**HONORS EXAM 2012 REAL ANALYSIS**

1. Prove or disprove: If {pn}^{∞}_{n=1} is a sequence of polynomials and ∑p_{n} → f uniformly on R as n → ∞, then f is a polynomial.

2. Let C = {f: [0, 1] → [0, 1] | f is continuous}, the set of continuous maps from the interval [0, 1] to itself. Define a metric d on C by d(f, g) = max_{x}_{∈}_{[0,1]}|f(x) - g(x)|. Let C_{i} and C_{s} be the sets of injective and surjective elements, respectively, of C. Prove or disprove the following:

(a) C_{i} is closed in C.

(b) C_{s} is closed in C.

(c) C is connected.

(d) C is compact.

3. Define a sequence of functions f_{1}, f_{2}, . . . :[0, ∞) → R by f_{n}(x) = sin(x/n)/x +(1/n). Discuss the convergence of {f_{n}} and {f'_{n}} as n → ∞.

4. Recall the Intermediate Value Theorem:

Let f: [a, b] → R be a continuous function, and y any number between f(a) and f(b) inclusive. Then there exists a point c ∈ [a, b] with f(c) = y.

(a) Prove the Intermediate Value Theorem.

(b) Prove or disprove the following converse to the Intermediate Value Theorem:

If for any two points a < b and any number y between f(a) and f(b) inclusive, there is a point c ∈ [a, b] such that f(c) = y, then f is continuous.

(c) Prove or disprove the following fixed-point theorem:

Let g: [0, 1] → [0, 1] be continuous. Then there exists a fixed point x ∈ [0, 1] (that is, a point x such that g(x) = x).

5. This question deals with the Riemann integral.

(a) Let S be the unit square [0, 1] × [0, 1]. Define f: S → R by setting

For each of the following integrals, compute its value or show that it does not exist:

_{0}∫^{1}(_{0}∫^{1}f(x, y) dx)dy, _{0}∫^{1}(_{0}∫^{1}f(x, y) dy)dx, ∫_{S}f(x, y).

(b) What conditions on f would guarantee that all three integrals exist and are equal?

(c) Give an example of a function g and a domain D such that ∫_{D}|g| exists but ∫_{D}g does not.

6. Let f: R^{2} → R^{2} be smooth (C^{∞}) and suppose that

∂f_{1}/∂x = ∂f_{2}/∂y, ∂f_{1}/∂y = - (∂f_{2}/∂x).

(These are the Cauchy-Riemann equations, which arise naturally in complex analysis.)

(a) Show that Df(x, y) = 0 if and only if Df(x, y) is singular, and hence f has a local inverse if Df(x, y) ≠ 0. Show that the inverse function also satisfies the Cauchy-Riemann equations.

(b) Give an example showing that the statement in part (a) (f has a local inverse if Df(x, y) ≠ 0) may be false if f does not satisfy the Cauchy-Riemann equations.

7. Let M be a compact 2-manifold in R^{2}, oriented naturally; give the boundary ∂M the induced orientation. Let f: R^{2} → R be a smooth (C^{∞}) function such that f(x) = 0 for any x ∈ ∂M.

(a) Prove that

∫_{M}f · (∂^{2}f/∂x^{2} + ∂^{2}f/∂y^{2}) dx ∧ dy = -∫_{M}((∂f/∂x)^{2} + (∂f/∂y )^{2}) dx ∧ dy.

(b) Deduce from (a) that if, in addition, f is harmonic on M (that is, ∂^{2}f/∂x^{2} + ∂^{2}f/∂y^{2} = 0 on M), then f(x) = 0 for any x ∈ M.

8. (a) (i) Let f be the polar coordinate map given by (x, y) = f(r, θ) = (r cos θ, r sin θ). Compute f^{∗}(dx), f^{∗}(dy), and f^{∗}(dx ∧ dy).

(ii) Compute ∫_{C }xy dx, where C = {(x, y)| x^{2} + y^{2} = 1, x ≥ 0, y ≥ 0}, the portion of the unit circle in the first quadrant, oriented counter-clockwise.

(b) Let M be a manifold, possibly with boundary. A retraction of M onto a subset A is a smooth (C^{∞}) map φ: M → A such that φ(x) = x for all x ∈ A. (For example, the map φ(x) = x/||x|| is a retraction of the punctured plane R^{2}\{(0, 0)} onto the unit circle S^{1}.) Prove the following theorem:

There does not exist a retraction from the plane R^{2} onto the unit circle S^{1}.

(Hint: Consider the 1-form x dy - y dx/x^{2} + y^{2}, which is defined in an open set containing S^{1}.)

9. (a) Let ω_{1} and ω_{2} be differential forms defined on the same domain.

(i) If ω_{1} and ω_{2} are closed, must ω_{1} ∧ ω_{2} also be closed? If ω_{1} ∧ ω_{2} is closed, must ω_{1} and ω_{2 }also be closed?

(ii) If ω_{1} and ω_{2} are exact, must ω_{1} ∧ ω_{2} also be exact? If ω_{1} ∧ ω_{2} is exact, must ω_{1} and ω_{2} also be exact?

(b) Show that every closed 1-form on the punctured space R^{3}\{(0, 0, 0)} is exact.

Should he have done anything differently : Evaluate Harvey's decision to bring Hopwood Manufacturing to the university from the perspective of human resource management. Did he make a mistake? Should he have done anything differently? |

Ranking democrat on the house commerce committee : CA1-16 (Economic Consequences) Presented below are comments made in the financial press. Instructions Prepare responses to the requirements in each item. (a) Rep. |

Calculate the total rental cost and total buying cost : Calculate the total rental cost and total buying cost. (Round your intermediate calculations and final answers to the nearest whole dollar.) |

Tony contributes land with a basis and value : Tony contributes land with a basis and value of $180,000 in exchange for a 40% interest in the calendar year of XYZ, LLC, which is treated as a partnership. In 2015, the LLC generates $600,000 of ordinary taxable income. Because LLC needs capital .. |

Prove the intermediate value theorem : Let f: [a, b] → R be a continuous function, and y any number between f(a) and f(b) inclusive. Then there exists a point c ∈ [a, b] with f(c) = y. Prove the Intermediate Value Theorem |

Demonstrate a critical awareness of real ethical issues : MBALN707 -Business Ethics - Examine the ethical environment in which McDonalds operates and consider the impact this has upon its business behaviour and performance. |

Various marketable securities and other investments : Company was formed in 1971 for the purpose of acquiring Blackacre which consisted of 1,500 acres of unimproved real property in Sussex County, Delaware. |

Explain the benefits of implementing your recommendations : The Board of Directors of Windsor Memorial Hospital has hired you to be their zero-based budget consultant. Specify how Windsor Memorial Hospital can implement a zero-based budget and provide your recommendations to the Board of Directors of the H.. |

Prove that 1+z 1-z bi is pure imaginary for zec z1 and z : Prove that 1+z/1-z=bi is pure imaginary, for zec, z= 1 and z-z=1 |

## Responsible for controlling the weight of a box of cerealAs a quality analyst you are also responsible for controlling the weight of a box of cereal. The Operations Manager asks you to identify the ways in which statistical quality control methods can be applied to the weights of the boxes. |

## Probability that the right headlightI recently had to replace both front headlights on my car. The life expectancy of my headlights follows an exponential distribution with a MTBF of 1500 hours. That is, the expected number of hours until failure is 1500 hours. For the purposes of t.. |

## Maximize the angle theta subtended"A painting in an art gallery has height h and is hung so that its lower edge is a distance d above the eye of an observer. How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so a.. |

## Natural law school of jurisprudenceWhich of the following is most consistent with the Natural Law School of jurisprudence? |

## Formulate the optimal solutionCan someone teach me how to formulate the Optimal solution for this problem without using Excel? |

## Determine whether the relation is reflexiveFor each property either prove that the property holds or give a counter-example (or reason) demonstrating that the property does not hold. |

## Two standard deviation of the meanWhat percentage of western states would you expect to have property crime rates between 2646 and 4048? |

## Compute the order of g and describe its structureHonors Examination 2013: Algebra. Let G be the group with presentation - (x, y, z | x2y2 = x2z2 = y2z2 = xyx-1y-1 = xzx-1z-1 = yzy-1z-1 = 1). Compute the order of G and describe its structure |

## Improper integral to evaluate the integralProblem 1: Use the definition of an improper integral to evaluate the integral below: |

## Determine the optimal product mixFormulate a linear programming model to determine the optimal product mix that will maximize profit. Transform this model into standard form. |

## Compute b1 and b0Compute B1 and B0 (for this I am having trouble finding the covariance and Sx if you could please write that out in detail) |

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd