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2 Pages

### sub

Course: MATH 132, Fall 2009
School: University of Iowa
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Word Count: 414

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A Denition: subbasis S for a topology on X is a collection of subsets of X whose union equals X. Lemma: If S is a subbasis for a topology on X, then B = {n Si | Si S} is a basis for a topology. i=1 Note that B is a collection of subsets of X (Si X implies n Si X). i=1 (1) Show for each x X, there is at least one basis element B containing x. Proof: Take x X. Find B B such that x B. We will rst show that S...

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A Denition: subbasis S for a topology on X is a collection of subsets of X whose union equals X. Lemma: If S is a subbasis for a topology on X, then B = {n Si | Si S} is a basis for a topology. i=1 Note that B is a collection of subsets of X (Si X implies n Si X). i=1 (1) Show for each x X, there is at least one basis element B containing x. Proof: Take x X. Find B B such that x B. We will rst show that S B: Suppose that S1 S. Then S1 = 1 S1 . Thus S1 is a n=1 nite intersection of elements of S since 1 is a nite number. Hence S1 B. Since S is a collection of subsets of X whose union equals X, there exists an S S B such that x S. Hence, there is at least one basis element B containing x. (2) Show that if x B1 B2 where B1 , B2 B, then there exists B3 B that such x B 3 B 1 B2 . 1 The topology T generated by a basis B is dened as follows: U is open if and only if for all x U , there exists B B such that x B U . Negation: U is NOT open in T i the following is false: for all x U , there exists B B such that x B U. U is NOT open in T i there exists an x U , such that the following is false: there exists B B such that x B U . U is NOT open in T i there exists an x U , such that for all B B the foll...

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University of Iowa - MATH - 132
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University of Iowa - MATH - 133
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San Diego State - INCLASS - 240
How life can change in a blink of an eye.A Biography BY Kristen LohmanA Biography BY Kristen LohmanJoanie Rae Smith has spent her high school years in a small town where she grew up. The teenage bully that she was, she still managed to be the
University of Iowa - MATH - 133
Randell's Submanifolds (2.3) = Boothby's Regular submanifold (III.5): If K is a submanifold of M , then K has the subspace topology. If ( , U ) is a chart for M such that (U ) = m (-, ) 1 m k and (U K) = 1 (-, ) k+1 {0} , then ( |U K , U K), |U
University of Iowa - MATH - 133
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University of Iowa - MATH - 133
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University of Iowa - MATH - 133
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University of Iowa - MATH - 133
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University of Iowa - MATH - 133
B is bilinear if B(cv1 + dv2 , w) = cB(v1 , w) + dB(v2 , w) B(v, cw1 + dw2 ) = cB(v, w1 ) + dB(v, w2 ) Thus B(v1 , w1 ) + (v2 , w2 ) = B(v1 + v2 , w1 + w2 ) = B(v1 , w1 + w2 ) + B(v2 , w1 + w2 ) = B(v1 , w1 ) + B(v1 , w2 ) + B(v2 , w1 ) + B(v2 , w2 )
University of Iowa - MATH - 133
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University of Iowa - MATH - 133
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University of Iowa - MATH - 133
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University of Iowa - MATH - 133
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San Diego State - INCLASS - 240
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VACATIONSara ZunigaBOOK COVER.indd 14/9/08 2:39:23 PM
San Diego State - DESIGN - 240
V C TO A A INSr Zn a aa u i g
San Diego State - DESIGN - 240
San Diego State - DESIGN - 240
University of Iowa - MATH - 150
\magnification 1200\parskip 9pt\parindent 0pt\hsize 7.4truein\hoffset -0.5truein\vsize 9.6trueinEx: Solve the recurrance relation: $h_n + h_{n-2} = 0$$h_0 = 3, h_1 = 5$$q^n + q^{n-2} = 0$$q^2 + 1 = 0$$q = \pm i$$h_n = c_1 i^n + c_2 University of Iowa - MATH - 150 7.4: Generating Functions g(x) = h0 + h1 x + h2 x2 + . is the generating function for the sequence h0 , h1 , h2 , . Ex: The generating fn for the sequence 2, 3, 4, 0, 0, 0, . is g(x) = 2 + 3x + 4x2 Ex: The generating function for the sequence 1, 1, 1 University of Iowa - MATH - 150 7.6: A geometry example Thm 7.6.1: Let hn = the number of ways of dividing a convex polygonal region with n + 1 sides into triangular regions by inserting diagonals which do not intersect in the interior of the polygonal region. Define h1 = 1. Then h University of Iowa - MATH - 151 2.3: Communication NetworkQuestion: What (and who) are the minimum number of people you need to give a message in order for the message to reach everyone one else in the network.Answer: Find a vertex basis.Defn: A vertex basis of$G = (V, A)
University of Iowa - MATH - 151
\magnification 2200\parindent 0pt\parskip 5pt\nopagenumbers\hsize 7.5truein\vsize 9.9 truein\hoffset -0.45truein\def\u{\vskip -7pt}\def\v{\vskip -5pt}\input ./././PAPERS/GOLD/psfig2.4: Digraphs and matricesRecall if $A = (a_{ij})$ and
San Diego State - DESIGN - 240
Untitled-3 14/9/08 2:15:52 PM
University of Iowa - MATH - 151
&lt;?xml version=&quot;1.0&quot; encoding=&quot;UTF-8&quot;?&gt;&lt;Error&gt;&lt;Code&gt;NoSuchKey&lt;/Code&gt;&lt;Message&gt;The specified key does not exist.&lt;/Message&gt;&lt;Key&gt;4ee54efa23a4a4d41bd912b7025f1647e12e7d69.txt&lt;/Key&gt;&lt;RequestId&gt;B2BC79CD71220871&lt;/RequestId&gt;&lt;HostId&gt;eJqdwIyKvSHa1JuzRAxsKH7nu7In
University of Iowa - MATH - 151
\magnification 2400\parindent 0pt\parskip 8pt\nopagenumbers\hsize 7.5truein\vsize 9.9 truein\hoffset -0.45truein\def\u{\vskip -7pt}\def\v{\vskip -5pt}\input ./././PAPERS/GOLD/psfigSuppose $D^*$ is the condensation of $D= (V, A)$. L
San Diego State - DESIGN - 240
San Diego State - DESIGN - 240
SARA ZUNIGAVACATIONVACATIONBOOKCOVER VERSION 2.indd 14/14/08 2:30:49 PM
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'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:02:43.894 20020308-1-001_133740.jpg [001] Chapter 7- Local Area Networks (LAN): The Basics002: 00:02:43.894 00:00:03.277 20020308-1-002_133743.jpg [002] Acknowledgement00
Cal Poly - SESSION - 20020309
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:05:24.878 20020309-1-001_134126.jpg [028] Wireless Topology002: 00:05:24.878 00:00:54.167 20020309-1-002_134220.jpg [029] Wireless Topology003: 00:06:19.046 00:08:26.199
Cal Poly - SESSION - 20020208
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:00:46.433 20020208-2-001_144123.jpg [027] Mobile Telephone002: 00:00:46.433 00:03:21.152 20020208-2-002_144444.jpg [028] Mobile Telephone003: 00:04:07.585 00:00:09.339 20
Cal Poly - SESSION - 20020309
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:04:17.789 20020309-2-001_145730.jpg [039] Round Robin Protocols002: 00:04:17.789 00:04:06.175 20020309-2-002_150136.jpg [040] Round Robin Protocols003: 00:08:23.964 00:03
Cal Poly - SESSION - 20020125
&lt;?xml version=&quot;1.0&quot; encoding=&quot;UTF-8&quot;?&gt;&lt;Error&gt;&lt;Code&gt;NoSuchKey&lt;/Code&gt;&lt;Message&gt;The specified key does not exist.&lt;/Message&gt;&lt;Key&gt;3318be69bf697b7785f830d09d14ddc8aa14feda.txt&lt;/Key&gt;&lt;RequestId&gt;CD486CD5F798E35A&lt;/RequestId&gt;&lt;HostId&gt;txBY3THD8q7WvWEE1Em/Vz+UUUjY
Cal Poly - SESSION - 20020309
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:05:24.878 20020309-1-001_134126.jpg [028] Wireless Topology002: 00:05:24.878 00:00:54.167 20020309-1-002_134220.jpg [029] Wireless Topology003: 00:06:19.046 00:08:26.199
Cal Poly - SESSION - 20020208
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:13:54.871 20020208-1-001_140030.jpg [001] The Media: Conducted and Wireless002: 00:13:54.871 00:00:02.933 20020208-1-002_140033.jpg [002] Acknowledgement003: 00:13:57.804
Cal Poly - SESSION - 20020125
' Slide timing and title information ' Slide#: Start_Time Duration File_Name Title 01: 00:00:00.000 00:01:01.000 20020125-1-001_134639.jpg [001] Data Communications and Computer Networks02: 00:01:01.000 00:00:16.000 20020125-1-002_134740.jpg [002]
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Cal Poly - SESSION - 20020322
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UNC - ECON - 001
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Cal Poly - SESSION - 20020413
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:06:23.992 22020413-2-001_145611.jpg [029] The Internet Protocol (IP)002: 00:06:23.992 00:01:22.179 22020413-2-002_145734.jpg [030] The Internet Protocol (IP)003: 00:07:46
San Diego State - DESIGN - 240
KimonoSteelSteelKimono.indd 14/14/08 8:02:12 AM
Cal Poly - SESSION - 20020223
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:00:55.330 20020223-1-001_080526.jpg [001] + No Title +002: 00:00:55.330 00:00:01.902 20020223-1-002_080527.jpg [002] Acknowledgement003: 00:00:57.232 00:01:44.339 20020
Cal Poly - SESSION - 20020222
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:07:49.457 20020222-1-001_135003.jpg [001] + No Title +002: 00:07:49.457 00:00:02.652 20020222-1-002_135006.jpg [002] Acknowledgement003: 00:07:52.109 00:03:14.300 20020
Cal Poly - SESSION - 20020413
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:06:23.992 22020413-2-001_145611.jpg [029] The Internet Protocol (IP)002: 00:06:23.992 00:01:22.179 22020413-2-002_145734.jpg [030] The Internet Protocol (IP)003: 00:07:46
Cal Poly - SESSION - 20020309
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:04:17.789 20020309-2-001_145730.jpg [039] Round Robin Protocols002: 00:04:17.789 00:04:06.175 20020309-2-002_150136.jpg [040] Round Robin Protocols003: 00:08:23.964 00:03
Cal Poly - SESSION - 20020322
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:09:38.339 20020322-2-001_150740.jpg [020] Switches002: 00:09:38.339 00:00:37.105 20020322-2-002_150818.jpg [021] Switches003: 00:10:15.445 00:00:24.792 20020322-2-003_150
Cal Poly - SESSION - 20020208
'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:00:46.433 ise-el536-20020208-2-001_144123.jpg [027] Mobile Telephone002: 00:00:46.433 00:03:21.152 ise-el536-20020208-2-002_144444.jpg [028] Mobile Telephone003: 00:04:07
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