text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
gain of k2 be W. Then we can write the following set of operator equations:
Equivalent to H (yes/no)? Yes
W = k2(k1X − (1 + k1)RY)
Y = RW
Eliminating W, we have:
Y = Rk2(k1X − (1 + k1)RY)
= k1k2RX − k1k2R2Y − k2R2Y
which is equal to the operator equation for the original system.
RR++k1k2k1XY−RRR++k1k2k1XY− Chap... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
describes the same system as the state machine.
y[n] = 2y[n − 2] + 3x[n − 2]
The important thing to see here is that the values in the state are (y[n−2], y[n−1], x[n−2], x[n−1]),
so that the output is 2y[n − 2] + 3x[n − 2].
5.8.7 On the Verge
For each difference equation below, say whether, for a unit sample input... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
Yes
positive/alternate/oscillate Oscillates
We first write the operator equation:
Y + RY + 5R2Y = X
And the system function
Y
X
=
1
1 + R + 10R2
Find the roots of the polynomial in z = 1/R:
Z2 + Z + 10 = 0
−1 ±
Z =
√
1 − 100
2
Z = 0.5 ± 4.97j
The magnitude of the poles is 5, which is greater than 1, so it diverg... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
negative, so the system will alternate positive and
negative signs.
5.8.8 What’s Cooking?
Sous vide cooking involves cooking food at a very precise, fixed temperature T (typically, low
enough to keep it moist, but high enough to kill any pathogens). In this problem, we model the
behavior of the heater and water bat... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
] + ci[n]
Starting with this form of the operator equation, taken from the derivation above, and then
rearranging terms
T (1 − k1R) = cI − k2RT (1 − k1R)
T = cI + k1RT − k2RT (1 − k1R)
We get an equation that’s easy to convert to the difference equation above.
2. Let the system start at rest (all signals are zero).... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
converge to 0, nor to diverge. The other pole at −0.5 will generate a component
51015(cid:45)505010015020051015(cid:45)505010015020051015(cid:45)505010015020051015(cid:45)505010015020051015(cid:45)505010015020051015(cid:45)5050100150200 Chapter 5 Signals and Systems
6.01— Spring 2011— April 25, 2011
230
with altern... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
of eight linear, time-invariant systems. Match them with the dominant
pole for each system (remember that the system may have more than one pole).
A. This signal is alternating in sign and converging. Each magnitude is about 0.75 of the mag
nitude of the previous sample. So, we’d expect the dominant pole to be about... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
2011
231
E. This signal is converging and oscillating. The period seems to be 8. So, we’d expect a pole at
angle ±pi/4. The magnitude is a bit tricky to estimate. It seems to get from 2 to about 0.1 in 8
steps, so it’s something like 0.7. That corresponds well to pole 6.
F. This signal is converging and oscillatin... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
52.010203040(cid:45)0.50.51.01.52.010203040(cid:45)0.50.51.01.52.010203040(cid:45)5050100150102030400.51.01.52.0 Chapter 5 Signals and Systems
6.01— Spring 2011— April 25, 2011
232
The system function for the larger system can be written as
H0 =
Y0
X0
=
H1
1 + K0RH1
.
Assume that H1 = H1B = Y1 as shown below. ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
negative real axis causes the unit sample
response to alternate. Thus there are no values of KO and KB for which there is monotonic
decay.
+H1RK0X0Y0−+RRKBX1Y1 MIT OpenCourseWare
http://ocw.mit.edu
6.01SC Introduction to Electrical Engineering and Computer Science
Spring 2011
For information about citing these m... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/1268f3289b19d628e9be3bd2ecfb4f44_MIT6_01SCS11_chap05.pdf |
Introduction to Robotics, H. Harry Asada
1
Chapter 9
Force and Compliance Controls
A class of simple tasks may need only trajectory control where the robot end-effecter is
moved merely along a prescribed time trajectory. However, a number of complex tasks, including
assembly of parts, manipulation of tools, and wa... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
i.e. a position,
while the latter controlled variable is a force in the z direction. Namely, two types of control loops
are combined in the hand control system, as illustrated in Figure 9.1.2.
z
O
y
Fz
x
Figure 9.1.1 Robot drawing a line with a pencil on a sheet of paper
Department of Mechanical Engineering
Mass... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
. Harry Asada
3
The key question is how to assign a control mode, position control or force control, to
each of the axes in the C-frame in such a way that the control action may not conflict with the
geometric constraints and physics. M. Mason addressed this issue in his seminal work on hybrid
position/force contro... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
0
=
0
=
0
In the statics domain, forces and torques are specified in such a way that the quasi-static
condition is satisfied. This means that the peg motion must not be accelerated with any
unbalanced force, i.e. non-zero inertial force. Since we have assumed that the process is friction-
less, no resistive force act... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
following assumptions and rule:
• Each C-frame axis must have only one control mode, either position or force,
• The process is quasi-static and friction less, and
• The robot motion must conform to geometric constraints.
In general, the axes of a C-frame are not necessarily the same as the direction of a separate ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
c
T
∆
p
c
(9.1.3)
(9.1.4)
c
a
,
p
F
c
∆⊥
∆⊥
since
by definition. For the infinitesimal displacement
p
pδ , its component in the constraint space must be zero:
F
a
displacement
∆ a
is in a static equilibrium, the virtual work must vanish for all virtual displacements
p δ=
p
becomes a virtual displacement, and eq... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
.1.2 summarizes the above results.
Table 9.1.2 Mason’s Principle of Hybrid Position/Force Control
Natural Constraints
Artificial Constraints
Kinematic
c V∈= p(cid:5)
0
arbitrary
c
∈p(cid:5):
a V
a
Static
a V∈= F0
a
∈F:
c V
c
arbitrary
The reader will appreciate Mason’s Principle when considering the following exe... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
general corrupted with sensor noise and the C-frame may be misaligned. Therefore, the position
signal may contain some component in the constraint space, and some fraction of the force signal
may be in the admissible motion space. These components are contradicting with the natural
constraints, and therefore should ... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
,
(cid:7)
e
p
and
(cid:7)
e
f
, are in the C-frame, hence they must be converted to
the joint space in order to generate control commands to the actuators. Assuming that the
positional error vector is small and that the robot is not at a singular configuration, the position
feedback error in joint coordinates is gi... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
.2.2)
The components of the compliance matrix, or the stiffness matrix, are design parameters
to be determined so as to meet task objectives and constraints. Opening a door, for example, can
be performed with the compiance illustrated in Figure 9.2.1. The trajectory of the doorknob is
geometrically constrained to t... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
.2.2 Compliance control synthesis
Now that a desired compliance is given, let us consider the method of generating the
desired compliance. There are multiple ways of synthesizing a compliance control system. The
simplest method is to accommodate the proportional gains of joint feedback controls so that
desired rest... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
∆
=
p
(9.2.4)
The necessary condition for joint feedback gain Kq to generate the endpoint stiffness Kp is given
by
K
q =
T
JKJ
p
assuming no friction at the joints and linkage mechanisms.
Proof
Using the Jacobian and the duality principle as well as eq.(4),
T
T
τ
=
JKJpKJFJ
p
=∆
=
p
∆⋅
q
T
Using eq.(5), the above... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
q
3
+
2
sk
21
ssk
211
2
ck
22
cck
212
+
10
(9.2.9)
(9.2.10)
(9.2.11)
Note that the joint feedback gain matrix Kq is symmetric and that the matrix Kq degenerates when
the robot is at a singular configuration. If it is non-singular, then
qJp
=∆=∆
JK
1
−
q
τ
=
JK
q
T
1
−
(
JKJJFJ
=
T
p
1
−
)
T
FKFJ
=
=
CF
(9.2.12)
... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/127c560e6052cb02ed3f7adc8d3c1512_chapter9.pdf |
18.03 Class 11, Feb 26, 2010
Second order linear equations:
Physical model, solutions in homogeneous case.
Characteristic polynomial, distinct real roots.
[1] Springs and masses
[2] Dashpots
[3] Second order linear equations
[4] Solutions in homogeneous case: Superposition I
[5] Exponential solutions: characteri... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/1283839fb9d8c815f9362c3703b80878_MIT18_03S10_c11.pdf |
discussing on Monday. So we get
mx" + kx = F_ext .
I displayed a weight on a rubber band. This is not a spring, as you
usually think of one, but it behaves like one, at least in a range.
Lay a rubber band laid out on a table. Fix the right end of it and set x = 0
where the left end is in a relaxed state, then the... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/1283839fb9d8c815f9362c3703b80878_MIT18_03S10_c11.pdf |
x' = 0 , F_Dash(x') = 0
If x' < 0 , F_Dash(x') > 0
The simplest way to model this behavior (and one which is valid in general
for small x' , by the tangent line approximation) is
F_fric(x) = -bx b > 0 the "damping constant."
This is "linear damping." Altogether the equation is
mx" + bx' + kx = F_ext
Diagramma... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/1283839fb9d8c815f9362c3703b80878_MIT18_03S10_c11.pdf |
ages. Maybe the
honey in the dashpot gets stiffer with time.
Most of the time we will assume that the coefficients
are CONSTANT: the timescale of their variation is much longer than
the timescale of the dynamical variable x .
But the external force F_ext(t) can certainly vary (maybe sinusoidally).
We can see phys... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/1283839fb9d8c815f9362c3703b80878_MIT18_03S10_c11.pdf |
OTHER FUNDAMENTAL FACT TO MEMORIZE!
In fact, you showed that any sinusoid of circular frequency omega,
x = a cos(omega t) + b sin(omega t) = A cos(omega t - phi) (*)
is also a solution. In fact these are the only solutions, because
x(0) = a
x'(0) = omega b
and so you can solve (uniquely) for a and b to give an... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/1283839fb9d8c815f9362c3703b80878_MIT18_03S10_c11.pdf |
_2)
= c_1(mx_1" + bx_1' + kx_1) + c_2(mx_2" + bx_2' + kx_2)
= c_1 ( 0 ) + c_2 ( 0 ) = 0 .
[5] The equation mx" + bx' + kx = 0 , for m, b, k constant,
is a lot like x' + kx = 0 , which has as solution x = e^{-kt}
(and more generally multiples of this). It makes sense to try for
exponential solutions of (*): e^{r... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/1283839fb9d8c815f9362c3703b80878_MIT18_03S10_c11.pdf |
so the roots are r = -1 and r = -4 . The corresponding exponential
solutions are e^{-t} and e^{-4t} .
By superposition, the "linear combination"
x(t) = c_1 e^{-t} + c_2 e^{-4t}
is a solution as well. This is the general solution.
Suppose we know also that x(0) = 2 and x'... | https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/1283839fb9d8c815f9362c3703b80878_MIT18_03S10_c11.pdf |
ESD.33 -- Systems Engineering
Session #4
Requirements Engineering
Pat Hale
1
Purpose
As stated in ISO/IEC 15288:2008: The purpose of the
Stakeholder Requirements Definition Process is to define
the requirements for a system that can provide the
services needed by users and other stakeholders in a defined
env... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/1292005d75bbb1badef7f9416fca2a1d_MITESD_33SUM10_lec04a.pdf |
Provide clear visibility across teams into requirements
allocation and cross-functional interactions.
• Easily and quickly assess the impact of changes to
requirements.
• Provide data for early and thorough validation and
verification of requirements and design artifacts.
• Avoid unpleasant downstream surprise... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/1292005d75bbb1badef7f9416fca2a1d_MITESD_33SUM10_lec04a.pdf |
and appropriate for system
hierarchy level
• Feasible
• Consistent (traceable) with requirements
above and below in the system hierarchy
8
Requirements Baselines
INCOSE Systems Engineering Handbook version 3.2 January 2010
Image by MIT OpenCourseWare.
9
Product Commercialization
Phases 1-6:
Requirements B... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/1292005d75bbb1badef7f9416fca2a1d_MITESD_33SUM10_lec04a.pdf |
Analysis)
System/Program Milestones
Milestone I
Milestone II
Milestone III
Milestone IV
Functional Baseline
Allocated Baseline
Product Baseline
Updated Product Baseline
System Management Plan
System Specification
(TYPE A)
Development, Process, Product, Material
Specifications (TYPES B, C, D and E)
Proces... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/1292005d75bbb1badef7f9416fca2a1d_MITESD_33SUM10_lec04a.pdf |
(Operational Model)
4.4
4.5
Incorporation of
Modification's)
4.6
feedback
System Evaluation
(Field Assessment
11
R
E
T
I
R
E
M
E
N
T
Adapted from Systems Engineering lecture slides at University of Witwatersrand, Johannesburg, South Africa (Dr. R. Siriram)
Requirements Hierarchy & Traceabi... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/1292005d75bbb1badef7f9416fca2a1d_MITESD_33SUM10_lec04a.pdf |
required
- Approval by Systems team/Chief Systems
Engineer
Post-PAS:
- Changes/new requirements proposed by any
development team member
- Review by Subsystem team with appropriate
representation from other affected teams
- Decision by Subsystem team leader with
primary design responsibility
- Weekly ... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/1292005d75bbb1badef7f9416fca2a1d_MITESD_33SUM10_lec04a.pdf |
pas
pas
pas
pas
pas
pas
SUBASSEMBLY
FILE FORMAT
INTERFACE
TEST
Image by MIT OpenCourseWare.
18
Formatting PAS in Word
for Export into Doors
Keep heading styles in the following form:
Heading 1, Heading 2, Heading 3, …,etc.
Notes:
– Framemaker documents use Sec-#, Sec-#.#,
Sec-#.#.#. Export Framemaker documents... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/1292005d75bbb1badef7f9416fca2a1d_MITESD_33SUM10_lec04a.pdf |
Product
Contract
Integrated
Test Plan
PFS
PASSS
?
Level 4
Detailed
Test Plans
Test
Suites
Test
Cases
?
Library
Test Plans
?
Host
Applications
?
?
Test
Suites
Network
Infrastructure
Requirements Links
Test Links
Req’ts-Test Links
26
Module Schema
27
MIT OpenCourseWare
http://ocw.mit.edu... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/1292005d75bbb1badef7f9416fca2a1d_MITESD_33SUM10_lec04a.pdf |
LECTURE 4
gcft and Quadratic Reciprocity
Last time, we reduced non-degeneracy and bimultiplicativity of the Hilbert
symbol (·, ·) to showing that for all quadratic extensions L/K, with K a local field,
NL× ⊆ K × is a subgroup of index 2. We showed that this holds for unramified
extensions and when p = char(OK/p) is odd (... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/1295695fd5558404de56e7a8cbec88df_MIT18_786S16_lec4.pdf |
2. We have (cid:98)Z = (cid:81)
Zp, where p ranges over all primes, and (cid:98)Z×R ⊆
AQ embedded as a subring, in which we may diagonally embed any n (cid:54)= 0. Similarly,
(cid:98)Z× = (cid:81)
Z×
Q . However, 2 and 1/2 won’t be in (cid:98)Z× ×R× as they
p , and (cid:98)Z× ×R× ⊆ A×
aren’t in Z×
2 , and the same hold... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/1295695fd5558404de56e7a8cbec88df_MIT18_786S16_lec4.pdf |
pairing by
y (cid:55)→
(cid:89)
(xp, y)p,
p
where we are regarding y as a p-adic unit, and (·, ·)p denotes the Hilbert symbol
at p, i.e., on Qp. Now, it’s not even clear a priori that this infinite product is
well-defined, and for this we introduce the following lemma:
Lemma 4.3. (xp, y)p = 1 for all but finitely many p.
... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/1295695fd5558404de56e7a8cbec88df_MIT18_786S16_lec4.pdf |
the product is 1.” The word for such “conspiracies” is “reciprocity
law.”
Proof (of Claim). First of all, since the map is invariant under multiplica-
tion by squares, we can assume x = ±p1 · · · pr and y = ±q1 · · · qs. Then bimulti-
plicativity implies that we can take x ∈ {−1, 2, p} and y ∈ {−1, 2, q}, where p and
... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/1295695fd5558404de56e7a8cbec88df_MIT18_786S16_lec4.pdf |
4. Thus, we have reduced to elementary
congruence conditions, and this is precisely the statement of quadratic reciprocity.
(cid:3)
Remark 4.6. Quadratic reciprocity allows for efficient computation of Le-
gendre symbols via successive reduction.
Proof (of Quadratic Reciprocity). Regard the Legendre symbol as a
map
(cid:... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/1295695fd5558404de56e7a8cbec88df_MIT18_786S16_lec4.pdf |
know by design that G2 ∈ Q, but now we’d like to know which (in fact, we will
see that it is either p or −p).
Suppose that χ : k× → C× is a multiplicative character, and ψ : k → C× is an
additive character, where K is any finite field. Let
Gχ,ψ :=
(cid:88)
x∈k×
χ(x)ψ(x).
Remark 4.7. As a fun analogy, the gamma function i... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/1295695fd5558404de56e7a8cbec88df_MIT18_786S16_lec4.pdf |
×
ψ(0)
(cid:88)
1
y∈k×
= χ(1) + 1 ·
= 1 + #k×
= #k,
since (cid:80)
w∈k× χ(w) = 0 similarly.
(cid:3)
Now, we’d like to know what Gχ−1,ψ−1 is for ψ corresponding to a power of ζp
and χ the multiplicative Legendre character. We have
Gχ−1,ψ−1 =
(cid:19)
p−1
(cid:88)
n=1
(cid:18) n
p
ζ −n
p
18
4. GCFT AND QUADRATIC RECIPRO... | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/1295695fd5558404de56e7a8cbec88df_MIT18_786S16_lec4.pdf |
q
p
= (−1)
p−1
2
q−1
2 ,
(cid:3)
MIT OpenCourseWare
https://ocw.mit.edu
18.786 Number Theory II: Class Field Theory
Spring 2016
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/1295695fd5558404de56e7a8cbec88df_MIT18_786S16_lec4.pdf |
15.081J/6.251J Introduction to Mathematical
Programming
Lecture 7: The Simplex Method III
1 Outline
• Finding an initial BFS
• The complete algorithm
• The column geometry
• Computational efficiency
• The diameter of polyhedra and the Hirch conjecture
2 Finding an initial BFS
• Goal: Obtain a BFS of Ax = b, x ... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/12b1c7e885991696c294953b2497739c_MIT6_251JF09_lec07.pdf |
2
Slide 3
Slide 4
Slide 5
6. Drive artificial variables out of the basis: If lth basic variable is artifi
If all elements = 0 ⇒ row redundant.
cial examine lth row of B−1A.
Otherwise pivot with
0 element.
=
1
�
Slide 6
Slide 7
Slide 8
2.2 Example
min
s.t.
x1 + x2 + x3
x1 + 2x2 + 3x3
−x1 + 2x2 + 6x3
4x2... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/12b1c7e885991696c294953b2497739c_MIT6_251JF09_lec07.pdf |
1/3
0
0
1
0
1
1/3
0
1
x1 =
x2 = 1/2
x7 =
0
x3 = 1/3
0
1
0
0
0
0
0
1
0
0
x1 x2 x3
x4
x5
x6 x7
0
0
0
2
2
1/2 1/2 −1/2
0
0
x8
1
1/2
0 −3/4 1/4
0
1
0 −1
1/3
0
x1 x2 x3
0 −3/4
1
0
0
1/3
1/4
−1
0
x4
∗
1/2
∗
0
∗
1
x1 =
x2 =
1/2
x3 =
1/3
∗
1
0
0
∗
0
1
0
0 −3/4
1
1/3
... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/12b1c7e885991696c294953b2497739c_MIT6_251JF09_lec07.pdf |
and tableau obtained from Phase I be the initial basis
and tableau for Phase II.
2. Compute the reduced costs of all variables for this initial basis, using the
cost coefficients of the original problem.
3. Apply the simplex method to the original problem.
3.1 Possible outcomes
1. Infeasible: Detected at Phase I.
... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/12b1c7e885991696c294953b2497739c_MIT6_251JF09_lec07.pdf |
7
Slide 18
Slide 19
Slide 20
z
z
4
x2
.
.
.
D
b .
initial basis
6
.
.
1
next basis
7
.
.
5
.
2
3
.
.
.
8
optimal basis
.
b
x3
x2
x1
x1
(a)
(b)
5
B
E
F
C
G
H
I
• The feasible set has 2n vertices
• The vertices can be ordered so that each one is adjacent to and has l... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/12b1c7e885991696c294953b2497739c_MIT6_251JF09_lec07.pdf |
.
.
.
.
.
.
.
.
.
.
. .
)
• Hirsch Conjecture: Δ(n, m) ≤ m − n.
• We know that
Δu(n, m) ≥ m − n +
n
5
� �
Δ(n, m) ≤ Δu(n, m) < m1+log2 n = (2n)log m
2
6
Slide 21
Slide 22
Slide 23
Slide 24
(
a
)
(
b
MIT OpenCourseWare
http://ocw.mit.edu
6.251J / 15.081J Introduction to Mathematical Programmi... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/12b1c7e885991696c294953b2497739c_MIT6_251JF09_lec07.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Lecture 10
Fall 2018
CONTINUOUS RANDOM VARIABLES
Contents
1. Continuous random variables
2. Examples
3. Expected values
4. Joint distributions
5. Independence
6. Radon-Nikodym derivative
Readings: For a less technical version of this material, but with mo... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
on (R,
B
b
a
R
R
R
1
The reader should revisit Section 4 of the notes for Lecture 5.
1
We note that fX should be more appropriately called “a” (as opposed to
“the”) PDF of X, because it is not unique. For example, if we modify fX at a
finite number of points, its integral is unaffected, so multiple densities ... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
calculus: If FX (x)
is continuous and differentiable everywhere except countably many points of R,
then
FX (x) =
x
′
FX (t)dt
−∞
Z
This provides a simple rule to find PDF from CDF in most cases of practical
interest.
Remark: The fact that a random variable X is continuous has no bearing on the
into R. In fact, ... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
, 2]
∪
We present here a number of important examples of continuous random vari-
ables.
2
2.1 Uniform
This is perhaps the simplest continuous random variable. Consider an interval
[a, b], and let
FX (x) =
0,
(x
1,
−
a)/(b
−
a,
x
a), a < x
x > b.
≤
b,
≤
It is easy to check that FX satisfies th... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
Exp(λ) and write
, for x
−
≥
−λx
Exp(λ).
X
∼
which stand for ”distributed as ...”)
(Recall notation =
d and
∼
The exponential distribution can be viewed as a “limit” of a geometric distri-
e−λδk ,
bution. Indeed, if we fix some δ and consider the values of FX (kδ) = 1
for k = 1, 2, . . .. Check that this is P(... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
> x + t)
P(X > x)
=
−λ(x+t)
e
e−λx
−λt
= e = P(X > t).
Exponential random variables are often used to model memoryless arrival
processes, in which the elapsed waiting time does not affect our probabilistic
model of the remaining time until an arrival. For example, suppose that the
time until the next bus arri... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
as
Exp(0) as X = +
X = 0 a.s., and X
Exp(+
with +
a.s..
∞}
∞}
∞
∞
∼
∼
∞
2.3 Normal distribution
Perhaps the most widely used distribution is the normal distribution which is
R and σ > 0, and
also called Gaussian distribution. It involves parameters µ
the density
∈
N (µ, σ2) :
fX (x) =
X
∼
1
−
e
√
σ 2... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
−
e
∞
(x−µ)2
2σ2 =
1
Z
2πσ −∞
√
∈
e− 2 = 1.
2
z
∞
1
√
Z
2π −∞
We use the notation N (µ, σ2) to denote the normal distribution with parameters
µ, σ. The distribution N (0, 1) is referred to as the standard normal distribu-
tion; a corresponding random variable is also said to be standard normal.
There is no cl... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
one parameter t and
X
∼
Ca(t) :
fX (x) =
t
1
π t2 + x2
, x
R
∈
It is an exercise in calculus to show that
PDF. The corresponding distribution is called a Cauchy distribution.
f (t)dt = 1, so that fX is indeed a
∞
−∞
R
Semigroup property of Cauchy: Let X1 ∼
X2. Then
Ca(t1), X2 ∼
Ca(t2) and X1 ⊥
⊥
X1 + X2... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
c > 0, for some
β/xα ,
parameters α, c > 0. In this case, the CDF is given by FX (x) = 1
x
c, and FX (x) = 0, otherwise. In order for X to be a continuous random
variable, FX cannot have a jump at x = c, and we therefore need β = cα and
cα/xα . The corresponding density is of the form
FX (x) = 1
≥
−
≥
−
fX (... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
x
|
R
R
∞
−∞
Practically all of the results developed for discrete random variables carry
over to the continuous case. Many of them, e.g., E[X + Y ] = E[X] + E[Y ],
have the exact same form. We list below two results in which sums need to be
replaced by integrals.
Proposition 1. Let X be a nonnegative random variab... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
Lecture 8.
Note that for this result to hold, the random variable g(X) need not be con-
tinuous.
4
JOINT DISTRIBUTIONS
Definition 1. Given a pair of random variables X and Y , defined on the
same probability space, their joint distribution PX,Y is a probability measure
on (R
) defined as
R,
×
B × B
PX,Y [B] , P[... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
(x, y) =
∂2F
∂y∂x
(x, y) = fX,Y (x, y).
Similar to what was mentioned for the case of a single random variable, for
any Borel subset B of R2 , we have
PX,Y [B] =
Z
B
fX,Y (x, y) dx dy =
Z
R2
1B (x, y)fX,Y (x, y) dx dy.
(3)
Furthermore, if B has Lebesgue measure zero, then PX,Y (B) = 0.
We observe that by (3... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
x = y
R2
∈
|
Proposition 2 has a natural extension to the case of multiple random vari-
ables.
Proposition 3. Let X and Y be jointly continuous with PDF fX,Y , and sup-
pose that g : R2
R is a (Borel) measurable function such that g(X) is
integrable. Then,
→
E[g(X, Y )] =
∞ ∞
Z
Z
−∞ −∞
g(u, v)fX,Y (u, v) du dv.... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
of any right-continuous function x
Then for any a
f (x). Let Ra =
Q : f (x) < a
R it follows
x
{
7→
∈
∈
f (x) < a
{
}
=
[ \ [
ǫ1>0 ǫ2>0 r∈Ra−ǫ1
ǫ2, r] .
[r
−
(4)
0, so that resulting operations are countable. If (4) holds then
ǫ>0 to mean union over arbitrary sequence of
f (x) < a
{
}
Here we write (abus... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
: R ! R is equivalent to usual
continuity when topology on the domain is refined declaring sets [a, b) open.
9
we infer that
Thus, we have
φ(x, y) < b
(x, y)
Lb
∈
⇒
φ(x, y)
b .
≤
⇒
(x, y) : φ(x, y) < a
{
}
=
La−ǫ
[
ǫ>0
which is a countable combination of measurable rectangles.
5
INDEPENDENCE
Recall t... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
)FY (y).
(d) If fX , fY , and fX,Y are corresponding marginal and joint densities, then
fX,Y (x, y) = fX (x)fY (y), for all (x, y) except possibly on a set that has
Lebesgue measure zero.
≤
∈
}
}
The proof parallels the proofs in Lecture 6, except for the last condition,
for which the argument is simple when the d... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
-Nikodym derivative dLeb on R. Similarly, X and Y are
exists on R2 , etc. One simple consequence of (1)
jointly continuous if
is that X cannot be a continuous random variable if PX has atoms, namely if
P[X = a]
R. However, as “singular” example in Section 4
shows the absence of atoms is not sufficient for continu... | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
.436J / 15.085J Fundamentals of Probability
Fall 2018
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms | https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/12b27e5630be533e1385ac142df8a70a_MIT6_436JF18_lec10.pdf |
MIT 3.016 Fall 2005
c
� W.C Carter
Lecture 4
21
Sept. 14 2005: Lecture 4:
Introduction to Mathematica III
Simplifying and Picking Apart Expression, Calculus, Numerical Evaluation
A great advantage of using a symbolic algebra software package like Mathematica r�
is
that it reduces or even eliminates errors that... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/12e8d3a26bee4e04f880c9a203959fbb_lecture_04.pdf |
Operations on Polynomials
Expand, Factor, Coefficients
Operations on Rational Expression
Together, Apart, Numerator
Mathematica r�
is very fastidious about simplifying roots of numbers. Unless, it is
specified otherwise, Mathematica r� makes no assumptions about whether a variable is
real, complex, positive, or neg... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/12e8d3a26bee4e04f880c9a203959fbb_lecture_04.pdf |
: it allows you to tackle solutions that
would be very onerous otherwise.
Mathematica r� Example: Lecture04
Solving Equations
Solve[] and its resulting rules
Sometimes, no closed form solution is possible. Mathematica r� will try to give you rules
(in perhaps a very strange form) but it really means that you don’... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/12e8d3a26bee4e04f880c9a203959fbb_lecture_04.pdf |
24
1. You will want to save your work.
2. You will want to modify your old saved work
3. You will want to use your output as input to another program
4. You will want to use the output of another program as input to Mathematica r
� .
You have probably learned that you can save your Mathematica r� notebook with a m... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/12e8d3a26bee4e04f880c9a203959fbb_lecture_04.pdf |
editing a package file with an editor. By doing this, you will see some of internal
structure of Mathematica r� and good examples of professional programming. | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/12e8d3a26bee4e04f880c9a203959fbb_lecture_04.pdf |
Figure by MIT OCW.
Figure by MIT OCW. | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/12fba3de95a582e553b097c52e6aead9_lec01t_note.pdf |
12. Instability Dynamics
12.1
Capillary Instability of a Fluid Coating on a Fiber
We proceed by considering the surface tension-induced
instability of a fluid coating on a cylindrical fiber.
Define mean thickness
h∗
=
λ
1
λ 0
h(x)dx
(12.1)
Local interfacial thickness
h(x) = h∗
+ ǫ cos kx
(12.2)
Volume con... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/130219f256b3470af82db2d059ee6e1b_MIT18_357F10_Lecture12.pdf |
ds =
≈
1
2
1 (r + h∗)ǫ2k2λ < (r + h0)λ.
So the inequality holds provided (r + h∗)λ + 4
from (12.3):
Substitute for h∗
+ ǫ cos kx)(1 + ǫ2k2 sin2 kx)1/2dx = (r + h∗)λ +
⇒
(r + h∗
J (
λ
0
f
f
)
]
ǫ2k2 sin2 kx
f
1/2
dx 1 +
[
λ
0
2π(r + h)ds < 2π(r + h0)λ?
1 (r + h∗)ǫ2k2λ.
4
ǫ2
1
4 r + h0
−
+
1
4
(r + h∗)ǫ2k2
... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/130219f256b3470af82db2d059ee6e1b_MIT18_357F10_Lecture12.pdf |
45
12.2. Dynamics of Instability (Rayleigh 1879)
Chapter 12. Instability Dynamics
12.2 Dynamics of Instability (Rayleigh 1879)
Physical picture: Curvature pressure induced by perturbation drives Couette flow that is resisted by
viscosity
−
where dp is the gradient in curvature pressure, which i... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/130219f256b3470af82db2d059ee6e1b_MIT18_357F10_Lecture12.pdf |
Fastest growing mode when dβ
dk
ǫ cos kx
h
3
hx =
k4
−
−
⇒ dt
i
8
(r+h0)2
= 0 = 2k
ǫk sin kx, hxx =
σh
dǫ = βǫ where β = 0
3µ
−
ǫ2k cos kx, ht = dǫ cos kx
2
k
(r+h0)2
k4
dt
3
h
−
i
4k∗
3 so
−
is the most unstable wavelength for the viscous mode.
λ∗ = 2√2π (r + h0)
(12.13)
Note:
•
•
•
Recall that for classic Ra-P on a c... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/130219f256b3470af82db2d059ee6e1b_MIT18_357F10_Lecture12.pdf |
, so that viscous effects are negligible.
The driving curvature force is thus resisted principally by fluid
inertia. Assume dynamics is largely 2D (true for a planar film,
h).
or for bubble burst for r(t)
Retraction of a Planar Sheet
Note: Force/ length acting on the rim may be calculated exactly
via Frenet-Serret
≫
F C =... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/130219f256b3470af82db2d059ee6e1b_MIT18_357F10_Lecture12.pdf |
2.19)
The first term is the surface energy released per unit length, the 2nd term the K.E. of the rim, and the
3rd term the energy required to accelerate the rim. Now we assume v is independent of x (as observed in
experiments), thus
v2dx = xv2 and the force balance becomes 2σx = ρhxv2
⇒
is the retraction speed (Taylor-... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/130219f256b3470af82db2d059ee6e1b_MIT18_357F10_Lecture12.pdf |
comments on soap film rupture.
≫
≪
∼
∼
(
)
1. for dependence on geometry and influence of µ, see
Savva & Bush (JFM 2009).
2. form of sheet depends on
h =
√
O
µ
√2hρσ
.
3. The growing rim at low
scalloping of the retracting rim
O
h is subject to Ra-Plateau
rim pinches off
⇒
⇒
into drops
4. At very high speed, air-... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/130219f256b3470af82db2d059ee6e1b_MIT18_357F10_Lecture12.pdf |
010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/130219f256b3470af82db2d059ee6e1b_MIT18_357F10_Lecture12.pdf |
03/11/13, Eikonal Equations, Superposition of EM Waves
Lecture Note (Nick Fang)
Outline:
‐ Connection of EM wave to geometric optics
‐ Path of Ligh
t in an Inhomogeneous Medium
‐
Superposition of waves, coherence
A. High
Frequency Limit, connection to Geometric Optics:
How can we obtain Geometric optics pictur... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/13134562742a242caad5fd92fcb27dfa_MIT2_71S14_lec10_notes.pdf |
To see the connection to geometric optics, we
decompose the field E(r, ) into two forms: a
fast oscillat
k(cid:2868) (cid:3404) (cid:2033)/(cid:1855)(cid:2868) and a slowly varying envelope
ing component exp(ik0),
Slowly varying envelope
E0(r)
E=E0 (r(cid:1318))exp(ik0)
Example of decomposition of E
field into ... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/13134562742a242caad5fd92fcb27dfa_MIT2_71S14_lec10_notes.pdf |
673)
(cid:3404) (cid:2013)(cid:4666)(cid:1876), (cid:1878)(cid:4667) (cid:3404) (cid:1866)(cid:2870)
(cid:4666)(cid:1876), (cid:1878)(cid:4667)
E0
(cid:1487)Φ
H0
3
2
1
Observatio
n (not proof):
The above equation yields: |(cid:1487)Φ|(cid:2870) (cid:3404) (cid:1866)(cid:2870), or |(cid:1487)Φ| (cid:3404) (cid:1866... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/13134562742a242caad5fd92fcb27dfa_MIT2_71S14_lec10_notes.pdf |
, follows exactly the gradient
the path of light in a
B. Path of Light in an Inhomogeneous Medium
‐
Example
1: 1D probl
em
s (Gradient index waveguide
s, Mirage
Effe
cts)
best
le of
The
this
known examp
kind
near a seashore, and we heard of
the e
decr
e
henc
ith en
d
ses w
increa
p
we
w can
in
ure
ict
W... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/13134562742a242caad5fd92fcb27dfa_MIT2_71S14_lec10_notes.pdf |
)(cid:2870)(cid:4666)(cid:1876)(cid:4667)
Since there is the index in independent of z, we may assume the slope of phase
change in z direction is linear:
This allows us to find
(cid:3105)(cid:2957)
(cid:4672)
(cid:3105)(cid:3053)
(cid:4673)=
C(const)
(cid:2034)Φ
(cid:2034)(cid:1876)
(cid:3404) (cid:3493)(cid:1866... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/13134562742a242caad5fd92fcb27dfa_MIT2_71S14_lec10_notes.pdf |
cid:3095)(cid:3004)
(cid:3041)(cid:3116)√(cid:3080)
.
x
x of
e
Ind
refr
action
n(cid:4666)x(cid:4667)
dx
dz
z
ation:
t
ted
Observ
:
optical ray. To check that we start by
the constant C
is rela
o the original “launching” ang
le
of the
If we assume C=(cid:1866)(cid:4666)(cid:1876)(cid:2868)(cid:4667)(cid:1855)... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/13134562742a242caad5fd92fcb27dfa_MIT2_71S14_lec10_notes.pdf |
1855)(cid:1867)(cid:1872)(cid:2010)
‐ Other
popul
ar examples
Luneberg
:
Lens
The Luneberg lens is
focal point
the cent
at the r
h
in
ear s
of th
er, t e gradient index function can be written as:
e sphere. For a sphere of
h
u
omogeneous sphere t
rface
hat brings a collimated beam of light to a
radius R with... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/13134562742a242caad5fd92fcb27dfa_MIT2_71S14_lec10_notes.pdf |
y, such device gained new interests in in phase
r by R. K. Luneberg,
ersity, Providence,
eberg lens was quickly
mmunications as well
d array
rs in solar energy
Left:
Picture of an Optical Luneberg
spherical retro‐reflector on Meteor‐3M
Right:
parallel ray
focus to a point on th
e of the sphere.
e edg
Ray Sche... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/13134562742a242caad5fd92fcb27dfa_MIT2_71S14_lec10_notes.pdf |
68)(cid:1864)(cid:1857)(cid:1876) (cid:1855)(cid:1867)(cid:1866)(cid:1862)(cid:1873)(cid:1859)(cid:1853)(cid:1872)(cid:1857)(cid:4667)
‐ Complex numbers simply
optics!
‐ Coherence
5
MIT OpenCourseWare
http://ocw.mit.edu
2SWLFV
Spring 2014
For information about c... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/13134562742a242caad5fd92fcb27dfa_MIT2_71S14_lec10_notes.pdf |
Engineering Risk Benefit Analysis
1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82
ESD.72J, ESD.721
Introduction
George E. Apostolakis
Massachusetts Institute of Technology
Spring 2007
Introduction
1
Risk-Benefit Tradeoffs
Introduction
2
Dealing with Uncertainty
• Risk management of large technological
syst... | https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/13386d646845d04ae28295d24b33e99b_intro.pdf |
6.096 Introduction to C++
Massachusetts Institute of Technology
January 24th, 2011
John Marrero
Lecture 9 Notes: Advanced Topics I
1
Templates
We have seen that functions can take arguments of specific types and have a specific return
type. We now consider templates, which allow us to work with generic types. Thr... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/13882f48b16656374a40a0d103a3b51c_MIT6_096IAP11_lec09.pdf |
The identifier can be used in any way inside the function template, as long as the code makes
sense after identifier is replaced with some type.
It is also possible to invoke a function template without giving an explicit type, in cases where
the generic type identifier is used as the type for a parameter for the fu... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/13882f48b16656374a40a0d103a3b51c_MIT6_096IAP11_lec09.pdf |
()
So, for example, getX could have been declared in the following way:
template <typename T>
T Point<T>::getX() { return x; }
assuming a prototype of T getX(); inside the class definition.
We can also define different implementations for a single template by using template
specialization. Consider the followin... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/13882f48b16656374a40a0d103a3b51c_MIT6_096IAP11_lec09.pdf |
);
floatac.set(3, 3.5);
cout << intac.get(2) << endl;
cout << floatac.get(3) << endl;
T set(const int i, const T val) { elts[i] = val; }
T get(const int i) { return elts[i]; }
return 0;
20
21 }
This program prints out 3 and 3.5 on separate lines. Here, one instance of the ArrayContainer
class works on a 5-element ar... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/13882f48b16656374a40a0d103a3b51c_MIT6_096IAP11_lec09.pdf |
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28 }
In this example, we create an integer set and insert several integers into it. We then create an
iterator corresponding to the set at lines 14 and 15. An iterator is basically a pointer that
provides a view of the set. (Mos... | https://ocw.mit.edu/courses/6-096-introduction-to-c-january-iap-2011/13882f48b16656374a40a0d103a3b51c_MIT6_096IAP11_lec09.pdf |
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