text stringlengths 16 3.88k | source stringlengths 60 201 |
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on this machine? We can try to
understand this in terms of the input/output equations. From the definition of the increment
machine, we have
o[t] = i[t] + 1 .
And if we connect the input to the output, then we will have
mFeedback(m)omFeedback2(m)oi Chapter 4 State Machines
6.01— Spring 2011— April 25, 2011
140
i... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
a machine whose output alternates between true
and false.
IncrDelay(0)Counter Chapter 4 State Machines
6.01— Spring 2011— April 25, 2011
141
4.2.3.1 Python Implementation
Following is a Python implementation of the feedback combinator, as a new subclass of SM that
takes, at initialization time, a state machine. ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
that we just computed and using it as input for getNextValues. This
will generate the next state of the feedback machine. (Note that throughout this process inp is
ignored—a feedback machine has no input.)
def getNextValues(self, state, inp):
(ignore, o) = self.m.getNextValues(state, ’undefined’)
(newS, ignore) = se... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
: 9
Step: 3
Feedback_96
Cascade_97
Increment_98 In: 9 Out: 11 Next State: 11
Delay_99 In: 11 Out: 9 Next State: 11
Step: 4
Feedback_96
Cascade_97
Increment_98 In: 11 Out: 13 Next State: 13
Delay_99 In: 13 Out: 11 Next State: 13
...
[3, 5, 7, 9, 11, 13, 15, 17, 19, 21]
(The numbers, like 96 in Feedback_96 are not ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
of numbers (appearing simultaneously) as input, and immediately
generates their sum as output.
Chapter 4 State Machines
6.01— Spring 2011— April 25, 2011
143
Figure 4.8 Machine to generate the Fibonacci sequence.
class Adder(SM):
def getNextState(self, state, inp):
(i1, i2) = splitValue(inp)
return safeAdd(i1, i... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
Delay_106 In: 1 Out: 1 Next State: 1
Adder_107 In: (1, 1) Out: 2 Next State: 2
Step: 2
Feedback_100
Cascade_101
Parallel_102
Delay_103 In: 3 Out: 2 Next State: 3
Delay(1)Delay(0)FibonacciDelay(1)+ Chapter 4 State Machines
6.01— Spring 2011— April 25, 2011
144
Cascade_104
Delay_105 In: 3 Out: 2 Next State: 3
D... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
is the completely passive machine, whose output is always in
stantaneously equal to its input. It is not very interesting by itself, but
sometimes handy when building things.
class Wire(SM):
def getNextState(self, state, inp):
return inp
Exercise 4.8.
Use feedback and a multiplier (analogous to Adder) to make a ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
sm.FeedbackAdd(sm.R(0), sm.Wire())
makes a machine whose output is the sum of all the inputs it has ever had (remember that sm.R is
shorthand for sm.Delay). You can test it by feeding it a sequence of inputs; in the example below,
it is the numbers 0 through 9:
>>> newM.transduce(range(10))
[0, 0, 1, 3, 6, 10, 15, 2... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
Here is how to do it in Python; we take advantage of having defined counter machines to abstract
away from them and use that definition here without thinking about its internal structure. The
initial values in the delays get the series started off in the right place. What would happen if we
started at 0?
fact = sm.Ca... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
State Machines
6.01— Spring 2011— April 25, 2011
147
Feedback2_6
Cascade_7
Multiplier_8 In: (3, 2) Out: 6 Next State: 6
Delay_9 In: 6 Out: 2 Next State: 6
Step: 3
Cascade_1
Feedback_2
Cascade_3
Increment_4 In: 4 Out: 5 Next State: 5
Delay_5 In: 5 Out: 4 Next State: 5
Feedback2_6
Cascade_7
Multiplier_8 In: (4, 6) O... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
of as actions) is input to the plant. This is shown schematically in
figure 4.10. For example, when you build a Soar brain that interacts with the robot, the robot (and
the world in which it is operating) is the “plant” and the brain is the controller. We can build a
coupled machine by first connecting the machines in... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
the robot’s velocity, v, and whose output sequence is the values of its distance to the wall, d.
Finally, we can couple these two systems, as for a simulator, to get a single state machine with no
inputs. We can observe the sequence of internal values of d and v to understand how the system
is behaving.
In Python, ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
world, and run it:
>>> wallSim = coupledMachine(WallController(), WallWorld())
>>> wallSim.run(30)
[5, 4.4000000000000004, 3.8900000000000001, 3.4565000000000001,
3.088025, 2.77482125, 2.5085980624999999, 2.2823083531249999,
2.0899621001562498, 1.9264677851328122, 1.7874976173628905,
1.6693729747584569, 1.568967028544... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
conditional combinator that runs two machines in parallel, but
decides on every input whether to send the input into one machine or the other. So, only one of
the parallel machines has its state updated on each step. We will call this switch, to emphasize the
fact that the decision about which machine to execute is ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
= state
(ns1, o1) = self.m1.getNextValues(s1, inp)
(ns2, o2) = self.m2.getNextValues(s2, inp)
if self.condition(inp):
return ((ns1, ns2), o1)
else:
return ((ns1, ns2), o2)
Exercise 4.11.
What is the result of running these two machines
m1 = Switch(lambda inp: inp > 100,
Accumulator(),
Accumulator())
m2 = Mux(lambd... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
execute sm1 or sm2. Ultimately, the state of the If machine will be a pair of values:
the first will indicate which constituent machine we are running and the second will be the state
of that machine. But, to start, we will be in the state (’start’, None), which indicates that the
decision about which machine to exec... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
ifState == ’runningM1’:
(newS, o) = self.sm1.getNextValues(smState, inp)
return ((’runningM1’, newS), o)
else:
(newS, o) = self.sm2.getNextValues(smState, inp)
return ((’runningM2’, newS), o)
4.3 Terminating state machines and sequential compositions
So far, all the machines we have discussed run forever; or, at least... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
��fth step, it generates the sum of the numbers
it has seen as inputs, and then terminates. It looks just like the state machines we have seen before,
with the addition of a done method. Its state consists of two numbers: the first is the number of
times the machine has been updated and the second is the total input ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
None, None, None, 15]
Now we can define a new set of combinators that operate on TSMs. Each of these combina
tors assumes that its constituent machines are terminating state machines, and are, themselves,
terminating state machines. We have to respect certain rules about TSMs when we do this. In
particular, it is not... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
machine in a done state (so, for example, nobody asks for its output, when its
done), unless the whole repeat machine is done as well.
def advanceIfDone(self, counter, smState):
while self.sm.done(smState) and not self.done((counter, smState)):
counter = counter + 1
smState = self.sm.startState
return (counter, s... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
def getNextValues(self, state, inp):
return (True, self.c)
def done(self, state):
return state
>>> a = CharTSM(’a’)
>>> a.run(verbose = True)
Start state: False
In: None Out: a Next State: True
[’a’]
See that it terminates after one output. But, now, we can repeat it several times.
>>> a4 = sm.Repeat(a, 4)
>>> a4.run(... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
smList
self.startState = (0, self.smList[0].startState)
self.n = len(smList)
The initial state of this machine is the value 0 (because we start by executing the 0th constituent
machine on the list) and the initial state of that constituent machine.
The method for advancing is also similar that for Repeat. The only di... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
’a’), CharTSM(’b’), CharTSM(’c’)])
>>> m.run()
Start state: (0, False)
In: None Out: a Next State: (1, False)
In: None Out: b Next State: (2, False)
In: None Out: c Next State: (2, True)
[’a’, ’b’, ’c’]
Even in a test case, there is something unsatisfying about all that repetitive typing required to
make each individu... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
l Next State: (10, False)
In: None Out: d Next State: (10, True)
[’H’, ’e’, ’l’, ’l’, ’o’, ’ ’, ’W’, ’o’, ’r’, ’l’, ’d’]
We can also see that sequencing interacts well with the Repeat combinator.
>>> m = sm.Repeat(makeTextSequenceTSM(’abc’), 3)
>>> m.run(verbose = True)
Start state: (0, (0, False))
In: None Out: a Nex... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
se
quential machine into its list of machines, and the last Boolean is the state of the CharTSM that is
being executed, which is an indicator for whether it is done or not.
4.3.3 RepeatUntil and Until
In order to use Repeat, we need to know in advance how many times we want to execute the
constituent TSM. Just as ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
done. This is appropriate in some situations; but in other cases, we
would like to terminate the execution of a TSM if a condition becomes true at any single step of
the machine. We could easily implement something like this in any particular case, by defining
a special-purpose TSM class that has a done method that t... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
1 Out: None Next State: (2, 1)
In: 2 Out: None Next State: (3, 3)
In: 3 Out: None Next State: (4, 6)
In: 4 Out: 10 Next State: (0, 0)
In: 5 Out: None Next State: (1, 5)
In: 6 Out: None Next State: (2, 11)
In: 7 Out: None Next State: (3, 18)
In: 8 Out: None Next State: (4, 26)
In: 9 Out: 35 Next State: (0, 0)
I... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
0 Out: None Next State: (False, (1, 0))
In: 1 Out: None Next State: (False, (2, 1))
In: 2 Out: None Next State: (False, (3, 3))
In: 3 Out: None Next State: (False, (4, 6))
In: 4 Out: 10 Next State: (False, (5, 10))
[None, None, None, None, 10]
However, if we change the termination condition, the execution will be termi... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
(range(20), verbose = True)
Start state: (False, (0, (0, 0)))
In: 0 Out: None Next State: (False, (0, (1, 0)))
In: 1 Out: None Next State: (False, (0, (2, 1)))
In: 2 Out: None Next State: (False, (0, (3, 3)))
In: 3 Out: None Next State: (False, (0, (4, 6)))
In: 4 Out: 10 Next State: (False, (1, (0, 0)))
In: 5 Out: None... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
description, see the Infrastructure Guide, which documents the io and util modules
in detail. The io module provides procedures and methods for interacting with the robot; the
util module provides procedures and methods for doing computations that are generally useful
(manipulating angles, dealing with coordinate fr... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
):
return (None, io.Action())
def setup():
robot.behavior = StopSM()
robot.behavior.start()
def step():
robot.behavior.step(io.SensorInput(), verbose = False).execute()
In the following sections we will develop two simple machines for controlling a robot to move a
fixed distance or turn through a fixed angle. Then we wi... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
its current
heading is, so it computes the desired heading (by adding the desired change to the current head
ing, and then calling a utility procedure to be sure the resulting angle is between plus and minus
π), and returns it and the current heading. Otherwise, we keep the thetaDesired component
of the state, and ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
x
and y coordinates when it starts, and drive straight forward until the distance between the initial
position and the current position is close to the desired distance. The state of the machine is the
robot’s starting position and its current position.
class ForwardTSM (SM):
forwardGain = 1.0
distTargetEpsilon = 0... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
a class that describes a machine that takes as input a series of pairs of goal points (ex
pressed in the robot’s odometry frame) and sensor input structures. It generates as output a
series of actions. This machine is very nearly a pure function machine, which has the following
basic control structure:
•
•
If the... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
.angleEps):
# Pointing in the right direction, so move forward
r = robotPoint.distance(goalPoint)
if r < self.distEps:
# We’re there
return (True, io.Action())
else:
return (False, io.Action(fvel = r * self.forwardGain))
else:
# Rotate to point toward goal
headingError = util.fixAnglePlusMinusPi(\
-0.7560.904-0... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
If the robot is close enough to the subgoal point, then it is time to change the state. We increment
the side length, pick the next direction (counter clockwise around the cardinal compass direc
tions), and compute the next subgoal point. The output is just the subgoal and the sensor input,
which is what the driver ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
systems. It will turn out that there are some systems
that are conveniently defined using this discipline, but that for other kinds of systems, other
disciplines would be more natural. As you encounter complex engineering problems, your job
is to find the PCAP system that is appropriate for them, and if one does not e... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
understanding of its modus operandi. (Now he is reluctant to
ride it for fear some new facet of its operation will appear, contradicting the algorithms given.)
We often fail to realize how little we know about a thing until we attempt to simulate it on a
computer.”
The Art of Computer Programming, Donald E., Knuth,... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
Machines
6.01— Spring 2011— April 25, 2011
167
class MySM(sm.SM):
startState = (0,0)
def getNextValues(self, state, inp):
(x, y) = state
y += inp
if y >= 100:
return ((x + 1, 0), y)
return ((x, y), y)
def done(self, state):
(x, y) = state
return x >= 3
The most important step, conceptually, is deciding what the state... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
myNewSM = sm.Repeat(Sum(), 3)
4.6.2 Practice problem: Inheritance and State Machines
Recall that we have defined a Python class sm.SM to represent state machines. Here we consider
a special type of state machine, whose states are always integers that start at 0 and increment by
1 on each transition. We can represent... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
subclass of CountingStateMachine called AlternateZeros. Instances of
AlternateZeros should be state machines for which, on even steps, the output is the same as
the input, and on odd steps, the output is 0. That is, given inputs, i0, i1, i2, i3, . . ., they generate
outputs, i0, 0, i2, 0, . . ..
class AlternateZero... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/063daea1b8a3573d2aff0f0b96d390da_MIT6_01SCS11_chap04.pdf |
Method of Green’s Functions
18.303 Linear Partial Differential Equations
Matthew J. Hancock
Fall 2006
We introduce another powerful method of solving PDEs. First, we need to consider
some preliminary definitions and ideas.
1 Preliminary ideas and motivation
1.1 The delta function
Ref: Guenther & Lee
10.5, Myint-U... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
three properties,
δ (x, y) =
0,
,
∞
(
(x, y) = 0,
(x, y) = 0,
δ (x, y) dA = 1,
Z
Z
f (x, y) δ (x
Z
Z
a, y
−
−
b) dA = f (a, b) .
1.2 Green’s identities
Ref: Guenther & Lee
8.3
§
Recall that we derived the identity
(G
∇ ·
F + F
· ∇
Z
D
Z
G) dA =
(GF)
C
Z
nˆdS
·
(1)
for any scalar function G and vecto... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
(x, y) be a fixed arbitrary point in a 2D domain D and let (ξ, η) be a variable
point used for integration. Let r be the distance from (x, y) to (ξ, η),
Considering the Green’s identities above motivates us to write
r = (ξ
q
x)2 + (η
y)2 .
−
−
2G = δ (ξ
∇
−
−
G = 0 on C.
x, η
y) = δ (r)
in D,
(5)
2
... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
’s function for a 2D domain D, we first find the simplest function
2v = δ (r). Suppose that v (x, y) is axis-symmetric, that is, v = v (r).
that satisfies
Then
∇
2
v = vrr +
∇
1
r
vr = δ (r)
For r > 0,
Integrating gives
vrr +
1
r
vr = 0
v = A ln r + B
For simplicity, we set B = 0. To find A, we integrate over ... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
For 3D domains, the fundamental solution for the Green’s function of
the Laplacian is
1/(4πr), where r = (x
ξ)2 + (y
η)2 + (z
ζ)2 .
−
The Green’s function for the Laplacian on 2D domains is defined in terms of the
−
−
−
q
corresponding fundamental solution,
G (x, y; ξ, η) =
ln r + h,
1
2π
h is regular,
∇
2h... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
ln r =
1
2π
ln (ξ
G =
2
y)
2
−
−
(ii) Half plane D =
(cid:3)
. We find G by introducing what is called an
(x, y) : y > 0
}
{
y) corresponding to (x, y). Let r be the distance from (ξ, η) to
(cid:2)
“image point” (x,
(x, y) and r ′ the distance from (ξ, η) to the image point (x,
−
y),
−
We add
r = (ξ
q
−
x)2 + ... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
G (x, y; ξ, η) for the Laplacian operator in the
upper half plane, for (x, y) = (√2, √2).
for (ξ, η)
∈
G =
1
2π
D. Writing things out fully, we have
ln r + h =
1
2π
ln r
1
− 2π
′
ln r =
1
2π
ln
r
r ′ =
1
4π
ln
(ξ
(ξ
−
−
2
2
x) + (η
y)
x)2 + (η + y)2
−
(7)
→ −∞
as (ξ, η)
√2, √2 .
G (x, y; ξ, η) is ... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
∂η
=
1
π (ξ
−
y
x)2 + y2
η=0
(cid:12)
(cid:12)
(cid:12)
(cid:12)
The solution of the BVP (6) with F = 0 on the upper half plane D can now be
written as, from (6),
u (x, y) =
G
f
∇
·
nˆdS =
C
Z
∞
y
π −∞ (ξ
Z
f (ξ)
x)2 + y
−
dξ,
2
which is the same as we found from the Fourier Transform, on page 13 of f... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
terms on the
2 = ∂2/∂ξ2+∂2/∂η2 of each
r.h.s. are regular for (ξ, η)
D, and hence the Laplacian
2G = δ (r).
of these terms is zero. The Laplacian of the first term is δ (r). Hence
Thus (8) is the Green’s function in the upper half plane D.
∇
∇
∈
For (ξ, η)
∈
C = ∂D (the boundary),
0
G
f
∇
·
nˆdS =
C
Z
f (0, ... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
=0
(cid:12)
(cid:12)
∂G
(cid:12)
(cid:12)
∂η η=0
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:0)
−π (x
−
(cid:0)
The solution of the BVP (6) with F = 0 on the upper right quarter plane D and
boundary condition u = f can now be written as, from (6),
(cid:1)
(cid:0)
(cid:1)
u (x, y) =
=
f
Z
∇
C
4yx
G nˆdS
·
∞
η... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
x ′ )2 + (η
y ′)2 .
−
6
[DRAW] Using the law of cosines, we obtain
r 2 = ρ˜2 + ρ2
r ′2 = ρ˜2 +
−
1
ρ2 −
2ρρ˜cos θ˜
(cid:16)
cos θ˜
(cid:16)
ρ˜
ρ
2
−
−
θ
θ
(cid:17)
(cid:17)
ξ2 + η2 and θ, θ˜ are the angles the rays (... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
)
2ρρ˜cos θ˜
(cid:16)
θ
−
−
(cid:17)
θ
(cid:17)
−
Note that
G
·
∇
nˆ =
∂G
∂ρ˜
u (ρ, θ) =
1
2π
2π
0
Z
1 + ρ2
−
+
1
4π
2π
1
ln
0
Z
0
Z
=
1
2π 1 + ρ2
ρ˜=1
−
ρ2
1
−
2ρ cos θ˜
(cid:16)
θ
−
(cid:17)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
ρ2
1
−
2ρ cos θ˜
(cid:16)
ρ˜2 + ρ2
−
−
ρ2ρ˜2 + 1
˜
f θ
d˜
... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
3 Conformal mapping and the Green’s function
Conformal mapping allows us to extend the number of 2D regions for which Green’s
2u can be found. We use complex notation, and let
functions of the Laplacian
α = x + iy be a fixed point in D and let z = ξ + iη be a variable point in D (what
we’re integrating over). If D ... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
D. Since
can write w (z) = (z
w (z) is 1-1, w ′ (z) > 0 on D. Thus n = 1. Hence
−
∈
|
|
|
|
|
and
where
r =
h =
w (z) = (z
α) H (z)
−
G =
1
2π
ln r + h
x)2 + (η
y)2
−
−
z
|
1
2π
−
ln
α
= (ξ
|
q
H (z)
|
|
Since H (z) is analytic and nonzero in D, then (1/2π) ln H (z) is analytic in D and
2h = 0 in ... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
to α = x + iy, also in the upper half plane, than to α∗ = x
iy, in the
lower half plane. Thus for z
< 1. The Green’s
D/∂D,
function is
w (z)
|
z
|
α∗
−
−
−
=
∈
α
/
z
|
|
|
|
G =
1
2π
ln
w (z)
|
|
=
1
2π
ln
z
|
z
|
α
∗|
α
|
−
−
=
ln
1
2π
r
r ′
which is the same as we derived before, Eq. (7).
8
�
... | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
problems,
such as the Wave Equation and the Heat Equation.
9 | https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-fall-2006/064b7e5d9e3296ab2219e44c53f7a1b5_greensfn.pdf |
Bifurcations: baby normal forms.
Rodolfo R. Rosales, Department of Mathematics,
Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139
October 10, 2004
Abstract
The normal forms for the various bifurcations that can occur in a one dimensional dynamical
system (x_ = f (x; r)) are derived via local approxi... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
. . . .
Problem 2: formal expansion to reduce to normal form.
. . . . . . . . . . . . . . .
Problem 3: What about problem 3.2.6 in the book by Strogatz? . . . . . . . . . .
3
3
4
4
4
5
6
6
6
7
8
8
10
11
11
11
1
Rosales Bifurcations: baby normal forms.
4 Pitchfork bifurcations.
Introduction of the re(cid:13)ection symm... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
,
the normal forms we develop here still apply. Thus:
Consider some critical point x = x0 (occurring for a value r = r0) i.e.
f (x0; r0) = 0: Then ask:
When is (x0; r0) = 0 a bifurcation point? A necessary condition is:
fx(x0; r0) = 0:
(1.2)
Why? Because otherwise the implicit function theorem would tells us that: In a... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
0) = 0, the most generic situation is
that where fr(0; 0) = 0 and fxx(0; 0) = 0. By appropriately re-scaling x and r in (1.1) | if needed,
we can thus assume that:
f (0; 0) = fx(0; 0) = 0;
fr(0; 0) = 1;
and fxx(0; 0) =
2:
(cid:0)
(2.3)
Remark 3 For arbitrary dynamical systems (such as (1.1)), we have to be careful abou... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
2) apply for (x; r; h) = (0; 0; 0) | here h small and nonzero produces an \arbitrary"
(smooth) perturbation to the dynamical system in (1.1). Consider now the system of equations:
f (x; r; h) = 0;
and
fx(x; r; h) = 0:
(2.4)
2These are points such that there is a neighborhood in (x; r) space where there is no other crit... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
\perturbations" of the form
dx
dt
= f (x; r) + h
d2x
dt2 :
(2.7)
What \small" means in this case is not easy to state (and we will not even try here). However, this
example should make it clear that: when talking about structural stability, for the concept to
even make sense, the dynamical system must be thought as bel... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
fact", with the answer being (basically) \because it works". Namely,
after we have (cid:12)gured out what is going on, we can explain why the scaling in (2.9) is the right one
to do. As follows: at a Saddle Node bifurcation | say, at (x; r) = (0; 0) | a branch of critical
point solutions | say x = X1(r) | turns \back" ... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
equation (2.10).
Furthermore: (cid:8)(0) = 1 and (cid:9)(0; 0) = 1 | thus: X
x and T
(cid:25)
(cid:25)
t close to the origin.
3Note that this is the reason that this type of bifurcation is also known by the name of turning point bifurcation.
Rosales Bifurcations: baby normal forms.
7
IMPORTANT: the de(cid:12)nition fo... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
that with this de(cid:12)nition (2.11) yields
dX
dT
X 2.
= r
(cid:0)
QED.
Problem 1 Implement a reduction to normal form, along lines similar to those used in theo-
rem 1, by formally expanding the coordinate transformation up to O((cid:15)2) | where (cid:15) is as in equation
(2.9). To do so, write the dynamical syste... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
points that goes through the bifurcation point.
Taking successive derivatives of the identity f ((cid:31)(r); r)
0, and evaluating them at r = 0 (where
(cid:17)
(cid:31) = f = fx = 0), we obtain:
f
r(0; 0) = 0 and frr(0; 0) =
2
(cid:22) fxx(0; 0)
(cid:0)
(cid:0)
2 (cid:22) fxr(0; 0);
(3.16)
(0). As before, we assume th... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
2; r2 x; r3):
(3.19)
We now assume4 that both r and x are small, of size O((cid:15)), where 0 < (cid:15)
keeping up to terms of O((cid:15)2) on the right (leading order) in (3.19), we obtain the equation:
(cid:28)
1. Then,
dx
dt
= r x
(cid:0)
x2 + a r2 = (x
(cid:0) (cid:0)
(cid:27) r) (x
1
(cid:27) r);
2
(cid:0)
where ... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
; 0) is an isolated critical point. As explained in remark 4, such
points are (generally) of little interest.
On the other hand, 1 + 4 a = 0 would lead to a double root of the right hand side in (3.19) (at
leading order). In principle this can be interpreted as a \limit case" of the transcritical bifurcation,
where the... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
(cid:0)
(cid:0)
(cid:27)2) + r h(X; r);
where h is some non-singular function. We note now that:
g((cid:27)p; 0) = 0 and gX((cid:27)p; 0) = ((cid:27)p
(cid:27)q) = 0;
(cid:0)
(3.24)
(3.25)
(3.26)
6
Rosales Bifurcations: baby normal forms.
10
where
p; q
=
1; 2
. Then the implicit function theorem guarantees that there ... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
assume that G is \generic", so that its derivatives do not vanish. In particular, we normalize the
(cid:12)rst order derivatives so that Gr(0; 0) =
(cid:0)
Gx(0; 0) = 1 | this normalization is consistent with
the one used in (3.17), where we must take a = 0.
At this point we can invoke the implicit function theorem, th... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
Assume that equation (3.27), and the normalizations immediately below it, apply.
Then, for x and r both small and O((cid:15)) | where 0 < (cid:15)
1 | implement a reduction to normal
(cid:28)
form, by formally expanding the coordinate transformation up to two orders in (cid:15). To do so, write
the dynamical system in ... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
a x3 + O(x4);
where R = 0 and a are constants. Introduce now a transformation (expanded) of the form
x = X + b X 3 + O(X 4);
(3.34)
(3.35)
6
Rosales Bifurcations: baby normal forms.
12
where b is a constant. Then show that b can be selected so that the equation for X has the form
dX
dt
= R X
(cid:0)
X 2 + O(X 4):
(3.3... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
Without any real loss of generality, assume that f (x; r) is an odd function
of x. Then (cid:31) = 0, X =
x, and (1.1) becomes:
(cid:0)
dx
dt
= x g((cid:16); r); where
(cid:16) = x2:
(4.37)
Rosales Bifurcations: baby normal forms.
13
The bifurcation condition (1.2) yields g(0; 0) = 0. Other than this, we assume that
g... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
in terms of a small parameter 0 < (cid:15)
1. This scaling should be consistent with the normal
form for the equation.7 It is very important since it assures that the ordering in the expansions is kept
(cid:28)
straight, without higher order terms being mixed with lower order ones. Third
Introduce expansions
for R = R(... | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
THE MODULI SPACE OF CURVES
1. The moduli space of curves and a few remarks about its
construction
The theory of smooth algebraic curves lies at the intersection of many branches
of mathematics. A smooth complex curve may be considered as a Riemann sur
face. When the genus of the curve is at least 2, then it may al... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
with a fixed point free involution i such that S/i is an
Enriques surface E. To be very concrete let C be the normalization of the plane
curve defined by the equation y2 = p(x) where p(x) is a polynomial of degree 2g + 2
with no repeated roots. The hyperelliptic involution is given by (x, y) ⊂� (x, −y).
Let Q1, Q2, Q... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
) over S coarsely represents the functor F if there is a
natural transformation of functors � : F � HomS (�, X(F )) such that
(1) �(spec(k)) : F (spec(k)) � HomS (spec(k), X(F )) is a bijection for every
algebraically closed field k,
(2) For any S-scheme Y and any natural transformation � : F � HomS (�, Y ),
there ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
. These are ordinary double
points of the surface. We can resolve these singularities by blowing up these points.
Figure 1. Quartics specializing to a double conic.
We now make a base change of order 2. This is obtained by taking a double cover
branched at the exceptional curves E1, . . . , E8. The inverse image of... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
it is an amazing fact that the moduli space of curves
can be compactified by allowing curves that have only nodes as singularities.
Definition 1.5. Consider the tuples (C, p1, . . . , pn) where C is a connected at
worst nodal curve of arithmetic genus g and p1, . . . , pn are distinct smooth points
of C. We call the ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
scheme parameterizes the locus of n-canonical curves
of genus g. The group P GL((2n − 1)(g − 1)) acts on K. The coarse moduli scheme
may be constructed as the G.I.T. quotient of K under this action. The proof that
this construction works is lengthy. Below we will briefly explain some of the main
ingredients. We begi... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
,
leaving you to read [Gr], [Mum2], [K], [Se] and the references contained in those
accounts for complete details.
Let us first concentrate on the case X = Pn and S = Spec(k), the spectrum of a
field k. A subscheme of projective space is determined by its equations. The poly
nomials in k[x0, . . . , xn] that vanish ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
polynomial (and not on the subscheme) that
works simultaneously for the ideal sheaf of every subscheme with a fixed Hilbert
polynomial.
Theorem 2.3. For every polynomial P , there exists an integer mP depending only
on P such that for every subsheaf I ≥ O with Hilbert polynomial P and every
integer k > mP
Pn
(1) ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
you. Let us begin with a sketch of the proof of the theorem.
Definition 2.4. A coherent sheaf F on Pn is called (Castelnuovo-Mumford) m-
regular if H i(Pn , F (m − i)) = 0 for all i > 0.
Proposition 2.5. If F is an m-regular coherent sheaf on Pn , then
(1) hi(Pn , F (k)) = 0 for i > 0 and k + i → m.
(2) F (k) is gen... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
the assumption that F is m regular for i → 0. We conclude that F is m + 1 regular.
Hence by induction k regular for all k > m. This proves item (1).
Consider the commutative diagram
H 0(F (k − 1)) ∗ H 0(OPn (1))
H 0(F (k − 1))
g
H 0(F (k))
u
v
H 0(FH (k − 1)) ∗ H 0(OH (1))
f
H 0(FH (k))
The map u is surj... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
+1
�
i=0
m
i
� �
Assuming the result by induction, we get an integer m1 depending only on the
coefficients a1, . . . , an such that IH has that regularity. Considering the long exact
sequence associated to our short exact sequence, we see that H i(I(m)) is isomorphic
to H i(I(m + 1) as long as i > 1 and m > m1 − i. ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
scheme structure and that this subscheme represents
the Hilbert functor. For this purpose we will use flattening stratifications.
Recall that a stratification of a scheme S is a finite collection S1, . . . , Sj of locally
closed subschemes of S such that
S = S1 � · · · � Sj
is a disjoint union of these subschemes.
Pr... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
can conclude that there is an
integer m such that if l → m, then
and
H i(Pn(s), F (s)(l)) = 0
βS�F (l) ∗ k(s) � H 0(Pn(s), F (s)(l))
6
is an isomorphism, where βS denotes the natural projection to S.
Next one observes that (1 × f )�F is flat over T if and only if f �βS�F (l) is locally
free for all l ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
rest of this section we will abbreviate G(P (mP ), H 0(Pn , OPn (mP ))) simply
by G. β�
2 T (−mP ) where T is the tautological bundle on G is an idea sheaf of OPn ×G.
Let us denote the corresponding subscheme by Y . The flattening stratification of
OY over G gives a subscheme HP of G corresponding to the Hilbert poly... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
proper. This is done by checking the valuative criterion of properness. This follows
from the following proposition [Ha] III.9.8.
7
Proposition 2.8. Let X be a regular, integral scheme of dimension one. Let p ⊗ X
be a closed point. Let Z ≥ Pn
X−p be a closed subscheme flat over X − p. Then there
exists a unique c... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
the Hilbert scheme of hypersurfaces of degree d in Pn
is isomorphic to P(n+d)−1 .
Example 2.15 (The Hilbert scheme of conics in P3). Any degree 2 curve is nec
essarily the complete intersection of a linear and quadratic polynomial. Moreover,
the linear polynomial is uniquely determined. We thus obtain a map
d
Hilb... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
homology ring. The cohomology ring of a projective bundle
over a smooth variety is easy to describe in terms of the chern classes of the bundle
and the cohomology ring of the variety.
8
Theorem 2.18. Let E be a rank n vector bundle over a smooth, projective variety
ci(E)ti . Let α denote the
X. Suppose that the ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
3)) �
=
Z[h, α]
< h4 , α 3 + 4hα 2 + 10h2α + 20h3 >
The class of the locus of conics interseting a line is given by 2h + α. This can be
checked by a calculation away from codimension at least 2. Consider the locus of
planes in P3� that do not contain the line l. Over this locus there is a line bundle
that associa... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
.21. Generalize the previous discussion to conics in P4 . Calculate the
numbers of conics that intersect general 11 − 2i − 3j planes, i lines and j points.
Example 2.22 (The Hilbert scheme of twisted cubics in P3). The Hilbert poly
nomial of a twisted cubic is 3t + 1. This Hilbert scheme has two components.
A gener... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
. Calculate the number of twisted cubics that are tangent to 12
general quadric hypersurfaces in P3 . (Hint: There are 5,819,539,783,680 of them.)
Towards the end of the course we will see how to use the Kontsevich moduli space
to answer these questions.
Unfortunately, Hilbert schemes are often unwieldy schemes to ... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
C/S commutes with base change.
(2) If S = Spec k where k is an algebraically closed field and C is the normal
ization of C, then �C/S may be identified with the sheaf of meromorphic
differentials on C that are allowed to have simple poles only at the inverse
image of the nodes subject to the condition that if the point... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
. �E/k(
no sections for any n → 2. By Serre duality, it follows that H 1(C, � C/k ) = 0. To
show that when n → 3, �C/k is very ample, it suffices to check that �C/k separates
points and tangents.
C/k
∗n
�
�
∗n
∗n
Exercise 3.1. Check that when n → 3, �C/k separates points and tangents.
∗n
4. Stable reduction
Stabl... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
.2. Fix a smooth curve C of genus g → 2. Let p ⊗ C be a fixed point
and let q be a varying point. More precisely, we have the family C × C � C with
two sections χp : C � C × C mapping a point q to (q, p) and χq : C � C × C
mapping q to (q, q). All the fibers are stable except when p = q. To obtain a stable
family, we... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
of doing steps 2
and 3 is to take a branched cover of the surface X branched along the reduction of
the divisor forming the central fiber modulo p. Repeat steps 2 and 3 until all the
components occuring in the central fiber appear with multiplicity 1.
�
Step 4. Contract the rational components of the central fiber tha... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
· Cj → 0 for all i �= j and Ci · C = 0 for all i.
(3) If K is the canonical class, then the arithmetic genus of Ci is given by the
genus formula as
C 2 + Ci · K
.
2
(4) The intersection matrix Ci ·Cj is a negative definite symmetric matrix. The
aiCi with the property that Z 2 = 0 are
1 +
i
only linear combinat... | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
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