text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
results) we have
0,8
n/¯
4
n/
2
h
= Y ξ(0)
ξn,p
n,p
W = Y W (0)Y †
These redefinitions are easily implemented in Q0
1,5 . They simply take
[ξ W ΓlC0(P +)W †ξ(u)] → [ξ
n,p
(d)
n,p
(d)(0)
n,p
W (0)ΓlC0(P +)W (0)†ξ(u)(0)
n,p
]
(11.6)
where we used the fact that Y commutes with the wilson coefficient C0(P +). Thi... | https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/5a73e8bb8f7f637a9b3eb58c5e733e6b_MIT8_851S13_SCETIIApplicat.pdf |
Programming Languages
Copyright
c
Nancy Leveson, Sept. 1999
As difficult to discuss rationally as religion or politics.
Prone to extreme statements devoid of data.
Examples:
"It is practically impossible to teach good programming to
students that have had a prior exposure to BASIC; as
potential programmers the... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
Selection by position (long parameter lists)
Defaults and implicit type conversion
Attempts to interpret intentions or fix errors
Meaning of features should be precisely defined (not
dependent on compiler.
Relationship between PL and Correctness (3)
Copyright
c
Nancy Leveson, Sept. 1999
Understandability
"Th... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
between PL and Correctness (6)
Copyright
c
Nancy Leveson, Sept. 1999
General
High-level languages take many decisions out of
programmer’s hands.
One reason they are so fiercely resented by
experienced programmers.
Language should restrict programmer to decisions
that really matter.
Decisions should be recorded ... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
c
Nancy Leveson, Sept. 1999
Cites experiments ("atheoretical" ) that evaluate only current
programming practice.
More interesting question: Can we elucidate underlying
psychological principles to allow generalization of results to
other classes of information structure in programming?
Hypothesis 1: If one langua... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
ijkstra’s guarded command:
if hard: boil
if not hard, juicy: fry
if not hard, not juicy: chop roast
�
Green: Program Creation
Programs as plans.
Copyright
c
Nancy Leveson, Sept. 1999
Role expressiveness: Outcome of a programmer’s effort
is a structure in which each part plays some role vis-a-vis
the progr... | https://ocw.mit.edu/courses/16-355j-software-engineering-concepts-fall-2005/5ab720f0c9da832dd1ce8737ad21131a_cnotes9.pdf |
The Invention Machine
The Invention Machine
Computational adaptation of TRIZ,
Computational adaptation of TRIZ,
Value Engineering and
Value Engineering and
the Semantic Web
the Semantic Web
Thanks to Invention Machine and
Dr. Mikhail Verbitsky for materials and
consultation and
SDM04 students who participated
Cou... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
Today’’s Problems:
Today
1) System does not cool well enough
2) Sensor is not accurate - leads to overheating
3) Customers complain about Noisy Fan
4) Management pressure to reduce costs
© Speller 2007, Engineering Systems Division, Massachusetts Institute of Technology
7
Step 1: Value Equation Development
Problem-Sta... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
. Patterns of technical systems evolution are
repeated across different industries. Systems
are being developed in the directions of:
(i) increased ideality; (ii) increased degree of
flexibility
3. Best innovations use scientific effects from
different fields
© Speller 2007, Engineering Systems Division, Massachuse... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
7, Engineering Systems Division, Massachusetts Institute of Technology
Unstructured
Text
22
Sample content:
“Or the Curie temperature can be controlled by
using two or more rare earth elements and adjusting
the composition ratio between them.”
Subject
Action
Object
Earth
elements
control
Curie temperature
1. What ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
7, Engineering Systems Division, Massachusetts Institute of Technology
31
•
•
•
•
•
Conclusions
Major steps of innovative design: (i) diagnostics of the
current design; (ii) identification of the ideal design; (iii)
moving current architecture closer to the ideal
Traditional TRIZ tools: Physical Effects, Matrix of
... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
/contradiction pairs
© Speller 2007, Engineering Systems Division, Massachusetts Institute of Technology
34
• The mechanics of the TRIZ principles make us think
about what are the effects that have been separated,
and the contradiction therefore eliminated
It allows going into detail, because the separation of the ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
concept creation1
The process of concept creation is the coupling of an intent-function pair with a form that performs it. There
are several ways to do this, and some of us just have it as a talent. However, when trying to do this
commercially, it is not feasible to trust in a group of artists that will invent when t... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
architecture of the semantic TRIZ itself is possible to see that the intent is to
systematically approach the invention process, with a solution that has all the advantages of the intended
effect, and none of the disadvantages of the counter effects, looking through the broadest possible field of
concepts.
The form ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/5ac7834e631541ae97d1783412aed765_triz_ve.pdf |
3.044 MATERIALS PROCESSING
LECTURE 5
General Heat Conduction Solutions:
∂T = ∇ · k∇T, T (¯x, t)
∂t
Trick one: steady state ∇2T = 0, T (x)
Trick two: low Biot number ∂T = α h(Ts
∂t
− Tf ), T (t)
Transient:
- semi-infinite
- infinite series: book, analytical
- graphical solutions
- computers, numerical, finite elements
Exam... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/5b02d3e73eaf3afa6ec9564595714beb_MIT3_044S13_Lec05.pdf |
S PROCESSING
3
cp = 0.12
kJ
kg K
h ≈ 10 − 1000
T0 = 1200K
Tf = 300K
T ≈ 750K
Solve for h (the only changable variable): h ≈ 260 W
m2 K
- oil bath w/ standoff/air gap
- big fans
- other gases
Example 2: Thermal Spray Coatings / Plasma Spray
Specific Example:
oxyacetylene torch: T = 2700◦C
powder: Ni alloy MAR-M200, r = 2 ... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/5b02d3e73eaf3afa6ec9564595714beb_MIT3_044S13_Lec05.pdf |
8s
recall: v = 100 m
s
⇒ distance travelled efore melting = 18m
b
How to decrease t and therefore decrease distance travelled?
- preheat the powder T0 ↑
- better plasma? Tf ↑
- smaller R → plausible but costs a lot of money
- change material
- change h → but h is already pretty large
Arc Melter to increase plasma tempe... | https://ocw.mit.edu/courses/3-044-materials-processing-spring-2013/5b02d3e73eaf3afa6ec9564595714beb_MIT3_044S13_Lec05.pdf |
6.801/6.866: Machine Vision, Lecture 11
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 ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
• Arcane grammar is used for legal purposes - “comprises”, “apparatus”, “method”, etc.
• References of other patents are often included - sometimes these are added by the patent examiner, rather than the patent
authors
• Most patents end with something along the lines of “this is why our invention was necessary” or “th... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
This approximates derivatives in a coordinate system rotated 45 degrees (x , y0). The derivatives can
0
be approximated using the Kx0 and Ky0 kernels:
→ Kx0 =
→ Ky =
0
0 −1
0
−1
1
0
0 −1
∂E
∂x0
∂E
∂y0
• Sobel Operator: This computational molecule requires more computation and it is not as high-re... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
an image, we can estimate the brightness gradient using some of the filters
defined above.
2. Compute Brightness Gradient Magnitude and Direction: Using the CORDIC algorithm, we can estimate the
brightness gradient magnitude and direction. The CORDIC algorithm does this iteratively through a corrective feedback
mecha... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
the procedure
above.
Some formulas for this system:
1. G0 = G2 + G2 (gradient estimation)
x
q
2. Gθ = tan
−1 Gy
Gx
y
(gradient estimation)
3. R0 = max(|Gx|, |Gy|) (octant quantization)
4. S0 = min(|Gx|, |Gy |) (octant quantization)
At a high level, the apparatus discussed in this patent is composed of: ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
of an edge in a discrete, and therefore, sampled image, and since the edge in the case of u(x) is infinitely thin, we
will not be able to find it due to sampling. In Fourier terms, if we use a perfect step function, we introduce artificially high
(infinite) frequencies that prevent us from sampling without aliasing effect... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
by the same location as the inflection
point of u(x) and the maximum of ru(x):
2
Gradient2 of “Soft” Unit Step Function, ru(x)
0.1
5 · 10−2
)
x
(
u
2
r
0
−5 · 10−2
−0.1
−6
−4
−2
2
4
6
0
x
For those curious, here is the math behind this specific function, assuming a sigmoid for u(x):
1. u(x) = 1+exp (−x)
... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
Next, we will look at the Sobel operator. For this analysis, it will be helpful to recall the following result from Taylor Series:
f (x + δx) = f (x) + δxf 0(x) +
(δx)2
2!
f 00(x) +
(δx)3
3!
f 000(x) +
(δx)4
24
f (4)(x) + ... =
∞
X
i=0
(δx)if (i)(x)
i!
, where 0! = 1
Δ
Let us first consider the simple two-pixe... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
f (x)
δx
+
2
f (x)−f (x−δx)
δx
= f 0(x) +
(δx)2
6
f 000(x) + ...
Now we have increased the error term to 3rd order, rather than 2nd order! As a computational molecule, this higher-order filter
Sobel operator looks like dE → Kx = (−1 0 1). But we can do even better! So long as we do not need to have a pixel at our... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
with minimal error by estimating it at the point at the center of these 2D
filters.
Estimating these individually in each dimension requires 3 operations each for a total of 6 operations, but if we are able to
take the common operations from each and combine them either by addition or subtraction, this only requires ... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
most common padding technique, but there are other techniques as well, such as wraparound padding.
• This approach avoids the half-pixel (in which we estimate an edge that is not on a pixel) that was cited above.
• Smoothing/averaging is a double edge sword, because while it can reduce/remove high-frequency noise by fil... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
Supression, http://justin-liang.com/tutorials/canny/#suppression
3. Padded Convolution, https://medium.com/@ayeshmanthaperera/what-is-padding-in-cnns-71b21fb0dd7
7
MIT OpenCourseWare
https://ocw.mit.edu
6.801 / 6.866 Machine Vision
Fall 2020
For information about citing these materials or our Terms of Use, visit:... | https://ocw.mit.edu/courses/6-801-machine-vision-fall-2020/5b03361b8fe3ccb51a0ce3c685dd88ae_MIT6_801F20_lec11.pdf |
Optics Overview
MIT 2.71/2.710
Review Lecture p-1
What is light?
• Light is a form of electromagnetic energy – detected through its
effects, e.g. heating of illuminated objects, conversion of light to
current, mechanical pressure (“Maxwell force”) etc.
• Light energy is conveyed through particles: “photons”
– ba... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
10
Review Lecture p-5
c=λν
“Dispersion relation”
(holds in vacuum only)
Light in matter
light in vacuum
Speed c=3×108 m/sec
light in matter
Speed c/n
n : refractive index
(or index of refraction)
Absorption coefficient 0 Absorption coefficient α
energy decay coefficient,
after distance L : e–2αL
E.g. vacuu... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
Lecture p-8
Monochromatic, spatially coherent
light
1/ν
λ
• nice, regular sinusoid
• λ, ν well defined
• stabilized HeNe laser
good approximation
• most other cw lasers
rough approximation
• pulsed lasers & non-
laser sources need
more complicated
description
Incoherent: random, irregular waveform
MIT 2.7... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
follows the
symmetric path POP’.
P ′′
mirror
P’
θ
θ
P
MIT 2.71/2.710
Review Lecture p-14
The law of refraction
reflected
θ
θ
n
n′
incident
refracted
θ′
n
sin
θ
′=
n
′
θ
sin
Snell’s Law of Refraction
MIT 2.71/2.710
Review Lecture p-15
Optical waveguide
n≈
1.00
n
=1.51
TIR
n
=1.5105
n
=1.51... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
0
Huygens principle
Each point on the wavefront
acts as a secondary light source
emitting a spherical wave
The wavefront after a short
propagation distance is the
result of superimposing all
these spherical wavelets
optical
wavefronts
MIT 2.71/2.710
Review Lecture p-22
Why imaging systems are needed
• Eac... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
agnification
Energy
conservation
M
a
s
i− =
s
o
M M x
a
=
1
Imaging condition: ray-tracing
image
(virtual)
1st FP
object
thin lens (+)
2nd FP
c
h
i
e
f
r
a
y
• The ray bundle emanating from the system is divergent; the virtual
image is located at the intersection of the backwards-extended rays
• The ... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
we will see later;
diffraction introduces undesirable artifacts in the image.
MIT 2.71/2.710
Review Lecture p-35
Field of View (FoV)
φ
FoV=angle that the chief ray from an object can subtend
towards the imaging system
MIT 2.71/2.710
Review Lecture p-36
Numerical Aperture
medium of
refr. index
n
θ
θ: half... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
in the image
MIT 2.71/2.710
Review Lecture p-40
Diffraction limited resolution
)
s
t
i
n
u
y
r
a
r
t
i
b
r
a
(
y
t
i
s
n
e
t
n
i
t
h
g
i
l
object
spacing
δx
lateral coordinate at image plane (arbitrary units
)
Point objects “just
resolvable” when
δx ≈ 22.1 λ
(NA)
Rayleigh resolution
criterion
MIT 2.71/2.7... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
10
Review Lecture p-44
Fresnel diffraction formulae
x
y
x´
y´
z
(
x
,
g
in
)
y
exp i
2π
z
∫
g
λ
in
g out (
′
x ,
′
z
;
)
y
=
1
i λ
z
x
′
,
yx
′
(out
g
)
( x ′ − x ) + ( y ′ − y )
λ z
2
2
x d d
y
( , )
x
y
exp
i π
y
z
)
(
v u
,
G
in
= exp i
2π
z ... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
y
g
)
2
)
(
x
,
t y
=
x
,(
y
)
g
1
impulse response
convolution
=
(
′
′
x
y
g
,
3
∗
x
y
,(
)
g
2
output
amplitude
)
=
x h
,(
y
)
Fourier
transform
Fourier
transform
(≡
plane wave
spectrum
)
)
(
v u
,
G
2
transfer function
multiplication
v u
),(
G
3
v u
),(
G
2
=
v... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/5b0a3acf433dcdadf931bc6223ff4fe6_wk2_a.pdf |
Machine learning for Pathology
Andrew H Beck MD PhD
CEO @ PathAI
6.S897/HST.956: Machine Learning for Healthcare. MIT.
March 19, 2019
© source unknown. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
1
Pat... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
Creative Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
Beck ... Koller. Science Translational Medicine 2011
9
Even today, the anatomic path lab has been
largely unchanged for routine diagnostics
© sources unknown. All rights reserved.
This content is... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
34-1241.
© Springer Nature. All rights reserved. This content is excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
12
Discordance among pathologists is common in interpretation of
melanocyti... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
is reproducible and objective
• Efficient – massive parallelization for speedy processing
• Exploratory - learn relationships in a purely data-driven
manner
16
What AI can’t do for pathology
Replace pathologists!
Proprietary & Confidential
17
A diagnosis/detection example:
Breast cancer ... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
Successfully applied deep learning approach to pathology
Our team won the Camelyon challenge in 2016, demonstrating outstanding initial performance in pathology
L
A
M
R
O
N
R
O
M
U
T
Whole Slide Image
Training Data
Deep Model
I
N
A
R
T
T
S
E
T
Whole Slide Image
Image Patches
Deep Model from Training
Tumor Proba... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
/help/faq-fair-use/
24
Pathologist + PathAI
© 2016 PathAI - Confidential & proprietary.
Do not distribute.
Proprietary & Confidential
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license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
25
... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
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29
Why is this a good application for AI?
• Exhaustive analysis is benefici... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
more information, see https://ocw.mit.edu/help/faq-fair-use/
34
Can we do better?
• Deep learning is data hungry
• Need 10s of thousands of precise
cell annotations
First, we need the
data
Proprietary & Confidential
35
Board-certified
training data
Working with pa... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
, see https://ocw.mit.edu/help/faq-fair-use/
Proprietary & Confidential
42
Predictive features guided by biomedical priors
Immune cell (lymphocyte) detection
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license. For more information, see https://ocw.mit.edu/he... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
)
Total number of plasma cells in epithelial/stroma interface (80um)
Area (mm2) of epithelial/stroma interface (80um) target positive cancer cells on target stain
Area (mm2) of epithelial PDL-1 positive macrophages on target stain
Necrosis area on target stain
Proportion of tumor infiltrating lymphocytes engaged b... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
umor Epithelium
51
Melanoma Cell Map
© source unknown. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
TCGA-EE-A2GL, Malignant Melanoma
Lymphocytes: Green
Macrophages: Orange
Plasma Cells: Blue
Fibroblast... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
Gene Set Name
REACTOME_IMMUNE_SYSTEM
REACTOME_ADAPTIVE_IMMUNE_SYSTEM
PID_TCR_PATHWAY
REACTOME_IMMUNOREGULATORY_INTERACTIONS_BETWEEN_A_
LYMPHOID_AND_A_NON_LYMPHOID_CELL
KEGG_PRIMARY_IMMUNODEFICIENCY
PID_IL12_2PATHWAY
PID_CD8_TCR_PATHWAY
KEGG_CELL_ADHESION_MOLECULES_CAMS
KEGG_CYTOKINE_CYTOKINE_RECEPTOR_INTERACT... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
excluded from our Creative Commons
license. For more information, see https://ocw.mit.edu/help/faq-fair-use/
• Same pipeline for any solid tumor
type
• Contrast to traditional approach:
hand-crafted algorithms.
Proprietary & Confidential
56
Extensive Slide Search & Data Standardization
Pro... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
to
practice
Analyze samples,
quantified & visual
results delivered
We can execute process in 4 – 8 weeks for new assays
Proprietary & Confidential
62
AI in medicine
Some closing thoughts
• ML in the real world:
•... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
health care
delivery. The pathologist as medical information specialist.”
(Arch Pathol Lab Med. 1987)
67
MIT OpenCourseWare
https://ocw.mit.edu
6.S897 / HST.956 Machine Learning for Healthcare
Spring 2... | https://ocw.mit.edu/courses/6-s897-machine-learning-for-healthcare-spring-2019/5b28c3a5a47a51faf2c97f17d2a8f5a1_MIT6_S897S19_lec12.pdf |
7.91 / 20.490 / 6.874 / HST.506
7.36 / 20.390 / 6.802
C. Burge Lecture #9
Mar. 6, 2014
Modeling & Discovery
of Sequence Motifs
1
Modeling & Discovery of Sequence Motifs
• Motif Discovery with Gibbs Sampling Algorithm
• Information Content of a Motif
• Parameter Estimation for Motif Models (+ others) ... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
w.mit.edu/help/faq-fair-use/.
Source: Ericsson, A. O., L. O. Faria, et al. "TcZFP8, A Novel
Member of the Trypanosoma Cruzi CCHC Zinc Finger Protein
Family with Nuclear Localization." Genetics and Molecular
Research 5, no. 3 (2006): 553-63.
CX2CX4HX4C
Zinc finger (DNA binding)
Ericsson et al. Genet. Mol. Res. 2006
... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
.1
0.1
0.1 …
0.0 0.5
Background (-)
Pos Generic
A
C
G
T
0.25
0.25
0.25
0.25
S = S1 S2 S3 S4 S5 S6 S7 S8 S9
Odds Ratio: R =
P(S|+) = P-3(S1)P-2(S2)P-1(S3) ••• P5(S8)P6(S9)
P(S|-) = Pbg(S1)Pbg(S2)Pbg(S3) ••• Pbg(S8)Pbg(S9)
Background model homogenous, assumes independence
7
... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
thought of calling it ‘information’,
but the word was overly used, so I decided to call it ‘uncertainty’. When I
discussed it with John von Neumann, he had a better idea. Von Neumann
told me, ‘You should call it entropy, for two reasons. In the first place your
uncertainty function has been used in statistical mech... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
ttcgatttcaagagttcaaaacgtg
cccgataggactaataaggacgaaacgagggcgatcaatg
ttagtacaaacccgctcacccgaaaggagggcaaatacct
agcaaggttcagatatacagccaggggagacctataactc
gtccacgtgcgtatgtactaattgtggagagcaaatcatt
...
...can be posed as an alignment problem
12
Approaches to Motif Finding
• Enumerative (‘dictionary’)
-... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
Θ
s k,A
1,
k
× Θ
2,
s k,A +1
k
× ... × Θ
8,
s k,A + 7
k
×θB,
s k ,Ak=8
× ... ×θB, L
ks
= “actactgtatcgtactgactgattaggccatgactgcat”
Motif location
kA
Lawrence et al. Science 1993
16
The Gibbs Sampling Algorithm In Words I
Given N sequences of length L and desire... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
The Gibbs Sampling Algorithm In Words, II
Given N sequences of length L and desired motif width W:
5) Sample a starting position in seq 1 based on this
probability distribution and set a1 to this new position.
6) Choose a sequence at random from the set (say, seq 2).
7) Make a weight matrix model of width W from ... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
ggcaattgtaaaacgacggcaatgttcg
cgtattaatgataaagaggggggtaggaggtcaactcttc
aatgcttataacataggagtagagtagtgggtaaactacg
tctgaaccttctttatgcgaagacgcgagggcaatcggga
tgcatgtctgacaacttgtccaggaggaggtcaacgactc
cgtgtcatagaattccatccgccacgcggggtaatttgga
tcccgtcaaagtgccaacttgtgccggggggctagcagct
acagcccgggaatatagacgcgtttggagtgcaaacat... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
Gibbs Sampler Summary
• A stochastic (Monte Carlo) algorithm for motif finding
• Works by ‘stumbling’ onto a few motif instances, which
bias the weight matrix, which causes it to sample more
motif instances, which biases the weight matrix more,
… until convergence
• Not guaranteed to converge to same motif every ... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
atgcctcctctgccgattcggcgagtgatcg
gatggggaaaatatgagaccaggggagggccacactgcag
ctgccgggctaacagacacacgtctagggctgtgaaatct
gtaggcgccgaggccaacgctgagtgtcgatgttgagaac
attagtccggttccaagagggcaactttgtatgcaccgcc
gcggcccagtgcgcaacgcacagggcaaggtttactgcgg
ccacatgcgagggcaacctccctgtgttgggcggttctga
gcaattgtaaaacgacggcaatgttcggtcgccta... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
Practical Motif Finding
• MEME is a classic method
Deterministic - like Gibbs, but uses expectation maximization
Bailey & Elkan 1995 paper is posted.
Run MEME at:
http://meme.nbcr.net/meme/
The Fraenkel lab’s WebMotifs combines
AlignACE (similar to Gibbs), MDscan, MEME, Weeder, THEME
Described in Romer et ... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
, qC = qG = 1/8
Suppose: pC = 1. H(q) - H(p) < 2
But RelEnt D(p||q) = log2(1/(1/8)) = 3
Which one better describes frequency of C in background seq?
* Alternate names: “Kullback-Leibler distance”, “information for discrimination”
35
... | https://ocw.mit.edu/courses/7-91j-foundations-of-computational-and-systems-biology-spring-2014/5b2e28907ea0396609fd2cbccd2dd462_MIT7_91JS14_Lecture9.pdf |
Chapter 4
Nonlinear equations
4.1 Root finding
Consider the problem of solving any nonlinear relation g(x) = h(x) in the
real variable x. We rephrase this problem as one of finding the zero (root)
of a function, here f (x) = g(x) − h(x). The minimal assumption we need on
f, g, h is that they’re continuous.
We have... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
LINEAR EQUATIONS
In practice, this iteration is stopped once f (mk) gets small enough. Let
x ∗ be the unknown root. The error obeys
|x ∗ − mk| ≤ |bk − ak| = 2−k|b0 − a0|.
Every step of the bisection discovers a new correct digit in the binary
expansion of x ∗ .
The advantage of the bisection method is that it is ... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
77
408
Starting with x0 = 1, we get x1 = 3 = 1.5, x2 = 17 = 1.4167...,
= 1.4142157... The true value of 2 is 1.4142135...
x3 =
Convergence is very fast, when it occurs. Assume that f '' is continuous,
and that f ' (x) = 0 in some neighborhood of the root x ∗ (large enough
so that all our iterates stay in this neig... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
∗)/f ' (x ∗)) as n → ∞. Hence the sequence is bounded
'' (x
|En+1| ≤ CE2 ,n
|En| ≤
(CE0)2k .
where C > 0 is some number (which depends on f but not on n.) It
follows that
1
C
We say the method “converges quadratically” because the exponent of
En is 2. The number of correct digits is squared at each iteration. I... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
.
If we do not know the derivative, we cannot set up Newton’s method,
but we can approximate it by replacing the derivative by (let fn =
f (xn))
f [xn−1, xn] =
fn − fn−1
.
xn − xn−1
Hence we define xn+1 by
xn+1 = xn −
fn
.
f [xn−1, xn]
The geometrical idea is to replace the tangent line at xn by the secant
... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
and xn. Evaluat
ing this relation at the root x = x ∗ , we get
0 = fn + f [xn−1, xn](x ∗ − xn) +
1
f '' (ξ)(x ∗ − xn)(x ∗ − xn−1).
2
On the other hand the definition of xn+1 gives
0 = fn + f [xn−1, xn](xn+1 − xn).
Subtracting these two equations we get
En+1 =
'' (ξ)
1
f
2 f [xn−1, xn]
EnEn−1.
Again, thanks... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
isection, but slower than Newton’s method.
The secant method inherits the problem of Newton’s method: it only
converges when the starting guesses x0 and x1 are sufficiently close to
the root.
5
CHAPTER 4. NONLINEAR EQUATIONS
We can also set up Newton’s method in several dimensions. A system of
nonlinear equations... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
is that we can fit the tangent
plane to each of the surfaces y = fi(x1, . . . , xn) in Rn+1 , find the line at
the intersection of all these planes, and check where this line intersects the
(hyper)plane y = 0.
Newton’s method is still quadratically convergent in multiple dimensions,
and special care must still be ta... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
.
4.2 Optimization problems
Another important recurring problem in science and engineering is that of
finding a minimum or a maximum of a function F (x). A point x ∗ is a local
minimum when F (y) ≥ F (x ∗) for all y in a neighborhood of x ∗ . It is a global
minimum when F (y) ≥ F (x ∗) for all y. We write
min F (x... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
∂F . We get
∂xi
xn+1 = xn − [VVF (xn)]−1 VF (xn).
The matrix VVF of second partial derivatives of F is called the Hessian. In
index notation,
(VVF )ij =
∂2F
∂xi∂xj
.
Compare Newton’s method with simple gradient descent:
xn+1 = xn − αVF (xn),
for some sufficiently small scalar α. Gradient descent is slower but ... | https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf |
Lecture # 17
Solar Thermal Energy
Ahmed Ghoniem
April 6, 2020
Renewables: Some characteristics and specifics.
Historical Trends …
Solar Thermals: Concentrators and Plants
Renewable Sources and Their Utilization
Biomass
Geothermal
Solar
Wind/Wave
Chemical
Thermal
photo
Kinetic
Combustion
Gasification... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
Saturated steam is generated at the receiver tower, fed directly to the turbine, or some stored in hot
water tank for extending the hours of operation. The receiver is a forced circulation radiant boiler
receiving ~ 55 MWt of concentrated solar radiation. Storage capacity is 20 MWht, sufficient to
operate the turbin... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
% of total power used in commercial applications was from natural sources (wind and
water). By 1911, all but 2% of power was generated from burning coal and harnessing steam.
“Within a few generations at most, some other energy than that of combustion of fuel must be
relied upon to do a fair share of the work of the... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
.
10
© Ausra, Inc. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/fairuse.
11
Solar Thermal Electric Generation Stations (SEGS) 1985-2002
Modern plants 2006-2014
Nine SEGS Plants in the
Mojave Desert (350MW)
US DOE
Image... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
• Average ~ 300 W/m2 (strong function of location)
© Source unknown. This content is excluded from our Creative Commons license.
For more information, see https://ocw.mit.edu/fairuse.
In London, solar
intensity, average over
the year is ~ 100 W/m2
from MacKay
© Ahmed F. Ghonie
m
13
... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
ors:
q = β I − ˆ( − Ta )
h Tc
q net flux collected by a fluid passing through the collector
I
β fraction absobed, depends on orientation & transmissivity < 0.8
hˆ overallheat transfer coefficient
Tc
Irradiance < 1 kW / m 2
environment T
collector T
Ta
at q = 0
(Tc )max = Ta +
β I
hˆ
for high (Tc )max , hˆ must be ... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
lost from the small area
of the collector only.
• Concentration Ratio CR is the ratio between irradiance on the collector (at the focal point
of the concentrator) and incident irradiance, I, is (also the area ratio):
CR = 107.5
Dm
F
for cylindrical
2Dm
⎛
⎞
= 11560
F⎜
⎟
⎝
⎠
Dm : mirror dimension, F : focal length
f... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
Solar field efficiency:
Upper temperature limits of
HT Oils
HT Steels
HT Alloys
C = 2500
Dish
C: concentration ratio
Heliostat
C = 1000
C = 700
C = 3
Vacuum
C = 1
Flat
C = 1
C = 1
C = 200
Trough
C = 40
C = 80
From Winter “Solar Power Plants” Greenhunt,M.Sc. Thesis, p. 35
20
... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
Solar thermal Electric Power systems
Source: US DOE 2005
Table courtesy of DOE.
© Ahmed F. Ghoniem
Source: US DOE
24
Parabolic-Trough Technology
Developed by Luz Int., and installed in Kramer Junction in 1991, company
failed commercially in 92 (low NG prices), but plant is still in operation.
Image ... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
29
Power Tower Technology
Image courtesy of DOE.
Image courtesy of DOE.
© Ahmed F. Ghoniem
30
Dispatchable Power Requires Storage
Image courtesy of DOE.
31
2009, Near Lancaster, CA
© Source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more informa... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
ocw.mit.edu/fairuse.
36
Operating Hybrid Combined Cycle Solar Plant
© DCSP. All rights reserved. This content is excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/fairuse.
37
Table and figure © SolarPACES. All rights reserved. This con... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
Solar Chimney
the Hydroelectric Power for the desert”
r the desert”
Vch =
ΔT
T
2gH ch
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Operated in Spain, 1982-89
From Encyclopedia of Physical Science and Technology, 2000
Article by J Schlaich and W Schiel
Figures 1 and 2 © Source unk... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
. Ghoniem, Journal of Hydrogen Energy, 40(7): 2939-2949, 2015
45
Solar Fuels?
Novel, looping based reformer
Parabolic
Solar Collector
Solar Radiation
Solar Window
Receiver Reactor
Solar Radiation
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permis... | https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/5b41ce8e948ad3b93015bb1827897026_MIT2_60s20_lec17.pdf |
Lecture 4: Stochastic
Thinking and Random
Walks
(cid:1010)(cid:856)(cid:1004)(cid:1004)(cid:1004)(cid:1006)(cid:3)(cid:62)(cid:286)(cid:272)(cid:410)(cid:437)(cid:396)(cid:286)(cid:3)(cid:1008)
1
Relevant Reading
Pages 235-238
Chapter 14
6.0002 LECTURE 4
2
The World is Hard to Under... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/5b4ec0ef29910b74e6a4bbd5ccf4dda5_MIT6_0002F16_lec4.pdf |
as well treat
the world as inherently
unpredictable
Predictive nondeterminism
6.0002 LECTURE 4
7
Stochastic Processes
An ongoing process where the next state might depend on
both the previous states and some random element
def rollDie():
""" returns an int between 1 and 6"""
def rollDie(... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/5b4ec0ef29910b74e6a4bbd5ccf4dda5_MIT6_0002F16_lec4.pdf |
if
impossible, and 1 if guaranteed.
If the probability of an event occurring is p, the
probability of it not occurring must be
When events are independent of each other, the
probability of all of the events occurring is equal to a
product of the probabilities of each of the events
occurring.
6.0002 LECTURE 4 ... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/5b4ec0ef29910b74e6a4bbd5ccf4dda5_MIT6_0002F16_lec4.pdf |
15
Output of Simulation
Actual probability = 0.0001286
Estimated Probability = 0.0
Actual probability = 0.0001286
Estimated Probability = 0.0
How did I know that this is what would get printed?
Why did simulation give me the wrong a... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/5b4ec0ef29910b74e6a4bbd5ccf4dda5_MIT6_0002F16_lec4.pdf |
= [0]*366
for p in range(numPeople):
birthDate = random.choice(possibleDates)
birthdays[birthDate] += 1
return max(birthdays) >= numSame
6.0002 LECTURE 4
19
Approximating Using a Simulation
def birthdayProb(numPeople, numSame, numTrials):
numHits = 0
for t in range(n... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/5b4ec0ef29910b74e6a4bbd5ccf4dda5_MIT6_0002F16_lec4.pdf |
/help/faq-fair-use/.
Chart
6.0002 LECTURE 4
22
Another Win for Simulation
Adjusting analytic model a pain
Adjusting simulation model easy
def sameDate(numPeople, numSame):
possibleDates = 4*list(range(0, 57)) + [58]\
+ 4*list(range(59, 366))\
+ 4*list(range(180, 270))
birthdays = [0]*366
for... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/5b4ec0ef29910b74e6a4bbd5ccf4dda5_MIT6_0002F16_lec4.pdf |
Walk
Robert
Brown
1827
Louis
Bachelier
1900
Albert
Einstein
1905
Images of Robert Brown and Albert Einstein are in the public domain. Image of Louis
Bachelier © unknown. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/5b4ec0ef29910b74e6a4bbd5ccf4dda5_MIT6_0002F16_lec4.pdf |
6.045: Automata, Computability, and
Complexity
Or, Great Ideas in Theoretical
Computer Science
Spring, 2010
Class 4
Nancy Lynch
Today
• Two more models of computation:
– Nondeterministic Finite Automata (NFAs)
• Add a guessing capability to FAs.
• But provably equivalent to FAs.
– Regular expres... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/5b6d125bc94344e53285a1cdbdb33371_MIT6_045JS11_lec04.pdf |
� ∪ {ε }.
The result is a set of states.
– q0 ∈ Q, is the start state, and
– F ⊆ Q is the set of accepting, or final states.
Formal Definition of an NFA
• An NFA is a 5-tuple ( Q, Σ, δ, q0, F ), where:
– Q is a finite set of states,
– Σ is a finite set (alphabet) of input symbols,
– δ: Q × Σε → P(Q) is t... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/5b6d125bc94344e53285a1cdbdb33371_MIT6_045JS11_lec04.pdf |
two additions:
– Allow δ(q, a) to specify more than one successor state.
– Add ε-transitions.
• Formally, an NFA is a 5-tuple ( Q, Σ, δ, q0, F ),
where:
– Q is a finite set of states,
– Σ is a finite set (alphabet) of input symbols,
– δ: Q × Σε → P(Q) is the transition function,
Σε means Σ ∪ {ε }.
– q0 ∈ Q, i... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/5b6d125bc94344e53285a1cdbdb33371_MIT6_045JS11_lec04.pdf |
– Then after another 0: { a, b }
– After 1: { a, c }
– After final 0: { a, b }
• Since neither a nor b is accepting, M does not
accept 0010.
0
0
0
{ a } Æ { a, b } Æ { a, b } Æ { a, c } Æ { a, b }
1
0,1
ε
ε
a
Example 2
0
1
b
e
c
f
1
0
d
g
• L(M) = { w | w ends with 01 or 10 }
• Computations for... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/5b6d125bc94344e53285a1cdbdb33371_MIT6_045JS11_lec04.pdf |
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