Download DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE

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DE–2004
11
DISTANCE EDUCATION
M.Phil. (Mathematics) DEGREE EXAMINATION,
MAY 2011.
COMMUTATIVE ALGEBRA
(upto 2006 Batch)
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
(5  20 = 100)
1.
2.
3.
4.
5.
6.
7.
f
g
(a)
If 0  M   M  M   0
~ M   M  .
M
(b)
Prove that if P is a projective module, then there exists a free module F
such that P  F is free.
(a)
Show that the tensor product of two modules exists and it is unique upto
isomorphism.
(b)
Let M and N be free modules with bases { x1 ,....., xn } and { y1 ,..... ym } ,
respectively. Then show that M  N is free with basis { xi  yi } .
(a)
Show that M is faithfully flat if and only if M is flat and for each
maximal ideal m of R, mM  M.
(b)
State and prove Chinese Remainder theorem.
(a)
If R is an Artinian ring prove that the nil radical N(R) is nilpotent.
(b)
Show that Rs is flate R-module.
(a)
Show that the nil radical of R extends to the nil radical of Rs.
(b)
State and prove Hilbert’s basis theorem.
(a)
State and prove the structure theorem for artinian rings.
(b)
State and prove Jordan Hölder theorem.
(a)
Prove that the ring of integers in an algebraic number field is Dedekind
domain.
is
a
1
split
exact
sequence
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prove
that
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8.
(b)
Let R be a finitely generated k-algebra, k filed, and m a maximal ideal of
R. Then prove that R/m is a finite extension of k.
(a)
Let
R
be
a
Noetherian
domain
in
which
every
non-zero prime ideal is maximal. Prove that the following statements are
equivalent.
(i)
R is Dedekind domain;
(ii)
Rp is discrete valuation ring for every non-zero prime ideal, P of R;
(iii) Every primary ideal of R is a power of a prime ideal.
(b)
Show that any R-module M can be embedded in an injective R-module.
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DE-2005
12
DISTANCE EDUCATION
M.Phil.(Mathematics) DEGREE EXAMINATION, MAY 2011.
MEASURE THEORY
(upto 2006 Batch)
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
(5  20 = 100)
Each question carries 20 marks.
1. (a)
Let f : x  [ 0,  ] be measurable then there exists simple measurable
function Sn on X then show that
(b)
(i)
0  S1  S2  .....  f
(ii)
Sn ( x )  f ( x ) as n   for every x  X .
Prove that the sums and products of measurable function into [0,  ] are
measurable.
2.
State and prove monotone convergence theorem.
3.
Suppose f and g  L1 (  ) and  ,  are complex numbers then prove that
f  g  L1 (  ) and  (f  g )du    fdu    gdu.
X
4.
(a)
X
X
State and prove Holder's and minkowski's inequalities.
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(b)
If P and Q are conjugate exponents 1  P   and if f  p (  ) and
g  q (  ) then prove that fg  L1 (  ) and fg  f
p
g
q.
5.
Show that Lp (  ) is a complete metric space for 1  p   and for every positive
measure  .
6.
(a)
If X is a locally compact Housdroff space then prove that Co(x ) is the
completion of CC (X ) relative to the metric defined by the supremum
norm f  sup f ( x ) .
xx
(b)
If 1  p   and fn  is a cauchy sequence in Lp (  ) with limit f thene
prove that fn  has a subsequence which converges pointwise almost
everywhere
to f (x ).
7.
State and prove Radon-Nikodyin theorem.
8.
(a)
If E SXJ then prove that E X EJ and E y  S for every x  X and
yY .
(b)
State and prove Fubini theorem.
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DE-2006
13
DISTANCE EDUCATION
M.Phil. (Mathematics) DEGREE EXAMINATION, MAY 2011.
TOPOLOGICAL VECTOR SPACES
(Upto 2006 Batch)
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
(5  20 = 100)
1. If X is a topological vector space with a countable local base, prove that there is a
metric d on X such that
(a)
d is compatible with the topology of X
(b)
the open balls centered at 0 are balanced, and
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(c)
d is invariant : d( x  z, y  z )  d( x, y) for x, y, z  X .
Further prove that if X is locally convex, then d can be chosen so that all
open balls are convex.
2.
(a)
(b)
Suppose A is a convex absorbing set in a vector space X. Prove the
following :
(i)
 A ( x  y)   A ( x )   A ( y)
(ii)
 A (tx )  t A ( x ) if t  0.
(iii)
 A is a seminorm if A is bounded
(iv)
If B  { x :  A ( x )  1} and C  { x :  A ( x )  1} then B  A  C and
 B   A  C .
Suppose  is a separating family of seminorm on a vector space X.
Associate to each p   and each positive integer n, the set
1

V ( p, n )  x : p( x )   . Let β be the collection of all finite intersections
n

of the sets V ( p, n ) . Prove that β is a convex balanced local base for a
topology  on X, which turns X into a locally convex space such that
3.
4.
5.
(i)
every p   is continuous and
(ii)
a set E  X is balanced if and only if every p   is bounded on
E.
(a)
State and prove the category theorem.
(b)
State and prove open mapping theorem.
(a)
If X and Y are topological vector spaces, K is a compact convex set in
X ,  is a collection of continuous linear mappings of X into Y, and the
orbits ( x )  {  x :    } are bounded subsets of Y, for every x  K ,
prove that there is a bounded set B  Y such that  ( K )  B for every
.
(b)
If f is a continuous linear functional on a subspace M of a locally convex
space X, prove that there exists   X * such that   f on M.
(a)
Suppose A and B are disjoint, nonempty, convex sets in a topological
vector space X. Prove that the following :
(i)
If A is open, there exists   X * and   R such that Re
 x    Re  y for every x  A and for every y  B
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(ii)
6.
7.
If A is compact, B is closed and X is locally convex then there exist
  X *  1  R,  2  R such that Re  x   1   2  Re  y for every
x  A and for every y  B .
(b)
If X is a vector space and X ' is a separating vector space of linear
functionals on X, prove that X ' -topology  ' makes X into a locally convex
space whose dual space is X'.
(a)
State and prove the Krein-Milman theorem.
(b)
If X is a locally convex space and H is the convex hull of a totally bounded
set E  X prove that H is totally bounded.
(a)
If
to
X
and
each
Y
are
T  (X, Y )
normed
spaces
prove
there
corresponds
that
a
unique T *   (Y * , X * ) that satisfies  Tx, y*    x, T * y*  for all
x  X and y*  Y * . Also prove that T *  T .
(b)
8.
(a)
If X and Y are Banach spaces and if T   (X, Y) prove
following equivalent
(i)
R (T ) is closed in Y.
(ii)
R (T * ) is weak*-closed in X * .
(iii)
R (T * ) is norm-closed in X * .
that the
Suppose
(i)
X is a Frechet space.
(ii)
Y is a complemented subspace of X.
(iii) G is a compact group which acts as a group of continuous linear
operators on X.
(iv)
TS (Y )  Y for every y  G .
Prove that there is a continuous projection Q of X onto Y which
commutes with every TS .
(b)
If P is a continuous projection in a topological vector space prove that
X  R( P )  N ( P ). Conversely, if X is an F-space and if X  A  B
prove that the projection P with range A and null space B is continuous.
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DE–2007
14 (a)
DISTANCE EDUCATION
M.Phil. (Mathematics) DEGREE EXAMINATION, MAY 2011.
FUNDAMENTALS OF DOMINATIONS IN GRAPH
(Upto 2006 onwards)
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
(5  20 = 100)
1. (a)
2.
3.
4.
State and prove ORE theorem.
(b)
Show that a graph G is bipartite if and only if it has no odd cycle and that
the chromatic number of a bipartite graph is 2.
(a)
Show that for a graph G with even order and no isolated vertices,
n
if and only if the components of G are the cycles C 4 or the
 G  
2
corona H  K , for any connected graph H.
(b)
State and prove Nieminen theorem.
n 
Show that a connected graph G satisfies  G     if G  g i for some i  1 to
2
4.
(a)
Show that if a graph G has no isolated vertices then  G    2 G  .
(b)
Show that for any graph G,
 G 
2
5.
(a)
 irG    G   2irG   1 .
Show that for any strongly perfect graph G,
 0 G   G   IRG 
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(b)
6.
7.
8.
Show that for any graph G, IRG   IR, G  .
Show that if T is a tree of order n  2 then
3n
1
2
(a)
RT  
(b)
F T   8n  2  4 .
Show that the k-cube Qk has a perfect d-cube if and only if
(a)
kd
(b)
k  2 j  1 and d  1
(c)
k  23 and d  3 .
State and prove the necessary and sufficient conditions for a graph G to be a
member of UER.
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DE–2008
14 B
DISTANCE EDUCATION
M.Phil. (Mathematics) DEGREE EXAMINATION, MAY 2011.
DATA STRUCTURES AND ALGORITHM
(upto 2006 Batch)
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
(5  20 = 100)
1. (a)
2.
Write notes on classes and objects.
(10)
(b)
Write a program that over loads ++ and – – operators.
(10)
(a)
Explain the concept of polymorphism.
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(10)
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3.
4.
5.
6.
7.
8.
(b)
Explain the class declaration of arrays. Describe the implementation of
such arrays with examples.
(10)
(a)
Extend the class polynomial for performing addition, subtraction and
division of polynomials using overloaded operators.(10)
(b)
What is the use of head nodes in the linked lists?
(10)
(a)
Explain implementation of stacks using arrays. (10)
(b)
Write a recursive algorithm for finding factorial of a number. Convert
this into a non recursive form using stack.
(10)
(a)
Explain circular queue with an example.
(b)
Explain the operations on queues with example. (10)
(a)
Discuss binary tress.
(b)
Write a non recursive algorithm for post-order traversal on a binary tree.
(10)
(a)
Explain binary search tress.
(b)
Write an example which takes a binary tree and swaps the left and right
children of every node. Find the computation time of your algorithm.
(10)
(a)
Explain Kruskal’s algorithm.
(b)
Write a procedure to determine whether or not G is an undirected graph.
(10)
(10)
(10)
(10)
(10)
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