text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
2
µ
2
µ
3
µ
2
µ
3
00
µ
3
01
µ
3
02
µ
3
µ
1
03
µ
1
13
23
2.852 Manufacturing Systems Analysis
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Copyright �2010 Stanley B. Gershwin.
c
Equivalence
Transfer Line Production Rate
Assembly System Production Rate
Production rate = rate of flow of material into M1
1
3
= µ1
p(n1, n2)
n1=0
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/460acb49eb2c6f02fa3a7ba41481f026_MIT2_852S10_a_d_systems.pdf |
µ
1
µ
1
10
20
03
3
02
3
01
3
00
2.852 Manufacturing Systems Analysis
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Copyright �2010 Stanley B. Gershwin.
c
Equivalence
Equal n¯1
Assembly System Production Rate
Therefore
T
A = n¯1
n¯1
2.852 Manufacturing Systems Analysis
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Copyright �2010 Stanley B. Gershwin.
c
Equivalence
Asse... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/460acb49eb2c6f02fa3a7ba41481f026_MIT2_852S10_a_d_systems.pdf |
1, n2) =
n2
p(n1, n2)
n1=0 n2=0
� �
�
n1=0
n2=0
� �
�
N = 2
1
N = 3
2
µ
1
µ
2
µ
3
µ
1
µ
1
23
13
µ
2
µ
3
µ
2
µ
3
µ
1
µ
1
12
µ
2
µ
3
µ
2
µ
1
11
µ
1
µ
3
22
21
µ
2
µ
3
µ
2
µ
3
03
µ
3
02
µ
3
01
µ
3
µ
1
00
µ
1
10
20
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Copyright �201... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/460acb49eb2c6f02fa3a7ba41481f026_MIT2_852S10_a_d_systems.pdf |
parameters; that is, µi = µi , i = 1, ..., kM and
′ Nb = Nb , b = 1, ..., kB .
′
′
◮ There is a subset of buffers Ω such that for j 6∈ Ω, u (j) = u(j) and
d (j) = d(j); and for j ∈ Ω, u (j) = d(j) and d (j) = u(j). That is,
there is a set of buffers such that the direction of flow is reversed in the
two networks.
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/460acb49eb2c6f02fa3a7ba41481f026_MIT2_852S10_a_d_systems.pdf |
(0) and s ′(0) are related as follows: nj
′(0) = nj (0)
for j 6∈ Ω, and nj (0) = Nj − nj (0) for j ∈ Ω.
′
◮ Then
P ′ (n ′ (0)) = P(n(0))
′ (n ′ (0)) = n¯b(n(0)), for j 6∈ Ω
n¯b
′ (n ′ (0)) = Nb − n¯b(n(0)), for j ∈ Ω
n¯b
2.852 Manufacturing Systems Analysis
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Copyright �2010 Stanley B. Gershwin.
c
Equiva... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/460acb49eb2c6f02fa3a7ba41481f026_MIT2_852S10_a_d_systems.pdf |
B
1
µ
2
N
2
B
1
B
2
N
2
N
1
µ
3
µ
2
µ
3
µ
2
µ
1
N
1
N
2
µ
1
µ
3
2.852 Manufacturing Systems Analysis
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Copyright �2010 Stanley B. Gershwin.
c
Equivalence
Equivalence classes of four-machine systems
Representative members
2.852 Manufacturing Systems Analysis
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Copyright �2010 Stanley B... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/460acb49eb2c6f02fa3a7ba41481f026_MIT2_852S10_a_d_systems.pdf |
Basic Network Metrics and Operations
• Meshness ratio
• Degree correlation
– Joint degree distribution
– K-nearest neighbors
– Pearson degree correlation
• Rich club metric
• Degree-preserving rewiring
• Generating a graph that has a specified degree sequence
• Finding Pearson degree correlation
• Finding commu... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
Meshness of Random Networks
2/16/2011 Basic Network Metrics © Daniel E Whitney 1997-2010
4/45
CAIDA Paper on Internet Structure
• Nice review and comparison of many metrics
• Follows up early 2000s papers purporting to find the
structure of the internet
• Shows that there are three ways to do this, each
approxi... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
1 2 2
2 3 1
2 3 2
2 3 2
3 1 3
4 2 3
4 2 3
2
2
5
2
2
5
x = 2.2
y = 2.2
2/16/2011 Basic Network Metrics © Daniel E Whitney 1997-2010
7/45
node
x
y
1
1
2
2
2
3
4
4
5
5
average
2
2
3
3
3
1
2
2
2
2
2.2
3
2
1
2
2
3
3
3
2
2
x = 2.2
y = 2.2
pearson
-0.676753
Calculating x-bar
x =
sum of co... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
for Pearson (symmetric)
r =
∑(x − x)(y − y)
∑(x − x)2 ∑ (y − y)2
Look at numerator, ignore xbar for the moment
∑(xiy j )= xiδij y j = x Ax
'
'
δij = 1 if i links to j
δij = 0 if i does not link to j
Essentially the calculation is a quadratic form.
Pearsondir does the calculation for asymmetric networks
9/45
2/... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
11 Basic Network Metrics © Daniel E Whitney 1997-2010
12/45
Degree Distribution for V8 Engine
2/16/2011 Basic Network Metrics © Daniel E Whitney 1997-2010
13/45
K Nearest Neighbors for V8
Missing marks indicate
that there are no nodes
with that degree
2/16/2011 Basic Network Metrics © Daniel E Whitney 1997-201... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
© Daniel E Whitney 1997-2010
20/45
Degree-preserving Pair-wise Rewiring
• Picks two pairs of nodes at random and swaps their links
so that each node retains its nodal degree
• Usually used to randomize a network
– Rewire at random, a lot
• Can also be used to change a network’s degree correlation
or clustering ... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
5
2/16/2011 Basic Network Metrics © Daniel E Whitney 1997-2010
24/45
Finding Communities
• Big topic in social network analysis
• Many algorithms exist, based on different principles,
several in UCINET
• Recent one based on network flow by Newman and
Girvan: M. E. J. Newman and M. Girvan, Phys. Rev. E 69,
0261... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
(the same
% format as specified in UCINET).
QRecord2, dendrogramRecord, and MarkCut
A1=load('TEST.txt');
outputFileName1='Q_resultTEST';
outputFileName2='dendrogramTEST';
outputFileName3='CutSequenceTEST';
m=max(max(A1(:,1:2)));
% This code builds the adjacency matrix from the edgelist in TEST.txt
% You can change the ... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
00000e+00 1.0000000e+00 1.0000000e+00
2.0000000e+00 1.0000000e+00 1.0000000e+00
3.0000000e+00 1.0000000e+00 1.0000000e+00
4.0000000e+00 2.0000000e+00 2.0000000e+00
5.0000000e+00 3.0000000e+00 2.0000000e+00
6.0000000e+00 4.0000000e+00 2.0000000e+00
Q is based on density of
Links inside groups compared
To links between ... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
nodes 1 - 3 are in community 1 while 4 - 6 are isolates in communities 2 - 4 respectively. In column 3 we see that nodes 1 - 3 are
in community 1 while nodes 4 - 6 are in community 2.
1
1
1
6
5
2
3
6
5
2
3
6
5
2
3
4
4
4
Q = -0.18367
Q = 0.04081
Q = 0.20408
2/16/2011 Basic Network Metrics © Daniel E Whit... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
Erdos-Gallai theorem tests if a degree sequence is
“graphic” (routine isgraphic.m)
• Generating the graph is fraught and often ends up
incomplete or disconnected, or else it has some self-loops
and multiple edges between nodes
2/16/2011 Basic Network Metrics © Daniel E Whitney 1997-2010
31/45
Random Graph Reali... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
7-2010
33/45
random_graph.m
% Random graph construction routine with various models
% Gergana Bounova, October 31, 2005
function [adj] = random_graph(N,p,E,distribution,fun,degrees)
% INPUTS:
% N - number of nodes
% p - probability, 0<=p<=1
% E - fixed number of edges
% distribution - probability distributio... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
with permission.
2/16/2011 Basic Network Metrics © Daniel E Whitney 1997-2010
35/45
Example Calls to random_graph
random_graph(10)
random_graph(10,0.1,20)
random_graph(10,0,0,'normal')
random_graph(10,0,0,'custom',@mypdf)
degs = [3 1 1 1];
random_graph(10,0,0,'custom',@mypdf,degs)
2/16/2011 Basic Network Metrics © D... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
stop);
Nseqabs=abs(Nseq); %protect against negative values
Nseqint=int16(Nseqabs); %Volz routine requires integers
dlmwrite('degdist.txt',Nseqint,'\t') %Volz routine requires tab delimited input
!java -jar RandomClusteringNetwork.jar degdist.txt 100 .001 output.txt % n = 100, desired clust =
% if you use 0.0 for desire... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
not
%protect against disconnecting the network or isolating nodes.
%
% Inputs:
% nv - number of nodes
% p - rewiring probability
% Kreg - initial node degree of for regular graph (use 1 or even numbers)
%
% Output:
% G is a structure implemented as data structure in this as well as other
% graph theory algori... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
0
42/45
SFNG
• Text from the “read me:”
• B-A Scale-Free Network Generation and Visualization
• By Mathew Neil George
• The *SFNG* m-file is used to simulate the B-A algorithm and returns scale-
free networks of given sizes.
• Here is a small example to demonstrate how to use the code. This code creates
a seed ... | https://ocw.mit.edu/courses/esd-342-network-representations-of-complex-engineering-systems-spring-2010/4619e02fd83344d1ff1d22a6b9ca72d6_MITESD_342S10_lec06.pdf |
Lecture 3
Semiconductor Physics (II)
Carrier Transport
Outline
• Thermal Motion
• Carrier Drift
• Carrier Diffusion
Reading Assignment:
Howe and Sodini; Chapter 2, Sect. 2.4-2.6
6.012 Spring 2009
Lecture 3
1
1. Thermal Motion
In thermal equilibrium, carriers are not sitting still:
• Undergo collisions wi... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4692cf463514c4f47d8629fab7d41c15_MIT6_012S09_lec03.pdf |
and the
velocity is randomized:
net velocity�
in direction �
of field
τc
The average net velocity in direction of the field:
time
v = vd = ±
τ c = ±
qE
2m n,p
qτ c
2m n,p
E
This is called drift velocity [cm s-1]
Define:
µn, p =
qτ c
2m n,p
Then, for electrons:
and for holes:
≡ mobility [cm 2 V−1 s... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4692cf463514c4f47d8629fab7d41c15_MIT6_012S09_lec03.pdf |
∝∝∝∝carrier concentration
∝∝∝∝carrier charge
Drift current densities:
drift = −qnvdn = qnµ n E
Jn
drift = qpvdp = qpµ p E
J p
Check signs:
E
vdn
-
drift
Jn
E
vdp
+
drift
Jp
x
x
6.012 Spring 2009
Lecture 3
7
Total Drift Current Density :
drift
J
drift
= J n + Jp
drift
= q n( µ n + pµ p )E
Has ... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4692cf463514c4f47d8629fab7d41c15_MIT6_012S09_lec03.pdf |
1
Doping (cm-3)
6.012 Spring 2009
Lecture 3
9
Numerical Example:
Si with Nd = 3 x 1016 cm-3 at room temperature
µn ≈ 1000 cm 2 / V • s
ρρρρn ≈ 0.21Ω • cm
n ≈ 3X1016 cm −3
Apply E = 1 kV/cm
vdn ≈ −106 cm / s << vth
drift ≈ qnvdn = qnµnE = σσσσE =
Jn
E
ρρρρ
drift ≈ 4.8 ×103 A / cm 2
Jn
Time to drift t... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4692cf463514c4f47d8629fab7d41c15_MIT6_012S09_lec03.pdf |
1]
D measures the ease of carrier diffusion in response to a
concentration gradient: D ↑ ⇒ Fdiff ↑
D limited by vibration of lattice atoms and ionized dopants.
6.012 Spring 2009
Lecture 3
12
Diffusion Current
Diffusion current density =charge ××××carrier flux
diff = qDn
Jn
dn
dx
diff = −qDp
J p
dp
dx
Che... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4692cf463514c4f47d8629fab7d41c15_MIT6_012S09_lec03.pdf |
dp
dx
J total = J n + J p
6.012 Spring 2009
Lecture 3
15
What did we learn today?
Summary of Key Concepts
• Electrons and holes in semiconductors are mobile
and charged
– ⇒⇒⇒⇒ Carriers of electrical current!
• Drift current: produced by electric field
drift ∝ E
J
drift
J
∝
dφ
dx
• Diffusion current: p... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/4692cf463514c4f47d8629fab7d41c15_MIT6_012S09_lec03.pdf |
224 Chapter4. WavenumberIntegrationTechniques
perimentally [23], which should be kept in mind when comparing synthetic and exper-
imental reflection data.
4.5 Wavenumber Integration
Integral transform techniques such as wavenumber integration is an important mod-
eling tool in all disciplines dealing with wave propagati... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
icular displacement or stress component;
is the associated wavenumber
kernel. The order of the Bessel function is "#(cid:10)%$ , except for the horizontal displace-
ment and shear stress, Eqs. (4.36) and (4.39), where "&(cid:10)%’ . The numerical evaluation
of this integral is complicated by the following features, whi... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
WavenumberIntegration 225
of modal singularities only the relatively smooth continuous spectrum remains, being
less susceptible to discretization problems. Further, with the field required at only a few
receivers, the integration, and the associated sampling of the kernel, and therefore the
solution of the depth-separat... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
evaluation of the acoustic field at a large number of receiver
ranges. The FFT technique is also well suited to illustrate the discretization problem
because of the direct analogy to periodic solution to cylindrical problems. Then the
more direct numerical integrations schemes, based on either fixed or adaptively deter-
... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
wavefield at very short ranges and is therefore neglected.
Next we replace
(cid:8) by its asymptotic form [30],
(cid:3)(cid:18)(cid:17)(cid:20)(cid:19)
(cid:5)(cid:8)(cid:7)(cid:10)(cid:9)
(cid:19)(cid:8)(cid:20)(cid:22)(cid:21)
(cid:3)(cid:18)(cid:17)(cid:20)(cid:19)
)(+*
,/.
(cid:5)(cid:8)(cid:7)(cid:10)(cid:9)
’&
(ci... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
than the
Bessel function, particularly in terms of computation time. Since the numerical im-
plementations used in underwater acoustics are almost without exception based on the
fast-field approximation, we will focus on the evaluation of this integral in the follow-
ing. It should be pointed out, however, that the trun... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
(cid:1)
(cid:27)
(cid:0)
(cid:28)
&
(cid:30)
(cid:3)
(cid:24)
*
,
(cid:8)
-
,
(cid:16)
(cid:28)
(cid:19)
4.5. WavenumberIntegration 227
Instead, the wavenumber axis is truncated, allowing for numerical quadrature with-
out the complication of a semi-infinite integration interval. The reason for this ap-
proach being ap... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
, we can take advantage
of the oscillatory nature of the exponential function in Eq. (4.96). Thus, for
it
, even for source and receiver
will ensure the convergence of the integral for
at the same depth where the kernel alone is non-integrable. Therefore, the contribu-
tion to the integral beyond a certain wavenumber
(... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
)(cid:17)
(cid:1)(cid:25)(cid:8)
(cid:1)(cid:25)(cid:8)
(cid:28)(cid:31)(cid:30)!
(cid:6)(cid:5)(cid:8)(cid:7)
(cid:14)(cid:13)
(4.98)
It is well known from the discretization of time–frequency transforms that under-
sampling in one domain causes aliasing (wrap-around) in the other domain (see e.g.,
Ref. [31], Sec. 3.... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
0)
(cid:3)(cid:30)(cid:0)(cid:31)(cid:6)
(cid:2))(cid:3)*(cid:17)(cid:20)(cid:19)
(cid:17)(cid:20)(cid:19)
(cid:27)(cid:29)(cid:17)(cid:20)(cid:19)
15
(cid:1)(cid:9)(cid:8)(cid:11)(cid:10)
(cid:1)(cid:25)(cid:8)
32
(cid:30)4"
(4.101)
(cid:3)(cid:30)(cid:0)(cid:31)(cid:6)
(cid:1)(cid:9)(cid:8)
represents the entire field... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
:2)
(cid:10)
(cid:23)
(cid:10)
&
(cid:7)
(cid:12)
(cid:15)
(cid:16)
2
(cid:17)
(cid:28)
(cid:30)
(cid:19)
(cid:15)
"
$
5
4.5. WavenumberIntegration 229
In this form it is clear that adding (cid:0)
multiple of
yields a periodic solution,
to the range (cid:0) simply adds an integer
to the argument of the exponential fun... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
boundaries,
(cid:16)(cid:0)(cid:18)(cid:17)(cid:20)(cid:19)
(cid:13)(cid:10)
(cid:17)(cid:9)(cid:19)
(cid:1)(cid:7)(cid:0)
5(cid:31)5
. This property is the true culprit of the aliasing problem.
(cid:17)(cid:20)(cid:19)
(cid:1)(cid:9)(cid:8)
(cid:11)(cid:0)
(cid:8)(cid:1)(cid:7)(cid:0)
(cid:2)(cid:4)(cid:3)*(cid:17)(ci... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
these components are ignored here, the positive wavenumbers will contribute at
small negative ranges, as indicated in the figure, yielding a non-vanishing field which
will also wrap into the current window at (cid:0)
also will have to appear as negatively propagating waves at
in the neighbor window (cid:1)
.
(cid:13)(cid... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
:10)
(cid:25)
(cid:8)
5
(cid:6)
$
(cid:6)
5
5
(cid:8)
5
(cid:10)
(cid:0)
(cid:0)
(cid:11)
(cid:1)
(cid:27)
(cid:1)
(cid:23)
(cid:1)
(cid:27)
(cid:0)
(cid:10)
(cid:0)
(cid:11)
(cid:6)
(cid:1)
(cid:10)
(cid:25)
(cid:8)
5
(cid:8)
(cid:0)
(cid:10)
$
(cid:4)
(cid:10)
(cid:10)
$
$
(cid:6)
(cid:0)
(cid:10)
(cid:10)
’
(c... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
curve near the origin. The discrete wavenumber integration yields the
periodic result shown in the lower frame by a dashed curve, approximating the correct con-
tinuous result shown as a solid curve. The discrete result is a superposition of the ’true’ field
produced by the mirror sources in all the range windows.
(cid:... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
which is attenuated enough to
reduce the periodic multiples to insignificance.
from both sides of the actual interval and therefore also from ranges smaller than (cid:0)
If
Because of the two-sided nature of the discrete Fourier transform, aliasing occurs
(cid:16)(cid:0)(cid:18)(cid:17)(cid:20)(cid:19) .
. are wrapped i... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
)
(cid:6)
(cid:2)
(cid:6)
(cid:10)
(cid:23)
(cid:4)
(cid:1)
(cid:27)
(cid:0)
"
(cid:9)
(cid:19)
&
(cid:5)
(cid:24)
,
,
.
&
(cid:7)
(cid:12)
(cid:15)
(cid:16)
(cid:15)
(cid:6)
(cid:28)
(cid:30)
(cid:19)
(cid:9)
(cid:15)
(cid:17)
"
$
2
(cid:17)
(cid:28)
(cid:30)
,
-
(cid:8)
(cid:12)
(cid:6)
232 Chapter4. WavenumberInteg... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
)
(cid:25)(cid:17)
(cid:4)(cid:19)
(cid:1)(cid:15)
(cid:5)(cid:14)(cid:13)
(cid:9)(cid:18)(cid:9)
"%$
"%$
"%$
(4.109)
(cid:17)(cid:20)(cid:19)
(cid:17)(cid:20)(cid:19)
(cid:3)(cid:2)
(cid:8)(cid:0)
(cid:16)(cid:0)(cid:18)(cid:17)(cid:20)(cid:19)
(cid:1)(cid:0)(cid:18)(cid:17)(cid:20)(cid:19)
Therefore, for
the upper ha... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
��nite contribution from the
small wavenumber components, and more inportantly the numerical artifact of the dis-
continuity of the kernel at
factor introduced by the
FFP approximation forces the kernel to vanish, the derivatives remain discontinuous,
and an artificial backward propagating field will result, which will b... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
4. WavenumberIntegrationTechniques
peak in Fig. 4.6(a) clearly introduces errors of up to 2 dB in the predicted transmission
loss, with the largest errors at longer ranges. However, even at short ranges the aliasing
introduces errors in the modal interference pattern.
4.5.5 Complex Contour Integration
(cid:14)(cid:23)
... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
particularly in cases where the field is needed only at relatively short ranges.
The aliasing problem can, however, be eliminated by moving the integration con-
tour out into the complex plane. According to Cauchy’s theorem the integral in the
complex plane between two points is invariant to a change in the integration ... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
(cid:4)
(cid:17)
0
(cid:17)
(cid:25)
(cid:17)
(cid:8)
(cid:17)
(cid:10)
(cid:17)
(cid:25)
(cid:8)
(cid:17)
0
(cid:19)
(cid:8)
4.5. WavenumberIntegration 235
B r a n c h c u t f o r k z =
2
2
mk -kr
-k m
*
C
1
m
P o l e s
k
*
C
2
k r
C
3
k
m i n
k m a x
Fig. 4.7. Complex integration contours for evaluation of wavenumbe... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
:3)
(cid:19)
0
(cid:19)
(cid:8)
(cid:6)
(cid:10)
(cid:23)
(cid:28)
(cid:5)
(cid:3)
(
(cid:9)
(cid:9)
(cid:19)
(cid:8)
(cid:19)
(cid:8)
(cid:10)
&
(cid:7)
(cid:12)
(cid:15)
(cid:16)
(cid:15)
(cid:25)
(cid:9)
(cid:5)
(cid:6)
(cid:28)
(cid:30)
(cid:19)
(cid:9)
(cid:15)
"
$
2
(cid:17)
(cid:15)
(cid:25)
(cid:9)
(cid:5)
(cid... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
) . On the other hand, signals wrapped around
(cid:8) . As was the case
also for the
(cid:0)(cid:3)(cid:2)(cid:5)(cid:4)
The explanation for this is as follows. The contour offset moves the integration
path away from singularities such as branch points and modes resulting in a similar
but smoother integration kernel. I... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
vertical contours.
Figure 4.8 illustrates the effects of using the complex integration contour for eval-
uation of the field in the Pekeris waveguide treated above. Figure 4.8(a) shows the
magnitude of the kernel of the summation in Eq. (4.114) for two contour offsets de-
fined by Eq. (4.115). The solid curve is the kern... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
)
(cid:0)
(cid:28)
(cid:3)
(cid:19)
(cid:5)
(cid:10)
(cid:11)
(cid:0)
(cid:19)
(cid:28)
(cid:10)
(cid:11)
(cid:1)
(cid:27)
(cid:3)
(cid:24)
(cid:25)
’
(cid:8)
(cid:19)
(cid:28)
(cid:25)
(cid:17)
(cid:8)
(cid:6)
(cid:6)
(cid:17)
(cid:25)
(cid:17)
(cid:10)
’
$
(cid:1)
(cid:10)
’
(cid:1)
(cid:10)
’
$
(cid:1)
(cid:10)
’
(c... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
Chapter4. WavenumberIntegrationTechniques
(cid:17)(cid:20)(cid:19)
out to a range of approximately 10 km, after which the result shows increasing errors.
These errors appear in spite of the fact that the maximum range for this sampling is
(cid:14) km. This is due to the fact that even though we have removed
(cid:8) , t... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
Field Program (FFP) approach described above has gained popularity because
of its efficiency in producing field estimates at a large number of ranges. However,
being based on the large argument asymptotic of the Hankel functions it is associated
with errors for small arguments (cid:17)(cid:9)(cid:19)
(cid:0) of the Besse... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
23)
4.5. WavenumberIntegration 239
an FFT for every receiver range, but, more importantly, it requires a numerical separa-
tion parameter which is not easily selected. The so-called Fast Hankel Transform [25]
is very efficient for relatively smooth kernels, but not well-suited for the rapidly vary-
ing kernels of waveg... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
26)(cid:0)
(cid:3)*(cid:17)(cid:20)(cid:19)
(cid:27)(cid:29)(cid:17)(cid:20)(cid:19)
3(cid:11)
(4.116)
(cid:5)(cid:18)(cid:7)(cid:10)(cid:9)
(cid:5)(cid:14)(cid:13)(cid:15)(cid:9)
(cid:1)(cid:0)(cid:3)(cid:2)
(cid:3)(cid:6)(cid:5)
for (cid:2)
time-dependence corresponds to outgoing waves and
where
to
incoming waves, bo... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
:9)
$
(cid:8)
2
"
$
(cid:13)
(cid:19)
(cid:28)
&
(cid:30)
(cid:5)
(cid:24)
,
,
(cid:9)
$
240 Chapter4. WavenumberIntegrationTechniques
0
10
−1
10
−2
10
−3
10
|
)
r
k
(
J
|
m
−4
10
−5
10
−6
10
−7
10
0
10
20
30
40
50
kr
60
70
80
90
100
Fig. 4.9. Error of far-field approximation of (cid:0)
approximation, Dashed: Error.
(c... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
3) it is only necessary to cor-
rect the contributions corresponding tovalues of
. This is performed
in a numerically stable manner by a Hanning weighted average of the contributions of
the exact Bessel function and the approximate FFP kernel,
(cid:9)(cid:7) KR (cid:10)
for
(cid:3)(cid:18)(cid:17)(cid:20)(cid:19)
(cid:... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
.121) can be evaluated very efficiently. First of all, with the wavenumber
and range sampling constrained by Eq. (4.107), all values of the exponentials are com-
puted as part of the FFT evaluation of Eq. (4.118). Secondly, the Bessel functions will
only be needed for a limited number of discrete values of the argument,... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
this part of the range window.
even, antisymmetric for "
(cid:0)(cid:3)(cid:2)(cid:5)(cid:4)
(cid:17)(cid:20)(cid:19)
The performance of this ’Fast Hankel Transform’ is illustrated by Fig. 4.11, which
shows the evaluation of the Hankel transform
(cid:3)(cid:30)(cid:0)(cid:31)(cid:6)
(cid:17)(cid:20)(cid:19)
(cid:3)(cid... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
5
(cid:1)
(cid:3)
(cid:23)
(cid:23)
(cid:0)
(cid:8)
(cid:6)
(cid:4)
(cid:10)
(cid:0)
(cid:1)
(cid:1)
(cid:10)
(cid:27)
(cid:1)
(cid:23)
(cid:0)
(cid:12)
(cid:14)
(cid:16)
(cid:28)
(cid:9)
(cid:17)
(cid:4)
(cid:22)
(cid:16)
(cid:0)
(cid:8)
(cid:4)
(cid:10)
2
(cid:3)
(cid:0)
(cid:1)
(cid:8)
(cid:13)
(cid:25)
(cid:17)
(ci... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
.121)
the array, often requiring a wavenumber sampling which is fixed for all frequencies,
which is not optimal.
As a consequence broad-band and discrete array computations are in general opti-
mally being performed by direct numerical quadrature schemes such as the trapezoidal
rule integration. This scheme approximates... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
, for example they do not have to use an
integration contour parallel to the real axis such as the contour C
in Fig. 4.7, but can
instead apply an ’exact’ Cauchy contour such as the one shown by the dashed line in
Fig. 4.7, totally eliminating the contributions from the two vertical contour sections.
This is particular... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
rule integration, for the omni-
directional field components, "
$ :
(cid:2)(cid:4)(cid:3)(cid:30)(cid:0)
(cid:2)(cid:4)(cid:3)*(cid:17)(cid:25)(cid:19)
(cid:3)*(cid:17)(cid:20)(cid:19)(cid:26)(cid:0)
(cid:17)(cid:20)(cid:19)
(cid:3)(cid:18)(cid:17)(cid:20)(cid:19)(cid:26)(cid:0)
(cid:10)(cid:13)(cid:12)
(cid:17)(cid:25)... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
:27)
(cid:8)
(cid:1)
(cid:3)
’
$
(cid:27)
(cid:8)
(cid:8)
.
(cid:1)
(cid:1)
’
$
(cid:27)
(cid:15)
(cid:0)
(cid:15)
(cid:1)
$
(cid:27)
$
(cid:0)
(cid:0)
(cid:1)
$
(cid:27)
(cid:0)
(cid:10)
(cid:1)
$
(cid:27)
(cid:27)
(cid:1)
(cid:1)
$
4.5. WavenumberIntegration 245
4.5.8 Filon Integration
While the trapezoidal rule int... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
23)
(cid:8) .
scheme is exact for linear variations of the kernel (cid:21)
In the present case (cid:24)
(cid:17)(cid:20)(cid:19) , i.e., the exponent is inherently a linear
function of (cid:17)(cid:25)(cid:19) . For the equidistant sampling given in Eq. (4.97), it is easily shown that
the Filon integration scheme leads... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
)(cid:2)
(cid:17)(cid:20)(cid:19)
(cid:4)(cid:2)
(4.131)
(cid:17)(cid:9)(cid:19)
Here it is interesting to note that Eq. (4.130) is identical to Eq. (4.108) except for the
simple change in integration weight from (cid:23)
, basically applying a sinc-function
squared to the field amplitude vs range. The summation can aga... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
13)
5
(cid:2)
246 Chapter4. WavenumberIntegrationTechniques
(a)
(b)
k1
k 2
k1
k 2
Fig. 4.12. Adaptive evaluation of wavenumber integral.
, the Filon scheme requires a sampling which is approxi-
inversely proportional to
mately inversely proportional to
. However, they considered seismic reflectivity
problems characteri... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
These integration problems may be overcome by an adaptive selection of the wave-
number sampling. Shown in Fig. 4.12 is an example of such an adaptive scheme, de-
veloped by Krenk et al. [34]. Here, the kernel is first sampled on a coarse wavenumber
grid, which is then subsequently subdivided by bisection, until a stabl... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
the same issues of windowing and sampling must be properly addressed.
Since the evaluation of the frequency integral is common to all numerical approaches
solving the Helmholtz equation, the associated numerical issues will be deferred to
Chap. 8. However, we present time-domain calculations also here (Sec. 4.8.4), due... | https://ocw.mit.edu/courses/2-068-computational-ocean-acoustics-13-853-spring-2003/469b7802dd486b13460b7edaf403f9ed_wavint.pdf |
Lecture 1
Overview of some probability
distributions.
In this lecture we will review several common distributions that will be used often throughtout
the class. Each distribution is usually described by its probability function (p.f.) in the case
of discrete distributions or probability density function (p.d.f.) i... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
in this case for any a
expectation of �(X) is defined by
∞
R we have P(X = a) = 0. Given a function � :
R, the
X �
E�(X) =
�
�(x)p(x)dx.
�
−�
Notation. The fact that a random variable X has distribution P will be denoted by
P.
Normal (Gaussian) Distribution N(�, π2). Normal distribution is a continuous dis
X... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
)/π. This, of course,
−
EY =
�
�
−�
y
1
≤2ϕ
2 y
2 dy = 0
e−
since the integrand is an odd function. To compute the second moment EY 2 , let us first note
1
is a probability density function, it integrates to 1, i.e.
that since �
2
y
2
2� e−
If we integrate this by parts, we get,
�
−�
�
1 =
1
≤2ϕ
2 y
d... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
π2 is a variance of a normal distribution. Let us
recall (without giving a proof) that if we have several, say n, independent random variables
N (�i, π2) then their sum will also have a normal distribution
Xi, 1
i
n, such that Xi �
≈
≈
X1 + . . . + Xn �
N(�1 + . . . + �n, π1
2 + . . . + π2 ).n
Normal distribut... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
.
Bernoulli Distribution B(p). This distribution describes a random variable that can
. The distribution is described by a probability
0, 1
}
{
take only two possible values, i.e.
function
=
X
p(1) = P(X = 1) = p, p(0) = P(X = 0) = 1
−
p for some p
[0, 1].
∞
It is easy to check that
EX = p, Var(X) = p(1
−... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
0 is
�
E(�) then
P(X
∼
t) = P(X
[t,
→
∞
)) =
t
�
�
�e−
�xdx = e−
�t .
Given another s > 0, the conditional probability that X will exceed level t + s given that it
will exceed level t can be computed as follows:
P(X
t + s X
|
∼
∼
t) =
P(X
t + s, X
∼
P(X
t)
∼
�t = e−
�(t+s)/e−
∼
= e−
t)
=
P(X
... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
.
Poisson distribution could be used to describe the following random objects: the number
of stars in a random area of the space; number of misprints in a typed page; number of
wrong connections to your phone number; distribution of bacteria on some surface or weed
in the field. All these examples share some common ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
this interval into a large number n of small equal subintervals of length T /n and denote
by Xi the number of random objects in the ith subinterval, i = 1, . . . , n. By the first property
above,
EXi =
�T
.
n
On the other hand, by definition of expectation
EXi =
kP(Xi = k) = 0 + P(Xi = 1) + αn,
0
k
�
�
k
2 kP(... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
λ]. This distribution has probability density function
1
λ , x
[0, λ],
∞
0, otherwise.
p(x) =
�
Matlab review of probability distributions.
Matlab Help/Statistics Toolbox/Probability Distributions.
Each distribution in Matlab has a name, for example, normal distribution has a name
’norm’. Adding a suffix defines ... | https://ocw.mit.edu/courses/18-443-statistics-for-applications-fall-2006/46afee1cbe70839c128ef68b2a6b1e16_lecture1.pdf |
MIT 6.972 Algebraic techniques and semidefinite optimization
February 28, 2006
Lecturer: Pablo A. Parrilo
Scribe: ???
Lecture 6
Last week we learned about explicit conditions to determine the number of real roots of a univariate
polynomial. Today we will expand on these themes, and study two mathematical objects o... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
• Sylvester matrix: If p(x0) = q(x0) = 0, then we can write the following (n + m) × (n + m) linear
system:
⎡
pn
pn−1
p
n
. . .
qm qm−1
qm
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
. . . p1
.
.
.
p0
.
.
.
.
.
.
. . . q0
.
.
.
.
.
.
. . .
⎤
⎡
n+m−1
x
⎥
0
⎥
n+m−2
x
⎥
0 ⎢
⎥
⎢
. . .
⎥
⎢
⎥
⎢
p1 p0
⎥
⎢
... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
⎥
⎥
⎥
⎥
⎥
⎦
= 0.
This implies that the matrix on the lefthand side, called the Sylvester matrix Sylx(p, q) associated
to p and q, is singular and thus its determinant must vanish. It is not too difficult to show that
the converse is also true; if det Sylx(p, q) = 0, then there exists a vector in the kernel of Sylx(p,... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
be shown that all these constructions are equivalent. They define exactly the same polynomial,
called the resultant of p and q, denoted as Resx(p, q):
Resx(p, q) = det Sylx(p, q)
n
= p m det q(Cp)
n
= (−1)nm q det p(Cq )
m
m= pn q n det(Cp ⊗ Im − I
m
n ⊗ Cq ).
The resultant is a homogeneous multivariate polynomia... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
.
Definition 2. The discriminant of a univariate polynomial p(x) is defined as
Disx(p) := (−1)n(n−1)/2 Resx p(x),
1
pn
�
�
.
dp(x)
dx
Similar to what we did in the resultant case, the discriminant can also be obtained by writing a
natural condition in terms of the roots αi of p(x):
Disx(p) = p 2n−2
n
�
(αj ... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
two univariate polynomials p(x, y0), q(x, y0). If y0 corresponds to the ycomponent of a root, then
these two univariate polynomials clearly have a common root, hence their resultant vanishes.
Therefore, to solve (2), we can compute Resx(p, q), which is a univariate polynomial in y. Solving
this univariate polynomia... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
One of the main reasons why nonnegativity conditions about polynomials are difficult is because these
sets can have a quite complicated structure, even though they are always convex.
Recall from last lecture that we have defined Pn ⊂ Rn+1 as the set of nonnegative polynomials of
degree n. It is easy to see that if p(x)... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
+ b. For what values of a, b does it
hold that p(x) ≥ 0 ∀x ∈ R? Since the leading term x4 has even degree and is strictly positive, p(x) is
strictly positive if and only if it has no real roots. The discriminant of p(x) is equal to 256 b (a − b)2 .
2
Here is a slightly different example, showing the same phenomenon. ... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
511.52a-1-0.50.511.52b-3-2-1123a-1123b0220444-3-2-1123a-1123b Figure 3: A threedimensional convex set, described by one quadratic and one linear inequality, whose
projection on the (a, b) plane is equal to the set in Figure 1.
One has to do with its algebraic structure, and the other one with convexity, and in parti... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
projection on the plane (a, b) is exactly
the one discussed in Example 5 and Figure 1.
The presence of “extraneous” components of the discriminant inside the feasible set is an important
roadblock for the availability of “easily computable” barrier functions. Indeed, every polynomial that
vanishes on the boundary o... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
how useful this approach is in practical
optimization problems.
1A face of a convex set S is a convex subset F ⊆ S, with the property that x, y ∈ S, 1
F is exposed if it can be written as F = S ∩ H, where H is a supporting hyperplane of S.
2 (x + y) ∈ F ⇒ x, y ∈ F . A face
65
−20246024012345abt Figure 4: The dis... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
Selfconcordant barriers for cones generated by Chebyshev systems. SIAM J.
Optim., 12(3):770–781 (electronic), 2002.
[KM02] C. Y. Kao and A. Megretski. A new barrier function for IQC optimization problems. In
Proceedings of the American Control Conference, 2002.
[RG95] M. Ramana and A. J. Goldman. Some geometric re... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/46bf636ecc6195991e396703bfe1a1ae_lecture_06.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.917 Topics in Algebraic Topology: The Sullivan Conjecture
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Steenrod Operations (Lecture 2)
The objective of today’s lecture is to introduce the Steenrod operations a... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
�n,
by permuting the tensor factors.
One of the most important examples of an F2-module spectrum is the cochain complex
C ∗(X; F2)
of a topological space X. The cohomology groups of this F2-module spectrum are simply the cohomology
groups of X. The cohomology H∗(X; F2) has the structure of a graded commutative rin... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
nth extended power
of V is given by the homotopy coinvariants
⊗
Vh
Σ
n
n
.
This is a complex which we will denote by Dn(V ).
Remark 3. In concrete terms, Dn(V ) may be computed in the following way. Let M denote the vector
space F2, with the trivial action of Σn. Choose a resolution
. . .
→
P −1
→ →
P 0 M
b... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
over the field F2. Then A has an underlying F2-module spectrum,
which is equipped with a symmetric multiplication.
∞
Our goal in this lecture is to study the consequences of the existence of a symmetric multiplication on a
complex V .
Notation 8. Let n be an integer. We let F2[−n] denote the complex which consists o... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
(unique) isomorphism
H∗(RP ∞; F2) � F2[t],
where the polynomial generator t lies in H1(RP ∞; F2). We have a dual description of the homology
H (RP ∞; F2): this is just a one-dimensional vector space in each degree m, with a unique generator which
we will denote by xm.
∗
Definition 9. Let V be a complex, and let v ∈... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
respect to the multiplication on
V . This is why the operations Sqi are called “Steenrod squares”.
Example 11. Let X be a topological space, and let V = C ∗(X; F2) be the cochain complex of X, equipped
with its usual symmetric multiplication. Then Definition 9 yields operations
Sqi : Hn(X; F2)
→
Hn+i(X; F2).
These ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
(v + v�) = Sq
k
(v) + Sq
(v�) ∈ Hn+k D2(V ).
In particular, if V is equipped with a symmetric multiplication, we have
Sqk(v + v�) = Sqk(v) + Sqk(v�) ∈ Hn+k V.
Proof. If k > n, then both sides are zero and there is nothing to prove. If k = n, then
k
Sq
k
(v + v�) = (v + v�)2 = Sq
k
(v) + Sq
(v�) + (vv� + v�v).
Sinc... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/46c2e62c9fe4730096ed92d58d85783b_lecture2.pdf |
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