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Maryland - BMGT - 110
Chapter 2How Economics Affects Business: The Creation and Distribution of WealthCPS questionsChapter 02: How Economics Affects Business: The Creation and Distribution of Wealth 2-11. Which is the study of how to employ resources to produce goods and s
Maryland - BMGT - 110
Chapter 5Choosing a Form of Business OwnershipCPS questionsChapter 05: Choosing a Form of Business Ownership5-11. Which is usually the easiest form of business to start and end?a. b. c. d. Sole proprietorship Limited partnership Corporation Cooperat
Maryland - BMGT - 110
Chapter 8Adapting Organizations to Today's MarketsCPS questionsChapter 08: Adapting Organizations to Today's Markets8-111. Which of the following does organizing a business begins with?a. b. c. d. Acquiring the necessary resources Choosing the best
Maryland - BMGT - 110
Chapter 6Entrepreneurship and Starting a Small BusinessCPS questionsChapter 06: Entrepreneurship and Starting a Small Business6-11. Entrepreneurs' _ is an example of their importance to our economy.a. impact on the political process b. ability to cr
Maryland - BMGT - 110
STUDY QUIZ #3Student: _1. Which of the following describes the marketing era immediately after the development of mass production assembly lines? A. production era B. selling era C. marketing era D. customer relationship era2. A firm gathers _ data by
University of Maryland - COMM - 107
COMM107 Oral Communication: Principles and PracticeNonverbal CommunicationSteven D. CohenNonverbal CommunicationThree key characteristics: Sensitive to the relationship between sender and receiver Have meaning based on their context Part of, not sepa
University of Maryland - COMM - 107
COMM107 Oral Communication: Principles and PracticeIntrapersonal CommunicationSteven D. CohenIntrapersonal Communication The process of "internally communicating with yourself" Selftalk Learning sequence Prelinguistic/preconceptual Linguistic Concep
University of Maryland - COMM - 107
COMM107 Oral Communication: Principles and PracticeListeningSteven D. CohenListening Important for learning, work, life Listening is active Listening doesn't just involve the earsThe Listening Processon epti Rece sag s MeAtt en tio nSpeakerFe e
University of Maryland - BMGT - 367
BMGT 367 Personal Marketing PortfolioSpring 2012 | Section 0201Throughout the semester, BMGT 367 students will complete a Personal Marketing Portfolio (PMP). The assignments in the PMP will correspond with the material presented in class and steps each
University of Maryland - BMGT - 367
BGMT 367 Identifying Your Skill SetAgenda Review Last Week Clip of the Week Focusing on YOU Assessing your Strengths Where to go for Help Turning Strengths to Marketable Skills Mapping your Skills to the Job Description Looking Ahead Writing your Resu
South Carolina - MATH - 554
South Carolina - MATH - 554
South Carolina - MATH - 554
Math 554- 703 I - Analysis I Existence of Square RootsTheorem. If a is a nonnegative real number, then there exists a unique positive real number such that 2 = a. We use the notation a := . Lemma. Positive square roots are unique. Proof. Suppose not. If
South Carolina - MATH - 554
South Carolina - MATH - 703
COMPACT SETS IN METRIC SPACES NOTES FOR MATH 703ANTON R. SCHEPIn this note we shall present a proof that in a metric space (X, d) a subset A is compact if and only if it is sequentially compact, i.e., if every sequence in A has a convergent subsequence
South Carolina - MATH - 703
Complex Variables Notes for Math 703. Part I Updated Fall 2011 Anton R. SchepCHAPTER 1Holomorphic (or Analytic) Functions1. Definitions and elementary properties In complex analysis we study functions f : S C, where S C. When referring to open sets in
South Carolina - MATH - 703
Complex Variables Notes for Math 703. Updated Fall 2011 Anton R. SchepCHAPTER 1Holomorphic (or Analytic) Functions1. Definitions and elementary properties In complex analysis we study functions f : S C, where S C. When referring to open sets in C and c
South Carolina - MATH - 703
Homework 1, Additional Problem. 1 (1) Let 1 < p < a real number and let q be defined by 1 = p + 1 . q 1 a. Let f (t) = p tp + 1 - t. Show (by means of calculus), that f (t) 0 for all t 0. q p q a b. Show that ab ap + bq for all a, b > 0. (Hint: Take t = b
South Carolina - MATH - 703
Homework 2, Additional Problem. (1) Let (X, d) be a metric space and let A X be a non-empty subset. Define d(x, A) = infcfw_d(x, y) : y A. a. Prove d(x, A) = 0 if and only if x A. b. Show that |d(x, A) - d(y, A)| d(x, y), for all x, y X.1
South Carolina - MATH - 703
Homework 4, Additional Problem. (1) Let (X, d) be a compact metric space and f : X X a mapping such that d(f (x), f (y) < d(x, y) for all x = y. a. Show that there exists x0 X such that f (x0 ) = x0 . (Hint: Consider infcfw_d(x, f (x) : x X, show that it
South Carolina - MATH - 703
Homework 7 1. Express in the form a + bi. a. (1 + i)20 . b.1-2i . 2+i2. Solve z 2 - 4z + (4 + 2i) = 0. 3. Describe the sets whose points satisfy the following relations. Which of these sets are regions (i.e., open and connected sets)? a. |z + i| 1. b. d
South Carolina - MATH - 703
(1)(2)(3)(4)(5)Homework 8. Prove that if z = x + iy and f (z) = (|xy|), then the real part and imaginary part of f satisfy the Cauchy-Riemann equations at z = 0, but f is not differentiable at z = 0. Let G C be an open and connected set and let f : G
South Carolina - MATH - 703
Homework 9. (1) Let cn > 0 in R. Prove that cn+1 cn+1 lim n cn lim n cn lim . lim cn cn In particular, if limn cn+1 exists, then limn n cn = limn cn+1 . cn cn (2) Let an 0 and bn 0. Assume that both (an ) and (bn ) are bounded sequences. (a) Prove that li
South Carolina - MATH - 703
Homework 10 dz, using a branch of log z, where is the join of the line segments (1) Evaluate [1 - i, 1 + i], [1 + i, -1 + i],and [-1 + i, -1 - i], starting at 1 - i and traversing the curve once (see figure 1).1 zFigure 1. (2) Compute2ecos t [cos(sin
South Carolina - MATH - 703
Homework 11 (1) Evaluate (without parametrizing, but using Cauchy's Integral Theorem) for a. (t) = 1 + eit (0 t 2). b. (t) = -i + eit (0 t 2). c. (t) = 2eit (0 t 2). d. (t) = 3i + 3eit (0 t 2). (2) Let C with | = 1. Compute2 01 1+z 2dzdt 1 - 2 cos t +
South Carolina - MATH - 703
Homework 13. (1) Let G be open and connected and f, g analytic on G such that f (z)g(z) = 0 for all z G. Prove that either f (z) = 0 for all z G or g(z) = 0 for all z G. (2) (Quals '02) Let f, g : cfw_z : |z| < 1 C be analytic functions such that |f (z)|
South Carolina - MATH - 703
Homework 14. (1) (Schwarz's lemma) Let f be a holomorphic function on B(0, 1) with |f (z)| 1 for all |z| < 1 and f (0) = 0. a. Define f1 (z) = f (z) for z = 0 in B(0, 1). Prove that z = 0 is a removable z singularity of f1 . b. Prove that |f1 (z)| 1 on B(
South Carolina - MATH - 703
Math 703 Course Outline Fall 2011 TTH 2:00 -3:15A Second Course in Mathematical Analysis by: J. and H. Burkill Professor: Anton R. Schep Office: LeConte 300C Webpage: http:/www.math.sc.edu/~schep/math703-2011.html Phone: 7-6190 Email: schep@math.sc.edu O
South Carolina - MATH - 703
Solutions for HW 1 Problem 10;5. (1) y f (AC A) y = f (x) for some x AC A y f (A) for some A C y AC f (A). (2) Let y f (AC A). This implies y = f (x) for some x AC A. Hence y f (A) for all A C, i.e., y AC f (A) Assume now f is injective. Let y AC f (A). T
South Carolina - MATH - 703
Solutions for HW 2 Problem 25:8. Let xn x, yn y in the normed space V and n , n in the scalars. Then we have n xn - x = n xn - n x + n x - x |n | xn - x + |n - | x . Now (n ) is bounded, so |n | xn - x 0, and similarly |n - | x 0. Hence we have n xn - x 0
South Carolina - MATH - 703
Solutions for HW 3 Problem 47: 4. Sets consisting of one point are obviously connected. Let E R \ Q contain at least two points a < b. Then there exist a rational number r with a < r < b. Let G1 = (-, r) and G2 = (r, ). Then G1 , G2 are disjoint open sets
South Carolina - MATH - 703
Solutions for HW 4 Problem 55: 7. For each n pick xn Fn . We first show that (xn ) is a Cauchy sequence. Let > 0. Then there exists N such that (Fn ) < for all n N . Let n, m N and assume n < m. Then (xn , xm ) (Fn ) < . Hence (xn ) is a Cauchy sequence.
South Carolina - MATH - 703
Solutions for HW 5 Problem 66: 3. Define: 1 fn (x) = 2nx - 1 01 for n < x 1 2 1 for n < x n 2 for 0 x n .Then fn B(, 1), but (fn ) has no uniformly convergent subsequence, as (fn , fm ) = 1 for all n = m, where denotes the uniform metric. Problem 66: 5.
South Carolina - MATH - 703
Solutions for HW 5 Problem 107: 2. Let M = | + | and let > 0. Then there exists N such that supx |fn (x) - f (x)| < 2M for all n N and supx |gn (x) - g(x)| < 2M for all n N . Now sup |fn (x)+gn (x)-f (x)-g(x)| | sup |fn (x)-f (x)|+| sup |gn (x)-g(x)| < +
South Carolina - MATH - 703
Homework 7 1. Express in the form a + bi. a. (1 + i)20 . i Solution: 1 + i = 2e 4 . Hence (1 + i)20 = 210 e5i = -210 . b.1-2i . 2+iSolution:1-2i 2+i=1-2i 2-i 2+i 2-i=-5i 5= -i.2. Solve z 2 - 4z + (4 + 2i) = 0. Solution: z 2 - 4z + (4 + 2i) = (z -
South Carolina - MATH - 703
Solutions Homework 8. (1) Prove that if z = x + iy and f (z) = (|xy|), then the real part and imaginary part of f satisfy the Cauchy-Riemann equations at z = 0, but f is not differentiable at z = 0. v v Solution: u(x, y) = |xy| and v(x, y) = 0. Hence x (x
South Carolina - MATH - 703
Solutions Homework 9. (1) Let cn > 0 in R. Prove that cn+1 cn+1 lim n cn lim n cn lim . lim cn cn Solution: Let A = lim cn+1 and let > 0. If A = , then there is nothing to prove, cn so assume A < . Then there exists N such that cn+1 < A + for all n N . Th
South Carolina - MATH - 703
Homework 10 dz, using a branch of log z, where is the join of the line segments (1) Evaluate [1 - i, 1 + i], [1 + i, -1 + i],and [-1 + i, -1 - i], starting at 1 - i and traversing the curve once (see figure 1).1 zFigure 1. Solution: Let F (z) = log |z|
South Carolina - MATH - 703
Solutions homework 11 1 (1) Evaluate (without parametrizing, but using Cauchy's Integral Theorem) 1+z2 dz for a. (t) = 1 + eit (0 t 2). 1 1 1 1 Solution: f (z) = 1+z2 = 2i ( z-i - z+i ). Hence both z = i and z = -i are outside the curve and thus f (z) dz
South Carolina - MATH - 703
Solutions Homework 12 (1) (Quals 1995) Let f be an entire function on C and assume that |f (z)| A|z|k + B for some constants A, B, integer k and all z C. Prove that f is a polynomial. Solution: Let R > 0 and R = Reit , 0 t 2. From Cauchy's integral formul
South Carolina - MATH - 703
Solutions homework 13. (1) Let G be open and connected and f, g analytic on G such that f (z)g(z) = 0 for all z G. Prove that either f (z) = 0 for all z G or g(z) = 0 for all z G. Solution: Assume neither f (z) = 0 for all z G or g(z) = 0 for all z G. The
South Carolina - MATH - 703
Solutions Homework 14. (1) (Schwarz's lemma) Let f be a holomorphic function on B(0, 1) with |f (z)| 1 for all |z| < 1 and f (0) = 0. a. Define f1 (z) = f (z) for z = 0 in B(0, 1). Prove that z = 0 is a removable z singularity of f1 . Solution: Since f is
South Carolina - MATH - 703
Math 703 Course Outline Fall 2011 TTH 2:00 -3:15A Second Course in Mathematical Analysis by: J. and H. Burkill Professor: Anton R. Schep Office: LeConte 300C Webpage: http:/www.math.sc.edu/~schep/math703-2011.html Phone: 7-6190 Email: schep@math.sc.edu O
South Carolina - MATH - 703
Math 703 Course Outline Fall 2011 TTH 2:00 -3:15A Second Course in Mathematical Analysis by: J. and H. Burkill Professor: Anton R. Schep Office: LeConte 300C Webpage: http:/www.math.sc.edu/~schep/math703-2011.html Phone: 7-6190 Email: schep@math.sc.edu O
South Carolina - MATH - 555
Math 555/704I Course OutlineSpring 2011 Text : A First Course in Real Analysis Springer, Undergraduate Texts in Mathematics by: Sterling K. Berberian Supplemented with notes.Professor : Anton R. Schep Office : LeConte 300C Email : schep@math.sc.edu Web
South Carolina - MATH - 555
Homework 1. (1) Prove that [-1, 1) is not compact by using the definition of a compact set (to get credit for the problem, use the definition and not any theorems about compact sets). (2) What is an interior point? Prove that1 4is an interior point of (
South Carolina - MATH - 555
Homework 3, Additional Problems. (1) Let (X, d) be a metric space. a. Let Ei X (i cfw_1, , n) be a finite collection of subsets of X. Prove that n Ei = n Ei . i=1 i=1 b. Let Ei (i I) now be an arbitrary collection of subsets of X. Prove that iI Ei iI Ei a
South Carolina - MATH - 555
Extra problems Homework 4. (1) Let f, g : R R be uniformly continuous functions. Assume that both f and g are bounded. Prove that the product f g is uniformly continuous. (2) A function f : R R is periodic, if there exists a c R such that f (x + c) = f (x
South Carolina - MATH - 555
Extra problems Homework 7. (1) Let f : [0, 1 R be a continuous function. Prove that 1 lim n nn 1f (k/n) =k=1 0f (x) dx.(Hint: Use uniform continuity to show that for > 0 there exists a > 0 such that S() - s() < for all subdivisions with norm N () < .
South Carolina - MATH - 555
Homework 9. Prove that f (x) = limn fn (x) exists for all x R. Does (fn ) (1) Let fn (x) = converge uniformly to f ?x2n . 1+x2n(2) Define fn : [0, 1] [0, 1] by fn (x) = xn (1 - x). Prove that fn converges uniformly to 0. (3) Prove that nx + sin(nx2 ) n
South Carolina - MATH - 555
Solutions homework 1. (1) Prove that [-1; 1) is not compact by using the definition of a compact set (to get credit for the problem, use the definition and not any theorems about compact sets).1 Proof: Let On = (-1, 1 - n ). Then [-1; 1) n On , but [-1;
South Carolina - MATH - 555
Solutions homework 2. Page 32 Problem 6: Clearly d(x, y) 0 and d(x, y) = 0 if and only if x = y. Hence property (i) holds. It is also clear that d(x, y) = d(y, x), so it remains to show that the triangle inequality holds. Let x, y, z X. If x = y, then d(x
South Carolina - MATH - 555
Solutions homework 3. Page 69 Problem 10: a. Assume A and B are neigborhoods of x. Then there exist r1 > 0 such that Ur1 (x) A and r2 > 0 such that Ur2 (x) B. let r = mincfw_r1 , r2 . Then r > 0 and Ur A B. Hence A B is a neighborhood of x. b. Let Br (c)
South Carolina - MATH - 555
Solutions homework 4. Page 108 Problem 4: Let (xn ) be a Cauchy sequence and let > 0. Then there exists > 0 such that |x - y| < implies |f (x) - f (y)| < . For this there exist N such that |xn - xm | < for all n, m N . Hence |f (xn ) - f (xm )| < for all
South Carolina - MATH - 555
Solutions homework 5. Page 128 Problem 3: Using the substitution y = -x we get g(y) - g(-c) f (-y) - f (c) f (x) - f (c) = =- . y - (-c) y - (-c) x-c Now letting y -c- is the same as letting x c+ , from which the problem follows. Page 128 Problem 4. As g(
South Carolina - MATH - 555
Solutions homework 6. Page 141 Problem 2. If f g, then mk (f ) = infcfw_f (x) : xk-1 x xk mk (g) = infcfw_g(x) : xk-1 x xk and thus sf () sg () for any subdivision . By definition this implies thatb bfa ag.The corresponding inequalities for the upp
South Carolina - MATH - 555
Solutions homework 7. Page 182 Problem 1. The answer is 2 times the sum of the geometric series with c = 1 , 3 1 so the answer is 2 1- 1 = 3. 3 Page 182 Problem 4. For n 1 we have s2n+2 = s2n + a2n+1 + a2n+2 s2n (since a2n+1 +a2n+2 0), so (s2n ) is an inc
South Carolina - MATH - 555
Page 188 Problem 6. follows now fromn 1 1 xSolutions homework 8. dx = 2 n - 2 as n . The divergence of the seriesn+11 A simpler proof follows from the fact that diverges. n Page 188 Problem 7. From ln x x - 1 it follows that ln(1 + n) (1 + n) - 1 = n.
South Carolina - MATH - 555
Solutions homework 9. Prove that f (x) = limn fn (x) exists for all x R. Does (fn ) (1) Let fn (x) = converge uniformly to f ? Solution: There are 3 cases: |x| < 1, |x| = 1, and |x| > 1. In case |x| < 1, then x2n 0 as n , which implies that fn (x) 0 as n
South Carolina - MATH - 555
Math 555/704I Course OutlineSpring 2011 Text : A First Course in Real Analysis Springer, Undergraduate Texts in Mathematics by: Sterling K. Berberian Supplemented with notes.Professor : Anton R. Schep Office : LeConte 300C Email : schep@math.sc.edu Web
South Carolina - MATH - 554
Extra Credit problems, MATH 554/703I 1. Let (an ) be a sequence of real numbers such that for some = 0 we have an - = 0. n an + lim What can you say about (an ). 2. Let , > 0. Prove thatnlimnn + n = maxcfw_, .3. Assume an a in R. Put sn = a1 +an . Pr