)
(cid:0)
Ny(u; v) = xv(u; v)zu(u; v)
... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
; v) do not necessarily exactly involve the factors
s
(cid:0)
v. See [16] for details for how to exclude trivial solutions.
y(u; v) and z(s; t)
x(u; v), y(s; t)
(cid:0)
u and t
(cid:0)
(cid:0)
(cid:0)
13.3.3 Tracing algorithm
Finding the starting points for tracing the self-intersection curve is very similar to the sam... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
(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:16) det... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
.
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 same symbols
are mapped to the same points in the o(cid:11)set surface. Figure (b) shows the mapping... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
; 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, M , N as coe(cid:14)cients of the (cid:12)rst and second fundamental
forms of the surface.
If we assume further that x and y axes are directed along the
... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
order approximation in the nonparametric form given by
(cid:11) = and
(cid:0)
(cid:0)
Its corresponding parametric form is
z =
1
2
((cid:11)x2 + (cid:12)y2)
r(x; y) = [x; y;
((cid:11)x2 + (cid:12)y2)]T
1
2
(13.51)
(13.52)
In the sequel we assume that d > 0, (cid:12) > 0 and (cid:11)
(cid:12) without loss of generality.... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
Parabolic Point
Table 13.1: Four types of explicit quadratic surfaces according to (cid:11) and (cid:12)
2 ((cid:11)x2 +(cid:12)y2). (a) Top left: Hyperbolic paraboloid
3, (cid:12) = 1). (b) Top right: Elliptic paraboloid ((cid:11) = 1, (cid:12) = 3). (c) Bottom left: Paraboloid
Figure 13.14: Explicit quadratic surface... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
2
p((cid:12)d)2(cid:0)1
(cid:12)
2 = 1
(13.54)
(cid:19)
(cid:18)
(cid:12) < d < 1
(cid:18)
2. An o(cid:11)set of an elliptic paraboloid (0 < (cid:11) < (cid:12)) self-intersects only in the y-direction
(cid:11) and self-intersects in both x and y-directions when 1
when 1
(cid:11) < d. The self-
intersection curve which... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
all directions, when 1
(cid:18)
(cid:19)
(cid:18)
(cid:11) < d.
), and its corresponding curve in the
(cid:12) = 1
The self-intersection curve is a point (0; 0; ((cid:12)d)2(cid:0)1
xy-plane is a circle (see Figure 13.15 (e)) given by
2(cid:12)
x2 + y2 =
((cid:12)d)2
(cid:12)
1
(cid:0)
2
!
p
(13.57)
4. An o(cid:11)set... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
-0.17
-0.50
-0.33
0.00
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... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
: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 with unit normal vector given by
Nij =
1
8
8
ni
i=1
X
(13.60)
3. Check dev... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
. Computer Aided
Geometric Design, 3(1):15{43, May 1986.
[6] R. T. Farouki and C. A. Ne(cid:11). Algebraic properties of plane o(cid:11)set curves. Computer Aided
Geometric Design, 7(1 - 4):101{127, 1990.
[7] R. T. Farouki and C. A. Ne(cid:11). Analytic properties of plane o(cid:11)set curves. Computer Aided
Geometric ... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
. Self-intersections of o(cid:11)sets of quadratic surfaces: Part I, explicit surfaces.
Engineering with Computers, 14:1{13, 1998.
[16] T. Maekawa, W. Cho, and N. M. Patrikalakis. Computation of self-intersections of o(cid:11)sets
of B(cid:19)ezier surface patches. Journal of Mechanical Design, Transactions of the ASME... | https://ocw.mit.edu/courses/2-158j-computational-geometry-spring-2003/3a77ce2173073840510f548601cd36b4_lecnotes13.pdf |
modeling using o(cid:11)set surfaces.
Journal of OMAE, Transactions of the ASME., 110(3):287{294, 1988.
[24] B. Pham. O(cid:11)set curves and surfaces: a brief survey. Computer-Aided Design, 24(4):223{
229, April 1992.
[25] T. Rausch, F.-E. Wolter, and O. Sniehotta. Computation of medial curves on surfaces.
In T. Goodm... | 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 |
• 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) using FFT
• evaluation at fixed point in O(n) time
•
Random test... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/3b0b8e79c6dd4381ed65ab37f7076165_n10.pdf |
s 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[E1 | E2] + Pr[E2]
3
�
�
�
Small problem:
• degree n poly can generate huge... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/3b0b8e79c6dd4381ed65ab37f7076165_n10.pdf |
• 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
• endogenous: leaf nodes empty.
• balanced if depth O(log n)
• average case.
•
worst case
Tree balancing
•
rotations
• implementing operati... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/3b0b8e79c6dd4381ed65ab37f7076165_n10.pdf |
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
– Suppose some z between inserted before y
– Then y in left subtree of z, x in right, so not ancestor
– Thus, y before every z
• Suppose y... | 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 |
��
⎠ + ρ ⋅g⋅h
power 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
... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
2
2⋅
=
⎛ Vj ⎞
⎜
⎝ VA ⎠
g h
2
V
if h = 0
ηi =
⎠ =
⎞
− 1⎟
2⋅⎜
⎝
⎛ VJ
VA
2
⎛ Vj ⎞
⎟ − 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 ... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
m_dot VA
⋅
⋅⎢
⎜
⎣⎝
V
j
V
A
⎞
− 1⎟
⎠
1
− CD 2
⋅
⎤
⎥
⎦
net thrust power
PT_net = Tnet ⋅VA =
⎛
⎡
2
m_dot VA
⋅
⋅⎢
⎜
⎣⎝
V
j
V
A
⎞
− 1⎟
⎠
1
− CD
⋅
2
⎤
⎥
⎦
delta p across pump must be increased to account for losses ...
we'll assume separate inlet and outlet losses
assume internal losses are ... ~ 1/2*ρ*v... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
⎛
2 ⎢
⋅ρ⋅VA ⋅ ⎜
⎢
⎝ VA ⎠
⎣
⎟ − 1 + 2⋅
g h
⋅
2
VA
⎤
2
Vj ⎞
⎛
⎥
+ Kin + Kout ⋅⎜
⎟
⎝ VA ⎠ ⎥
⎦
ideal pump power is ...
Ppi =
poj
m_dot⋅
−
pop
ρ
=
2
⎡
Vj
⎞
⎛
⎢
1
2
m_dot VA
⋅
⋅
⋅
⎟
⎜
⎢
2
VA
⎠
⎝
⎣
− 1 + 2⋅
g h
⋅
2
V
+
Kin Kout
+
⋅
⎛
⎜
⎝
Vj
⎞
⎟
VA
⎠
2⎤
⎥
⎥
⎦
define ηp such that
η = p
Ppi ... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
⎛
⎢
⋅
⎜
⎢
⎝
⎣
⎞
− 1 −
⎟
⎠
⎡
⎛
2
m_dot VA
⋅
⋅
⎜
⎢
⎣
⎝
Vj
VA
CD⋅
⎤
1
⎥
2
⎦
ηreal
=
PT_net
Pp
=
2
m_dot VA
⋅
ηp
1
⋅
2
2
⋅
⎡
Vj
⎞
⎛
⎢
⎟
⎜
⎢
VA
⎠
⎝
⎣
− 1
+
2
⋅
⎡
⎢
Kin
⎢
⎣
+
g h
⋅
2
VA
+
Kout
⎛
⋅
⎜
⎝
Vj
VA
⎞
⎟
⎠
⎤
⎤
2
⎥
⎥
⎥
⎥
⎦
⎦
for a different form ... multiply numerator and
denomina... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
/Vj ...
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
⋅
(
)
... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
⎡
⎜
⎢
⎣
⎝
Vj
VA
⎞
− 1 −
⎟
⎠
⋅CD
⎤
1
⎥
⎦
2
2
Vj
⎞
⎟
VA
⎠
− 1
+
2
⋅
+
Kin
+
⎛
Kout
⋅
⎜
⎝
Vj
⎞
⎟
VA
⎠
⋅g h
2
VA
=
2
1 − t
1 − w
⋅
η
p
=ηD
1 − 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_... | https://ocw.mit.edu/courses/2-611-marine-power-and-propulsion-fall-2006/3b26401197d2c93c660289c51fa8e7bb_04waterjet.pdf |
1 − t
1 − w
⎞
⎟ − CD⋅
− 1
⎠
1⎤
⎥
2⎦
this is the form previously
2⋅ηp ⋅
⋅ηp ⋅
⎡⎛ Vj
⎜
⎢
VA
⎝
⎣
2
⎛ Vj ⎞
⎜
⎝ VA ⎠
⎟ − 1 + 2⋅
+ K
g h
⋅
2
VA
and ... with CD = 0 and assuming h = 0 (i.e. head loss is small compared to other terms ...
ηD =
1 − t
1 − w
⋅ηp ⋅
2⋅μ ⋅(1 − μ)
2
1 + Kout − μ ⋅(1 − Kin)
this is ... | 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 |
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 AD .
The other direction is more complex, and... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
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.
• The input to ZD is given to the simulated Z, and the output from Z is given as the
output of ZD .
82
6
6
output
?
input
?
Z
6 6 6
ZD
Q
Q
}ZZ Q Q
Z
Q
Z
Q
... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
either AD or SD to the
dummy environment ZD as the real adversary (as indicated in the diagram above).
83
First, notice that the output of ZD with AD running protocol Π will be indistinguishable
from the output of ZD with SD and functionality F:
ExecΠ,AD ,ZD ≈ IdealF
SD ,ZD
However, ignore the solid box for ZD ... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
model which will be used in the main UC
theorem. This execution model is much like previous ones, except that now parties can
access multiple copies of the functionality F. In particular, the hybrid model of protocol Π
with ideal functionality F and environment Z is system of ITMs as follows:
• The initial TM is th... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
“corrupted” on the subroutine tape of Z.
– At each subsequent activation, P sends its entire state to Z.
– A assumes all write privileges of P .
– The reaction of F to corruption messages is left up to the discretion of the definer
of F. Such messages are also written on the subroutine tape of Z, however.
4 The Com... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
H, we will rely on the security of
protocol P . Since P securely realizes F, we know that there exists a simulator SP such that
no environment can distinguish between P /AD in the real setting and F/SP in the ideal
setting:
We will use SP
D ,
P,A Z
to construct the adversary H as follows:
Ideal
F
P ,
S Z
≈
Exec... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
ection from copies of F to instances of
SP . Thus, when a copy of SP wishes to send a message to its copy of F (which it
presumes to be the only one in the world) H routes it to the appropriate copy among
the many copies of F available. It is essential for this that the sid identifiers given
to the various execution... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
i, letting the first i executions of P be replaced by the
simulator SP and the functionality F. The rest of the execution of Π remains unchanged.
For the example of Figures 3 and 4, “hybrid” 1 would be as shown in Figure 5.
Thus, if Z can distinguish between “hybrid” 0 (the execution of QP and AD in the real
setting... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
dummy adversary
A, then the simulated Z� sees exactly the ith “hybrid” model. If the external protocol is
simulated by the simulator SP and functionality F, then the simulated Z� sees exactly the
i + 1st “hybrid” model. Hence, A�� will distinguish the protocol P from simulation SP with
advantage at least e/m, contr... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
in a modular
way. To design a protocol that securely realizes F:
• Break F into subtasks F1, F2,. . . Fn.
• Design protocols Πi that securely realize each Fi.
• Design a protocol Π that securely realizes F assuming ideal access to the Fi function
alities.
• Compose the protocols Πi with protocol Π to achieve a p... | https://ocw.mit.edu/courses/6-897-selected-topics-in-cryptography-spring-2004/3b2e382e61c2740c5174cd29a3d51cd3_l8.pdf |
Chapter
3:
Collecting
Data
population
is
a
collection
of
objects,
items,
humans/animals
(“units”)
about
A
which
information
is
sought.
A
sample
is
a
part
of
the
population
that
is
observed.
parameter
is
a
numerical ... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
know
what
you
can
calculate
and
what
you
can’t
calculate!
You
can’t
calculate
anything
from
the
population
if
you
only
have
a
sample.
sampling
frame
is
a
list
of
all
units
in
a
finite
population.
Often,
we
... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
possible
with
these
sampling
methods.
To
avoid
bias,
sample
.
sampling)
randomly
from
the
population.
quota
A
wi
bei
simple
random
sample
(SRS)
of
size
n
from
a
population
of
size
N
is
drawn
th
ng
c
eplacement
so
... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
(some
races
are
rarer
than
others)
involves
dividing
the
population
into
homogeneous
eful
when
you
want
This
ons
as
well
as
on
the
wh
on
subpopulati
ole
po
is
us
n.
g
Multistage
cluster
samplin
each
stage.
Useful
... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
every
kth
unit.
Useful
for
sampling
items
coming
off
assembly
lines.
MIT OpenCourseWare
http://ocw.mit.edu
15.075J / ESD.07J Statistical Thinking and Data Analysis
Fall 2011
For information about citing the... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/3b37951fcbab6644a27511d7616576a3_MIT15_075JF11_chpt03.pdf |
Lecture 7
Quantum Mechanical Measurements.
Symmetries, conserved quantities, and the labeling of states
Today’s Program:
1. Expectation values
2. Finding the momentum eigenfunctions and the dispersion relations for free particle.
3. Commutator and observables that commute
4. Symmetries and conserved quantities –... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
linear combination of
n :
u x
x
C u x
n n
n
Where the coefficients of the expansion are just as they are in the geometrical analogy
projections of the function
un
direction given by the inner products between the
onto the
function and the normalized basis function
:un
C u
n
n
As... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
non-degenerate): When the physical quantity a is measured on a
system in the normalized state
the probability P a n of obtaining the non-degenerate
t
eigenvalue an of the corresponding observable is:
n
P a u
n
2
u x dx C * u * xu x dx C
C * u * x
s
n
n
s
s
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
C
x, t0
Substituting this into the equation above we get:
Aˆ x,t x,t * ˆ x
x A dx
ˆ
A
n
* *
n n
A C x dx
C u x ˆ u
n n
n
n
C u x C Au x dx
n
C u x C a u x dx
n n n
* *
n n
ˆ
* *
n n
n
C *C a u * x
s
n
n
n
s
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
1 2ik0x4i0t
x, t e
e
2
2
ik0 xi0t
1
2
e
3
e
1
2
What is the average momentum of this particle?
Using a definition of the expectation value we can write:
pˆ x,t pˆ x,t * x,t pˆ x,t dx
1 ik0xi0t
1 2ik0x4i0t
e
e
2
2
... | https://ocw.mit.edu/courses/3-024-electronic-optical-and-magnetic-properties-of-materials-spring-2013/3b767a0fdb4bfc53ac253546e002d8a7_MIT3_024S13_2012lec7.pdf |
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