text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 19
Biomaterials processing by microfluidic patterning... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
/ 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 21
Biomaterials processing by stencils
> Stencils
• Use PDMS stamps as dry resists
... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 23
Material properties and coupled domains
> The basic functionality of many MEMS devi... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
, have
highly predictable and repeatable constitutive properties
> Most microelectronic materials, however, exhibit some degree of
process dependence in their material properties, especially
deposited or thermally formed thin films
> Some properties, like thin-film residual stress, can be wildly
dependent on deposi... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 28
Examples of constitutive properties... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 30
Tensor properties
> Properties that involve directions, either the relative directions of
applied vector loads and vector responses, or the orientation of
loads and/or responses relative to internal (crystalline) axes,
require tensors for their specification
> Exa... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
.
> One course goal is acquiring domain knowledge, which gives
you some insight into which material properties are important in
a given situation.
> Today: a sneak preview of what you might worry about
> A useful resource:
• A previous year’s assignment involved looking up material
properties for many MEMS materials.... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
Breakdown Strength
> The maximum electric field that an insulating material in the gap
between two flat electrodes can withstand without suffering
dielectric breakdown
> Depends on the size of the interelectrode gap
> Importance: high voltage actuators, maximum performance
> Can vary with film composition, defect den... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
wafer, in the absence of
external loading.
> Two flavors:
• Intrinsic stress: related to structure
• Thermal stress: accumulated from a change in temperature
> Residual stress is a VERY VARIABLE PROPERTY, and must be
measured.
> Can play games, such as adjusting deposition conditions to
ensure that intrinsic and t... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
Pyrex) to Si
• Thermal stress in a film that is deposited at high T
> CTE is tabulated, and one of the less variable material
properties.
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Mass... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
ank tensor, as is the stress σ
> The piezoresistive effect is described by a fourth-rank
tensor
] J
⋅σ⋅Π+ρ=E
[
e
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Te... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
3
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 42
Piezoresistivity in Sili... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
idene fluoride)
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 44
Piezoelect... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
in the x or y direction = d31*Electric field in the z direction
• Strain in the z direction = d33*Electric field in the z direction
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachuset... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
icients depend on exact PZT material, on underlying
material, on frequency, and on electric field
• - d31 is in the ballpark of 100 pC/N to several hundred pC/N
• d33 is in the ballpark of several hundred pC/N to 1000 pC/N
• Again, expansion along the field corresponds to contraction
in the transverse direction
Cite ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
w.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
CL: 6.777J/2.372J Spring 2007, Lecture 5 - 50
The bottom line
> You need to use test structures to characterize materials with
variable properties
> Measuring electrical properties requires electrical test devices
> Measuring mechanica... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
77J/2.372J Spring 2007, Lecture 5 - 52
Membrane Load-Deflection Example
> The problem: residual stress and stiffness of membranes affect
their deflection under load
> Example: pressure sensors
d
t
2a
> Approach:
Pressure (p)
• Apply different pressures and measure resulting deflections
• Fit to an energy-based model... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
Test Structures." Journal of
Microelectromechanical Systems 6, no. 2 (1997): 107-118.
> Weakness: surface micromachined beams often have some
support compliance
Cite as: Carol Livermore, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (ht... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/511de50fb6460b31fd47ee2f9e4958a0_07lecture05.pdf |
5)!(cid:17)(cid:9)(cid:18)(cid:10)!(cid:26)(cid:29)#$(cid:26)(cid:29)(cid:13)(cid:16)$(cid:25)(cid:5)(cid:9)!(cid:20)
(cid:30).(cid:29)(cid:29).(cid:29)(cid:30)(cid:10)
+++/(cid:18)(cid:14)(cid:17)(cid:20)(cid:8)(cid:4)(cid:5)(cid:17)(cid:3)(cid:4)(cid:5)(cid:7)(cid:8)(cid:18)(cid:9)(cid:23)(cid:5)/(cid:12)(cid:7)(cid:... | https://ocw.mit.edu/courses/ids-410j-modeling-and-assessment-for-policy-spring-2013/5124bdeef9b2df6208ee15c8e0de758f_MITESD_864S13_lecture8.pdf |
(cid:1)Currently, in the UNFCCC negotiation process, the concrete
environmental consequences of the various positions are not clear to
all of us. There is a dangerous void of understanding of the short and
long term impacts of the espoused …unwillingness to act on behalf of the
Parties.(cid:2)(cid:3)(cid:4)
– Chri... | https://ocw.mit.edu/courses/ids-410j-modeling-and-assessment-for-policy-spring-2013/5124bdeef9b2df6208ee15c8e0de758f_MITESD_864S13_lecture8.pdf |
31)(cid:6)/000(cid:20)
+++/(cid:18)(cid:14)(cid:17)(cid:20)(cid:8)(cid:4)(cid:5)(cid:17)(cid:3)(cid:4)(cid:5)(cid:7)(cid:8)(cid:18)(cid:9)(cid:23)(cid:5)/(cid:12)(cid:7)(cid:6)(cid:10)
11
C-ROADS Scientific Review Panel
• Dr. Robert Watson - Department for Environment, Food and
Rural Affairs (DEFRA) and former chai... | https://ocw.mit.edu/courses/ids-410j-modeling-and-assessment-for-policy-spring-2013/5124bdeef9b2df6208ee15c8e0de758f_MITESD_864S13_lecture8.pdf |
31)(cid:10)&%(cid:17)(cid:14)(cid:13)(cid:10)(cid:17)(cid:3)(cid:4)%(cid:17)(cid:9)(cid:12)(cid:3)(cid:10)
16
Energy and Climate System Overview
Economic
Social
Environmental
effects
Changes to:
Capital stocks (vehicles(cid:2))
Efficiency (mpg(cid:2))
Utilization (driving(cid:2))
Transportation
Changes to:
... | https://ocw.mit.edu/courses/ids-410j-modeling-and-assessment-for-policy-spring-2013/5124bdeef9b2df6208ee15c8e0de758f_MITESD_864S13_lecture8.pdf |
Bayesian Networks
Representation and Reasoning
Marco F. Ramoni
Children’s Hospital Informatics Program
Harvard Medical School
HST 951 (2003)
Harvard-MIT Division of Health Sciences and Technology
HST.951J: Medical Decision Support
Introduction
� Bayesian network are a knowledge representation
formalism for reasoning... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
(B,A) is not in N.
Undirected: if (A,B) is in N, then (B,A) is in N.
Note: The link — should be «
.
A
B
C
Characters:
D
E
Adjacent set: the nodes one step away from A:
Adj(A)={B|(A,B)˛ L}.
Path: The set of n nodes Xi from A to B via links:
Loop: A closed path: X1 = Xn.
Acyclic graph: A graph with no cycles.... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
of states p
i of its parents P.
A p(A)
A p(A)
0.3
Y
0.3
Y
0.7
O
0.7
O
E p(E)
E p(E)
L
0.8
0.8
L
0.2
H
0.2
H
A
E
I
I
A E
I
A E
L
L
Y
L
L
Y
H
L
Y
H
L
Y
L
H
Y
H
Y
L
H H
Y
H H
Y
L
L
O
L
L
O
H
L
O
H
O
L
L
O H
O H
L
O H H
O H H
p(I|A,E)
p(I|A,E)
0.9
0.9
0.1
0.1
0.5
0.5
0.5 ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
two variables are
independent, I do not need to model their interaction
but I can reason about them separately.
� In
this
form of
independence, however, a variable will
nothing about another variable, by design.
independence, called marginal
tell me
� There is another, more useful, form of independence,
whic... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
� C) = p(A|C) · p(B|C) · p(C).
C
A
B
HST 951
Markov Equivalence
� Different network structures may encode the same
conditional independence statements:
A and B are conditionally independent given C.
can be encoded by 3 different network structures.
�
In all these network structures, the information flow
running... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
0.3
0
0.7
1
B p(B)
0
1
0.6
0.4
E p(E)
0
1
0.1
0.9
A C p(C|A)
0
0
1
1
0.25
0.75
0.50
0.50
0
1
0
1
D F p(F|D)
0
0
1
1
0.80
0.20
0.30
0.70
0
1
0
1
A B D p(D|A,B)
0
0
0
0
1
1
1
1
0.40
0.60
0.45
0.55
0.60
0.40
0.30
0.70
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
0
4000
3500
3000
2500
2000
1500
1000
500
0
2
4
6
8
10
12
Decomposition
Decomposition: D breaks the BBN into two BBNs:
p(d)= S p(a)p(b)p(c|a)p(d|a,b)p(e)p(f|d)p(g|d,e)=.
= (S p(a)p(b)p(c|a)p(d|a,b)) (S p(e)p(f|d)p(g|d,e)).
Saving: We move from 64 to 23 + 23=16, and most of all
the terms move from 7 to ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
in each node.
HST 951
Algorithm
Input: A BBN with a set of variables X and a set of
evidential statements e = {A=a,B=b,…}.
Output: Conditional probability distribution p(X|e ) for
each non evidential variable X.
Initialization Step:
Each evidential variable X,
if x ˛ e p(x)=1, else p(x)=0.
if x ˛ e l(x)=1, els... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
- Bayesians are not just
dreamers after all.
HST 951
Multiply Connected BBN
When the BBN is a multiply connected graph.
The associated undirected graph contains a loop.
Each node does not break the network into 2 parts.
Information may flow through more than one paths.
Pearl’s Algorithm is no longer applicable.... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
. For each node X,
� a =p(x|e ) �
� Compute p(x|e)= a
/S
cp(x|e ,c1,…,cn)p(e |c1,…,cn)p(c1,…,cn),
xp(x).
HST 951
S
Complexity
� The computational complexity is exponential in the
size of the loop cutset, as we must generate and
propagate a BBN for each combination of states of
the loop cutset.
� The identifica... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
0
1
1
1
1
0.4
0.6
0.5
0.5
0.7
0.3
0.2
0.8
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
Example
� Loop cutset: {A}.
� p(B=0)=p(B=0|A=0)p(A=1) + p(B=0|A=1)p(A=1).
A
1.000
0.000
0
1
A
0.000
1.000
0
1
B
0.400
0.600
0
1
C
0.200
0.800
0
1
+
B
0.100
0.900
0
1
C
0.500
0.500
0
1
... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
The basic strategy (Lauritzen & Spiegelhalter 1988) is:
1. Convert a BBN in a undirected graph coding the
same conditional independence assumptions.
2. Ensure the resulting graph is decomposable.
3. This operation clusters nodes in locally
independent subgraphs (cliques).
4. These cliques are joint to each other ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
C
A
B
C
Moralize
1.Marry parents
2.Drop arrows
D
E
D
E
HST 951
Reading Independence
� The translation method via moralization reads the
conditional independence statements in BBN.
� DAGs cannot encode any arbitrary set of conditional
independence assumptions.
I(D,A|(B,C))
I(C,B|(A,D))
A
B
C
B
D
A ... | https://ocw.mit.edu/courses/hst-951j-medical-decision-support-spring-2003/5156626d6accc2b3cd04f15de95f0a38_lecture5.pdf |
Acceleration
Structures for Ray Casting
MIT EECS 6.837 Computer Graphics
Wojciech Matusik, MIT EECS
© ACM. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Hašan et al. 2007
1
Recap: Ray Tracing
trace ray
Intersec... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
. .
Can we reduce this?
9
Today
• Motivation
– You need LOTS of rays to generate nice pictures
– Intersecting every ray with every primitive becomes the
bottleneck
• Bounding volumes
• Bounding Volume Hierarchies, Kd-trees
For every pixel
Construct a ray from the eye
For every object in the scene
Fi... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
dx = 0 (ray is parallel) AND
Rox < X1 or Rox > X2 → no intersection
y=Y2
y=Y1
Rd
Ro
x=X1
x=X2
(The same
for Y and Z,
of course)
18
Find Intersections Per Dimension
• Basic idea
– Determine an interval along the ray for each dimension
– The intersect these 1D intervals (remember CSG!)
– Done!
y=Y2
y=... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
26
Find Intersections Per Dimension
• Calculate intersection distance t1 and t2
y=Y2
y=Y1
t2
t1
Rd
Ro
x=X1
x=X2
27
Find Intersections Per Dimension
• Calculate intersection distance t1 and t2
– t1 = (X1 - Rox) / Rdx
– t2 = (X2 - Rox) / Rdx
– [t1, t2] is the X interval
y=Y2
y=Y1
t2
t1
Rd
Ro
x=X1... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
2
:-)
33
Is there an Intersection?
• If tstart > tend → box is missed
tstart
tend
y=Y2
y=Y1
x=X1
x=X2
34
Is the Box Behind the Eyepoint?
• If tend < tmin → box is behind
y=Y2
y=Y1
tend
tstart
x=X1
x=X2
35
Return the Correct Intersection
• If tstart > tmin → closest intersection at tstart
• Els... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
3,z3) =
M (xmax,ymax,zmin)
(x2,y2,z2) =
M (xmin,ymax,zmin)
(x1,y1,z1) =
M (xmax,ymin,zmin)
(x0,y0,z0) =
M (xmin,ymin,zmin)
(x'min, y'min, z'min)
= (min(x0,x1,x2,x3,x4,x5,x6,x7),
min(y0,y1,y2,y3,y4,x5,x6,x7),
min(z0,z1,z2,z3,z4,x5,x6,x7))
43
Bounding Box of a Transform
Bounding box of transformed object... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
sphere
45
Bounding Volume Hierarchies
• If ray hits bounding volume,
must we test all primitives inside it?
– Lots of work, think of a 1M-triangle mesh
• You guessed it already, we’ll split the primitives in
groups and build recursive bounding volumes
– Like collision detection,
remember?
bounding
sphere
h... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
traverse
– binary tree (=simple structure)
• Disadvantages
– may be difficult to choose a good split for a node
– poor split may result in minimal spatial pruning
60
BVH Discussion
• Advantages
– easy to construct
– easy to traverse
– binary tree (=simple structure)
• Disadvantages
– may be difficult to ... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
& Stoll, IRT 2006
– Zhou et al., SIGGRAPH Asia 2008
Zhou et al.
© ACM. All rights reserved. This content is excluded from our Creative Commons
license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
68
Kd-tree Traversal - High Level
• If leaf, intersect with list of primitives
• If intersects b... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
,t_end)
else:
# case three: traverse both sides in turn
t_hit = self.frontSideNode.traverse(orig, dir, t_start, t)
if t_hit <= t: return t_hit; # early ray termination
return self.backSideNode.traverse(orig, dir, t, t_end)
74
Important!
travers(orig, dir, t_start, t_end):
#adapted from Ingo Wald’s thesis
#... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
(orig, dir,t_start,t_end)
else:
# case three: traverse both sides in turn
t_hit = self.frontSideNode.traverse(orig, dir, t_start, t)
if t_hit <= t: return t_hit; # early ray termination
return self.backSideNode.traverse(orig, dir, t, t_end)
76
Early termination is powerful
• If there is an intersection in th... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
is parallel to one axis
80
Important Details
Questions?
• For leaves, do NOT report
intersection if t is not in [tnear, tfar].
– Important for primitives that overlap multiple nodes!
• Need to take direction of ray into account
– Reverse back and front if the direction has negative
coordinate along the spli... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
87
Efficient Implementation
• Not so easy, need ability to sort primitives along the
three axes very efficiently and split them into two
groups
• Plus primitives have an extent (bbox)
• Extra tricks include smarter tests to check if a
triangle is inside a box
bbox of triangle
Node
88
Hard-core efficiency c... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/51937f370603b18f259f00e814f96a0c_MIT6_837F12_Lec14.pdf |
r
1:
P
1
8.3
1
4
re
u
Lect
inciples
of
Applied
Mathematics
Rodolfo
Rosales
Spring
2014
an
1
now
th
e
r
Mo
Con
is
Flu
q
x
ect
v
equal
sources
&
sink
-‐D.
a
w
a
l
ion
t
serva
or.
Us
s
u
Ga
e
s
s).
2-‐D
or
3-‐
s
in
e
th orem
t... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/51a55257964390fa4f1cdda67a3e5742_MIT18_311S14_Lecture4.pdf |
c
mi
[
es
v
wa
y
e
o
r
f
r
e
riv
nto
c
i
t
e
n
g
roma
ng
i
lect
ter
township.
E
XAMPLE:
Heat
flow
in
2-‐D
or
3-‐D.
Then,
ρ
=
r
c
v
T
=
conserved
stuff
(heat)
per
unit
mass
Wh
ere:
cv
=
c
e
t
of
material
a
... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/51a55257964390fa4f1cdda67a3e5742_MIT18_311S14_Lecture4.pdf |
thermal
diffusion.
nu
T),
and
kappa
=
heat
conductivity.
t
in
water,
sugar
in
coffee,
ink
in
water,
ETC.)
al
(S
ion
XAM
E
PLE:
ion
t
a
equ
sion
Diffu
eat
uat
q
e
h
s
a
me
a
S
placian
C
La
*
=
nu
t
C
Where
C
=
con
nu... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/51a55257964390fa4f1cdda67a3e5742_MIT18_311S14_Lecture4.pdf |
su
e
k
a
t
it
does
g
on
l
n
te
e
we
to
s
r
h
e
l
y
er
v
a
t
i
tart
w
s
m:
ma
f
o
b
blo
ll
s
ink,
R
=
f
b
o
blo
e
th
f
o
s
the
radius
at
i
α
(t)
s:
R
y
sa
s
si
sional
analy
n
a
co
ink,
a
R(
√(nu*t)
nd
a
t),
as
... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/51a55257964390fa4f1cdda67a3e5742_MIT18_311S14_Lecture4.pdf |
>
=
=
O(L)
=
/nu).
ime
t
t
evan
l
e
r
e
Also
th
eede
n
d
to
cool/heat
a
size
L
vessel.
1
r
1:
P
1
8.3
1
4... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/51a55257964390fa4f1cdda67a3e5742_MIT18_311S14_Lecture4.pdf |
ns:
does
stirring
help?
wh
y
why
?
ur
c
c
n
o
o
ti
c
s
c
e
nv
o
oe
d
ating
something
with
a
flame
in
the
absence
of
e
h
n
e
wh
n
e
p
ul
p
ha
d
o
w
at
wh
avity?
gr
At
room
temperature,
in
cm^2/sec
l
d
herma
T
Diffu
sion
ity
v
i
us
f
if
er... | https://ocw.mit.edu/courses/18-311-principles-of-applied-mathematics-spring-2014/51a55257964390fa4f1cdda67a3e5742_MIT18_311S14_Lecture4.pdf |
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
18. Consensus
Consensus (sometimes called ‘reliable broadcast’ or ‘atomic broadcast’) is a fundamental
building block for distributed systems. Informally, we say that several processes achieve
consensus if they all agree on some ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
input by defining some total order on the set of possible inputs.1 We have
already seen one application of this replicated state machine idea, in the code for transactions;
there the replication takes the form of redoing a sequence of actions that is remembered in a log.
Suppose, for example, that we want to build a... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
of this handout
explains a variety of ways to make a replicated state machine run efficiently: leases, transactions,
and batching.
Spec for consensus
Here is the spec for consensus; we have seen it already in handout 8 on history and prophecy
variables. The idea is that the outcome of consensus should be one and o... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
MODULE Consensus [V] EXPORT Allow, Outcome =
% data value to agree on
MODULE TermConsensus [V] EXPORT Allow, Outcome =
VAR outcome
: (V + Null) := nil
APROC Allow(v) = << outcome = nil => outcome := v [] SKIP >>
APROC Outcome() ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
before LateConsensus does (in the Agree action). We saw these specs in
handout 8 on generalized abstraction functions, where prophecy variables are explained.
In the code we have in mind, there are some processes, each with its own outcome variable
initialized to nil. The outcome variables are supposed to reach cons... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
that can communicate messages from one process to another. In particular,
the model must define what faults are possible. There are lots of ways to do this, and we have
space to describe only the models that are most popular and closest to reality.
There are two broad classes of models:
•
Synchronous, in which a n... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
faults: a faulty process makes arbitrary transitions; these are named after the
Byzantine Empire, famous for treachery. The motivation for this model is usually not fear of
treachery, but ignorance of the ways in which a process might fail. Clearly Byzantine failure
is an upper bound on how bad things can be.
Is co... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
+1 rounds of communication, and 2f bits of data
communicated.
For processors with Byzantine faults and digital signatures (so that a process can present
unforgeable evidence that another process sent it a message), consensus requires f+1
processes. Even if the network is fully connected, it takes f+1 rounds to reac... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
one vote for 11 and one for 12, so you can’t tell that 12 had a majority.
The Paxos algorithm: The idea
In the rest of this handout, we describe Lamport’s Paxos algorithm for coding asynchronous
consensus; Liskov and Oki independently invented this algorithm as part of a replicated data
storage system.6 Its heart i... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
with a sloppy timeout-based algorithm
for choosing a single leader. If the sloppy algorithm leaves us with no leader or more than one
leader for a time, the consensus algorithm may not terminate during that time. But if the sloppy
algorithm ever produces a single leader for long enough the algorithm will terminate, ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
is easier to explain.
It takes a total of 21/2 round trips for a successful round. If there’s only one leader that doesn’t
fail, Paxos reaches consensus in one round. If the leader fails repeatedly, or several leaders fight
it out, it may take arbitrarily many rounds to reach consensus.
The rounds are numbered (not... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
a single value by having at most one leader process per round,
and making the leader’s identity part of the round number. So N = [i, l], and leader l chooses
(i, l) for n, where i is an I that l has not used before, for instance, a local clock. The leader
keeps vn in a volatile variable; rather than resuming an old ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
IF VAR a, v | sn
a IS V}.majority
a = v => RET v [*] RET nil FI
For this to be an abstraction function, we need an invariant:
(I1)
Every successful round has the same value.
Handout 18. Consensus
7
Handout 18. Consensus
8
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
I... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
to this query from a majority of agents give the leader enough information to make
round n safe, as follows:
It looks back from n, skipping over rounds with no V state, since these must be dead
a = v
(remember that the reported state is a V or no). When it comes to a round n' with sn'
for some agent a, it chooses v... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
in a second round trip the leader commands everyone for round n. Each agent that is still
neutral in round n (because it hasn’t answered the query of a round later than n) accepts by
changing its state to vn in round n; in any case it reports its state to the leader. If the leader
collects vn reports from a majority... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
’t all have to be up at the same time. Therefore we want a single leader, who runs one
round at a time. If there are several leaders, the one running the biggest round will eventually
Handout 18. Consensus
9
Handout 18. Consensus
10
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
agents retransmit only in response to the leader’s retransmission.
A process acting as a leader uses messages to communicate with the same process acting as an
agent, so we describe the two roles of each process completely independently. In fact, the leader
need not be an agent at all.
The next section gives the al... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
outcome(a) >>
= << VAR l | allowed(l) := allowed(l) \/ {v} >>
THREAD LeaderActions(l) =
VAR
n
phase
reports
v: (V+Null) := nil
|
:= N{i := 1, l := l},
:= idle,
:= S{},
% leader state (volatile except n)
% last round started
% leader’s phase
% info about agents so far
% used iff phase = commanding
DO <<... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
Handout 18. Consensus
11
Handout 18. Consensus
12
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
[] % This round is dead if a majority has no state. Try another round.
Dead(reports, n) => phase := idle
>> OD
THREAD AgentActions(a) = % State is in sa and outcomea , which ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
v => RET v [*] RET nil FI
% The value of round n according to s': if anyone has v then v else nil.
===================Useful functions for the invariants===================
% We write xl for LeaderActions(l).x to make the formulas more readable.
FUNC IsTotal(le: (L, L) -> Bool) -> Bool =
% Is le a total order?
RE... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
% from neutral, and either to no (query) or to agree with the leader (command).
a IS V | sn
(ALL n | {a | s!a /\ sn
a}.size <= 1)
% (4) All the S's in the channel or in any reports agree with s.
( ALL s1 :IN {m | m IN UnreliableCh.q /\ m.x IS S | m.x} \/ {l | reportsl} |
(ALL a, n | s1!a /\ s1(a)!n /\ s1n
a # neutral... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
the value of round n, in messages, in s, or in a leader.
Optimizations
It’s possible to reduce the size of the leader and agent state and the messages transmitted to a few
bytes, and in many cases to reduce the latency and the number of messages sent by combining
rounds of the algorithm.
RET
{m, a, s1 | m IN Unre... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
here on
or just after accepting:
last
next . . .
vlast
neutral from here on
In a leader, there are two cases for reports.
•
•
If phase = querying, reports consists of the sa’s, for rounds less than n, transmitted by a
set of agents a. Hence it can be encoded as a set of ‘last state’ tuples (a, lasta, v). From
... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
run a separate instance of the algorithm for each
action, and combine the query/report messages for all the actions. Note that these action
numbers, which we are calling K’s, are not the same as the Paxos round numbers, the N’s; each
action has its own instance of Paxos and therefore its own set of round numbers. In... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
successful outcome was never broadcast. If leaders follow the rule of not starting
consensus on k+1 until a majority knows the outcome for k, then this can happen at most once. It
may be convenient for a new leader to start by getting consensus on a SKIP action in order to get
this complication out of the way before... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
state component. So there is a tradeoff between the cost of renewing a lease
and the time you have to wait for it to expire after a (possible) failure.
There are several variations:
•
•
If you issue the lease to some known set of processes, you can revoke it provided they all
acknowledge the revocation.
If you k... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
to be changed. There’s no reliable way to do this without running consensus.
In spite of this, leases are not completely irrelevant to updates. With a lease you can use a simple
read-write memory as an agent for consensus, rather than a fancier one that can do compare-and-
swap, since the lease allows you do the nece... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
(vlast, last, n) provided next <= n. A command for round n changes
the state to (vn, n, n) provided next = n. So we need a representation that allows us to
atomically test the current value of next and change the state in one of these ways. This is
possible if an N fits in a single memory cell that CAS can read and ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
way of doing this is to use a single master with passive agents that just implement
simple memory; usually these are disk drives that record redundant copies of the log. The
previous section on leases explains how to run Paxos with such passive agents. When a master
fails, the new one has to sort out the consensus o... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/51b01bc0fd94819cc4588d6de2ea43a6_18.pdf |
Massachusetts Institute of Technology
Department of Materials Science and Engineering
77 Massachusetts Avenue, Cambridge MA 02139-4307
3.205 Thermodynamics and Kinetics of Materials—Fall 2006
October 26, 2006
Lecture 1: Fields and gradients; fluxes; continuity equation; entropy production; driving forces and fluxes
1. ... | https://ocw.mit.edu/courses/3-205-thermodynamics-and-kinetics-of-materials-fall-2006/51c50576b319acaf26bcce70746fe3f8_lecture01_review.pdf |
course of a spontaneous process.
• The local entropy production can be expressed as a sum of terms, each of which is a product of a flux and a
conjugate “force” (see KOM Eq. 2.15).
• Familiar empirical laws are linear relationships between fluxes and their conjugate forces: Fourier’s law of
heat conduction, Fick’s la... | https://ocw.mit.edu/courses/3-205-thermodynamics-and-kinetics-of-materials-fall-2006/51c50576b319acaf26bcce70746fe3f8_lecture01_review.pdf |
18.700 JORDAN NORMAL FORM NOTES
These are some supplementary notes on how to find the Jordan normal form of a small
matrix. First we recall some of the facts from lecture, next we give the general algorithm for
finding the Jordan normal form of a linear operator, and then we will see how this works for
small matrices... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
. .
(2)
(3)
Since dim(V ) = n, it cannot happen that each dim(E(X−λi)k ) < dim(E(X−λi)k+1 ), for each
k = 1, . . . , n. Therefore there is some least integer ei ≤ n such that E(X−λi )ei = E(X−λi)ei+1 .
As was proved in class, for each k ≥ ei we have E(X −λi)k = E(X−λi)ei , and we defined the
generalized eigenspace... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
. . . , Egen defined by
λ1
λr
Egen = {v ∈ V |∃e, (T − λiIV )e(v) = 0},
λi
Date: Fall 2001.
1
(4)
2
18.700 JORDAN NORMAL FORM NOTES
give a direct sum decomposition of V . Moreover, we have dim(Egen) equals the algebraic
multiplicity of λi, mi.
λi
(B) The semisimple part S of T and the nilpotent part N of T ... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
T = T T �) iff T �
commutes with both S and N . Moreover T � commutes with S iff for each i = 1, . . . , r,
we have T �
→
(Egen) ⊂ Egen
.
λi
λi
(6) If (S�, N �) is any pair of a diagonalizable operator S� and a nilpotent operator N � such
that T = S� + N � and S�N � = N �S�, then S� = S and N � = N . We call the un... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
⎜
⎝
[S]
B,B
=
0m1×m2
λ1Im1
0m2×m1 λ2Im2
.
.
.
.
.
.
0mr ×m1 0mr ×m1
⎛
⎜
⎜
⎜
⎝
C (1)
0m2×m1
.
.
.
0m1×m2
C (2)
.
.
.
0mr ×m1 0mr ×m2
[N ]
B,B
=
. . . 0m1×mr
. . . 0m2×mr
.
.
. . . λr Imr
.
.
.
.
. . . 0m1×mr
. . . 0m2×mr
.
.
.
.
.
C (r)
.
. . .
⎞
⎟
⎟
⎠
,
⎞
⎟
⎟
⎟
⎠
.
(8)
(9)
Notice that D(i... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
0
. . . 1 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
(10)
In other words,
(11)
Jae1 = e2, Jae2 = e3, . . . , Jaea−1 = ea, Jaea = 0.
Notice that the powers of Ja are very easy to compute. In fact J a = 0a,a, and for d =
1, . . . , a − 1, we have
a
d e1 = ed+1, Ja
Ja
Notice that we have ker(Ja
d e2 = ed+2, . . . , Ja
d) = span(ea+1−d, e... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
⎟
⎠
,
(13)
We say that a basis B(i) puts T (i) in Jordan normal form if C (i) is in Jordan normal form.
, . . . , B(r) puts T in Jordan normal form if each B(i) puts T (i)
�
B(1)
�
We say that a basis B =
in Jordan normal form.
WARNING: Usually such a basis is not unique. For example, if T is diagonalizable, then... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
1 and ea1+1. We have already seen that a distinguishing
feature of e1 is that it is an element of ker(J a1 ) which is not in ker(J a1−1). If a2 = a1,
then this is also a distinguishing feature of ea1+1. But if a2 < a1, this doesn’t work. In
this case it turns out that the distinguishing feature is that ea1+1 is in k... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
is as in the previous section. For
each eigenvalue λi, choose any basis C for V and let A = [T ]C,C . Define B = A − λiIn. Let
< ku ≤ n be the distinct integers such that there exists a nontrivial primitive
1 ≤ k1
for Gkj . Then the
subspace Gkj . For each j = 1, . . . , u, choose a basis v[j]1, . . . , v[j]pj
des... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
characteristic polynomial cA(X) is a quadratic polynomial. The first dichotomy
is whether cA(X) has two distinct roots or one repeated root.
Two distinct roots Suppose that cA(X) = (X − λ1)(X − λ2) with λ1 = λ2. Then for
each i = 1, 2 we form the matrix Bi = A − λiI2. By performing GaussJordan elimination we
may fin... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
have
�
�
λ1
0
0
λ2
, A = P
�
λ1
0
0
λ2
�
P −1 .
[A]B,B =
Also S = A and N = 02×2.
Now we consider an example. Consider the matrix
�
A =
�
.
38 −70
21 −39
The characteristic polynomial is X 2 − trace(A)X + det(A), which is X 2 + X − 12. This factors
as (X + 4)(X − 3), so we are in the case discu... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
root: cA(X) = (X −λ1)2 .
Again we form the matrix B1 = A − λ1I2. There are two cases depending on the dimension
of Eλ1 = ker(B1). The first case is that dim(Eλ1 ) = 2. In this case A is diagonalizable. In
fact, with respect to some basis B we have
[A]B,B =
�
�
.
λ1 0
0 λ1
But, if you think about it, this means... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
, A = P
�
�
λ 0
1 λ
�
�
λ 0
1 λ
P −1 .
This is the one case where we have nontrivial nilpotent part:
S = λ1I2 =
, N = A − λ1I2 = B1 = P
�
�
λ 0
0 λ
�
0 0
1 0
�
P −1 .
Let’s see how this works in an example. Consider the matrix from the practice problems:
�
A =
�
.
−5 −4
1 −1
The trace of A is... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
1
0
−3
, A = P
�
−
3
1
0
−3
�
P −1 .
The semisimple part is just S = −3I2, and the nilpotent part is:
N = B1 = P
�
�
0 0
1 0
P −1 .
(32)
(33)
(26)
(27)
(28)
(29)
(30)
18.700 JORDAN NORMAL FORM NOTES
7
3.2. Threebythree matrices. This is basically as in the last subsection, except now there
a... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
this:
⎛
2
D = ⎝ −1
1
1
0 −1
⎞
⎠ .
1
−
2
3
(35)
(36)
Here we easily compute trace(D) = 6 and det(D) = 8. Finally to compute the coefficient t,
we set c = 2 and we get
det(2I2 − A) = det ⎝ −1
0
⎛
0 −1
1
⎞
1 −2 ⎠ = 0.
1 −1
(37)
Plugging this in, we get
(2)3 − 6(2)2 + t(2) − 8 = 0
(38)
or t = 12, i.e. cA(X... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/51c72a6989c6a79228158a69f7750c5a_normal.pdf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.