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### hw04

Course: M 310, Fall 2009
School: Sveriges...
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Word Count: 289

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310 Math Homework #4 Cover Sheet name: tutorial, check one: T9:30; T10:30; T11:30; R10:30; R11:30; R12:30. begin each problem on a new page &amp; clearly identify each question. use words to describe your procedures &amp; to interpret your results. put boxes around your nal results. due on friday 04 september at start of lecture. question # CONCEPT keywords &amp; MAIN formula/result #...

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Coursehero >> Other International >> Sveriges lantbruksuniversitet >> M 310

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310 Math Homework #4 Cover Sheet name: tutorial, check one: T9:30; T10:30; T11:30; R10:30; R11:30; R12:30. begin each problem on a new page & clearly identify each question. use words to describe your procedures & to interpret your results. put boxes around your nal results. due on friday 04 september at start of lecture. question # CONCEPT keywords & MAIN formula/result # 3.1.30 concept result # 3.2.14 # 3.2.15 # 3.3.12 # 3.4.21 # 3.4.23 Math 310 Second-Order Linear ODEs Homework %4 homework portfolios will also be graded on completeness & presentation. certain problems will be designated as practice problems; and although not subject to submission, will be assumed to have been covered for purposes of examinations. unless otherwise stated, numbered problems refer to Boyce/DiPrima, 7th edition. Section 3.1 practice: # 28, 29 #30 identify ALL possible solutions. Be systematic in your presentation is, that can you show that you have ALL possible solutions (just because you have found the ones in the back of the text is not a logical demonstration). Section 3.2 practice: # 5-10, 13, 23-26 (there are a lot of important concepts in this section.) #14/15 present as one combined probem. These are conceptual (as opposed to mechanical) questions, explain your reasons & conclusions clearly (you must be using the right keywords here). Section 3.3 practice: # 17-20 #12 again, you must clearly explain your ideas here. Section 3.4 practice: # 1-4, 7-10, 17-19 (im...

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UMiami - MTH - 309
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San Diego State - ART - 448
About type and type layersWhen you create type, a new type layer is added to the Layers palette. Important: Type layers aren't created for images in Multichannel, Bitmap, or Indexed Color mode, because these modes don't support layers. In these mode
San Diego State - ART - 448
Entering typeThere are three ways to create type: at a point, inside a paragraph, and along a path. Point type is a horizontal or vertical line of text that begins where you click in the image. Entering text at a point is a useful way to add a few w
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Apply anti-aliasing to a type layerAnti-aliasing produces smooth-edged type by partially filling the edge pixels. As a result, the edges of the type blend into the background.Anti-aliasing set to None (left), and Strong (right) When creating type
San Diego State - ART - 448
Select characters1. Select the Horizontal Type tool or the Vertical Type tool .2. Select the type layer in the Layers palette, or click in the text to automatically select a type layer. 3. Position the insertion point in the text, and do one of th
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University of Illinois, Urbana Champaign - MATH - 430
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University of Illinois, Urbana Champaign - MATH - 430
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University of Illinois, Urbana Champaign - MATH - 430
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University of Illinois, Urbana Champaign - MATH - 430
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University of Illinois, Urbana Champaign - MATH - 430
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University of Illinois, Urbana Champaign - MATH - 430
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University of Illinois, Urbana Champaign - MATH - 430
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University of Illinois, Urbana Champaign - MATH - 430
Homology and winding numbers22 SeptemberLet U be an open set, and let c= n be a one-chain in U . For P not in the support of c, dene W (c, P ) =n W (, P ).The main goal today is to prove Theorem . If c and d are closed 1-chains on an open s
University of Illinois, Urbana Champaign - MATH - 430
Cubical chains, simplicial chains, functoriality, homotopy invariance25 SeptemberLet X be a topological space. Fulton denes a rectangle in X to be a map : [0, 1] [0, 1] X. Let i be the inclusion of the four boundary unit intervals 1(t) = (t,
University of Illinois, Urbana Champaign - MATH - 430
Mayer-Vietoris I: the going-around map27 SeptemberLet U and V be open sets in a topological space X . Recall that the Mayer-Vietoris sequence in cohomology is an exact sequenceUV 0 H 0(U V ) H 0(U )H 0(V ) H 0(U V ) UV H 1(U V ) H 1(U )H
University of Illinois, Urbana Champaign - MATH - 430
Mayer-Vietoris II: exactness29 SeptemberProposition 1. The sequenceU U H1(U V ) H1(U ) H1(V ) V V (i,i)jj H1(U V ) H0(U V ) H0(U )H0(V ) H0(U V ) 0 jU jV D(iU ,iV )is exact. Lemma 1. is surject
University of Illinois, Urbana Champaign - MATH - 430
Mayer-Vietoris for cohomology2 OctoberWe have already used the following Proposition 1. The sequence 0 H 0(U V ) H 0U H0V H 0U V H 1U V H 1U H 1V H 1U V is exact. It is easy to see that Lemma 1. is injective. Also Lemma 2. Ker =
University of Illinois, Urbana Champaign - MATH - 430
Homology and cohomology duality4 OctoberLet P1, . . . , Pn be distinct points in C, and let U = C\{P1, . . . , Pn}. Let i be a closed curve in U which describes a counterclockwise circle around Pi,and encircles no other Pj . Let i be the one-form
University of Illinois, Urbana Champaign - MATH - 430
Covering maps9 October 00Denition 1. A covering map is a map p : Y X of topological spaces, with the property that each point of X has an open neighborhood N such that p1(N ) is a disjoint union of open sets, each mapped homemorphically by p onto
University of Illinois, Urbana Champaign - MATH - 430
Covering maps from group actions11 October 00Let G be a (topological) group and Y a space. Denition 1. A left action of G on Y is a continous map GX G such that(g,x)gx1. g (hx) = (gh)x.2. 1x = xThe orbit of y Y is the set Gy of points
University of Illinois, Urbana Champaign - MATH - 430
The fundamental group: pictures of homotopies13 October 00In this lecture we dene the fundamental group, which turns out to play the role in covering space theory of the galois group. Let I be the unit interval [0, 1]. Let X be a space. A path in
University of Illinois, Urbana Champaign - MATH - 430
The fundamental group: functoriality15 October 00Denition 1. The fundamental group of X at p is 1(X ; p) = [(I, 0, 1), (X, p, p)]. Proposition 1. If is a path from p to q , then conjugation by denes an isomorphism of groups 1(X ; p) = 1(X ; q )
University of Illinois, Urbana Champaign - MATH - 430
The fundamental group: the Hurewicz homomorphism20 October 00Let X be a space and p is a point of X . Suppose that : (I, 0, 1) (X, p, p). We denote by ( ) the class of in 1(X ; p) = [(I, 0, 1), (X, p, p)]. We may also consider as an element o
University of Illinois, Urbana Champaign - MATH - 430
The fundamental group and covering spaces23 October 00A lifting criterion. Let p : Y X be a covering, and p(y ) = x. Lemma 1. The map 1 p : 1 (Y ; y ) 1 (X ; x) is injective. Proof. Homotopy lifting and unique path lifting. Lemma 2. Let : I
University of Illinois, Urbana Champaign - MATH - 430
The universal covering space27 October 00Simply connected A space Y is simply connected if it is path connected, and if 1(Y ; y ) = 0 for some (and so necessarily all) y Y . Let X be a connected and locally path-connected space. A universal cover