B U Department of Mathematics
Spring 2006 Math 332 - Real Analysis II, Final, 29/5/2006, 15:00-17:30
Whatever theorem, proposition, lemma you use, you must be sure that it is applicable. More hint means more detail is required!
Full Name : Student ID :
ov
Spring 2006 Math 332 - Real Analysis II Quiz 3, 26/4/2006 Proposition: A C 1 map f : R2 R cannot be one-to-one. We prove above Proposition using Implicit Function Theorem as follows: (a) [5] We may assume that there exists a point (a, b) R2 for which Df (
Spring 2006 Math 332 - Real Analysis II Quiz 4, 17/5/2006 1. Is there a relation between countability and being measure zero in Rn ? That is, which of the following is true/false? Why? (a) A measure zero set is necessarily countable. (b) A measure zero se
Shakespeare script
1: son: John Baker
2: mother: bertha Baker
3: daughter: Carla Baker
4: daughters friend: Christina Jobson
Story - 1 comes late, 2 puts the phone down and starts questioning his lateness,
both of them starting arguing, 3 comes in and ope
MATH 331-REAL ANALYSIS: SUPPLEMENTARY EXERCISES ON THE
LEAST UPPER BOUND PROPERTY OF R
The following exercises were assigned to my calculus students last year. I recommend that
you work on them to review the Least Upper Bound Property of R. In case you no
Sample Final
Multiple Choice:
2
1. The bicycle industry is made up of 100 firms with the short-run cost curve T C(q) = 2 + ( q2 ) and 120
2
firms with the short-run cost curve T C(q) = ( q4 ). Suppose that the industry is perfectly competitive.
What is th
Sample Midterm I
(Budget Constraint, Preferences, Utility)
PART I - Multiple Choice:
1. If there are only two goods, if more of good 1 is always preferred to less, and if less of good 2 is
always preferred to more, then indifference curves
(a) slope downw
Sample Midterm II
(Choice, Demand, Labor Supply and Intertemporal Choice)
Part I
Multiple Choice:
1. If there are two goods and if income doubles and the price of good 1 doubles while the price of
good 2 stays constant, a consumers demand for good
(a) 1 w
REAL ANALYSIS I HOMEWORK 2
CIHAN
BAHRAN
The questions are from Stein and Shakarchis text, Chapter 1.
1. Prove that the Cantor set C constructed in the text is totally disconnected and
perfect. In other words, given two distinct points x, y C, there is a p
Spring 2006 Math 332 - Real Analysis II Quiz 2, 5/4/2006 1. Let f : Rn R be a C 2 function; xo , u Rn be xed. Consider h : R R dened as h(t) = f (x0 + ut). Calculate Dh(0) and D2 h(0) in terms of the partial derivatives of f . Show your work in full detai
Spring 2006 Math 332 - Real Analysis II Quiz 1, 20/3/2006 1. (MH, page 383, ex.1) Let A be an open subset of Rm ; f, g : A Rn be dierentiable; , R. Prove, using the denition, that the function h = f + g : A Rn is dierentiable and for x A, D(h)(x) = Df (x)
B U Department of Mathematics
Spring 2006 Math 332 - Real Analysis II, First Midterm, 17/4/2006, 17:00-19:30
Whatever theorem, proposition, lemma you use, you must be sure that it is applicable. Justify all your claims in your proofs.
Full Name : Student
B U Department of Mathematics
Spring 2006 Math 332 - Real Analysis II, Second Midterm, 22/5/2006, 17:00-19:30
Whatever theorem, proposition, lemma you use, you must be sure that it is applicable. Justify all your claims in your proofs.
Full Name : Student
i
1
MATH 224 FINAL- EXAM June 5, 20.10, 10:00-12:30
NAME and SURNAME: SIGNATUU:
ii Vi V LV III Total
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/10 /12 /80 /18 /16 /14
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(In ink)
1. Let V= Pi(x) be the real vector space of all polynomials of degree at most 2. Let T be a Iinear operatar
B U Department of Mathematics
Fall 2008 Math 321 - Algebra, First Midterm Exam, 14/11/2008, 17:00-19:00
I hope you enjoy the exam!
Full Name Student ID
: :
over 100
In what follows, Zn is the additive group with the modular addition; G, H and K are groups
B U Department of Mathematics
Fall 2008 Math 321 - Algebra, Second Midterm Exam, 16/11/2008, 13:00-15:00 Full Name Student ID : :
over 100
1. (a) [10pts] The set of all elements aba1 b1 for a, b G generates a subgroup of G. This subgroup is called the com
B U Department of Mathematics
Spring 2009 Math 322 - Algebra II, Final Exam, 24/5/2009, 11:30-13:00 Full Name Student ID : :
over 40
A 1-1 ring homomorphism between two elds is called a eld homomorphism. An onto eld homomorphism is called a eld isomorphis
Spring 2009 Math 322 - Algebra II Quiz 1 - Greatest Common Divisor Look at the computation below, displaying the Euclidean algorithm in the principal ideal domain Z: 96 = 6 14 + 12, |12| < |14| 14 = 1 12 + 2 , |2| < |12| 12 = 6 2 + 0
(2 is the gcd of 96 a
Spring 2009 Math 322 - Algebra II Quiz 2 - Vector Spaces Consider the ring R = (Z2 Z2 , +, ) with the obvious addition and multiplication. (1) Is R commutative ring with identity? An integral domain? A Euclidean domain? Principal ideal domain? Unique fact
Spring 2009 Math 322 - Algebra II Quiz 3 - Field Extensions
Prove that Q( 2, 3) Q( 2 + 3). Generalize your result (not an extensive generalization, just = one step ahead).
Here are some possible generalizations; choose one of them which you believe is tr
Spring 2009 Math 322 - Algebra II Quiz 4 - Minimal polynomials 1) Let F be a eld with prime eld Q. Then Irr(F, 3 2) 3. 2i/ 2) deg Irr(Q, e 5 ) =? 3) First show 5 Q[ 2, 3]; then conclude Irr(Q[ 2, 3], 5) = 2.
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B U Department of Mathematics
Fall 2008 Math 322 - Algebra II, First Midterm Exam, 24/3/2008, 15:00-17:00
Write in this box how dicult you nd the exam (1-40). 1: too hard; 40: too easy. over 40
Full Name Student ID
: :
In what follows, R[x] is the ring of
B U Department of Mathematics
Spring 2009 Math 322 - Algebra II, Second Midterm Exam, 21/4/2008, 17:00-19:00 Full Name Student ID : :
over 40
Below, Q(S ) denotes the smallest subeld of C containing Q and S C.
1. [10pts] Prove that Q( 2, 3) Q( 2 + 3). Ge