text stringlengths 16 3.88k | source stringlengths 60 201 |
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17.31)
�
u(0, ·)(Δ∞) = (2α)−1
u(Δ1, Δ∞)dΔ1, � Δ∞ ≤ Rn−1 .
ˆ
R
�
Use Cauchy’s inequality to show that this is continuous as a map on
Sobolev spaces as indicated and then the density of S(Rn) in H m(Rn)
to conclude that the map is well-defined and unique.
Problem 76. [Restriction by WF] From class we know that the ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
∗0 2αi
≡[(A − t − i∂)−1 − (A + t + i∂)−1]π, ϕ� −∩ µα,φ
in the sense of distributions – or measures if you are prepared to work
harder!
Problem 78. If u ≤ S(Rn) and ϕ∞ = ϕR + µ is, as in the proof of
Lemma 12.5, such that
show that
supp(ϕ∞) ∃ Css(u) = ∞
S(Rn) � π ◦−∩ πϕ∞ u ≤ S(Rn)
is continuous and hence (or otherw... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
x, p) ≤ WFsc(u)∃(Rn×Sn−1), (y, p) ≤ WFsc(v)∃(Rn×Sn−1)}
∞∞χ∞∞
∞∞χ∞∞|
∗ {(χ, q) ≤ Sn−1 × Bn; χ =
s∞χ∞ + s
|s∞χ∞ + s
, 0 → s ∞ , s → 1,
∞∞
(χ∞ , q) ≤ WFsc(u) ∃ (Sn−1 × Bn
), (χ∞∞
, q) ≤ WFsc(v) ∃ (Sn−1 × Bn)}.
Problem 82. Formulate and prove a bound similar to (17.36) for WFsc(uv)
when u, v ≤ S ∞(Rn) satisfy (12.50). ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
f is defined and satisfies
φu = f. Show that under this condition
f is defined using Prob
lem 84. What can you say about WFsc(u)? Why is it not the case that
φu = 0, even though this is true if u has compact support?
�
� | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/367cb3a939cc40b0d2dea20d2fd8f47b_problems.pdf |
6.801/6.866: Machine Vision, Lecture 19
Professor Berthold Horn, Ryan Sander, Tadayuki Yoshitake
MIT Department of Electrical Engineering and Computer Science
Fall 2020
These lecture summaries are designed to be a review of the lecture. Though I do my best to include all main topics from the
lecture, the lectures will... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
orthonormal
rotation matrices). Most notably, these are:
1. Composition of rotations:
o
p
o
q = (p, q)(q, q) = (pq − q · q, pq + qq + q × q)
2. Rotating vectors:
(cid:48)
o
r
∗
o
q
o
r
o
q
=
= (q2 − q · q)r + 2(q · r)q + 2q(q × r)
Recall from the previous lecture that operation (1) was faster than using orthonormal rot... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
)
2(qzqx − q0qy)
0
2(qxqy − q0qz)
y − q2
x + q2
0 − q2
q2
z
2(qzqy + q0qz)
0
2(qxqz + q0qy)
2(qyqz − q0qx)
y + q2
x − q2
0 − q2
q2
z
The matrix ¯QT Q has skew-symmetric components and symmetric components. This is useful for conversions. Given a
quaternion, we can compute orthonormal rotations more easily. Fo... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
works because at angles θ where cos
and vice versa.
(cid:17)
(cid:16) θ
2
is “bad” (is extremely sensitive), sin
(cid:17)
(cid:16) θ
2
is “good” (not as sensitive),
1.2 Quaternion Transformations/Conversions
Next, let us focus on how we can convert between quaternions and orthonormal rotation matrices. Given a 3 × 3 or... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
+ r22 − r33 = 4q2
y
r13 − r31 = 4q0qy
r32 + r23 = 4qyqz
r13 + r31 = 4qzqz
This system of four equations gives us a direct way of going from quaternions to an orthonormal rotation matrix. Note that this
could be 9 numbers that could be noisy, and we want to make sure we have best fits.
1.3 Transformations: Incorporating ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
)
i=1
||r(cid:48)
r,i||2
2
||r(cid:48)
r,i||2
2
(cid:17)
(cid:17)
− 2s
− 2s
n
(cid:88)
(cid:16)
i=1
n
(cid:88)
(cid:16)
i=1
(cid:17)
r,iR(r(cid:48)
r(cid:48)
l,i)
(cid:17)
r,iR(r(cid:48)
r(cid:48)
l,i)
+ s2
+ s2
n
(cid:88)
i=1
n
(cid:88)
i=1
||R(r(cid:48)
l,i)||2
2
||r(cid:48)
l,i||2
2 (Rotation preserves vector length... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
2
Issues with Symmetry
Symmetry question: What if instead of going from the left coordinate system to the right one, we decided to go from right
to left? In theory, this should be possible: we should be able to do this simply by negating translation and inverting our
rotation and scaling terms. But in general, doing th... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
R(r(cid:48)
r(cid:48)
l,i)
(cid:17)
+ s
i=1
n
(cid:88)
i=1
||r(cid:48)
l,i||2
2 (Rotation preserves vector lengths)
We then take the same definitions for these terms that we did above:
1. sr
∆= (cid:80)n
i=1
(cid:16)
||r(cid:48)
r,i||2
2
(cid:17)
2. D ∆= (cid:80)n
i=1
(cid:16)
r,iR(r(cid:48)
r(cid:48)
(cid:17)
l,i)
3. s... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
the variance/spread/size of the point
cloud in their respective coordinate systems.
We can deal with translation and rotation in a correspondence-free way, while also allowing for us to decouple rotation. Let us
also look at solving rotation, which is covered in the next section.
1.4 Solving for Optimal Rotation in Abs... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
J(
T
o
q
o
q
o
q
N
T o
q
o
dJ(
q)
o
q
d
=
=
=
o
q
= 0
N
T o
q
T
o
q
o
q
T
o
q
(
d
o
q
d
d
o
q
d
o
o
q
q)
N
T o
q −
T
o
q
N
o
q
T o
q)
(
o
q d
o
q
d
o
q
T o
q)2
T
o
q
(
o
q) = 0
N
(
o
q
2
T o
q)2
o
q
2N
T o
o
q
q
−
o
q
(
5
= 0
From here, we can write this first order condition result as:
o
q
o
q =
N
o
q
o
q
T
o
q
N
T o
... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
ormal matrices, which either require a complex Lagrangian (if we solve with
Lagrange multipliers) or an SVD decomposition from Euclidean space to the SO(3) group (which also happens to be a manifold).
This approach raises a few questions:
• How many correspondences are needed to solve these optimization problems? Recal... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
z
+
a14
a24
a34
6
But we also have to account for translation, which gives us another 3 unknowns, giving us 12 in total and therefore requiring at
least 4 non-redundant correspondences in order to compute the full general linear transformation. Note that this doesn’t have
any constraints as well!
On a prac... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
i.e. the matrix M is singular? Then using the formulas above we must have that the coefficient
c1 = 0. Then this problem reduces to:
λ4 + c2λ2 + c0 = 0
This case corresponds to a special geometric case/configuration of the point clouds - specifically, when points are coplanar.
1.4.3 What Happens When Points are Coplanar?
W... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
(cid:48)
r,i0
=
=
n
(cid:88)
i=1
n
(cid:88)
i=1
= 0
Therefore, when a point cloud is coplanar, the null space of M is non-trivial (it is given by at least Span({ˆn}), and therefore
M is singular. Recall that a matrix M ∈ Rn×d is singular if ∃ x ∈ Rd, x (cid:54)= 0 such that M x = 0, i.e. the matrix has a non-trivial
nu... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
(cid:18)
cos
o
q =
θ
2
, sin
(cid:19)
θ
2
ˆω
2. Perform an in-plane rotation. Now that we have the quaternion representing the rotation between these two planes, we can
orient two planes on top of each other, and then just solve a 2D least-squares problem to solve for our in-place rotation.
With these steps, we have a ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
is good enough accept it, and if it is
not, run another sample. Note that this step has different variations - rather than just immediately terminating once you
have a good fit, you can run this many times, and then take the best fit from that.
Furthermore, for step 3, we threshold the band from the fitted line/hyperplane ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
of rotations.
Why are we interested in this space? Many orientation problems we have studied so far do not have a closed-form
solution and may require sampling. How do we sample from the space of rotations?
1.6.1
Initial Procedure: Sampling from a Sphere
Let us start by sampling from a unit sphere (we will start in 3D,... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
: As we mentioned above, our goal is to generalize this from 3D to 4D. Cubes and spheres simply
become 4-dimensional - enabling us to sample quaternions.
1.6.3 Sampling From Spheres Using Regular and Semi-Regular Polyhedra
We saw the approach above requires discarding samples, which is computationally-undesirable becau... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
:0)− π
4
(cid:1) ˆx) = 1√
2
(1, −ˆx)
(cid:1) , sin (cid:0)− π
4
(cid:1) ˆy) = 1√
2
(1, −ˆy)
(cid:1) , sin (cid:0)− π
4
(cid:1) ˆz) = 1√
2
(1, −ˆz)
These 10 rotations by themselves give us 10 ways to sample the rotation space. How can we construct more samples? We can
do so by taking quaternion products, specifically, pr... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
2
= 1
2 =⇒ θ
2 = π
3 =⇒ θ = 2π
3
Therefore, we have produced a new rotation that we can sample from!
These are just a few of the pairwise quaternion products we can compute. It turns out that these pairwise quaternion products
produce a total of 24 new rotations from the original 10 rotations. These are helpful for ach... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/3684c9529d76a9a87fe3db7ae5e91f71_MIT6_801F20_lec19.pdf |
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| https://ocw.mit.edu/courses/8-322-quantum-theory-ii-spring-2003/36b3cccc5336969c0304126613a3121e_83223Lecture2.pdf |
Lecture 2
8.321 Quantum Theory I, Fall 2017
5
Lecture 2 (Sep. 11, 2017)
2.1 More Relevant Math
2.1.1 Inner Products
Last time, we discussed the concept of a maximally linearly independent set, which is a set {|αj(cid:105)}
of vectors that are linearly independent, and such that there exists no |β(cid:105) such that {|α... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
cid:105)) .
4. For all |α(cid:105) ∈ V ,
and if (|α(cid:105), |α(cid:105)) = 0, then |α(cid:105) is the zero ket, |α(cid:105) = 0.
(|α(cid:105), |α(cid:105)) ≥ 0 ,
Let’s consider some examples. First, consider V = Cn, whose vectors are of the form
|z(cid:105) =
z1
.. ,
.
zn
z1, . . . , z
n ∈ C .
One definiti... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
:107)|α(cid:105)(cid:107) .
Using this norm, we can normalize any nonzero ket by defining
which has
2.1.2 Dual Space
|˜α(cid:105) =
1
N
| (cid:105) ,
α
(|α˜(cid:105), |α˜(cid:105)) = 1 .
(2.10)
(2.11)
(2.12)
Now we introduce the dual space. The space V that we have described so far is the space of kets
|α(cid:105). The ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
(cid:105)} such that
(cid:104)φi|φj(cid:105) = δij ,
where δij is the Kronecker delta. We know that we can write any ket in the form
|α(cid:105) =
(cid:88)
i
ci
|φi(cid:105) .
(2.16)
(2.17)
If the |φi(cid:105) form an orthonormal basis, then we find that ci = (cid:104)φi|α(cid:105). Thus, if we have an orthonormal
basis... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
2.20)
We will primarily be interested in linear operators: a linear operator X satisfies the property
X(cα|α(cid:105) + cβ|β(cid:105)) = cαX|α(cid:105) + cβX|β(cid:105)
(2.21)
for all cα, cβ ∈ F and |α(cid:105), |β(cid:105) ∈ H. We can also define the notion of an anti-linear operator: an
anti-linear operator X satisfies
... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
I, Fall 2017
8
2.1.5 Operators as Matrices in a Given Basis
In a given basis, the action of an operator can be expressed by a matrix. To see this, we first define
the identity operator 1, which satisfies
1|α(cid:105) = |α(cid:105)
(2.27)
for all |α(cid:105) ∈ H. Given an orthonormal basis |{a(cid:48)}(cid:105) (notation f... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
cid:105), given any orthonormal basis |{a(cid:48)}(cid:105). This defines an
n × n matrix with complex entries corresponding to each operator X. For this reason, we will often
use the words “operator” and “matrix” interchangeably if the chosen basis is clear, even though
the concept of an operator is more fundamental.
2... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
for matrices. As we have seen, any operator has
the same information content as an n × n matrix, where n = dim H. If M is a matrix corresponding
to the operator X, then X † corresponds to the matrix found by conjugating the entries of the
transpose M T. This matrix is denoted as M †, and is also called the Hermitian co... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
dx
f2(x) =
∞
(cid:18)
dx
−
d
f ∗
dx 1 (x)
(cid:19)
f2(x) ,
−∞
telling us that
−∞
Thus,
(cid:28)
(cid:12)
d
(cid:12)
(cid:12)
(cid:12) dx
(cid:12)
(cid:12)
(cid:12)
(cid:12)
f1
(cid:29)
(cid:28)
f2
= −
f2
(cid:12)
(cid:12)
(cid:12)
(cid:12)
d
dx
(cid:12)
(cid:12)
(cid:12)
(cid:12)
f1
(cid:29)∗
.
(cid:19)†
(cid:18) d
dx
... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
are also interested in projection operators, which are operators A satisfying A2 = A. An
example of a projection operator is A = |α(cid:105)(cid:104)α|, for some |α(cid:105) ∈ H.
Lecture 2
8.321 Quantum Theory I, Fall 2017
10
2.1.8 Eigenstates and Eigenvalues
If, for some operator A and ket |α(cid:105) ∈ H, we have
A|... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
12)a(cid:48)(cid:11)
(cid:12) (cid:11)
(cid:12)
(cid:10)
(cid:12)A(cid:12)a(cid:48)
(cid:48)(cid:48)
a
(cid:48)(cid:48)(cid:12)
(cid:11) ,
(cid:10)a (cid:12)a(cid:48)
= (cid:48)
a
a(cid:48)(cid:48)(cid:1)∗(cid:10)a(cid:48)(cid:48)(cid:12)
(cid:0)
(cid:12)a(cid:48)
=
(cid:11) .
Comparing
Eqs. (2.48a) and (2.48b), we see... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
1 tells
us that we have
i.e., the normalized eigenkets are orthonormal. We can then decompose
(cid:104)ai|aj(cid:105) = δij ,
A =
(cid:88)
a
a
|a(cid:105)(cid:104)a| .
We can check that for some eigenket |b(cid:105), we have
A|b(cid:105) =
(cid:88)
a
a|a a
(cid:105)(cid:104) |b(cid:105) =
(cid:88)
a
a
|a(cid:105)δab = ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/36c32531fa12a2d99687cb3f0ac58502_MIT8_321F17_lec2.pdf |
Queueing Systems: Lecture 5
Amedeo R. Odoni
October 30, 2006
Lecture Outline
• A fundamental result for queueing networks
• State transition diagrams for Markovian
queueing systems and networks: examples
• Examples
• Dynamic queueing systems and viable
approaches
• Qualitative discussion of behavior
Reference:... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/36d90014e175b04466290678ce09bfbf_lec9.pdf |
neg. exp’l; μ 2
No queuing
space
No queuing
space
Note: The queuing system on the right may “block” the one on
the left.
Example 2: M/Ek/1 System, with
system capacity for total of N users
See distributed notes.
Example 3: Two Types of Users
and Non-Preemptive Priorities
Type 1 customers;
Poisson arrivals... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/36d90014e175b04466290678ce09bfbf_lec9.pdf |
?
2. Time to reach “steady state” is large for values of ρ
which are close to 1; therefore “steady state”
expressions may be very poor approximations when
intervals are relatively short
3. Approach does not take into consideration the
“dynamics” of the demand profile
The Two Viable Approaches
1. Simulation:
• Hi... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/36d90014e175b04466290678ce09bfbf_lec9.pdf |
Roth, “Approximate
Solutions for Multi-Server Queueing Systems with
Erlangian Service Times”, with M. Escobar and E. Roth,
Computers and Operations Research, 29, pp. 1353-1374,
2002.
Ingolfsson, A., E. Akhmetshina, S. Budge, Y. Li and X.
Wu, “A Survey and Experimental Comparison of Service
Level Approximation Me... | https://ocw.mit.edu/courses/1-203j-logistical-and-transportation-planning-methods-fall-2006/36d90014e175b04466290678ce09bfbf_lec9.pdf |
“Medicine and the Computer:
The Promise and Problems of Change”
(cid:216) Perceived problems
—W.B. Schwartz, NEJM 1970
(cid:216) Physician shortage and maldistribution
(cid:216) Ever-expanding body of knowledge, so that the physician
cannot keep up
(cid:216) Exploit the computer as an “intellectual”, “deductive”
i... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
Domain
Knowledge
Inference
Engine
9
Flowcharts
(cid:216) Good:
(cid:216) Simple
(cid:216) Easy to build
(cid:216) Bad:
(cid:216) Hard to deal with
(cid:216) missing data
(cid:216) out of sequence data
(cid:216) uncertainty
(cid:216) Hard to maintain
10
Mycin—Rule-based Systems
(cid:216) Task: Diagnosis a... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
(s), with
a suitable certainty
(cid:216) Backward chaining from goal to given facts
(cid:216) Dynamically traces out behavior of (what might be) a
flowchart
(cid:216) Information used everywhere appropriate
(cid:216) Single expression of any piece of knowledge
13
Explore Mycin’s Use of
Knowledge
** Did you us... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
state
yConditional independence
xP(s,t|d) = P(s|d)P(t|d)
z Bayes’ Rule updates disease probabilities
based on observing symptoms
z Next lecture’s large example
19
Taking the Present Illness—Diagnosis by
Pattern Directed Matching
Hypothesis
Facts about
Patient
20
PIP's Theory of Diagnosis
z From initial complaint... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
PRESENTING SYMPTOMS: EDEMA, ERYTHEMATOUS, PITTING, SYMMETRICAL,
WORSE-IN-EVENING, FIRST-TIME, FOR-DAYS AND MASSIVE. HE DOES NOT
HAVE DYSPNEA. HE HAS SOCIAL ALCOHOL CONSUMPTION. HE DOES NOT
HAVE JAUNDICE. IT IS NOT EXPLICITLY KNOWN WHETHER IN THE PAST HE
HAD PROTEINURIA, BUT HE HAS SMALL-POLICY LIFE INSURANCE, AND HE... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
ing
(cid:216) Several questioning strategies
26
QMR Scoring
(cid:216) Positive Factors
(cid:216) Evoking strength of observed Manifestations
(cid:216) Scaled Frequency of causal links from confirmed
Hypotheses
(cid:216) Negative Factors
(cid:216) Frequency of predicted but absent Manifestations
(cid:216) Importance ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
(cid:216) following a process
(cid:216) heuristics
38
The Surprisingly Normal pH
(cid:216) Diarrhea causes bicarbonate (alkali) loss
(cid:216) Vomiting causes acid loss
(cid:216) Therefore, normal pH is a manifestation of
{diarrhea + vomiting}!
39
Temporal Reasoning
(cid:216)Keeping track of multiple forms of tempor... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
(cid:216) other expected findings
(cid:216) reasonable interventions
(cid:216) Qualitative models
(cid:216) Combining associational and model-based
reasoning
44
Guyton's Model of
Cardiovascular Dynamics
45
Long's Clinical Model of Heart Failure
Predictions for Mitral Stenosis with Exercise
46
Heart Disease Model
V... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
doing
so in the future
49
State of Practice (today)
(cid:216) Low-hanging fruit (important & tastes good)
(cid:216) “one-rule” expert systems
(cid:216) data presentation
(cid:216) Knowledge Ł Data
(cid:216) Classification, regression, neural networks, rough
sets, fuzzy logic, Bayes nets, …
(cid:216) Integration into... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/371121f1a51f3c1903cee9fd86022412_lecture1.pdf |
Lecture #1 Instructor Notes
First off, welcome. I hope that these notes are interesting and helpful to you.
Also, please note that there is a set of “Comments” on each lecture, that go along with the
readings and the Instructor’s Notes here.
Let us start with the first question you should always ask in a course….. ... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/3719d6726de02bbc57b61939f0cd14c5_MIT2_682S12_lec01.pdf |
depth z. In a very simplified form, the soundspeed as a function of depth z (its main
dependence) is
This is the first equation in the Computational Ocean Acoustics book, and shows that the
soundspeed is very sensitive to temperature, weakly sensitive to salinity, and moderately (and
linearly) sensitive to depth. T... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/3719d6726de02bbc57b61939f0cd14c5_MIT2_682S12_lec01.pdf |
8
Continuous-Time
Fourier Transform
In this lecture, we extend the Fourier series representation for continuous-
time periodic signals to a representation of aperiodic signals. The basic ap-
proach is to construct a periodic signal from the aperiodic one by periodically
replicating it, that is, by adding it to itself s... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
aperiodic signal
remaining when the period goes to infinity.
Although the Fourier transform is developed in this lecture beginning
with the Fourier series, the Fourier transform in fact becomes a framework
that can be used to encompass both aperiodic and periodic signals. Specifical-
ly, for periodic signals we can def... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
,
pages 196-202
Continuous-Time Fourier Transform
MARKERBOARD
8.1(a)
Cw%-nwnsA -TWAe
Ferio. SAsDg
Nce pevb~e.t.
r.
T . 4,)
.4
FOURIER REPRESENTATION OF APERIODIC SIGNALS
(""N
-TO
(00,
x Mt
T1
TO
2
TRANSPARENCY
8.1
Representation of an
aperiodic signal as a
periodic signal with
the period increasing
to infinity.
x(t) ... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
Fourier Transform
+00
X(o) =
x(t) ei jWt dt
00
Fourier transform
- analysis
x(t) =
E X(kwo) ejkwo tWO
k=-0O
As To-- oo,
coo -* 0 1x t) - x (t), we doE -
+a(
x(t) = 2
X(o) e jcot dco
-2o
Inverse
Fourier transform
- synthesis
TRANSPARENCY
8.4
The analysis and
synthesis equations
associated with the
Fourier transform.
x... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
1
/
'it/I
Continuous-Time Fourier Transform
MARKERBOARD
8.1(b)
ExCapkt (Td' 4.'1)
4S
e.e
-i
-
e.
I
~ c*w-+ a4-30&
I~
V~V
~-~-
OCAvhawA&-Tt-&
foe~ar
Tpiwd
.
4
SIV
3is
X..
IS
Xvt~*-b
)
=i
Example 4.7: eat u(t) +-+
1
a+)j>
a > o
IX(MOI
1/aV2
TRANSPARENCY
8.8
An exponential time
function and its
Fourier transform.
[... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
The Fourier transform
of x(t) as the envelope
of the Fourier series
coefficients of 2(t).
As the period To
increases, the samples
become more closely
spaced. This
transparency shows
x(t) and its Fourier
transform.
[Transparency 8.5
repeated]
TRANSPARENCY
8.12
±(t) and its Fourier
series coefficients
with To = 4T1.
[Tra... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
periodic signal x(t) for
which one period is x(t)
- x(t) has a Fourier series
TRANSPARENCY
8.16
Summary of the
development of the
Fourier transform
from the Fourier
series. [The periodic
signal has been
corrected here to read
o(t), not x(t).]
- as period of 'x(t) increases,
xM(t)-.x(t) and Fourier series of
x(t)-- Four... | https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/3730bb94a928490f9726ff18b8760e67_MITRES_6_007S11_lec08.pdf |
Observers, state feedback
6.011, Spring 2018
Lec 10
1
Observers
2 System (“plant”)
x[n]
w[n]
q[n]
A, b, cT, d
y[n]
+
1[n]
3
A good model
x[n]
w[n]
[n
q[n]
A, b, cT, d
b
y[n]
+
y[n]
b
1[n]
4
Observer configuration
x[n]
w[n]
q[n]
A, b, cT
Plant
y[n]
+
Z[n]
y[n]
q[n]
[n... | https://ocw.mit.edu/courses/6-011-signals-systems-and-inference-spring-2018/374940065e1339ea37dd2aa3cf8c54bd_MIT6_011S18lec10.pdf |
1 2 3 4 5 6 7 8 MIT OpenCourseWare
http://ocw.mit.edu
6.890 Algorithmic Lower Bounds: Fun with Hardness Proofs
Fall 2014
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/378d4376b42f8454d18bbaa761b01fe2_MIT6_890F14_L03.pdf |
18.336 spring 2009
lecture 13
03/19/09
Initial Value Problems (IVP)
⎧
⎨
⎩
in Ω×]0, T [
on Ω × {0}
on ∂Ω×]0, T [
ut = Lu
u = u0
u = g
where L differential operator.
←
←
←
PDE
initial condition
boundary condition
⎫
⎬
⎭
Ex.: • L = �
2
→
Poisson equation
•Lu = b · �u
advection equation
•Lu = −�2(�2
biharm... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/379860bd755f63b2873eda4e2fc5a337_MIT18_336S09_lec13.pdf |
) by �u(t)
Approximate Lu by A �u (for linear problems) [FD, FE, spectral]
·
→ system of ODE:
•
In time:
·
d
dt
�u = A · �u
u(x, t) ≈
u(x, t + Δt) − u(x, t)
Δt
Approximate time derivative by step:
d
dt
→
unew(x) = u(x) + ΔtLu(x) = (I + ΔtL)u(x)
Need to know about ODE solvers.
Stationary problem:
[explicit E... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/379860bd755f63b2873eda4e2fc5a337_MIT18_336S09_lec13.pdf |
= one order less than LTE.
Higher Order Time Stepping
• Taylor Series Methods:
Start with EE, add terms to eliminate leading order error terms.
PDE
Lax-Wendroff
→
• Runge-Kutta Methods:
Each step = multiple stages
= f (yn + Δt
�
k1
aij kj )
. . .
kr
j
= f (yn + Δt
�
arj kj )
n+1
y
= yn + Δt
�
j
bj kj
j
But... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/379860bd755f63b2873eda4e2fc5a337_MIT18_336S09_lec13.pdf |
�
�
�
�
1 − Δt λi
�
< 1 always
·
2
|λi|
EE conditionally stable: Δt <
IE unconditionally stable
2
ρ(A)
Message: One step implicit is more costly than one step explicit.
, then implicit pays!
But: If ρ(A) large
�
�
��
stiffness
Ex.: Different time scales
−50
49
49 −50
A =
�
Solution: y(t) = e−t
�
, ˚y =
�
·... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/379860bd755f63b2873eda4e2fc5a337_MIT18_336S09_lec13.pdf |
6.241 Dynamic Systems and Control
Lecture 1: Introduction, linear algebra review
Readings: DDV, Chapter 1
Emilio Frazzoli
Aeronautics and Astronautics
Massachusetts Institute of Technology
February 2, 2011
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
1 / 22
Outline
1
Syllabus review
2
Linear Algeb... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
LTI) systems.
Robust Stability and Performance. Approaches to optimal and robust control
design.
Hopefully, the material learned in this course will form a valuable foundation for
further work in systems, control, estimation, identification, signal processing, and
communications.
E. Frazzoli (MIT)
Lecture 1: Intr... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
ative schedule
#
1
2
3
4
5
6
7
Date Topic
Feb 2, 2011
Introduction to dynamic systems and control.
Matrix algebra.
Feb 7, 2011 Least Squares error solutions of overdeter-
mined/underdetermined systems
Feb 9, 2011 Matrix Norms, SVD, Matrix perturbations
Feb 14, 2011 Matrix Perturbations
Feb 16, 2011 State... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
, 2011
Apr 27, 2011
May 2, 2011
May 4, 2011
May 9, 2011
May 11, 2011
Stability Robustness (MIMO)
Reachability
Reachability - standard and canonical forms,
modal tests
Observability
Minimality, Realization, Kalman Decomposi
tion, Model reduction
State feedback, observers, output feedback,
MIMO poles and ze... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
;
Associativity of : a(bv ) = (ab)v , ∀a, b ∈ F , v ∈ V ;
·
Distributivity of w.r.t. vector +: a(v + w ) = av + aw , ∀a ∈ F , v , w ∈ V ;
·
Distributivity of w.r.t. scalar +: (a + b)v = av + bv , ∀a, b ∈ F , v ∈ V ;
·
Normalization: 1v = v , ∀v ∈ V .
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
11 / 22
Ve... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
the nullspace of any m × n matrix.
The set of all linear combinations of a given set of vectors.
The intersection of two subspaces.
The union of two subspaces.
The Minkowski (or direct) sum of two subspaces.
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
14 / 22
Linear (in)dependence, bases
n vectors ... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
;
√
A matrix Q is Hermitian if Q � = Q, and positive definite if x �Qx > 0 for
x = 0. Then
x �Qx is a norm.
�x� =
√
For x ∈ Rn , �x�1 =
�
n |xi |, and �x�∞ = maxi |xi |.
1
R
:
For a continuous function f : [0, 1] →
�
�
1
0
�f �∞ = supt∈[0,1] |f (t)|, and �f �2 =
|f (t)|2dt
�1/2
.
E. Frazzoli (MIT)
Lecture... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
product and norms
An inner product induces a norm �x� =
�
�x, x�.
For example, define �x, y � = x �Qy with Q Hermitian positive definite.
For f , g continuous functions on [0, 1], let �f , g � =
� 1
0
f (t)g (t) dt
Cauchy-Schwartz inequality: |�x, y �| ≤ �x� �y �, ∀x, y ∈ V ,
with equality only if y = αx for some α... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
�m + δ�m0) ∈ M achieves a better solution than ˆm. In fact:
�y − mˆ − δ�m0�2 = �y − mˆ �2 − δ��y − mˆ , m0� − δ�m0, y − mˆ � + |δ|2�m0�2
= �y − mˆ �2 − |δ|2 − |δ|2 + |δ|2�m0�2 = �y − mˆ �2 − |δ|2 .
E. Frazzoli (MIT)
Lecture 1: Introduction
Feb 2, 2011
21 / 22
�
Linear Systems of equations
Consider the followin... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
there may be more than one way to complete a desired
task. If there is a solution xp (i.e., Axp = y ), then typically there are many other
solutions of the form x = xp + xh, where xh ∈ N (A) (i.e., Axh = 0). In this case it is
desired to find the solution than minimizes some cost criterion.
E. Frazzoli (MIT)
Lectur... | https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/37b7dadc2e54c4dd227b60ac24489b3e_MIT6_241JS11_lec01.pdf |
Lectures 11 and 12
Air Pollution and SI Engine Emissions
Atmospheric Pollution
• SMOG
O
||
O3
NO2
– Ozone Nitrogen dioxide
R-C-OONO2
PAN(Peroxyacyl Nitrate)
• TOXICS
– CO, Benzene, 1-3 butadiene, POM (Polycyclic organic Matters),
Aldehydes
Primary Pollutants:
Direct emissions from vehicles
CO, HC, NOx... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
g
(
x
O
N
0.1
1975
1977
1
1981
1994 TLEV
Euro 3
1997-2003 ULEV
Euro 4
Euro 5
PZEV
PZEV
1975 1980 1985 1990 1995 2000 2005 2010
1975 1980 1985 1990 1995 2000 2005 2010
0.01
Starting year of implementation
Starting year of implementation
Historic trend: Factor of 10
reduction every 15 years
At 28.5 mil... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
Significant amount in fuel rich condition
• Immediately following combustion, CO is in
chemical equilibrium with the burned gas
• During expansion, as the burned gas
temperature decreases, CO is ‘frozen’
– Empirical correlation
[CO][H O]
[CO ][H ]
2
2
2
3.7
4
CO is mostly an A/F equivalence ratio issue
... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
1
N2
1 1
2
k k NO
1 2
k k N O
2
1
k NO
k O
2
2
2
2
d[NO]
dt
[NO] 0
2k1
O
N2
k1 7.6x10 exp
13
38000
T(K)
P=15 bar
• O, O2, N2 governed by major
heat release reaction
– In equilibrium in the hot
burned ... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
amount of NO
production
• In reality, there is
mixing between the
layers
• Rate is non-linear in
temperature
12
Crank angle (deg)
© McGraw-Hill Education. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use.
6
... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
(FID)
Chemi-ionization process
Signal proportional to C atom concentration
• Emissions regulation: NMOG as g/mile
– EPA definition of HC
Normal gasoline CH1.85
Reformulated gasoline CH1.92
Compressed natural gas CH3.78
– Need speciation to detect CH4
8
HC Impact on smog formation
• Species depen... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
– Main combustion: very little HC except for
very lean/ dilute or very late combustion
(misfires/ partial burns)
Various mechanisms for HC to escape from main
combustion
– Cold start emissions (wall film) especially
important
10
SOURCES OF UNBURNED HC IN SI ENGINE
a) Crevices
b) Absorption and de... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
age (0.1%)
2.5%
Crankcase (0.7%)
- Recycled -
4.6%
5.1%
In-Cylinder Oxidation
Blow-by (0.6%)
- Recycled -
1/3 Oxidized
2/3 Oxidized
1.7%
Exhaust Oxidation (0.8%)
1/3
3.4%
2.3%
1.7%
1/3
1.5%
Unburned HC in Residual
(1.3%) - Recycled -
Engine- out HC (1.6%)
Fully Burned Exhaust
Tailpipe- out HC (0.1-0.... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
• Reduce crevice volume
• Keep liner hot
• Spark retard
– Higher burned gas temperature in the later
part of expansion stroke and higher
exhaust temperature
• Comprehensive cold start strategy
– Retard timing, fuel rich followed by exhaust
air injection
14
MIT OpenCourseWare
https://ocw.mit.edu
2.61 Intern... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/37be0f6e7601d3239a746095c1a6f91d_MIT2_61S17_lec11-12.pdf |
6.825 Techniques in Artificial Intelligence
What is Artificial Intelligence (AI)?
Lecture 1 • 1
If you're going to teach or take an AI course, it's useful to ask: "What's AI?"
It's a lot of different things to a lot of different people. Let's go through a few
things that AI is thought to be and situate them within... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
people's heads and then build computational models that mirror those kind
of processes.
A crucial question is to decide at what level to mirror what goes on inside
people's heads. Someone might try to model it a very high-level, for
example, dividing processing into high-level vision, memory, and cognition
modules... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
it when we see it. We'll give up on trying to decide what
intelligence is and spend our time thinking about rationality. What might it
mean to behave rationally? We'll get into that in more detail later.
4
6.825 Techniques in Artificial Intelligence
What is Artificial Intelligence (AI)?
•
•
Computational models... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
AI, because
you were writing down statements in a high-level language; and how could a computer
possibly understand that stuff? Well, you had to do work to make a computer understand the
high-level language and that was taken to be AI. Now that we understand compilers and
there's a theory of how to build compilers ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
this course.
6
Agents
Software that gathers information about an
environment and takes actions based on that
information.
• a robot
•
•
•
a web shopping program
a factory
a traffic control system…
Lecture 1 • 7
We're going to be talking about agents. This word used to mean “something
that acts.” Now, peop... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
we begin to formalize the problem of building
an agent?
• Make a dichotomy between the agent and its environment
• Not everyone believes that making this dichotomy is a
good idea, but we need the leverage it gives us.
percepts
agent
environment
actions
Lecture 1 • 9
Here's a robot and the world it lives in. T... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
inclined plane they will walk down the hill (if you get it balanced
right); so you don't need any computation at all to do that walking. So, the
computation, or intelligence or whatever, is in the design of the hardware.
On the other hand, you could build a great big contraption, as some
researchers have, with six ... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
and low-level
systems; but we're going to think of things rather more discretely and so
we're going to model the interaction between the agent and the environment
in discrete time, with a cycle taking place every one second or two seconds
or ten seconds or ten minutes. Time won't enter too much in the methods
we'l... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
taken by the agent.
Later on we'll talk in detail about the fact that these functions may not be
deterministic and they may not really be known. Suppose you wanted to
make a robot that could vacuum the hallways or something in this building.
You'd like not to have to completely specify how this building is laid out... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
Well, it's going to
turn out to be really quite hard. But, at this level of abstraction, it's
straightforward what we want to do. We want to put the program in the head
of the agent that does as well as it can, subject to this specification of how
the world works and what we want in the world.
17
Rationality
• A... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
.
• Assume I don’t like to get wet, so I bring an umbrella. Is
that rational?
• Depends on the weather forecast and whether I’ve heard
If I’ve heard the forecast for rain (and I believe it) then
it.
bringing the umbrella is rational.
•
Rationality omniscience
≠
• Assume the most recent forecast is for rain bu... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
the same as success. Imagine that I take my umbrella,
I know that it's nice and sunny out and I take the umbrella anyway, which
seems to be irrational of me. But, then, I use the umbrella to fend off a rabid
dog attack. You might say, well it was rational of her to take the umbrella
because it saved her from the ra... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
very well or very fast and so, for instance,
humans are irrational because they're bad at doing a variety of tasks; they
just can't compute the optimal response in certain circumstances. That we
know; there's no question; but yet, you might be able to argue that given our
squishy brains that's the best we can do.
... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
program. Given the
specification of an environment, we want to find the best possible mapping
from P* to A (sequences of percepts to actions) that, subject to our
computational constraints, does the best job it can as measured by our utility
function.
24
Issues
•
How could we possibly specify completely the
do... | https://ocw.mit.edu/courses/6-825-techniques-in-artificial-intelligence-sma-5504-fall-2002/37cc451f7925405c3cf9274863d488ba_Lecture1Final.pdf |
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