M.Sc. Mathematics - Final - Kuvempu University

KUVEMPU UNIVERSITY
OFFICE OF THE DIRECTOR
DIRECTORATE OF DISTANCE EDUCATION
Jnana Sahyadri, Shankaraghatta – 577 451, Karnataka
Phone: 08282-256426; Fax: 08282-256370; Website: www.kuvempuuniversitydde.org
E-mails: ssgc@kuvempuuniversity.org; info@kuvempuuniversitydde.org
TOPICS FOR INTERNAL ASSESSMENT ASSIGNMENTS (2012-13)
Course: M.Sc. MATHEMATICS (Final)
Note: Students are advised to read the separate enclosed instructions before beginning the writing of
assignments.
Out of 20 Internal Assignment marks per paper, 5 marks will be awarded for regularity (attendance) to
Counseling/ Contact Programme pertaining to the paper. Therefore, the topics given below are only for
15 marks each paper.
PAPER V: COMPLEX ANALYSIS
ANSWER ALL TOPICS
1. a) If and are the functions that satisfies the C-R equations at a point , then prove
with counter example that also satisfies the C-R equations at .
b) Find the most general linear transformation of the circle || into itself.
∞
2. a) Show that
using Morera’s theorem.
is analytic in the left half plane : 0!
b) How many roots of the equation " 8 $ 3 & 8 3 0 lie in the right half
plane? (Sketch the image by applying Argument principal).
3. a) Suppose is analytic in || ' 1 )*+ || ' 2 -. || 1, 01 2 0 & || '
3 -. || 1, 01 0 then prove that |0| ' √6.
b) Give the example of a divergent series of functions whose partial sums are bounded.
PAPER VI: TOPOLOGY
1. a) A finite product of compact spaces is compact.
b) Find a family of union closed subsets of the real line whose union is not closed.
2. a) The closure of a connected set is connected?
b) Every regular, second countable space is normal.
3. a) is complete but not totally bounded.
b) 0,1 is totally bounded but not complete.
PAPER VII: MEASURE THEORY AND FUNCTIONAL ANALYSIS
1. a) If || is measurable, is measurable ?

) Construct a sequence 6 7 8 of non-negative, Riemann integrable functions such that 7
increases monotonically to . What does this imply about changing the order of
integration & the limiting process?
c) Construct a monotone function on 90,1: which is discontinuous at each rational point.
2. a) Show tan > ? is absolutely continuous on .
b) Show that every subset of a separable normed linear space is separable.
c) State Contraction mapping theorem. Whether the map @: 90,1: A 90,1: such that @?
? & B ? 1 is a contraction map or not?
3. a) If closed and bounded sets are compact in a normed linear space C then show that it is
finite dimensional.
b) State Reisz lemma. Show that if D is finite dimensional proper subspace of a normed
space E , then there is ? E such that F? F 1 and ? , D 1.
c) State open mapping theorem and closed graph theorem. Give atleast two examples to show
that the closed graph theorem and open mapping theorem may not hold if the normed
spaces E and D are not Banach spaces.
PAPER VIII: NUMERICAL ANALYSIS
1. a) Find the solution of the system
4? B ? & B ? $ 2
B? 4? & B ? " 2
B? 4? $ B ? " 1
B? & B ? $ 4? " 1
using Gauss Seidel Method by performing 4 iterations.
b) Find the all Eigen values and the corresponding Eigen vectors for the following matrix
3 2 1
using Jacobi method H2 3 2I .
1 2 3
2. a) Find the cubic spline interpolation polynomial for the following data
x 0 1 2 3
f(x) -1 1 9 29
Given J′′ (0) = J′′ (3) = 1
b) Obtain the solution of the Boundary Value Problem ?KLL K′ B ?K 0, with K0 0,
K1 1.5 using Shooting method.
3. Obtain the solution of N 4N OO , 0 ' ? ' 4, subjected to the boundary conditions
N0, N4, 0, P 0 and the initial conditions N?, 0 4? B ? & using
Crank-Nicolson method, perform five iterations, choose + 1, Q R .
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