1111-~ljJA~ 102 & r if±JjiJ b]fA~

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1111-~ljJA~ 102 & r if±JjiJ b]fA~

[jij~-~~]

advanced calculus

1. Let the sequence {an} be given recursively by the formula

Show that {an} is monotone increase and bounded above. (20 points) 2. Show that the function

1

^1/x ¹

¹⁰

^t2

f(x)

=

- - d t - - - d t

0 1

+

t⁴ ^X 1

+

t⁴

is constant for x

>

0. (20 points) 3. Evaluate the double integral

J

₀²

JY 2

ex²dxdy. (20 points)

4. For x E JR³, let p(x) be a charge density that is continuous and such that p(x) = 0 for

llxll

2

2::

1. Show that the electrostatic potential, given by

is a convergent integral for each x E JR³. (20 points) 5. Evaluate

JRn

^jjxjj²^·^e-llxll²^dx. (20 points)

1

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Linear Algebra PhD Entrance Exam Date: Friday 03/05/2013 Work out all problems and no credit will be given for an answer without reason- ing.

1. (a) (5%) If Vis a vector space over F of dimension 5 and U and Ware subspaces of V of dimension 3, prove that U

n

WI= {0}. Generalize.

(b) (5%) Let V

=

IR³ and W ={(a, b, c) E V

I

^a+b

=

c}. Is W a subspace of V? If so, what is its dimension?

(c) (10%) Let V and W are vector spaces over a field F. Define a vector space isomorphism is a one-to-one linear transformation from V onto W. If V is a vector space over F of dimension n, prove that V is isomorphic as a vector space to pn = {(a1, a2, ... , an)

I

^ai^E

F}.

2. (a) (10%) Show that the linear transformation T: IR³ --+ IR³defined by

is invertible, and find a formula for its inverse.

(b) ( 10%) Let V be the vector space of 2 by 2 matrices over IR and let

Let T : V --+ V be the linear transformation defined by T(A) = AM- M A. Find a basis and the dimension of the kernel W of T.

3. (a) (5%) Prove that if A is a square matrix, then AAT and AT A have the same eigenvalues.

(b) (8%) Diagonalize the matrix

and compute Ak in terms of k.

(c) (7%) Let

[-3 5]

A= -2 4 '

1 2 3 6 7 8

4 5 9 10 A = 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Find det(A- h).

2-1

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4. (a) (10%) Find the minimal polynomial m(t) of

A=

[~ ~ I ^!]

(b) (10%) Find a Jordan canonical form and a Jordan basis for the given matrix:

A= [-;

3

~2 ~]

^.

-1 0 -1

5. (a) ( 10%) Find an orthogonal basis for the subspace spanned by the set { 1,

Jx,

^{x} of}

the vector space Cro,l] of continuous functions with domain 0 ::; x ::; 1, where the inner product is defined by (!,g) =

J

₀¹ f(x)g(x) dx.

(b) (10%) LetT be a linear operator on a finite dimensional inner product space V.

Show that there exists a unique linear operator T* on V such that (T(u), v)

=

(u, T*(v))

for every u, v E V.

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Real Analysis 1. Lebesgue outer measure in lR

(a) (5) State the definition of the Lebesgue outer measure.

(b) (25) Show that the Lebesgue outer measure of an interval is its length.

2. Lebesgue measurable real-valued function

(a) (5) State the definition of a Lebesgue measuable real-valued function.

(b) (25) Show that the pointwise a.e. limit of a sequence of Lebesgue measuable real-valued functions is again measurable.

3. Lebesgue integral in lR

(a) (5) State the definition of the Lebesgue integral of a nonnegative measurable function.

(b) (35) state and prove Fatou's Lemma.

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Algebra Exam

May 2013

Z

=

integers. Q

=

rational numbers. C

=

complex numbers.

1. (20 points) Determine whether each statement below is true or false. If true, prove the statement. If false, provide a counterexample.

(a) Every prime ideal of every commutative ring with unity is a maximal ideal.

(b) If Dis an integral domain, then D[x] is an integral domain.

(c) Let lF be a field. Every principal ideal of JF[x] is a maximal ideal.

(d) A ring homomorphism¢ : R-> R' carries ideals of R into ideals of R'.

2. (10 points) Let Hand K be subgroups of a group G. Show that the index of H nK in His at most equal to the index of K in G:

[H : H n K] ~ [ G : K].

3. Let ^Gbe a simple group of order 60.

(a) (8 points) Show that G has six Sylow 5-subgroups.

(b) (8 points) Show that G has ten Sylow 3-subgroups.

4. (10 points) Let p ::f. q be prime numbers. Prove that no group of order p²q is simple.

5. (10 points) Determine the Galois group of p(x) = x⁵- 4x

+

2 over Q.

6. What is the Galois group of p(x) = x³- x

+

4, considered over the ground fields (a) (5 points) z3,

(b) (5 points) R, (c) (8 points) Q.

7. Let X be a topological space. Consider the ring R(X) of continuous real-valued functions on X. The ring structure is given by point-wise addition and multiplication.

(a) (8 points) Show that for each x EX the set

Mx = {J ER(X) I f(x) = 0}

is a maximal ideal in R(X).

(b) (8 points) Show that if X is compact, that is, every open covers of X has a finite subcover, then every maximal ideal in R(X) is equal to Mx for some x EX.

This exam has 7 questions, for a total of 100 points.

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1. (20 pts) Let S be the elliptic paraboloid given by

S

=

{(x, y, z) E lR³¹2z

=

x²

+ l}.

Find the Gauss curvature K (x, y, ::) and the mean curvature H (x, y, z) of S.

2. (25 pts) Let a : I ___, JR³be a regular curve parametrized by arc length s with curvature k (s) and torsion T (s). Assume that T (s)

=I

0 and k' (s) =f 0 for all s E I. Show that a necessary and sufficient condition for a (I) to lie on a sphere is that

2 I 2 2

R

+

(R) T

=

const.,

where R = 1/k, T = 1/T, and R' is the derivative of R relative to s.

3. (15 pts) Let S be a regular orientable surface. Show that the mean curvature H at p E S is given by

1

~n·27r

H = - K.n (e) del 27r. 0

where ^K,n(B) is the normal curvature at ^palong a direction making an angle

e

with a fixed direction.

4. (15 pts) Let S be a connected and orientable regular surface. Prove that H² 2: K where K and H are the Gauss curvature and mean curvature of S, respectively. If H²= K for all p E S, what is S?

5. (25 pts) Show that the circular cylinder S¹x (0, 1) is a regular surface.

1111-~ljJA~ 102 ~~& r if±JjiJ b]f~~~~A~~~