text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
S9 refers to a particular situation
• Result function: a function that describes the new situation
resulting from taking an action in another situation.
• Result(MoveNorth, S1) = S6
Lecture 10 • 15
So we can say things like "at robot Room Six" in "situations 9". More ordinarily,
we would have said at(robot,room6)... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
objects] and
use them as predicate arguments.
• At(Robot, Room6, S9) where S9 refers to a particular situation
• Result function: a function that describes the new situation
resulting from taking an action in another situation.
• Result(MoveNorth, S1) = S6
• Effect Axioms: what is the effect of taking an action i... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
situation that results from doing the drop action in situation s.
You can see the power of the logical representation in these axioms. We’re able to
say things about a very large, or possibly infinite set of situations, without
enumerating them.
18
Situation Calculus
• Reify situations: [reify = name, treat them ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
objects] and
use them as predicate arguments.
• At(Robot, Room6, S9) where S9 refers to a particular situation
• Result function: a function that describes the new situation
resulting from taking an action in another situation.
• Result(MoveNorth, S1) = S6
• Effect Axioms: what is the effect of taking an action i... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
x.s. color(x,s) = color(x, Result(Grab, s))
•
∀
• Can be included in Effect axioms
Lecture 10 • 21
It turns out that there's a solution to the problem of having to talk about all the
actions and what doesn’t change when you do them. You can read it in the book
where they're talking about successor-state axioms. Y... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
in s0 and it’s not holding gold, but it is
holding rope, and so on.
24
Planning in Situation Calculus
• Use theorem proving to find a plan
• Goal state:
∃
• Initial state: At(Home, s0) Æ
s. At(Home, s) Æ Holding(Gold, s)
Holding(Gold, s0) Æ
¬
Holding(Rope, s0) …
• Plan: Result(North, Result(Grab, Result(Sou... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
of the construction of that situation what
the actions should be.
• First, move South, then Grab and then move North.
Lecture 10 • 26
Note that we’re effectively planning for a potentially huge class of initial states at
once. We’ve proved that, no matter what state we start in, as long as it satisfies
the initia... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
, and we're going to take advantage of some special properties of the
planning problem to do it fairly efficiently. So, what special properties of
planning can we take advantage of?
28
Special Properties of Planning
• Reducing specific planning problem to general
problem of theorem proving is not efficient.
• We... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
to make your plan for going to Tahiti actually in the
order that you're going to execute it, necessarily. Right? If you did that, you
might have to consider all the different taxis you could ride in, and that would
take you a long time, and then for each taxi, you think about, "Well, then how
do I...." You don't wa... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
planning algorithm.
We'll be very restrictive in the language that we can use to talk about goals and
states and actions and so on. We'll be able to relax some of those restrictions
later on.
32
STRIPS representations
Lecture 10 • 33
Now we're going to talk about something called STRIPS representation. STRIPS is... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
we don't say whether
Door 1 is open or closed, and so that means Door 1 could be either. And when
we make a plan, it has to be a plan that would work in either case.
34
STRIPS representations
• States: conjunctions of ground literals
• In(robot, r3) Æ Closed(door6) Æ …
• Goals: conjunctions of literals
• (impli... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
3) Æ Closed(door6) Æ …
• Goals: conjunctions of literals
• (implicit
r) In(Robot, r) Æ In(Charger, r)
∃
• Actions (operators)
• Name (implicit
):
∀
Go(here, the
re)
• Preconditions: conjunction of literals
– At(here) Æ path(here, there)
Lecture 10 • 37
The preconditions are also a conjunction of literals. A... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
very special way, we don't have to explicitly, have arguments that
describe the situation, because, implicitly, an operator description says, "If these
things are true, and I do this action, then these things will be true in the resulting
situation." But the idea of a situation as a changing thing is really built in... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
database and see if they were there or not. STRIPS is at one extreme of
simplicity in inference, and situation calculus is at the other, complex extreme.
Now, people are building systems that live somewhere in the middle. But it's
useful to see what the simple case is like.
39
Strips Example
Let's talk about a pa... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
there be a path between x and y. We’ll assume that you can get
from anywhere to anywhere else.
43
Strips Example
• Action
• Buy(x, store)
– Pre: At(store), Sells(store, x)
– Eff: Have(x)
• Go(x, y)
– Pre: At(x)
– Eff: At(y),
¬
At(x)
• Goal
• Have(Milk) Æ Have(Banana) Æ Have(Drill)
OK, now let's imagine th... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
go to any possible
location in our domain. There might be a lot of places you could go, which
would generate a huge branching factor. And then you can buy stuff. Think of
all the things you could buy -- right? – there’s potentially an incredible number
of things you could buy. So, the branching factor, when you hav... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
if I want this to be true in the last step,
what had to have been true on the next to the last step, so that "buy milk" would
put me in that circumstance? "Well, we have to be at a store, and that store has
to sell milk. And we also have to already have bananas and a drill.”
47
Planning Algorithms
• Progression p... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
,store)
At(store) Æ Sells(store,M) Æ Have(B) Æ
Have(D)
• Both have problem of lack of direction – what
action or goal to pursue next.
Lecture 10 • 49
But anyway, you're going to see that if you try to build the planner based on this
principle, that, again, you're going to get into trouble because it's hard to se... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
.
51
Plan-Space Search
• Situation space – both progressive and regressive
planners plan in space of situations
• Plan space – start with null plan and add steps to
plan until it achieves the goal
• Decouples planning order from execution order
Lecture 10 • 52
Now you can de- couple the order in which you do t... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
what actions before thinking about what
order to do the actions
• Means-ends analysis
– Try to match the available means to the current ends
Lecture 10 • 54
Plan-space search also lets us do means-end analysis. That is to say, that you can
look at what the plan is trying to do, look at the means that you have ava... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
This may just be a partial order (that is, it
doesn’t have to specify whether j comes before k or k comes before j), but it
does have to be consistent (so it can’t say that j comes before k and k comes
before j).
57
Partially Ordered Plan
• Set of steps (instance of an operator)
• Set of ordering constraints Si ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
"Step I achieved pre-
condition C for Step J." So, if I have to have money in order to buy the bananas,
then I might have a "go to the bank" action and a "buy bananas" action and I
would put, during the bookkeeping and planning, I would put this link in there
that says: The reason I have the go-to-the-bank action in... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
to
set things up.
63
Initial Plan
• Steps: {start, finish}
• Ordering: {start < finish}
• start
• Pre: none
• Effects: start conditions
• finish
• Pre: goal conditions
• Effects: none
And there's a special final action, "finish", that has as its pre-conditions the goal
condition, and it has no effect. So, ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
is complete iff every precondition of every
step is achieved by some other step.
• Si �c Sj (“step I achieves c for step j”) iff
• Si < Sj
For Step I to achieve C for step J, SI has to come before SJ, right? It's part of the
notion of achievement,
Lecture 10 • 68
68
Plan Completeness
• A plan is complete iff e... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
So, there’s no step that can
mess up our effect that could possible intervene between when we achieve it
with si and when we need it at sj.
70
Plan Completeness
• A plan is complete iff every precondition of every
step is achieved by some other step.
• Si �c Sj (“step I achieves c for step j”) iff
• Si < Sj
... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
10 • 72
For a plan to be consistent, it is enough for the temporal ordering constraints to be
consistent (we can’t have I before j and j before I) and for the variable binding
constraints to be consistent (we can’t require two constants to be equal).
72
PO Plan Example
Lecture 10 • 73
Let us just spend a few m... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
causal link that says we are going to use this effect of
have(Milk) to satisfy the precondition have(Milk) in the finish step.
Lecture 10 • 76
76
PO Plan Example
start
Sells(SM, M) Sells(SM,B) At(H)
At(x1) Sells(x1,M)
Buy (M,x1)
Have(M)
Have(M) Have(B)
finish
Lecture 10 • 77
Every causal link also implies ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
Have(B)
Have(M) Have(B)
finish
Lecture 10 • 80
Now, a relatively straightforward thing to do is satisfy sells(x1,Milk) by
constraining x1 to be the supermarket. We add a variable binding constraint,
saying that x1 is equal to the supermarket. And that allows us to put a causal
link between Sells(SM,M) in the eff... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
At(x3)
GO (x3,SM)
At(SM)
At(x1) Sells(x1,M)
At(x2) Sells(x2, B)
Buy (M,x1)
Have(M)
Buy (B,x2)
Have(B)
Have(M) Have(B)
finish
The effect of at(SM) can be used to satisfy both preconditions, so we add causal
links to the at preconditions and temporal links to the buy actions.
Lecture 10 • 83
83
PO Plan Exa... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
(H)
At(x3)
GO (x3,SM)
At(SM)
At(x1) Sells(x1,M)
At(x2) Sells(x2, B)
Buy (M,x1)
Have(M)
Buy (B,x2)
Have(B)
Have(M) Have(B)
finish
We can add a causal link from the At(home) effect of start to this precondition, and
we’re done!
Lecture 10 • 86
86
x1 = SM
x2 = SM
x3 = H
PO Plan Example
start
Sells(SM,... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/1184a975225bdbab3e3d215bf173bde1_Lecture10FinalPart1.pdf |
3.46 PHOTONIC MATERIALS AND DEVICES
Lecture 3: System Design: Time and Wavelength Division Multiplexing
Lecture
DWDM Components
CATV → 1 amp/1000 homes
Fiber (α = 0.2 dB/km)
core: d = 8 μm
clad: d = 125 μm
Δn = 0.5%
Dispersion control: D↓
Δn
⇒
o
r
o
r
Amplifier (EDFA)
SiO2 : Er (100 ppm)
More BW
Al2O3 d... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/1190b2ecfa7f8d9a1da501aff52c7a1f_3_46l3_sysdesign.pdf |
��
2 ⎣
⎢1− e
− πν ⎤
j2 T
⎥⎦
Fabry-Perot interferometer
H( )
ν = C
n
e−
j 2πν −Φ )
(
n
n
FSR =
1 C0
=
nL
T
T = 100 ps ⇒ FSR = 10 GHz
FSR = free spectral range
Tx
#1D
Tx
#2
Tx
#N
Terminal A
,λ λ
...,
2
1D
λ
N
Fiber
Amplifier
r
e
x
e
p
l
i
t
l
u
M
1
x
N
Gain
Equalizer
˜
λ
1D
Add/Drop
Dispersi... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/1190b2ecfa7f8d9a1da501aff52c7a1f_3_46l3_sysdesign.pdf |
3 of 4
Lecture
Notes
Splitters & Combiners
Out 1
L/2
R L
π
2=
/
Out 2
rectional
Di
couplers
In
Out 2
(a)
In
Out 1
(b)
(c)
A Fabry-Perot interferometer: (a) free-
space propagation, (b) waveguide
analog, and (c) transmission response.
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/1190b2ecfa7f8d9a1da501aff52c7a1f_3_46l3_sysdesign.pdf |
System Architecture
IAP Lecture 2
Ed Crawley
January 11, 2007
Rev 2.0
Massachusetts Institute of Technology © Ed Crawley 2007
1
Today’s Topics
(cid:122) Definitions - Reflections
(cid:122) Form
(cid:122) Function
(cid:122) Reference PDP - “In the Small”
Massachusetts Institute of Technology © Ed Crawley 2007
2
¿ Ref... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
language - nouns)
• Code
• Illustrations, schematics, drawings
• Each discipline has developed shorthand for
representing their discipline specific form
But the form is the actual suggested
physical/informational embodiment.
Massachusetts Institute of Technology © Ed Crawley 2007
7
Duality of Physical and Informati... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
:122) There is also a backward direction relation.
(cid:122) Usually it is only necessary to show one, and the other
is implicit.
Massachusetts Institute of Technology © Ed Crawley 2007
11
Structural Link Examples
Chair
Data
Disk
Is under
Is stored in
n
Table
Array
contacts
25
Blades
Spatial
(under)
Topological
(wit... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
Implementation Structure - Whistle - “List”
Bump
Channel
Ramp
Hole
Step
Cavity
Mech.
integral
Mech.
integral
Mech.
integral
Bump
Channel
Ramp
Hole
Step
Cavity
Mech.
integral
Mech.
integral
Mech.
Integral
Mech.
integral
Mech.
integral
Mech.
integral
Mech.
integral
Mech.
Integral
(cid:122) “N-squared” matri... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
of
form)?
Massachusetts Institute of Technology © Ed Crawley 2007
Figure by MIT OCW.
17
The Whole Product System
Figure by MIT OCW.
(cid:122) We usually architect form which is both a product and a system,
and which we designate the product/system
(cid:122) Often for the product/system to deliver value, it must be j... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
this is not a problem?
Massachusetts Institute of Technology © Ed Crawley 2007
20
Use Context
Figure by MIT OCW.
(cid:122) The whole product system fits within a use context, which included
the objects normally present when the whole product operates, but
not necessary for it to deliver value
(cid:122) The product/s... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
2)
by an interface
It is good practice to draw the system boundary clearly
on any product/system representation, and identify
interfaces explicitly
Massachusetts Institute of Technology © Ed Crawley 2007
23
Camera - Whole Product & Boundaries
Supplied with camera
Memory Card
Wrist Strap
Digital
Digital Camera
Batter... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
2)What are the boundaries?
(cid:122)What are the interfaces?
Massachusetts Institute of Technology © Ed Crawley 2007
26
Amp - Whole
Product System
Boundary and
Interfaces
(cid:122)
(cid:122)
In the matrix
representation,
the boundary
is between
rows and
columns
Interfaces are
clearly
identified in the
block... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
x e
t x
Implementation
Interfaces
1
r
o
t
i
s
e
r
x
t
t
resitor 1
resistor 2
op amp
+input interface
-input interface
output interface
ground interface
-5 V interface
+5 V interface
input circuit
output circuit
power supply
ground
t = touching, tangent
b = boundary
w = within, s = surrounding
ov = overlapping
da = at a... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
(cid:122)What is the use context?
(cid:122)What are the boundaries?
(cid:122)What are the interfaces?
Massachusetts Institute of Technology © Ed Crawley 2007
29
Bubblesort -
Whole Product
Boundary and
Interfaces
(cid:122) The structural
interfaces in
software determine
sequence and
nesting, and
compilation
(cid... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
Static Graphical User
Interface - Whole
Product?
(cid:122)What is the whole product system
(cid:122)What is the use context?
(cid:122)What are the boundaries?
(cid:122)What are the interfaces?
Massachusetts Institute of Technology © Ed Crawley 2007
Courtesy of ENPHO. Used with permission.
31
Form of Whole Product - ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
which exists, or
has the potential to exist
(cid:122) Objects + Structure (of form)
(cid:122)
Is a system attribute, created by the architect
(cid:122) Product/system form combines with other supporting
systems (with which it interfaces at the boundary) to
form the whole product system that creates value
Massachuset... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
Otto]
(cid:122) Process is a transformation (creation or change)
applied to one or more objects in the system
[Dori]
Massachusetts Institute of Technology © Ed Crawley 2007
40
Function is Associate with Form
(cid:122) Change voltage proportional to current
(cid:122) Change voltage proportional to charge
(cid:122) Re... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
_array)
for i=1 to length_of_array
for j=1 to length_of_array - i
if array[ j ] > array [ j+1 ] then
Conditionally
Exchange
Contents
Exchange
Contents
temporary = array [ j+1 ]
array[ j+1 ] = array [ j ]
array[ j ] = temporary
Sort from
small to
large
end if
end of j loop
end of i loop
return array
End of procedure... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
.
Massachusetts Institute of Technology © Ed Crawley 2007
47
Function = Process + Operand
(cid:122) Note that function consists of the action or
Figure by MIT OCW.
transformation + the operand, an object which is acted
on, or transformed:
– Change voltage
– React force
– Amplify signal
– Regulate shaft speed
– Conta... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
be created, destroyed, or altered:
– Image is captured
– Signal is amplified
– Array is sorted
(cid:122) You often do not supply the operand, and there may be
more than one operand to consider
(cid:122) The double arrow is the generic link, called “effecting”
(cid:122) A single headed arrow can be used to represent
“... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
External function at the
interface: supporting +
vehicle
product/system boundary
(cid:122) External function is
delivered across a
boundary - the value
related operand is
external to the
product/system
(cid:122) External function is
linked to the
delivery of benefit
(cid:122)What is the value related operand?
(... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
Massachusetts Institute of Technology © Ed Crawley 2007
610
58
Semantically Exact Representation with OPM
Operand
Processing
Instrument
Object
Function
Form
(cid:122) Architecture is made up of operands + processes
(functions) plus instrument object (form)
(cid:122) Examples:
– Image is captured by digital camera
– ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
Here
Person
There
Transporting
Person
Transporting
(cid:122) P yields or creates O
Entropy
Transporting
(cid:122) P consumes or destroys O
Energy
Transporting
(cid:122) O is an agent of P (agent)
Operator
Transporting
(cid:122) O is and instrument of P
Skateboard
Transporting
(cid:122) P occurs if O is in state A
Enoug... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
:122) What are the value related states that change?
(cid:122) What is the externally delivered function?
Massachusetts Institute of Technology © Ed Crawley 2007
Figure by MIT OCW.
65
Externally Delivered Function Emerges from
Internal Function
(cid:122) System function which emerges at the highest level is
the exte... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
i
i
hich is exec
w
y form
b
on,
uted
Massachusetts Institute of Technology © Ed Crawley 2007
Interfacing + Reacting +
Bridge and
ramp
loads
Carrying +
compression
Figure by MIT OCW.
610
69
Delivered and Internal Function - Whistle
Figure by MIT OCW.
Product/system boundary
Channel
Step
Hole
Bump
Ramp
Cavity
wall
I... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
array[ j ], array [ j+1 ])
end if
end of j loop
end of i loop
return array
End of procedure
Procedure exchange_contents(List array,
number i, number j)
temporary = array [ j+1 ]
array[ j+1 ] = array [ j ]
array[ j ] = temporary
return array
(cid:122) What is the value related operand and
(cid:122) What are the intern... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
DP
(cid:122) Customer-focused products
(cid:122) Competitive product designs
(cid:122) Team coordination
(cid:122) Reduce time to introduction
(cid:122) Reduce cost of the design
(cid:122) Facilitate group consensus
(cid:122) Explicit decision process
(cid:122) Create archival record
(cid:122) Customizable methods
Mass... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
07
79
Summary to Date
Operand
Processing
Instrument
Form
Architecture?
Form = Elements + Structure
Function = Process + Operand
Form in the instrument of function
Massachusetts Institute of Technology © Ed Crawley 2007
80 | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/119970d5c8944785abb7d2da18bd72d9_lec2.pdf |
THE DELTA-METHOD AND ASYMPTOTICS OF SOME ESTIMATORS
18.465, Feb. 24, 2005
√
The delta-method gives a way that asymptotic normality can be preserved under
nonlinear, but differentiable, transformations. The method is well known; one version of
it is given in J. Rice, Mathematical Statistics and Data Analysis, 2d. ed... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/11b322ac770ab24cabb34a45a6365d94_delta_asympt.pdf |
(|y−µ|)
as y → µ by definition of derivative. Thus
√
√
f (Yn) = f (µ) + f (cid:1)(µ)(Yn − µ) + op(|Yn − µ|),
so
√
√
n[f (Yn) − f (µ)] = f (cid:1)(µ) n(Yn − µ) + nop(1/ n).
√
√
The last term is op(1), so the conclusion follows.
�
Let’s say a distribution function F has a good median if F has a continuous densit... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/11b322ac770ab24cabb34a45a6365d94_delta_asympt.pdf |
and variance 1/[4(2k + 3)] = 1/[4(n + 2)].
This beta distribution is asymptotically normal with its mean and variance as n → ∞
or equivalently k → ∞. This fact is a special case of facts known since about 1920, but
lacking a handy reference, I’ll indicate a proof. Let y = x − (1/2), so |y| ≤ 1/2 where the
density i... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/11b322ac770ab24cabb34a45a6365d94_delta_asympt.pdf |
median m, and n = 2k + 1 odd,
the sample median mn = X(k+1) has the distribution of F ← (U(k+1)) because F ← is
monotonic (non-decreasing, and strictly increasing in a neighborhood of 2 ). We have
F ←(1/2) = m. √So by the delta-method theorem above, n(mn − m), being equal in
distribution to n(F ←(U(k+1)) − F ←(1/2)... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/11b322ac770ab24cabb34a45a6365d94_delta_asympt.pdf |
m) converging in distribution as n → ∞ to N (0, 1/(4f (m)2)), just
as when n is odd and as stated by Randles and Wolfe.
√
√
Next, let’s consider the Hodges-Lehmann estimator. In this case, beside assuming
F has a good median m, we’ll assume the distribution is symmetric around m.
(If a
distribution is symmetric aro... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/11b322ac770ab24cabb34a45a6365d94_delta_asympt.pdf |
,
we can assume m = 0, because subtracting m from all the observations makes m = 0 and
doesn’t change the distribution of the difference. So we can assume F is symmetric around
0.
Let G be the distribution function of X1 + X2. Then G has a density g given by the
� ∞
convolution of f with itself, g(x) = −∞ f (x − y)... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/11b322ac770ab24cabb34a45a6365d94_delta_asympt.pdf |
≤i<j≤n
Ψ(x − Xi − Xj).
For x = 0, bearing in mind that under symmetry around 0, −Xi−Xj is equal in distribution
to Xi + Xj, this becomes the U -statistic that Randles and Wolfe call U4 and is closely
related to the Wilcoxon signed-rank statistic. We get that n(U
2 ) converges in
distribution as n → ∞ to N (0, 1/3)... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/11b322ac770ab24cabb34a45a6365d94_delta_asympt.pdf |
n) plus smaller terms that don’t affect the
)
x
asymptotic distribution. So we will have, where again Zn is asymptotically N (0, 1),
√
√
1
= G(2x) + Zn/ 3n + op(1/
n).
√
√
(n)
(x)
V
If this equals 1/2 (within O(1/n2)), then
�
θˆHL = x =
1
G←
2
1
2
�
− (Zn/ 3n) + op(1/ n).
√
√
It follows by the delta-metho... | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/11b322ac770ab24cabb34a45a6365d94_delta_asympt.pdf |
. In the present case, since U
have a relatively simple U -process, but still, the argument was incomplete.
(n)
(x)
3 | https://ocw.mit.edu/courses/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005/11b322ac770ab24cabb34a45a6365d94_delta_asympt.pdf |
MIT 3.071
Amorphous Materials
3: Glass Forming Theories
Juejun (JJ) Hu
1
After-class reading list
Fundamentals of Inorganic Glasses
Ch. 3 (except section 3.1.4)
Introduction to Glass Science and Technology
Ch. 2
3.022 nucleation, precipitation growth and interface
kinetics
Topological const... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/11e638bcfa158b75e2c775ce38d5dc24_MIT3_071F15_Lecture3.pdf |
�� Thermodynamics of nucleation
G
W
Homogeneous
nucleation
Heterogeneous
nucleation
Size
Surface energy contribution
Energy barrier for nucleation
7
SSGSSGsmlVGTTSlsSGGG Kinetics of nucleation
G
W
Nucleation rate:
Size
8
SSGsmlVGTTSlsSGGGexpnBWRDkT... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/11e638bcfa158b75e2c775ce38d5dc24_MIT3_071F15_Lecture3.pdf |
� Kinetics of growth
Nucleus
Atom
Net diffusion flux:
10
exp1exp~expgBBBBRFFEkTkTGGEkTkT: 0, 0mgGTTR0: 0nTR Crystal nucleation and growth
Metastable
zone of
supercooling
Tm
Driving force:
supercooling
Both processes
are thermally
activated
11
Tim... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/11e638bcfa158b75e2c775ce38d5dc24_MIT3_071F15_Lecture3.pdf |
104
105-108
106-108
Vapor deposition
Up to 1014
Maximum glass sample thickness:
: thermal diffusivity
13
max~cTdR Glass formation from liquid
V, H
Supercooled
liquid
Liquid
Increasing
cooling rate
3
2
1
Glasses obtained at
different cooling rates
have different structures
With infinitely s... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/11e638bcfa158b75e2c775ce38d5dc24_MIT3_071F15_Lecture3.pdf |
hedra share corners, not edges, not faces
Maximize structure geometric flexibility
At least three corners are shared
Formation of 3-D network structures
Only applies to most (not all!) oxide glasses
Highlights the importance of network topology
19
Classification of glass network topology
Floppy / fle... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/11e638bcfa158b75e2c775ce38d5dc24_MIT3_071F15_Lecture3.pdf |
.532.4rrnrn r41222rxxx3122rxxx41222rxxx
Temperature-dependent constraints
The constraint number should be evaluated at the glass
forming temperature (rather than room temperature)
Silica glass SixO1-x
Bond stretching
O-Si-O bond angle
Isostatic condition
SiO2
n
o
i
... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/11e638bcfa158b75e2c775ce38d5dc24_MIT3_071F15_Lecture3.pdf |
constraints
25
#23BBr
Property dependence on network rigidity
Many glass properties exhibit extrema or kinks at the
rigidity percolation threshold
J. Non-Cryst. Sol. 185, 289-296 (1995).
26
2.4r Measuring glass forming ability
Figure of merit (FOM):
Tx : crystallization temperature
Tg : glass tra... | https://ocw.mit.edu/courses/3-071-amorphous-materials-fall-2015/11e638bcfa158b75e2c775ce38d5dc24_MIT3_071F15_Lecture3.pdf |
Session 3: Inventory Analysis
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Introduction: Slide 31
LCA: Methodology
• Goal & Scope Definition
– What is the unit of analysis?
– What materials, processe... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/11ecb61a034ff68951ffbccbf7368cde_lec11.pdf |
15 in "Life Cycle Assessment of Aluminum:
for the Worldwide Primary Aluminium Industry."
International Aluminium Institute. March 2003.
Inventory Data
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Introd... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/11ecb61a034ff68951ffbccbf7368cde_lec11.pdf |
soluble using linear algebra
• Scale all flows relative to interconnection flows
• Sum all equivalent flows
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Introduction: Slide 38
4
Product Production... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/11ecb61a034ff68951ffbccbf7368cde_lec11.pdf |
Solid Waste
Oil Combustion
HCl
Cu
600
600
3800
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Introduction: Slide 41
Environmental Data – Plant B
Summary
Products
Raw Material
Inputs/Outputs
... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/11ecb61a034ff68951ffbccbf7368cde_lec11.pdf |
ation – Diesel Fuel
Energy
Driving Conditions Energy Consumption
Long Haul
City Traffic
1
2.7
Units
MJ/tonne-km
MJ/tonne-km
Energy Production Emissions
Energy Production Emissions
Emissions (g/MJ fuel consumed)
Diesel
Substance
0.208
HC
1.3
NOx
78.6
CO2
Oil
0.018
0.15
79.8
Massachusetts Institute... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/11ecb61a034ff68951ffbccbf7368cde_lec11.pdf |
Introduction: Slide 48
8
Results – Total Inventory
A
Transport
B
C
u
f
/
g
300
250
200
150
100
50
0
HCl
Cu
Solid
Waste
(kg)
HC
NOx
CO2
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain ... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/11ecb61a034ff68951ffbccbf7368cde_lec11.pdf |
Randolph Kirchain
Introduction: Slide 51
Considering Energy from Electricity
Conversion Efficiency % (MJ out / MJ in)
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
Coal Thermal
Combined
Cycle NG
Fuel Oil
Thermal
Diesel
Generation
Gas Turbine
Massachusetts Institute of Technology
Department of Materials ... | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/11ecb61a034ff68951ffbccbf7368cde_lec11.pdf |
9,293
56,560
171,393
85,853
200,686
Ignoring efficiency of electrical conversion,
drastically alters energy picture!
Massachusetts Institute of Technology
Department of Materials Science & Engineering
ESD.123/3.560: Industrial Ecology – Systems Perspectives
Randolph Kirchain
Introduction: Slide 54
11 | https://ocw.mit.edu/courses/esd-123j-systems-perspectives-on-industrial-ecology-spring-2006/11ecb61a034ff68951ffbccbf7368cde_lec11.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.969 Topics in Geometry: Mirror Symmetry
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
MIRROR SYMMETRY: LECTURE 6
DENIS AUROUX
1. The Quintic 3-fold and Its Mirror
The simplest Calabi-Yau’s are hypersurface... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/11f2fffa337f9a6ec301c32b1ddb3d98_MIT18_969s09_lec06.pdf |
n + 2))
(4)
OPn+1 (−(n + 2)) X = Ωn
| ∼
X ⊗ OPn+1 (−d)| ⇒
X = Ωn
∼
X = O
if d = n + 2, i.e. our X is indeed Calabi-Yau.
Example. Cubic curves in P2 correspond to elliptic curves (genus 1, isomorphic
to tori), while quartic surfaces in P3 are K3 surfaces.
The quintic in P4 is the world’s most studied Calabi-Yau 3-... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/11f2fffa337f9a6ec301c32b1ddb3d98_MIT18_969s09_lec06.pdf |
. Restricting to X gives
(7)
(1 + h|X )5 = 1 + 5h|X + 10h2|X + 10h3|X = (1 + c1 + c2 + c3)(1 + 5h|X)
so c1 = 0, c2 = 10h2|X , c3 = −40h3|X . Thus,
(8)
χ(X) = −40h3
·
[X] = −40([line] ∩ [X]) = −40 5 = −200
·
We conclude that
(9)
h0 + h2 − h3 + h4 + h6 = 1 + 1 − dim H3(X) + 1 + 1 = −200
implying that dim H3 = 2... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/11f2fffa337f9a6ec301c32b1ddb3d98_MIT18_969s09_lec06.pdf |
· ·
5 +
+ x4
: x4) ∈ P4 fψ = x0
|
5 − 5ψx0x1x2x3x4 = 0}
(10) Xψ = {(x0 :
�
ai = 0}/(Z/5Z = {(a, a, a, a, a)}). Then
Let G = {(a0, . . . , a4) ∈ (Z/5Z)5
|
G ∼
= (Z/5Z)3 acts on Xψ by (xj )
�→ (xj ξaj ) where ξ = e2πi/5 (fψ is G-invariant
because
aj = 0 mod 5, and (1, 1, 1, 1, 1) acts trivially because the xj are... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/11f2fffa337f9a6ec301c32b1ddb3d98_MIT18_969s09_lec06.pdf |
1,2 = {x0 =
stabilizer (Z/5Z)2, so P012/G = {pt}.
x1
= x2 = 0, x3
5 + x5 = 0} with
4
The singular locus of Xψ/G is the 10 curves Cij = Cij/G ∼= P1 with C ij , C jk, C ik
meeting at the point P ijk.
Next, let Xψ
∨ be the resolution of singularities of (Xψ/G), i.e. Xψ
∨ π Xψ/G which is an isomorphism outside π−1(
... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/11f2fffa337f9a6ec301c32b1ddb3d98_MIT18_969s09_lec06.pdf |
six divisors, for a total of 60 divisors. Thus, X ∨ contains 100
∨) = 101.
new divisors in addition to the hyperplane section, so indeed h1,1(Xψ
Similarly, as we were only able to build a one-parameter family, h2,1(Xψ
∨) = 1,
giving us mirror symmetric Hodge diamonds:
5, x2
ψ
(11)
hij(X) = ⎜
⎜
⎝
⎛
0
1
0
1
0 101... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/11f2fffa337f9a6ec301c32b1ddb3d98_MIT18_969s09_lec06.pdf |
Last time, we saw a basis {ei} of
H 2(X, Z) by elements of the K¨ahler cone gives coordinates on the complexified
tiei, the parameter qi = exp(2πiti) ∈ C∗
K¨ahler moduli space: if [B + iω] =
gives the large volume limit as qi
0, Im (ti) → ∞. Physics predicts that
the mirror situation is degeneration of a large com... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/11f2fffa337f9a6ec301c32b1ddb3d98_MIT18_969s09_lec06.pdf |
= Z2 by
→
(the Dehn twist): observe that Ct
C ∪ {∞} by projection to x, and the branch points are ∞ plus the roots
of x3 + x2 − t. As t → 0, there is one root near −1 and two near 0, which
rotate as t goes around 0. Letting a be the line between the two roots
∗ ∈ Aut(Hn
�
�
1 1
0 1
4
DENIS AUROUX
near 0 an... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/11f2fffa337f9a6ec301c32b1ddb3d98_MIT18_969s09_lec06.pdf |
a constant value, so the ratio goes to +i∞. In
to 0 and
the latter case, q(t) is a holomorphic function of t, and goes around 0 once when
t does, i.e. it has a single root at t = 0. Thus, q is a local coordinate for the
family.
→
�
b y
a
Next time, we will see an analogue of this for a family of Calabi-Yau mani... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/11f2fffa337f9a6ec301c32b1ddb3d98_MIT18_969s09_lec06.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.969 Topics in Geometry: Mirror Symmetry
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
MIRROR SYMMETRY: LECTURE 3
DENIS AUROUX
Last time, we say that a deformation of (X, J) is given by
(1)
{s ∈ Ω0,1(X, T ... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/1232012395d30aca2bf98aacbb7ad036_MIT18_969s09_lec03.pdf |
− ∗ ∂∗
Δ = dd∗ + d∗d, � = ∂∂
∗
+ ∂
∗
∂
Every (d/∂)-cohomology class contains a unique harmonic form, and one can
show that � = 1 Δ. We obtain
2
(6)
dR
H k (X, C) ∼= Ker (Δ : Ωk(X, C) �) = Ker (� : Ωk �)
H p,q (X)
Ker (� : Ωp,q �) ∼
=
�
�
∼
=
∂
p+q=k
p+q=k
1
2
DENIS AUROUX
The Hodge ∗ operator gives an i... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/1232012395d30aca2bf98aacbb7ad036_MIT18_969s09_lec03.pdf |
)
1
0
0
1
0
0
h1,1
h2,1
h2,1
h1,1
0
0
1
0
0
1
Mirror symmetry says that there is another Calabi-Yau manifold whose Hodge
diamond is the mirror image (or 90 degree rotation) of this one.
There is another interpretation of the Kodaira-Spencer map H 1(X, T X) ∼=
H n−1,1 . For X = (X, Jt)t∈S a family of com... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/1232012395d30aca2bf98aacbb7ad036_MIT18_969s09_lec03.pdf |
t=0)(n−1,1) = 0
∂
Ωt |t=0)(n−1,1) + ∂(
∂
∂(
∂t
∂t
Thus, ∃[( ∂Ωt |t=0)(n−1,1)] ∈ H n−1,1(X).
∂t
For fixed Ω0, this is independent of the choice of Ωt. If we rescale f (t)Ωt,
∂
∂t
∂f
(f (t)Ωt) = Ωt + f (t)
∂t
∂Ωt
∂t
→
0, the former term is (n, 0), while for the latter, f (0) scales linearly
(12)
Taking t
with... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/1232012395d30aca2bf98aacbb7ad036_MIT18_969s09_lec03.pdf |
X
X
2. Pseudoholomorphic curves
(reference: McDuff-Salamon) Let (X 2n, ω) be a symplectic manifold, J a com
·
patible almost-complex structure, ω( , J ) the associated Riemannian metric.
Furthermore, let (Σ, j) be a Riemann surface of genus g, z1, . . . , zk ∈ Σ market
points. There is a well-defined moduli space Mg,... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/1232012395d30aca2bf98aacbb7ad036_MIT18_969s09_lec03.pdf |
⊗ u∗T X).
Σ
�
�
�
�
�
4
DENIS AUROUX
We can define a linearized operator
D∂ : W r+1,p(Σ, u∗T X) × T Mg,k → W r,p(Σ, Ω0,1 ⊗ U ∗T X)
Σ
(17) D∂ (v, j�) =
1
(�v + J�vj + (�vJ) · du · j + J · du · j�)
2
= ∂v +
1
2
(�vJ)du · j +
1
2
J · du · j�
This operator is Fredholm, with real index
(18)
indexRD∂ := 2d = 2�... | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/1232012395d30aca2bf98aacbb7ad036_MIT18_969s09_lec03.pdf |
is a Baire subset in J (X, ω), and for J ∈ J reg(X, β), M∗ (X, J, β) is smooth
(as an orbifold, if Mg,k is an orbifold) of real dimension 2d and carries a natural
orientation.
g,k | https://ocw.mit.edu/courses/18-969-topics-in-geometry-mirror-symmetry-spring-2009/1232012395d30aca2bf98aacbb7ad036_MIT18_969s09_lec03.pdf |
6.001 Structure and Interpretation of Computer Programs. Copyright © 2004 by Massachusetts Institute of Technology.
6.001 Notes: Section 5.1
Slide 5.1.1
In this lecture we are going to continue with the theme of
building abstractions. Thus far, we have focused entirely on
procedural abstractions: the idea of captu... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/1239c38467fc8018b19c808d121bcf5d_lecture5webhand.pdf |
Slide 5.1.4
We need to provide it with inputs of a specified type.
Slide 5.1.5
We know by the contract associated with the procedure, that if
we provide inputs of the appropriate type, the procedure will
produce an output of a specified type...
Slide 5.1.6
... and by giving the whole procedure a name, we create ... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/1239c38467fc8018b19c808d121bcf5d_lecture5webhand.pdf |
structure of try and the use of the special form if to control the evolution of this
procedure.
The method for improve simply incorporates the ideas from the algorithm, again with a procedure abstraction
to separate out idea of averaging from the procedure for improving the guess.
Finally, notice how we can build a... | https://ocw.mit.edu/courses/6-001-structure-and-interpretation-of-computer-programs-spring-2005/1239c38467fc8018b19c808d121bcf5d_lecture5webhand.pdf |
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