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### 508F06Ex1solns

Course: MATH 508, Fall 2010
School: UPenn
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Word Count: 849

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1 Math Exam 508 October 12, 2006 Jerry L. Kazdan 12:00 1:20 Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5 points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems (60 points, 15 points each). Closed book, no calculators but you may use one 3 5 card with notes. Part A: Examples (4 problems, 5 points each). Give an...

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1 Math Exam 508 October 12, 2006 Jerry L. Kazdan 12:00 1:20 Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5 points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems (60 points, 15 points each). Closed book, no calculators but you may use one 3 5 card with notes. Part A: Examples (4 problems, 5 points each). Give an example of an innite set in a metric space (perhaps R ) with the specied property. A1. Bounded with exactly two limit points. 1 Solution: The set {(1)n (1 + n ), n = 1, 2, 3, . . .} in R . A2. Containing all of its limit points. Solution: Lots of exmples: 1). The empty set. 4). The closed interval {0 x 1 in R} . 2). All of R . 3). The point {0} R . A3. Distinct points {xj , j = 1, 2, 3, . . .} with xi = xj for i = j that is compact. 1 Solution: The following subset of the real numbers: {0} { n , n = 1, 2, 3, . . .} . A4. Closed and bounded but not compact. Solution: The closed unit ball x 1 in 2 . The standard basis vectors e1 = (1, 0, 0, . . .), e2 = (0, 1, 0, 0, . . .), etc have no convergent subsequence. Another example: the real numbers {x R | 0 x 1} with the discrete metric: d(x, y ) = 1 for x = y , d(x, x) = 0. Part B: Classify sets (20 points) For each of the following sets, circle the listed properties it has: a) {1 + 1 n R, n = 1, 2, 3, . . .} closed bounded compact countable open closed bounded compact countable c) {(x, y ) R2 : 0 < y 1} open closed bounded compact countable d) {(x, y ) R2 : x = 0} open closed bounded compact countable b) {1} {1 + 1 n open R, n = 1, 2, 3, . . .} 1 e) {(x, y ) R2 : x2 + y 2 = 1} open closed bounded compact countable f) {(x, y ) R2 : x2 + y 2 1} open closed bounded compact countable g) {(x, y ) R2 : y > x2 } open closed bounded compact countable closed bounded compact countable h) {(k, n) R2 : k, n any positive integers} open Part C: Traditional Problems (4 problems, 20 points each) C1. In R , if an A and bn B , show that the product an bn AB . Solution: Let pn = an A 0, qn = bn B 0. Then an bn = (pn + A)(qn + B ) = pn qn + Aqn + Bpn + AB. Using that for convergent sequences xn and yn we know lim(xn + yn ) = lim xn + lim yn and lim(cxn ) = c lim xn , we see that it is enough to show that pn qn 0. Given > 0 (which we may assume satises < 1), pick N so that if n > N then |pn | < and |qn | < . Consequently |pn qn | < 2 < . a1 + + an be the sequence averages of (arithmetic n mean). If ak converges to A , show that the averages Cn also converge to A . C2. Given a real sequence {ak } , let Cn = Solution: Letting Bn = an A 0, I could reduce to the case A = 0. Instead, for variety I proceed directly. Note that Cn A = Given any (a1 A) + + (an A) a1 + + an A= n n > 0, pick N so that if n > N then |an A| < . Then write Cn A = (a1 A) + + (aN A) (aN +1 A) + + (an A) + . n n In Jn Now [n (N + 1)] n = for any n > N. n n We will show that by choosing n even larger, we can make |In | < . Since the sequence an A converges, it is bounded, so for some M we have |an A| < M . Thus for n sucientnly large |Jn | < |In | < Consequently, |Cn A| |In | + |Jn | < 2 . 2 NM <. n C3. Let Kj , j = 1, 2, . . . be compact sets in a metric space. Give a proof or counterexample for each of the following assertions. a) K1 K2 is compact. Solution: True. Since compact sets are closed, then K1 K2 is a closed subset of the compact set K1 , and hence compact. b) K1 K2 is compact. Solution: True. Let {U } be any open cover of K1 K2 . A nite number of these, say {V1 , . . . , Vk } , cover K1 , and {W1 , . . . , Wn } , cover K2 . Then {V1 . . . Vk W1 . . . Wn } is the desired nite cover of K1 K2 . c) j =1 Kj is compact. Solution: Counterexample. The non-negative real numbers {x 0} is the union of the compact sets (closed intervals) Kj = {j 1 x j ; j = 1, 2, . . .} . Since this set is not bounded, it is not compact. C4. In a complete metric space M , let d(x, y ) denote the distance. Assume there is a constant 0 < c < 1 so that the sequence xk satises d(xn+1 , xn ) < cd(xn , xn1 ) for all n = 1, 2, . . . . a) Show that d(xn+1 , xn ) < cn d(x1 , x0 ). Solution: Since d(x2 , x1 ) < cd(x1 , x0 ), then d(x3 , x2 ) < cd(x2 , x1 ) < c2 d(x1 , x0 ). Using this, d(x4 , x3 ) < cd(x3 , x2 ) < c3 d(x1 , x0 ). The induction to the general case is obvious. b) Show that the {xk } is a Cauchy sequence. Solution: Say n > k . Then using the previous part and that 0 < c < 1 d(xn , xk ) d(xn , xn1 ) + . . . + d(xk+1 , xk ) (cn1 + cn2 + + ck ) d(x1 , x0 ) (ck (1 + c + c2 + c3 + . . .) d(x1 , x0 ) = ck d(x1 , x0 ). 1c Pick N so that cN < . If n > k > N then d(xn , xk ) < 1c d(x1 , x0 ). c) Show that there is some p M so that limn xk = p . Solution: Since the metric space is complete, there is a point p in the metric space to which the Cauchy sequence xk converges. 3
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UPenn - MATH - 508
SignaturePrinted NameExam 2Math 508December 8, 2006Jerry L. Kazdan12:00 1:20Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points),Part B has 5 traditional problems (15 points each, so 75 points).Closed bo
UPenn - MATH - 508
Exam 2Math 508December 8, 2006Jerry L. Kazdan12:00 1:20Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points),Part B has 5 traditional problems (15 points each, so 75 points).Closed book, no calculators but
UPenn - MATH - 508
Exam 1Math 508October 16, 2008Jerry L. Kazdan10:30 11:50Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems(
UPenn - MATH - 508
Exam 2Math 508December 4, 2008Jerry L. Kazdan10:30 11:50Directions This exam has two parts, Part A has 10 True-False problems (30 points, 3 pointseach). Part B has 5 traditional problems (70 points, 14 points each).Closed book, no calculators or co
UPenn - MATH - 508
Exam 1Math 508October 14, 2010Jerry L. Kazdan9:00 10:20Directions This exam has three parts, Part A asks for 4 examples (20 points, 5 points each).Part B has 4 shorter problems (36 points, 9 points each. Part C has 3 traditional problems (45points,
UPenn - MATH - 508
Exam 2Math 508December 9, 2010Jerry L. Kazdan9:00 10:20Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15p
UPenn - MATH - 508
Math 508December 9, 2010Exam 2Jerry L. Kazdan9:00 10:20Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15p
UPenn - MATH - 508
Advanced Analysis: OutlineMath 508, Fall 2010Jerry L. KazdanThis outline of the course is to help you step back and get a larger view of what we havedone. Since this is only an outline, I will often not explicitly state the precise assumptionsneeded
UPenn - MATH - 508
Analysis ProblemsPenn MathJerry L. KazdanIn the following, when we say a function is smooth, we mean that all of its derivatives existand are continuous.These problems have been crudely sorted by topic but this should not be taken seriouslysince man
UPenn - MATH - 508
Some Classical InequalitiesFor all of these inequalities there are many methods. We give a sampling. 1.ARITHMETIC - GEOMETRIC MEAN and decide when equality holds. INEQUALITYIf cfw_ b j &gt; 0, prove the following (1)(b1 b2 bn )1/n b1 + b 2 + + b n . n
UPenn - MATH - 508
Basic DenitionsLet S Rn . and p Rn . S is bounded if it is contained in someball in Rn . S is a neighborhood of p if S containssome open ball around P . A point p is a limit point of S if everyneighborhood of p contains a point q S , where q = p .
UPenn - MATH - 508
Calculus ProblemsMath 504 505Jerry L. Kazdan1. Sketch the points (x , y) in the plane R2 that satisfy |y x| 2.2. A certain function f (x) has the property thatf (t ) dt = ex cos x + C . Find both f0and the constant C .cos x3. Compute limx0 cos 2
UPenn - MATH - 508
Math 508Jerry L. KazdanCompleteness of1Let 1 be the vector space of innite sequences of real numbers X = (x1 , x2 , . . .) with nite normX := =1 |x j | . Here we show this space is complete. The proof is a bit fussy.j(n)(n)S TEP 1: F IND A CANDID
UPenn - MATH - 508
Contracting Maps and an ApplicationMath 508, Fall 2010Jerry L. KazdanOne often effective way to show that an equation g(x) = b has a solution is to reduce theproblem to nd a xed point x of a contracting map T , so T x = x . For instance, assumeV is a
UPenn - MATH - 508
ConvolutionLet f (x) and g(x) be continuous real-valued functions for x R and assume that f or g is zerooutside some bounded set (this assumption can be relaxed a bit). Dene the convolution( f g)(x) :=Zf (x y)g(y) dy(1)Since f or g is zero outside
UPenn - MATH - 508
Numbers and Sets - exercises for enthusiasts 1.W. T. G.1. Let A be the sum of the digits of 44444444 , and let B be the sum of the digits of A.What is the sum of the digits of B ?nn2. Let x1 , . . . , xn be real numbers such that i=1 xi = 0 and i=1
UPenn - MATH - 508
Numbers and Sets - exercises for enthusiasts 2.W. T. G.1. Does there exist an uncountable family B of subsets of N such that for every A, B B(distinct) the intersection of A with B is nite?2. Is it possible to write the closed interval [0, 1] as a cou
UPenn - MATH - 508
The language and grammar ofmathematics1gate and multiply and render the sentences unintelligible.To illustrate the sort of clarity and simplicitythat is needed in mathematical discourse, let usconsider the famous mathematical sentence Twoplus two e
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 0: Rust RemoverD UE : These problems will not be collected.You should already have the techniques to do these problems, although they may take somethinking.1. Show that for any positive integer n , the nu
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 1D UE : Thurs. Sept. 16, 2010. Late papers will be accepted until 1:00 PM Friday.1. Let x0 = 1 and dene xk :=increasing.2. Show that 1 +3xk1 + 4, k = 1, 2, . . . . Show that xk &lt; 4 and that the xk are1
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 2D UE : Thurs. Sept. 23, 2010. Late papers will be accepted until 1:00 PM Friday.1. Let F be a eld, such as the reals or the integers mod 7 and x, y F . Here x means theadditive inverse of x .a) If xy = 0
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 3D UE : Thurs. Sept. 30, 2010. Late papers will be accepted until 1:00 PM Friday.1. Find all (complex) roots z = x + iy of z2 = i .2. Let xn &gt; 0 be a sequence of real numbers with the property that they co
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 4D UE : Thurs. Oct 7, 2010. Late papers will be accepted until 1:00 PM Friday.5n + 17.n+23n2 2n + 17. Calculate lim an .b) Let an := 2nn + 21n + 21. a) Calculate limn2. Investigate the convergence
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanBonus Problem for Set 41. Dene two real numbers x and y to be equal if |x y| is an integer. We write x y (mod 1) .Thus we have a topological circle whose circumference is one.Let be an irrational real number, 0 &lt; &lt; 1
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 5D UE : Thurs. Oct. 21, 2010. Late papers will be accepted until 1:00 PM Friday.k1. [Ratio Test] Let ak be a sequence of complex numbers. Ley s := lim sup aa+1 . By comparisonkwith a geometric series, sh
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 6D UE : Thurs. Oct. 28, 2010. Late papers will be accepted until 1:00 PM Friday.1. Give examples of the following:a) An open cover of cfw_x R : 0 &lt; x 1 that has no nite sub-cover.b) A metric space having
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 7D UE : Thurs. Nov. 4, 2010. Late papers will be accepted until 1:00 PM Friday.Note: We say a function is smooth if its derivatives of all orders exist and are continuous.1. Let f : [a, ) R be a smooth fun
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 8D UE : Thurs. Nov. 11, 2010. Late papers will be accepted until 1:00 PM Friday.Note: We say a function is smooth if its derivatives of all orders exist and are continuous.1. a) Let A(t ) and B(t ) be n n
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 9D UE : Thurs. Nov. 18, 2010. Late papers will be accepted until 1:00 PM Friday.Note: We say a function is smooth if its derivatives of all orders exist and are continuous.1. Let f (x) be a smooth function
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 10D UE : Tues. Nov. 30, 2010. Late papers will be accepted until 1:00 PM Wednesday.Note: We say a function is smooth if its derivatives of ball orders exist and are continuous.1. Find an integer N so thst
UPenn - MATH - 508
Math508, Fall 2010Jerry L. KazdanProblem Set 11D UE : NeverNote: We say a function is smooth if its derivatives of ball orders exist and are continuous.1. Partition [a, b] R into sub-intervals a &lt; x1 &lt; x2 &lt; &lt; xn = b . A function h(x) that isconstant
UPenn - MATH - 508
Math 508, Fall 2008Jerry KazdanTwo Inequalities for Integrals of Vector Valued FunctionsTheorem Let F : [a, b] Rn be a continuous vector-valued function. ThenZbF (t ) dt ZbF (t ) dtaawith equality if andR only if there is a continuous scalar val
UPenn - MATH - 508
NUMBERS AND SETS EXAMPLES SHEET 1.W. T. G.1. Let A, B and C be three sets. Give a proof that A \ (B C ) = (A \ B ) (A \ C ) usingthe criterion for equality of sets.2. The symmetric dierence AB of A and B is dened to be (A \ B ) (B \ A). (Thatis, it
UPenn - MATH - 508
M 2003NUMBERS AND SETS EXAMPLES SHEET 2W. T. G.1. Prove by induction that the following two statements are true for every positive integer n.(i) The number 2n+2 + 32n+1 is a multiple of 7.(ii) 13 + 33 + 53 + . . . + (2n 1)3 = n2 (2n2 1) .2. Suppose
UPenn - MATH - 508
NUMBERS AND SETS EXAMPLES SHEET 3.W. T. G.1. Solve (ie., nd all solutions of) the equations(i) 7x 77 (mod 40).(ii) 12y 30 (mod 54).(iii) 3z 2 (mod 17) and 4z 3 (mod 19).2. Without using a calculator, work out the value of 1710,000 (mod 31).3. Again
UPenn - MATH - 508
NUMBERS AND SETS EXAMPLES SHEET 4.W. T. G.1. Is there a eld with exactly four elements? Is there one with six elements?2. Let F be a eld. Prove that (1)(1) = 1 in F. [1 is of course dened to be theadditive inverse of the multiplicative identity. If yo
South Carolina - MKGT - 465
Financial Management 1In-Class ProblemsGROSS MARGINGross margin (or gross profit) is: Total sales revenue - total cost of goods soldor \$ 200 - \$ 100 = \$ 100 A s % = (100/200) x 100% = 50% On a per-unit basis, unit selling price - unit cost of good
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South Carolina - MKGT - 465
South Carolina - MKGT - 465
South Carolina - MKGT - 465
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Michigan State University - EC 340 - 340
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Michigan State University - EC 340 - 340
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Michigan State University - EC 340 - 340
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