text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
— where (cid:126)y is vector valued and A is a constant square matrix.
The (unique) solution to this problem is (cid:126)y = eA t (cid:126)y0, which clearly depends continuously (in
fact, analytically) on all the parameters in the problem: the entries of A and (cid:126)y0.
6. Consider the boundary value problem u(cid:4... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
solution as possible” (in some appropriate sense), which then yields a well posed problem.
(cid:90)
8. The initial value problem x˙ = 2
and x = t2. Thus, it is an ill-pos
ed problem.
(cid:113)
|x|, with x(0) = 0. This has at least two solutions:3 x = 0
2Later we will state a theorem that generalizes this to very genera... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
can have arbitrarily large
changes as t → ∞. Chaos is tied up to “fast” growth of the errors with time, not with errors going
from “infinitesimal” to finite in arbitrarily short times (which is what lack of continuity means). ♣
When a model is discretized and programed into a computer, the following things can happen:
9.... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
objective of this course is to describe the “topology” of these curves: how does the “phase
portrait” look like? That is: we are not seeking quantitative information, but mainly qualitative
information. This approach was proposed by Poincar´e when, towards the end of the 19-th century,
it became clear that “exact solut... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
other courses, but we must draw the line somewhere.
Check the Dynamical view of the world section by Strogatz, with the frontier in the dimension
versus nonlinearity diagram. Obviously the diagram is incomplete, and many things are missing,
but it illustrates well where the boundary between known and unknown roughly is... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
form in (1.1.2) by letting
(cid:126)yn = ((cid:126)xn, (cid:126)xn 1). Similarly for systems of the form (cid:126)x
−
n+1 = ((cid:126)xn, (cid:126)xn−1, (cid:126)xn
),−2
(cid:126)
F
etc.
(cid:126)
— How to reduce systems of the form (cid:126)xn+1 = Fn((cid:126)xn) to the form in (1.1.2) by introducing n as
an extra unk... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
erentiable. Let (cid:126)y0 = G((cid:126)x0). Then a unique inverse function (cid:126)x = F ((cid:126)y)
— with (cid:126)x0 = F ((cid:126)y0) — is defined in a neighborhood of (cid:126)y0, provided that the linearized problem
♣
δ(cid:126)y = A δ(cid:126)x is invertible (that is, A = “gradient of G at x0” is an invertibl... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
cation theory, which will be an important part of this
course. Another important theorem is the following:
3. Existence and uniqueness for the I.V. problem for ode. Consider the initial value problem:
d(cid:126)y
dt
(cid:126)
= F ((cid:126)y),
(cid:126)y(0) = (cid:126)y0, where F is Lipschitz continuous.
(cid:126)
(1.2... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
8
(cid:113)
|x|, with x(0) = 0. Clearly
6. Example of a differentiable function, with derivative not bounded near a point (hence not
(1.2.3)
f (x) = |x|a sin(1/x), where 1 < a < 2.
Lipschitz):
Then f (cid:48)(x) = −|x|a−2 cos(1/x) + a |x|a−1 sin(1/|x|) for x = 0, and f (cid:48)(0) = 0.
Another example is: f (x) = |x|a s... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
from a linearization, and when you cannot. The rule is:
Imagine some system of equations, for which you have one solution. Imagine also that you can
write a linearized system that solutions “infinitesimally close” to the one you know must satisfy.9
8It can be shown that, for existence, continuity is enough. It is the un... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
means that the property remains true when the linear system is perturbed slightly.
MIT OpenCourseWare
http://ocw.mit.edu
18.385J / 2.036J Nonlinear Dynamics and Chaos
Fall 2014
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/67b0710b3bc473a75d8f91b29954b0e9_MIT18_385JF14_SelectedLec.pdf |
6.088 Intro to C/C++
Day 5: Inheritance & Polymorphism
Eunsuk Kang & Jean Yang
In the last lecture...
Objects: Characteristics & responsibilities
Declaring and defining classes in C++
Fields, methods, constructors, destructors
Creating & deleting objects on stack/heap
Representation invariant
Today’s topics
Inhe... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/67b1aec3f2867734ec0fb33034c8b5c8_MIT6_088IAP10_lec05.pdf |
#include <string>
class MITPerson {
protected:
int id;
std::string name;
std::string address;
public:
MITPerson(int id, std::string name, std::string address);
void displayProfile();
void changeAddress(std::string newAddress);
};
Base type: MITPerson
access control
#include <string>
class MITPerson {
pro... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/67b1aec3f2867734ec0fb33034c8b5c8_MIT6_088IAP10_lec05.pdf |
, std::string address,
int course, int year);
void displayProfile();
void addClassTaken(Class* newClass);
void changeCourse(int newCourse);
};
Constructing an object of subclass
#include <iostream>
#include <vector>
#include "MITPerson.h"
#include "Class.h"
class Student : public MITPerson {
int course;
int y... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/67b1aec3f2867734ec0fb33034c8b5c8_MIT6_088IAP10_lec05.pdf |
address;
public:
MITPerson(int id, std::string name, std::string address);
void displayProfile();
void changeAddress(std::string newAddress);
};
class Student : public MITPerson {
int course;
int year;
// 1 = freshman, 2 = sophomore, etc.
std::vector<Class*> classesTaken;
public:
Student(int id, std::string n... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/67b1aec3f2867734ec0fb33034c8b5c8_MIT6_088IAP10_lec05.pdf |
: 6
Classes taken:
6.088
-----------------------------
Polymorphism
Polymorphism
Ability of type A to appear as and be used like another
type B
e.g.A Student object can be used in place of an
MITPerson object
Actual type vs. declared type
Every variable has a declared type at compile-time
But during runtime,... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/67b1aec3f2867734ec0fb33034c8b5c8_MIT6_088IAP10_lec05.pdf |
*
steve
Student object
vptr
&Student::displayProfile
&MITPerson::changeAddress
VTABLE
Note “changeAddress” is declared virtual in but not
overridden
Virtual destructor
Should destructors in a base class be declared as
virtual? Why or why not?
Virtual destructor
Should destructors in a base class be declared... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/67b1aec3f2867734ec0fb33034c8b5c8_MIT6_088IAP10_lec05.pdf |
088");
steve->addClassTaken(c1);
What will happen?
Type casting
MITPerson* steve =
new Student(911923, "Steve", "99 Cambridge St.", 18, 3);
Class* c1 = new Class("6.088");
steve->addClassTaken(c1); X
Can only invoke methods of the declared type!
“addClassTaken” is not a member of MITPerson
Type casting
MITPer... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/67b1aec3f2867734ec0fb33034c8b5c8_MIT6_088IAP10_lec05.pdf |
��ning abstract methods in C++
Use pure virtual functions
class BST {
public:
virtual ~BST() = 0;
this says that “find” is pure
(i.e. no implementation)
virtual void insert(int val) = 0;
virtual bool find(int val) = 0;
virtual void print_inorder() = 0;
};
this says that “find” is virtual
Defining abstract methods ... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/67b1aec3f2867734ec0fb33034c8b5c8_MIT6_088IAP10_lec05.pdf |
is called!
Pure virtual destructor
Can also define a destructor as pure.
class BST {
public:
virtual ~BST() = 0;
virtual void insert(int val) = 0;
virtual bool find(int val) = 0;
virtual void print_inorder() = 0;
};
But must also provide a function body.Why?
BST::~BST() {}
Until next time...
Homework #5 (due... | https://ocw.mit.edu/courses/6-088-introduction-to-c-memory-management-and-c-object-oriented-programming-january-iap-2010/67b1aec3f2867734ec0fb33034c8b5c8_MIT6_088IAP10_lec05.pdf |
18.465, March 8, 2005, revised May 2
Breakdown points of some 1-dimensional location estimators
Recall that a set C in a real vector space is called convex if for any x, y in C and
0 ≤ λ ≤ 1 we have λx + (1 − λ)y ∈ C. In the real line, a convex set is just an interval, a
half-line or the whole line. At a finite endp... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/684183e815da73f9e914941a3d94c2d9_brkdn_location.pdf |
(b) ρ(x) := (c2 + x2)1/2 − c for any real c > 0, and
(c) ρ(x) := x2 for |x| ≤ b and ρ(x) := c|x| − d for |x| > b where b > 0 and the other
constants are chosen to make ρ continuously differentiable. Then cb−d = b2 and 2b = c,
so d = b2 and for |x| > b, ρ(x) = b(2|x| − b).
Since Huber especially studied functions defi... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/684183e815da73f9e914941a3d94c2d9_brkdn_location.pdf |
t) < 0 < ψ(t) for some t > 0 since ψ is nonconstant. We will have ψ(t) → A
as t → +∞ for some A > 0. Examples of such functions ψ include the derivatives ρ(cid:1)(x)
of wide-sense Huber functions, where such derivatives are defined, with suitable choices
where they are not defined, specifically, ψ(0) = 0 in all cases, ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/684183e815da73f9e914941a3d94c2d9_brkdn_location.pdf |
∗∗
which is also finite since the sum is ≥ 0 for θ ≤ X(1) and < 0 for θ ≥ X(n) + t. We
∗
have θ ≤ θ
because of the monotonicity of ψ. In order to have a unique estimator,
the (location) M-estimator for the sample (based on ψ) will be defined, as for the sample
median, by
θˆ := θˆ(X1, . . . , Xn) :=
1
(θ ∗
2
It wil... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/684183e815da73f9e914941a3d94c2d9_brkdn_location.pdf |
particular choice of S, let M be the median of the sample, defined as X(k+1)
if n = 2k + 1 is odd, and (X(k) + X(k+1))/2 if n = 2k is even. Let MAD denote the
median absolute deviation, namely the median of |Xi − M |, and S = MAD/.6745, where
the constant 0.6745 is (to the given accuracy) the median of |Z| for a stan... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/684183e815da73f9e914941a3d94c2d9_brkdn_location.pdf |
(n − j)A < 0, or equivalently j > n/(2 − ε). Now θ > X(i) + κ for at least j
values of i is equivalent to θ > X(j) + κ. So we have θ ≤ X(j) + κ where j is the smallest
integer > n/(2 − ε).
(cid:1)
n
i=1
∗∗
Now if Yi = Xi for at least j values of i = 1, ..., n, we have Y(j) ≤ X(n), so
∗∗
θ (Y1, ..., Yn) ≤ X(n) + κ ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/684183e815da73f9e914941a3d94c2d9_brkdn_location.pdf |
/2, and Y = n−j X , so that Yi = Xi for at least j values of
i, then X(1) ≤ Yi ≤ X(n) for at least j values of i. Thus as noted above
(1)
Y(j) ≤ X(n)
and if MY
is the median of Y1, ..., Yn, then X(1) ≤ MY ≤ X(n). Also, |Yi − MY | ≤
X(n) −X(1) for at least j values of i, so MADY , the median of |Yi −MY |, satisfies ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/684183e815da73f9e914941a3d94c2d9_brkdn_location.pdf |
X by (1) and (2). So
n ψ((Yi − θ)/SY ) < 0 if Yi − θ < −κSY
i=1
∗∗θ (Y1, ..., Yn) ≤ X(n) + κKX .
Symmetrically, we have
∗θ (Y1, ..., Yn) ≥ X(1) − κKX .
So θˆ(Y1, ..., Yn) remains bounded for Y = n−j X and the breakdown point of θˆ is at least as
large as for the median. If S = 0, this doesn’t cause breakdown of ... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/684183e815da73f9e914941a3d94c2d9_brkdn_location.pdf |
6.012 Microelectronic Devices and Circuits Fall 2005
Lecture 11
Lecture 1 6.012 Overview
September 8, 2005
Contents:
1. The microelectronics revolution
2. Keys to the microelectronics revolution
3. Contents of 6.012
Reading assignment:
Howe and Sodini, Ch. 1
Announcement:
In Homework 1, need to use the... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/6845e4ed047070a1df081426e820ad2e_lec1.pdf |
thousand fold.
6.012 Microelectronic Devices and Circuits Fall 2005
Lecture 15
� Si digital microelectronics today
Take the cover off a microprocessor. What do you see?
Image removed due to copyright restrictions.
[Intel Pentium IV]
• A thick web of interconnects, many levels deep
• High density of very sm... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/6845e4ed047070a1df081426e820ad2e_lec1.pdf |
improvements in:
• system performance
• costperfunction
• powerperfunction
• system reliability
Experimental SOI IBM microprocessor. Image removed due to copyright restrictions.
6.012 Microelectronic Devices and Circuits Fall 2005
Lecture 110
� Clock speed
4order of magnitude improvement in 30 years... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/6845e4ed047070a1df081426e820ad2e_lec1.pdf |
Microelectronic Devices and Circuits Fall 2005
Lecture 116
90 nm NMOS
Courtesy of Intel Corporation. Used with permission.
[Picture from: http://www.intel.com/technology/silicon/micron.htm]
6.012 Microelectronic Devices and Circuits Fall 2005
Lecture 117
4. CMOS
CMOS = Complementary MetalOxideSemic... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/6845e4ed047070a1df081426e820ad2e_lec1.pdf |
manufacturing variations
• require little power consumption
IREF
+2.5 V
+1.0 V
RS
M2
M1
vs
VBIAS
+
−
+
−
−1.0 V
−2.5 V
Q4
M3
0 V
RL
= 1 k
+
vOUT
−
Ω
6.012 Microelectronic Devices and Circuits Fall 2005
Lecture 121
3. Contents of 6.012
Deals with microelectronic devices...
• semiconducto... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2005/6845e4ed047070a1df081426e820ad2e_lec1.pdf |
6.172
Performance
Engineering
of Software
Systems
LECTURE 11
Storage
Allocation
Julian Shun
© 2008-2018 by the MIT 6.172 Lecturers
!"##$
%&'&(!"#)*+)$#)*+,*-./01
1
!"##$
%&'&(!"#)*+)$#)*+,*-./01
STACKS
© 2008-2018 by the MIT 6.172 Lecturers
2
Stack Allocation
Array and pointer
#
used
!"
All... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
unused
Free x bytes
!"*+,*-.
! Allocating and freeing take "(1) time.
! Must free consistent with stack discipline.
! Limited applicability, but great when it works!
! One can allocate on the call stack using $%%&'$(),
but this function is deprecated, and the compiler is
more efficient with fixed-size frames.
© 2008... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
+$%,(
#$,-#+&%(&
© 2008-2018 by the MIT 6.172 Lecturers
13
13
Fixed-Size Allocation
used
used
used
Free list
!
used
"#$$
*
Allocate 1 object
!"#"$%&&'
"#$$%&%"#$$'()$*+,
#$+-#)%*,%
© 2008-2018 by the MIT 6.172 Lecturers
14
Fixed-Size Allocation
used
used
used
Free list
!
used
"#$$
%
Allocat... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
&'&"#$$(
"#$$&'&"#$$)*+$%,(
#$,-#+&%(&
!"#$%!&'(')*%%+
"#$$&'&%(&
© 2008-2018 by the MIT 6.172 Lecturers
19
Fixed-Size Deallocation
Free list
!
used
"#$$
%
used
used
used
Allocate 1 object
free object %
%&'&"#$$(
"#$$&'&"#$$)*+$%,(
#$,-#+&%(&
%)*+$%,&'&"#$$(
!"##$%$&'
© 2008-2018 by the MIT 6.172... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
1!.1 = .82 versus .5!.5 + .5!.5 = .5
© 2008-2018 by the MIT 6.172 Lecturers
23
!"##$
%&'&(!"#)*+)$#)*+,*-./01
VARIABLE-SIZE
HEAP ALLOCATION
© 2008-2018 by the MIT 6.172 Lecturers
24
Variable-Size Allocation
Binned free lists
! Leverage the efficiency of free lists.
! Accept a ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
Lists
Allocate " If bin k = #lg x$ is nonempty, return a
x bytes
block.
" Otherwise, find a block in the next larger
nonempty bin k% > k, split it up into blocks
of sizes 2k%-1, 2k%-2, …, 2k, 2k, and distribute
the pieces.
Example
x = 3 & #lg x$ = 2.
Bin 2 is empty.
0
1
2
3
4
!
© 2008-2018 by the MIT 6.... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
the page table would
degrade tremendously leading to disk
thrashing, since all nonzero memory must be
backed up on disk in page-sized blocks.
Goal of storage allocators
Use as little virtual memory as possible, and try
to keep the used portions relatively compact.
© 2008-2018 by the MIT 6.172 Lecturers
31
... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
##$
%&'&(!"#)*+)$#)*+,*-./01
GARBAGE COLLECTION
BY REFERENCE COUNTING
© 2008-2018 by the MIT 6.172 Lecturers
34
Garbage Collectors
Idea
∙ Free the programmer from freeing objects.
∙ A garbage collector identifies and recycles the
objects that the program can no longer access.
∙ GC can be built-in (Java, Python) ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
each object. If the count drops to 0, free the dead
object.
root
root
2
1
0
1
1
3
3
root
© 2008-2018 by the MIT 6.172 Lecturers
40
Reference Counting
Keep a count of the number of pointers referencing
each object. If the count drops to 0, free the dead
object.
root
root
2
1
0
1
0
1
2 ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
form a directed graph
G = (V, E). Live
objects are reachable
from the roots. Use
breadth-first search to
find the live objects.
FIFO queue 7
!"#$%! &"'($)
*!$%#""+%&(($)
&,-.#/ 0$12
3456363%78$&(2
9$3:;3$&,-.#/ 0$<2
=>*:3$%7$?0$#($)
6$0$@356363%7(2
!"#$%! &"' such that %68&(" A($)
*!$%&,-.#/ 00$<($)
&,-.#/ ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
#
$
)&"'
-"*.
© 2008-2018 by the MIT 6.172 Lecturers
52
Breadth-First Search
"
%
(
#
$
&
'
!
*
+
)
,
!
#
$
)&"'
-"*.
© 2008-2018 by the MIT 6.172 Lecturers
53
Breadth-First Search
"
%
(
#
$
&
'
!
*
+
)
,
!
#
$
)&"'
-"*.
© 2008-2018 by the MIT 6.172 Lecturers
54
Bread... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
172 Lecturers
59
Breadth-First Search
"
%
(
#
$
&
'
!
*
+
)
,
!
#
$
'
&
%
)&"'
-"*.
© 2008-2018 by the MIT 6.172 Lecturers
60
Breadth-First Search
"
%
(
#
$
&
'
!
*
+
)
,
!
#
$
'
&
%
(
)&"'
-"*.
© 2008-2018 by the MIT 6.172 Lecturers
61
Breadth-First Search
"
% ... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
MIT 6.172 Lecturers
66
Copying Garbage Collector
FROM space
next
allocation
live
dead
unused
© 2008-2018 by the MIT 6.172 Lecturers
67
Copying Garbage Collector
FROM space
next
allocation
live
dead
unused
© 2008-2018 by the MIT 6.172 Lecturers
68
Copying Garbage Collector
FROM space
nex... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
Example
FROM
TO
!"#$
%#&'
Remove an item from the queue.
© 2008-2018 by the MIT 6.172 Lecturers
76
Example
FROM
TO
!"#$
%#&'
Remove an item from the queue.
© 2008-2018 by the MIT 6.172 Lecturers
77
Example
FROM
TO
!"#$
%#&'
Enqueue adjacent vertices.
© 2008-2018 by the MIT 6.172 Lecturers
78
... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
© 2008-2018 by the MIT 6.172 Lecturers
83
Dynamic Storage Allocation
Lots more is known and unknown about
dynamic storage allocation. Strategies include
● buddy system,
● variants of mark-and-sweep,
● generational garbage collection,
● real-time garbage collection,
● multithreaded storage al... | https://ocw.mit.edu/courses/6-172-performance-engineering-of-software-systems-fall-2018/68624442da6047731a33b3590cefd030_MIT6_172F18_lec11.pdf |
Lecture # 12
Solar Photovoltaics
Ahmed F. Ghoniem
March 11, 2020
Solar resources, potential, progress, pricing ..
Semiconductor physics, p-n junction, bandgap, efficiency ..
Solar panels, fabrication, variety, farms, systems ….
1
Solar Energy is “Everywhere”.
Opportunities vary.
Distribution ne... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
othermal
Solar-thermal
The dotted line is the prediction based on new policies to be implemented
Source: historic data from IEA website (up to 2017)
prediction data from IEA world energy outlook 2018, P528
© IEA. All rights reserved. This content is excluded from our Creative Commons
license. For more information,... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
“energy balance”, although being “energy” in a scientific sense.
7
Estimated (in 2019) Levelized Cost of Electricity Generation Plants in 2023
U.S. Average Total System Levelized Cost
(
) Capacity Factor
No tax credits are assumed for renewables
All cases for utility scale plants
10... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
onie
m
9
© ITACA. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/fairuse.
The yearly variation of the mean total daily solar radiation (to... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
Intrinsic semiconductors have intermediate bandgap values ( <3 eV). They have average
number of valence electrons (4 in the case of silicon)
• When doped with other metal, they can increase or decrease the number of electron in their
valence band depending on the dopant.
12
... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
also move across the junction
towards the p-type.
© CRC Press, LLC. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/fairuse.
The illuminated p-n junction showing the
formation of electron-hole pair by the
adsorption of a photon
(EHP: ... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
remaining energy dissipating into heat.
• The photon-induced current, which is proportional
to the incident photon intensity, can move across
the junction or to an external circuit.
• Freed electron (and electron holes) could be
reabsorbed within the material unless the distance
between the junction and the circu... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
omo”
junction ~ 30% (the Shokley limit)
• Hetero or multi junction (layered homojunctions) could be used to overcome
this limit (semiconductor layers with different bandgaps can capture photons
with different wavelength).
© Source unknown. All rights reserved. This content is excluded from our Creative
Commons li... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
.
Chen, Physics of Solar Energy, Wiley, 2011
Also Aliza Khurram 2019 term paper on Mars Mission
20 The external current density-voltage, J-V, relation of an illuminated p-n junction is:
⎛ ε0V ⎞
⎝⎜
⎠⎟
nkT
⎛ e0V ⎞
⎝⎜
nkT
⎛
⎞
j = js − j0 ⎝⎜ exp
⎠⎟ ≈ js − j0 exp
⎠⎟ − 1
js : zero voltage (short circuit) current V =... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
irically determined)
Eg (T ) ~ Eg ( )0 −
⎛ αT 2 ⎞
⎝⎜ T + β
⎠⎟ : bandgap energy
∞
0
js = εo ∫ηλ (λ )φ (λ)dλ
ηλ : is the quantum efficiency
φ (λ); is the spectral flux
js is often measured experimentally
material
Eg(0) in eV
αx10-4 in ev K-1
β in K
Si
Ge
GaAs
1.1557
0.7412
1.5216
7.021
4.561
8.871
11... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
ides)
but not necessarily electronic grade.
• All methods start with a molten solar grade silicon (doped with
different impurities to produce the p or n semiconductor, or to
pacify some of the defects).
• Production of solar cells is energy intensive (with some pay-back
energy period).
Edge-define film-fed growt... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
Thin film technology
• Material with bandgap close to 1.5 eV should be
used to achieve higher efficiency (than Si), but
they are expensive (CdTi, GaAs, InP, Zn3P2, …)
• Can only be used in thin film form to be
economical
• Because of the thin film (few microns), material
should also have high optical absorption ... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
Cells (2-terminal,
Technologies
Thin-Film
O CIGS (concentrator)
e CIGS
0 CdTe
0 Amorphous
Si:H (stabilized}
(concentrator)
(non-concentrator)
(concentrator)
(non-concentrator)
or more (concentrator)
or more (non-concentrator)
PV
Emerging
0 Dye-sensitized
0 Perovskite
cells
cells
• Organic
tandem
.A. O... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
L.,_..J.____,1_....L._L.,___J__..J.___J_....1...._.1,_--L.,_.J._--1._....1,_
__J'--....1....-L--'--J._-'-___Jl
1975 1980 1985 1990 1995 2000 2005 2010 2015 2020
Image courtesy of NREL, DOE.
Shar
Solar
NREL / Konarka
U. Linz
Siemens
p
U.Toronto
(PbS-QD)
U.DresdenNREL
(ZnO/PbS-QD)
;:::-
�
�
29 PV Farms
Fixed ti... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
from our Creative
Commons license. For more information, see https://ocw.mit.edu/fairuse.
© Wiley. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/fairuse.
Photovoltaic Solar Energy, Reinder et al, Ed., Wiley, 217
31
... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/687db8b61f59473bf61d6a611b157f80_MIT2_60s20_lec12.pdf |
Material Flow Analysis
Jeremy Gregory
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Slide 1
What is Material Flow Analysis?
“Material flow analysis (MFA) is a systematic
assessment of the flows and ... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/687f1c8bbbc1d92f41b1213694ff26b0_lec14.pdf |
• Key processes and goods
– Inputs: water, food, building and transport materials
– Outputs: sewage, off-gas, solid waste
The first application of MFA?
• Santorio Santorio (1561-1636)
• Measured human input and output
• Output weighs much less
• Hypothesis: output of “insensible perspiration”
Massachusetts Inst... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/687f1c8bbbc1d92f41b1213694ff26b0_lec14.pdf |
Throughput of
sectors
e.g., production sectors,
chemical industry,
construction
c
regions
e.g., total throughput, mass
flow balance, total material
requirement
associated with substances, materials, products
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: In... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/687f1c8bbbc1d92f41b1213694ff26b0_lec14.pdf |
– System Boundary: Defines start and end of flows
•Process Chain Analysis: Defines processes using
accounting and balancing
– Mass balancing to determine inputs and outputs
– Modeling may be applied
•Evaluation
– May involve impact criteria
Massachusetts Institute of Technology
Department of Materials Science & ... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/687f1c8bbbc1d92f41b1213694ff26b0_lec14.pdf |
Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Slide 17
Type IIc: Anthropogenic Metabolism
Prehistoric
Breath
5.1
Image of caveman removed
due to copyright restrictions.
Modern
Off gas
19
Goods
6
Stock
~0
Excreta
0.... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/687f1c8bbbc1d92f41b1213694ff26b0_lec14.pdf |
Domestic Hidden Flows
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Slide 20
Type IIc: Economy-Wide Material Flows
Metrics (cont.)
Consumption
• DMC (Direct Materials Consumption)=
DMI - Exports
• ... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/687f1c8bbbc1d92f41b1213694ff26b0_lec14.pdf |
/c/yr)
Massachuse
Depa
tts Institute of Technology
rtment of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – S
ystems Perspectives
Randolph Kirchain
Slide 24
US Material Flows, 1990 (Mt)
244
Recycled
Inputs
1960
1921
249
112
260
629
Energy
Construction minerals
Industrial minerals
Metals
Fo... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/687f1c8bbbc1d92f41b1213694ff26b0_lec14.pdf |
: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Slide 29
Courtesy of World Resources Institute. Used with permission. Source: Matthews, E., et al. "The Weight of Nations." World Resources Institute, 2000.
Trends in TDO
Massachusetts Institute of Technology
Department of Materials Science & Engineeri... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/687f1c8bbbc1d92f41b1213694ff26b0_lec14.pdf |
Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Slide 35
Courtesy of World Resources Institute. Used with permission. Source: Matthews, E., et al. "The Weight of Nations." World Resources Institute, 2000.
NAS and DPO, 1996
Massachusetts Institute of Technology
Department of... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/687f1c8bbbc1d92f41b1213694ff26b0_lec14.pdf |
ESD.33 Systems Engineering
Lecture 5
Innova’on in Systems
Engineering
Qi Van Eikema Hommes
Customer-Centered Products
Creating Successful Products through Smart
Requirements Management.
Ivy F. Hooks and Kristin A. Farry
Sample Chapter Titles:
• Chapter 2: Why Johnny Cannot Write Requirements?
• Chapter 5: On... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
crea’vity
Innova’on in the context of the technical
systems
Architectural Innova’on
The route of innova’on management
© June 22, 2010
Qi Van Eikema Hommes
4
Know Your User (Stakeholder) Needs
© June 22, 2010
Qi Van Eikema Hommes
5
Class Discussion Ques’ons
• Do we always know what customers rea... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
put exis’ng ideas together in new combina’ons.
• Naturally tries out solu’on that departs from the status quo.
• Feels comfortable disagreeing with others.
• Habitually combines knowledge from seemingly disparate
fields.
• Perseveres through long dry spells of tedious
experimenta’on.
– I have not failed. ... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
Individuals’ crea’vity in systems design
Know your users’ needs
The characteris’cs of a crea’ve person
Structured Innova’on in Systems Design—TRIZ
Innova’on in large systems
Managing crea’vity
Innova’on in the context of the technical
systems
Architectural Innova’on
The route of inno... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
in Siberia
• 1954 – Released from the prison arer Stalin’s death.
Produced his first publica’on on TRIZ.
• 1989 – first TRIZ Associa’on in Russia
• 1999 – first TRIZ Ins’tute in US. Altshuller passed away.
Atshuller, 40 Principles Extended Edition, 2005
© June 22, 2010
Qi Van Eikema Hommes
13
TRIZ
• TRI... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
)(cid:11)(cid:12)(cid:13)(cid:14)(cid:5)
Conventional thinking: Additional function = additional
system
(cid:5)
Function 1
(cid:5)Function 2
Function 3
(cid:5)
System 1
System 2
System 3
TRIZ thinking: Adding function without increasing
resources (or even use no resource).
(cid:2)(cid:13) (cid:4)(cid:18)(cid:15)(cid:6)... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
© June 22, 2010
Qi Van Eikema Hommes
17
Ideality Example
ESD.33 2007 Dan Frey
The Russians launched an unmanned Lunar Probe
to the moon’s surface with the inten’on to
transmit TV pictures to the Earth. A projector
using a light bulb was designed to illuminate the
lunar surface ahead of the vehicle. However,... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
chamber
Protective coating
Acid
Specimens
© June 22, 2010
Qi Van Eikema Hommes
20
Courtesy of Daniel Frey. Used with permission.
Solu’on
ESD.33 2007 Dan Frey
Acid
Specimen
Transition to an Ideal
Solution: Chamber is absent
Ideal Solution:
Specimen-Chamber
Acid
Specimen
© June 22, 2010
Qi Van Eikema H... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
technical contradic’on
a. Which characteris’c needs to be improved?
b. Which characteris’c will deteriorate as a result?
3. Resolve the technical Contradic’on using the
TRIZ Principles
© June 22, 2010
Qi Van Eikema Hommes
24
The 39 Technical Characteris’cs
1 Weight of a mobile object
21 Power
2 Weight of... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
a technical system.
• They are Altshuller’s view of the guiding
principles of technical inven’on, arer
reviewing 200,000 patents.
© June 22, 2010
Qi Van Eikema Hommes
26
The 40 Principles
1 Segmentation
2 Extraction
3 Local quality
4 Asymmetry
5 Consolidation
6 Universality
7 Nesting
8 Counterweight ... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
students may form groups on their
own or think through this on their own.
© June 22, 2010
Qi Van Eikema Hommes
29
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:3)(cid:7)(cid:8)
ESD.33 2007 Dan Frey
© June 22, 2010
Qi Van Eikema Hommes
30
(cid:8)
Courtesy of Daniel Frey. Used with permission.
Class Discussion:
Y... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
of the technical
systems
Architectural Innova’on
The route of innova’on management
© June 22, 2010
Qi Van Eikema Hommes
34
Three Components of Crea’vity for Every Individual
Amabile, HBR, 1998
Expertise is, in a word,
knowledge - technical,
procedural, and intellectual.
Expertise
Creative-
thinking
sk... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
ema Hommes
37
Prac’ce 2: Freedom
• What does mean to give employees freedom
in their assignments? What are the benefits?
• As a manager, what can you do to ensure the
success of this approach?
© June 22, 2010
Qi Van Eikema Hommes
38
Prac’ce 3: Resources
• Does providing a lot of resources help improve
... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
Hommes
41
Prac’ce 6: Organiza’onal Support
• Put in place appropriate systems or
procedures and emphasize values that make it
clear that crea’ve efforts are a top priority.
– Properly reward crea’vity
– Encourage exchange of ideas and collabora’on
– Intrinsic mo’va’on increases when people are
aware that t... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
e
g
a
k
n
i
L
s
t
n
e
n
o
p
m
o
C
d
n
a
s
t
p
e
c
n
o
C
Image by MIT OpenCourseWare.
© June 22, 2010
Qi Van Eikema Hommes
Henderson and Clark, HBR, 1998
44
Ceiling Fan Example
© June 22, 2010
Qi Van Eikema Hommes
Image by MIT OpenCourseWare.
45
Radical Innova’on
Core concept and architecture are bo... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
ural
Innovation
Radical
Innovation
© June 22, 2010
Qi Van Eikema Hommes
50
Images by MIT OpenCourseWare.
Architectural Innova’on
• Successful product development requires:
– Component knowledge
– Architectural knowledge
• We need to be aware of innova’on that use
many exis’ng core design concepts i... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
Innova’on
• Hard to iden’fy the innova’on has
architectural implica’on, because the core
concept seems to be the same.
• Established organiza’ons have challenges to
change its old way to communica’on and
learning.
• New entrants with smaller organiza’on find it
easier to build the organiza’on knowledge
aroun... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
controlling distortion.
Henderson and Clark, 1998
© June 22, 2010
Qi Van Eikema Hommes
56
Image by MIT OpenCourseWare.
Leading Manufacturers
Share of Deflated Cumulative Sales (%) 1962-1986, by Generation, for the Leading
Optical Photolithographic Alignment Equipment Manufacturers*
Alignment Equipment
Firm
Contac... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
58
Four Criteria of the Value Proposi’on of an
Innova’on
• Customer Value
• Integrability
• Maturity
• Profit
Maniak, Midler, and Lenfle, 2010
© June 22, 2010
Qi Van Eikema Hommes
59
Courtesy of Remi Maniak, Christophe Midler, and Sylvain Lenfle. Used with permission.
Interplay between Innova’on and Dev... | https://ocw.mit.edu/courses/esd-33-systems-engineering-summer-2010/688d747edcfaea3428fa599b88c5a30d_MITESD_33SUM10_lec05.pdf |
18.212: Algebraic Combinatorics
Andrew Lin
Spring 2019
This class is being taught by Professor Postnikov.
March 15, 2019
Recall that we’ve been talking about ranked posets: given a poset P with a rank function ρ : P → Z≥ 0, we can construct
sets Pi = {x ∈ P | ρ(x) = i } of a given rank, and we let ri = |Pi | be t... | https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/688d7bea866de628c9cb1dceb9c1bb06_MIT18_212S19_lec17.pdf |
chains Ci = {x0 l · · · l x`, ρ(x`) = N − ρ(x0).
For example, here is a poset that has a symmetric chain decomposition:
Lemma 3
If P has a symmetric chain decomposition, then it is rank-symmetric, unimodal, and Sperner.
1
Proof. Each chain contributes 1 to some set of rank numbers which is symmetric about t... | https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/688d7bea866de628c9cb1dceb9c1bb06_MIT18_212S19_lec17.pdf |
n elements in a chain.
Fact 4
Then the Boolean lattice Bn = [2] × [2] × · · · × [2] (n times).
For example, [2] × [2] is a square, [2] × [2] × [2] is the 1-skeleton of a cube, and so on.
Theorem 5 (de Bruijn, 1948 + generalization)
Bn has a symmetric chain decomposition! More generally, [a] × [b] × · · · × [c] has... | https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/688d7bea866de628c9cb1dceb9c1bb06_MIT18_212S19_lec17.pdf |
at finite posets in general. Given any poset P , remember that we define M(P ) to be the
maximum number of elements in any antichain of P . Define m(P ) to be the minimum number of disjoint chains needed to
cover all elements of P .
Theorem 8 (Dilworth, 1950)
For any finite poset P , M(P ) = m(P ).
There’s also a dual... | https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/688d7bea866de628c9cb1dceb9c1bb06_MIT18_212S19_lec17.pdf |
they are conjugates: their Young diagrams are transposes of each other.
For example, consider the following poset:
Here, we can cover 2, 4, 5 elements with 1, 2, 3 antichains, so
λ = (2, 4 − 2, 5 − 4) = (2, 2, 1) =⇒
3
Meanwhile, we can cover 3, 5 elements with 1, 2 chains, so
µ = (3, 5 − 3) = (3, 2) =⇒
So Dilwo... | https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/688d7bea866de628c9cb1dceb9c1bb06_MIT18_212S19_lec17.pdf |
Secants and Tangents
We defined the tangent line as a limit of secant lines. We also know that as
∆x approaches 0 the secant’s slope ∆f
∆x approaches the slope of the tangent line.
How close to 0 does ∆x have to be for ∆f
∆x to be close to the slope of the tangent
line?
We’ll use the Secant Approximation mathlet to look... | https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/68a59e6c2e777ee29a9236d54f779a64_lec1ses1ex1-secants.pdf |
.
3. Let x = 0.75.
(a) Find ∆y
∆x when x = 0.75 and ∆x has the values:
−0.5, −0.25, 0.25, 0.5.
(b) Find the slope of the tangent line to the graph of f (x) at x = 0.75.
(c) Find a value of ∆x for which the value of ∆y
∆x is within 0.1 units of
the slope of the tangent line.
1
4. Compare your answers to the previous pr... | https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/68a59e6c2e777ee29a9236d54f779a64_lec1ses1ex1-secants.pdf |
≤ 0.10 is correct.
2. Now use the red slider to set x = 0.
(a) Find ∆y
∆x when x = 0 and ∆x has the values:
−0.5, −0.25, 0.25, 0.5.
∆x
∆y
∆x
−0.50 −0.88
−0.25 −0.97
0.25 −0.97
0.50 −0.88
2
(b) Find the slope of the tangent line to the graph of f (x) at x = 0.
m ≈ −1.
(c) Find a value of ∆x for which the value of ∆y
∆x... | https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/68a59e6c2e777ee29a9236d54f779a64_lec1ses1ex1-secants.pdf |
Was your answer to part (c) the same for each problem?
There was a much wider range of correct answers in (2) than in (1)
and (3). It’s likely that your answers to 1(c) and 3(c) were closer to
0 than your answer to 2(c) was.
Also notice that there was a wider range of positive correct answers
to 1(c) and a wider range ... | https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/68a59e6c2e777ee29a9236d54f779a64_lec1ses1ex1-secants.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.