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### hmwk08

Course: MATH 550, Fall 2009
School: CSU Northridge
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Word Count: 191

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550. Math Homework 8. Due 11/10/03 Problem 1. Let U, V be open subsets of R2 . Prove that the coboundary map : H 0 (U V ) H 1 (U V ) is a homomorphism of vector spaces. Problem 2. Let F : U V be a smooth map from the open set U R2 into the open set V R2 . In a previous homework set we showed that F induces a pull-back operator F that sends n-forms on V into n-forms on U. Prove that F induces a linear map...

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550. Math Homework 8. Due 11/10/03 Problem 1. Let U, V be open subsets of R2 . Prove that the coboundary map : H 0 (U V ) H 1 (U V ) is a homomorphism of vector spaces. Problem 2. Let F : U V be a smooth map from the open set U R2 into the open set V R2 . In a previous homework set we showed that F induces a pull-back operator F that sends n-forms on V into n-forms on U. Prove that F induces a linear map of vector spaces F : H nV H nU, for n = 0, 1. Moreover, prove if that F is a diffeomorphism, then F : H nV H nU is an isomorphism of vectors spaces. Problem 3. Prove that if U and V are connected, and H 1 (U V ) = 0, then U V is connected. Problem 4. Prove that if the open set U R2 can be written as an union U = U1 Un , where each U j is a c...

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CSU Northridge - MATH - 311
Math 311. Quiz 5 Due: Monday, October 20, 2008Name:Construction a Golden Triangle We have proved that an isosceles triangle with long side equal to 1 and short of 51 side equal to is a golden triangle. Problem 1 uses this information in order to
CSU Northridge - MATH - 53971
Math 311. Quiz 5 Due: Monday, October 20, 2008Name:Construction a Golden Triangle We have proved that an isosceles triangle with long side equal to 1 and short of 51 side equal to is a golden triangle. Problem 1 uses this information in order to
CSU Northridge - MATH - 53971
Math 550. Homework 9. Due 12/03/03Problem 1. Suppose that U = R2 \ {P1 , , Pn } is the complement of n points in the plane. Prove that the mapping that takes a closed 1-chain to W (, P1 ), , W (, Pn ) determines an isomorphism of H1 U with t
CSU Northridge - MATH - 550
Math 550. Homework 9. Due 12/03/03Problem 1. Suppose that U = R2 \ {P1 , , Pn } is the complement of n points in the plane. Prove that the mapping that takes a closed 1-chain to W (, P1 ), , W (, Pn ) determines an isomorphism of H1 U with t
CSU Northridge - MATH - 53971
Math 550. Homework 7. Due 10/29/03Problem 1. A vector X in Rn is called a probability vector if its coordinates are all nonnegative and add up to 1. An n n matrix is an stochastic matrix if its columns are probability vectors. Use the Brouwer xed
CSU Northridge - MATH - 550
Math 550. Homework 7. Due 10/29/03Problem 1. A vector X in Rn is called a probability vector if its coordinates are all nonnegative and add up to 1. An n n matrix is an stochastic matrix if its columns are probability vectors. Use the Brouwer xed
CSU Northridge - MATH - 53971
Math 623. Homework 5. Due 04/21/04Problem 1. Show that the hyperbolic distance d(z, w) between points z and w in the unit disk satises the following identity 1 zw Tanh d(z, w) = . 2 1 zw Problem 2. A circle in H2 centered at p with radius r is the
CSU Northridge - MATH - 623
Math 623. Homework 5. Due 04/21/04Problem 1. Show that the hyperbolic distance d(z, w) between points z and w in the unit disk satises the following identity 1 zw Tanh d(z, w) = . 2 1 zw Problem 2. A circle in H2 centered at p with radius r is the
CSU Northridge - MATH - 53971
Math 655. Homework 3. Due 2/26/03Problem 1 Let f be an analytic function on a connected open set U C. (1) Show that if f is real valued, then f is constant on U . (2) Show that if f has constant absolute value, then f is constant on U . Problem 2
CSU Northridge - MATH - 655
Math 655. Homework 3. Due 2/26/03Problem 1 Let f be an analytic function on a connected open set U C. (1) Show that if f is real valued, then f is constant on U . (2) Show that if f has constant absolute value, then f is constant on U . Problem 2
CSU Northridge - MATH - 53971
Math 655. Homework 1. Due 2/5/03Problem 1 Prove thatab &lt;1 1 abif |a| &lt; 1 and |b| &lt; 1. Prove also thatab =1 1 abif either |a| = 1 or |b| = 1. What exception must be made if |a| = |b| = 1? Problem 2 Show that the functions f (z) and f (z) a
CSU Northridge - MATH - 655
Math 655. Homework 1. Due 2/5/03Problem 1 Prove thatab &lt;1 1 abif |a| &lt; 1 and |b| &lt; 1. Prove also thatab =1 1 abif either |a| = 1 or |b| = 1. What exception must be made if |a| = |b| = 1? Problem 2 Show that the functions f (z) and f (z) a
CSU Northridge - MATH - 53971
Math 592D. Homework 4. Due: 4/28/051. Problem 5.3.2 2. Problem 5.4.2
CSU Northridge - MATH - 592
Math 592D. Homework 4. Due: 4/28/051. Problem 5.3.2 2. Problem 5.4.2
CSU Northridge - MATH - 53971
Math 592D. Homework 3. Due: 3/10/05Do at least 2, 3, 4 (not 4(e). Then 1 if possible. 5 and 6 are suggestions. 1. The approach to modeling age-structured populations in continuous time is similar to what we have done in discrete time. If u(a, t) is
CSU Northridge - MATH - 592
Math 592D. Homework 3. Due: 3/10/05Do at least 2, 3, 4 (not 4(e). Then 1 if possible. 5 and 6 are suggestions. 1. The approach to modeling age-structured populations in continuous time is similar to what we have done in discrete time. If u(a, t) is
CSU Northridge - MATH - 53971
Math 550. Homework 2 (Revised). Due 9/22/2003Problem 1 Let U be the union of two open sets U1 ,U2 , i.e., U = U1 U2 . Let f j be a smooth functions on U j , j = 1, 2, such that f1 (x) = f2 (x) for every x in U1 U2 . Prove that f (x) = is a smooth f
CSU Northridge - MATH - 550
Math 550. Homework 2 (Revised). Due 9/22/2003Problem 1 Let U be the union of two open sets U1 ,U2 , i.e., U = U1 U2 . Let f j be a smooth functions on U j , j = 1, 2, such that f1 (x) = f2 (x) for every x in U1 U2 . Prove that f (x) = is a smooth f
CSU Northridge - MATH - 53971
Math 550. Homework 1. Due 9/3/2003Problem 1 Let U be an open subset of the plane. Prove that U is connected if and only if every locally constant function on U is constant on U . Problem 2 Let = ydx xdy and let be the line segment from (0, 0) to
CSU Northridge - MATH - 550
Math 550. Homework 1. Due 9/3/2003Problem 1 Let U be an open subset of the plane. Prove that U is connected if and only if every locally constant function on U is constant on U . Problem 2 Let = ydx xdy and let be the line segment from (0, 0) to
CSU Northridge - MATH - 53971
Math 550. Homework 6. Due 10/22/03Denition 1. Two continuous mappings f , g : X Y are homotopic if there is a continuous mapping H : X [0, 1] Y such that H(x, 0) = f (x) and H(x, 1) = g(x) for all x in X. Problem 1. Let C and C be circles. (i) P
CSU Northridge - MATH - 550
Math 550. Homework 6. Due 10/22/03Denition 1. Two continuous mappings f , g : X Y are homotopic if there is a continuous mapping H : X [0, 1] Y such that H(x, 0) = f (x) and H(x, 1) = g(x) for all x in X. Problem 1. Let C and C be circles. (i) P
CSU Northridge - MATH - 53971
Math 550. Homework 5. Due 10/15/03Problem 1. Let : [a, b] R2 \ {0} be a continuous path. Prove that there are continuous functions r : [a, b] R+ (the positive real numbers) and : [a, b] R, so that (t) = (r(t) cos (t), r(t) sin (t), a t b.P
CSU Northridge - MATH - 550
Math 550. Homework 5. Due 10/15/03Problem 1. Let : [a, b] R2 \ {0} be a continuous path. Prove that there are continuous functions r : [a, b] R+ (the positive real numbers) and : [a, b] R, so that (t) = (r(t) cos (t), r(t) sin (t), a t b.P
CSU Northridge - MATH - 53971
Math 550. Homework 4. Due 10/01/2003Problem 1 Given a 1-form on an open set U, prove that the following are equivalent (i) d = 0; (ii)R R = 0 for all closed rectangles R contained in U; (iii) every point in U has a neighborhood such that = 0 f
CSU Northridge - MATH - 550
Math 550. Homework 4. Due 10/01/2003Problem 1 Given a 1-form on an open set U, prove that the following are equivalent (i) d = 0; (ii)R R = 0 for all closed rectangles R contained in U; (iii) every point in U has a neighborhood such that = 0 f
CSU Northridge - MATH - 53971
Math 550. Homework 3. Due 9/24/2003You have two options to choose from. Option A Generalize (that is, state and prove) the last lemma proved in class today (9/10) to an open set of the form U = R2 \ {P1 , , Pn }. Option B Prepare a set of lectur
CSU Northridge - MATH - 550
Math 550. Homework 3. Due 9/24/2003You have two options to choose from. Option A Generalize (that is, state and prove) the last lemma proved in class today (9/10) to an open set of the form U = R2 \ {P1 , , Pn }. Option B Prepare a set of lectur
CSU Northridge - MATH - 512
Math 512B. Homework 1. Due 1/30/08(Revised 1/27)Problem 1. In class we dened cos x and sin x for x in [0, ]. The values of cos x and sin x for x not in [0, ] are dened in two steps as follows: (1) If x 2, then set cos x sin x = cos(2 x), = sin
CSU Northridge - MATH - 53971
Math 512B. Homework 1. Due 1/30/08(Revised 1/27)Problem 1. In class we dened cos x and sin x for x in [0, ]. The values of cos x and sin x for x not in [0, ] are dened in two steps as follows: (1) If x 2, then set cos x sin x = cos(2 x), = sin
CSU Northridge - MATH - 512
Math 512B. Homework 3. Due 2/13/08The symbol lim f (x) means the limit of f (x) as x approaches . We say that lim f (x) = L if for every x x &gt; 0 there is a number M such that, for all x, if x &gt; M , then |f (x) L| &lt; . A similar denition applies t
CSU Northridge - MATH - 53971
Math 512B. Homework 3. Due 2/13/08The symbol lim f (x) means the limit of f (x) as x approaches . We say that lim f (x) = L if for every x x &gt; 0 there is a number M such that, for all x, if x &gt; M , then |f (x) L| &lt; . A similar denition applies t
CSU Northridge - MATH - 512
Math 512B. Homework 4. Due 2/20/08(Revised 2/18)Problem 1. (i) Prove that the remainder R2n+1,0,arctan (x) of degree 2n + 1 of the function arctan satises |R2n+1,0,arctan (x)| for |x| 1. (ii) Use the relation arctan x + arctan y = arctan x+y 1 x
CSU Northridge - MATH - 53971
Math 512B. Homework 4. Due 2/20/08(Revised 2/18)Problem 1. (i) Prove that the remainder R2n+1,0,arctan (x) of degree 2n + 1 of the function arctan satises |R2n+1,0,arctan (x)| for |x| 1. (ii) Use the relation arctan x + arctan y = arctan x+y 1 x
CSU Northridge - MATH - 512
Math 512B. Homework 9. Due 4/37/08Problem 1. Let z1 , z2 , z3 be three distinct nonzero complex numbers. Prove that the following are equivalent: (i) The points z1 , z2 , and z3 are the vertices of an equilateral triangle. (ii) The center of mass o
CSU Northridge - MATH - 53971
Math 512B. Homework 9. Due 4/37/08Problem 1. Let z1 , z2 , z3 be three distinct nonzero complex numbers. Prove that the following are equivalent: (i) The points z1 , z2 , and z3 are the vertices of an equilateral triangle. (ii) The center of mass o
CSU Northridge - MATH - 512
Math 512B. Homework 8. Due 4/16/08Problem 1. (i) Find the Fourier series of the following functions (they are 2-periodic, so only the values on (, ] are given). (a) f (x) = 0 if &lt; x 0 and f (x) = sin x if 0 &lt; x . (b) g(x) = x2 if &lt; x . (ii) Di
CSU Northridge - MATH - 53971
Math 512B. Homework 8. Due 4/16/08Problem 1. (i) Find the Fourier series of the following functions (they are 2-periodic, so only the values on (, ] are given). (a) f (x) = 0 if &lt; x 0 and f (x) = sin x if 0 &lt; x . (b) g(x) = x2 if &lt; x . (ii) Di
CSU Northridge - ICH - 530
Chondrichthyes: Cartilaginous FishesI. Chondrichthian characteristics &amp; groupsReviewSuperclass Gnathostomata - Jawed fishes Class Placodermi (plate-skinned) Class Acanthodii (spiny sharks) extinct extinctII. Holocephali III. Elasmobranchii A
CSU Northridge - ICH - 530
SwimmingI. Types of swimming II. Methods for generating propulsion III. Forces resisting movement IV. Adaptations to different swimming modes V. Relationships between swimming and ecologyFishes swim in lots of different ways. I. Types of swimming
CSU Northridge - ICH - 530
I. Lifetime spawning frequency Reproduction in FishesI. Lifetime spawning frequency II. Spawning cycles III. Modes of spawning IV. Sex change and mating systemsSemelparity spawn once Iteroparity spawn multiple times semelparous iteroparousWhy?
CSU Northridge - JMK - 7693
Norm Herr Types of Graphs Causes of Death in California, 2000 diseases of heart 68,533 malignant neoplasms 53,005 cerebrovascular diseases 18,090 chronic respiratory diseases 12,754 accidents 8,814 influenza &amp; pneumonia 8,355 diabetes 6,203 alzheimer
CSU Northridge - JMK - 7693
CSU Northridge - JMK - 7693
Assignment #Assignment NameDate AssignedDate DuePoints PossPts. Earned
CSU Northridge - LL - 656883
Molecular evolution of dinoflagellate luciferases, enzymes with three catalytic domains in a single polypeptideLiyun Liu, Therese Wilson, and J. Woodland Hastings* `Department of Molecular and Cellular Biology, Harvard University, Cambridge, MA 02
CSU Northridge - ENGL - 36711
Three Versions of Cdmons Hymn Cdmons Hymn occurs in a number of manuscripts, which vary in their spelling of the poem. The two earliest manuscripts, the Moore and the Leningrad versions date to the eighth century and give the poem in a Northumbrian d
CSU Northridge - ENGL - 400
Three Versions of Cdmons Hymn Cdmons Hymn occurs in a number of manuscripts, which vary in their spelling of the poem. The two earliest manuscripts, the Moore and the Leningrad versions date to the eighth century and give the poem in a Northumbrian d
CSU Northridge - ENGL - 2
EX74EX ER C ISE A NSW ER KEYCHAPTER 13 STYLISTIC TRANSFORMATIONS AND SENTENCE ANALYSIS Exercise 13.1 1. 2. 3. 4. 5. 6. 7. Ran is not a phrasal verb; through the airport is a prepositional ADVP. Type I: NP1 - MVint Ran through is a phrasal verb.
CSU Northridge - ENGL - 36711
EX74EX ER C ISE A NSW ER KEYCHAPTER 13 STYLISTIC TRANSFORMATIONS AND SENTENCE ANALYSIS Exercise 13.1 1. 2. 3. 4. 5. 6. 7. Ran is not a phrasal verb; through the airport is a prepositional ADVP. Type I: NP1 - MVint Ran through is a phrasal verb.
CSU Northridge - SK - 302
ExerciseAnswers Identifytheverbphrasesinthefollowingsentences.Statewhethertheyaremainverbphrases,present participlephrases,pastparticiplephrasesorgerunds. 1. Encouragedbyhercoachspraise,Melodyrepeatedherroutineonthebalancebeam. encouragedbyhercoa
CSU Northridge - SK - 36711
ExerciseAnswers Identifytheverbphrasesinthefollowingsentences.Statewhethertheyaremainverbphrases,present participlephrases,pastparticiplephrasesorgerunds. 1. Encouragedbyhercoachspraise,Melodyrepeatedherroutineonthebalancebeam. encouragedbyhercoa
CSU Northridge - ENGL - 36711
Old EnglishStressed Vowel Change _ _ _ _ _ _ _ _ _ _ _ _Middle EnglishModern Englishhen crft fr healf dop stn stap cyssan seofon glo dl hl_ _ _ _ _ _ _ _ _ _ _ _heen craft fir half dep ston stepe kisse seven gle del hol_ _ _ _ __ _ _ _
CSU Northridge - ENGL - 400
Old EnglishStressed Vowel Change _ _ _ _ _ _ _ _ _ _ _ _Middle EnglishModern Englishhen crft fr healf dop stn stap cyssan seofon glo dl hl_ _ _ _ _ _ _ _ _ _ _ _heen craft fir half dep ston stepe kisse seven gle del hol_ _ _ _ __ _ _ _
CSU Northridge - VCMTH - 00
High Achievement in Mathematics: Lessons from Three Los Angeles Elementary Schools by David Klein Commissioned by the Brookings InstitutionAugust 2000Abstract. This paper describes characteristics and academic policies of three low income element
CSU Northridge - CHEM - 355
Experiment 5: Phase diagram for a three-component system(Dated: July 24, 2008)I.INTRODUCTIONIt is sometimes necessary to know the mutual solubilities of liquids in a two-phase system. For example, you may need to know how much water is dissolv
CSU Northridge - SED - 48405
Author: Scott MakarchukPage 1All Contents Copyright.Video Cabling Cheat SheetSponsored by WWW.A2ZCABLES.COMAll contents copyright. For educational use only.Below are descriptions of six different types of video cabling. Generally speaking,
CSU Northridge - SED - 618
Author: Scott MakarchukPage 1All Contents Copyright.Video Cabling Cheat SheetSponsored by WWW.A2ZCABLES.COMAll contents copyright. For educational use only.Below are descriptions of six different types of video cabling. Generally speaking,
CSU Northridge - VCMTH - 00
Assessment For The California Mathematics Standards Grade 3Introduction: Summary of GoalsGRADE THREEBy the end of grade three, students deepen their understanding of place value and their understanding of and skill with addition, subtraction, mul
CSU Northridge - PSY - 321
1PSY 321/321L: Experimental Psychology and LabUnit: 3/1 Class Lecture Lab 1 (#11683) Lab 2 (#12058) Day T, Th T, Th T, Th Hours 11:00-12:15 12:30-1:45 2:00-3:15 Class Room SH 365 SH 382 SH 341 TA Kristin Hyde (Lab 1) Andrea Maloof (Lab 2)Instruc
CSU Northridge - AC - 53971
THREE DIMES OF TOPOLOGYA. CandelClass Notes for Math 262, Winter 95-96, The University of Chicago. Preface Chapter I. Topological spaces 1. The concept of topological space 2. Bases and subbases 3. Subspaces, unions and hyperspaces 4. Pseudometri
CSU Northridge - SF - 70713
Department of MathematicsProposed by Bernardo brego and Silvia FernndezNovember 13-20Prove that the deadline cannot be written as the sum of three cubes. That is, prove that the equation x3 + y3 + z3 = 11.20.2006 = 441, 320. has no solutions wi