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KARNATAKA STATE OPEN UNIVERSITY
Mukthagangotri, Mysore – 570 006
Department of Studies in Mathematics
Email: [email protected],
Date: 10.02.2015
THIRD SEMESTER M.Sc. MATHEMATICS
INTERNAL ASSIGNMENTS
(For 2013 – 2014 Batch)
Instructions:
¤ Answer all the questions and each question carry 02 marks.
¤ Assignment of each course shall be submitted separately.
¤ Assignment shall be in one’s own handwriting and be written in A4 size sheet.
¤ On the covering sheet of the assignment write your Name, Register Number and Course
¤ The facing sheet of the assignment shall be in a prescribed format, which may be
downloaded fromhttp://karnatakastateopenuniversity.in
¤ Using plastic sheets / spiral wires to bind assignment is strictly prohibited.
¤ Assignments shall be submitted on or before 15th of April 2015. The assignments
submitted after the last date will not be considered for internal assessment marks.
¤ All the assignments shall be put in a single cover and to be submitted to the address given
bellow by post or in person on or before the last date.
Lecturer
DOS in Mathematics
Karnataka State Open University
Mukthagangotri, Mysuru -560 006.
Answer All the Questions:
Department of Mathematics
Topology - Math 3.1
1) Define closure of a set. Show that a point ‘x’ belongs to the closure of a set
if and only if every open set G which contains x has a non empty intersection
with A.
2) Show that a bijective function f: XY is a homeomorphism if and only if
f(A0) = [f(A)] 0 ,  A X.
3) Prove that (X, ) is compact if and only if every family of closed sets having
finite intersection property has a non - empty intersection.
4) A countably compact metric space is totally bounded.
5) State and prove Urysohn’s lemma.
Measure and Integration- Math 3.2
1) If E [0,1) is a Lebesgue measurable set. Then show that for each y  [0,1),
the set E + y is Lebesgue measurable and m(E + y) = mE.
2) If f is a bounded function defined on a measurable set E with mE < ∞. Then
prove that f is measurable if and only if,

inf
  sup



f E
f
E
for all simple functions  and  .
3) State and prove Lebesgue Convergence Theorem.
4) If C is a contant, f and g are measurable functions then prove that f+c, cf,
f+g, f-g, f2 and fg are also measurable functions.
5) State and prove Radon - Nikodym theorem.
Functional Analysis - Math 3.3
1) Show that all completion of a given metric space are isometric.
2) State and prove Baire’s category theorem.
3) If (X, d) is a complete and totally bounded metric space. Then prove that X is
compact.
4) State and prove Stone - Weierstrass Theorem (in real case).
5) If X and Y are two Banach spaces and if T: XY is a linear transformation
from X into Y. Then prove that, T is continuous if and only if (T) is closed.
1)
2)
3)
4)
5)
Mathematical Modeling - Math 3.4
Explain the characteristics of mathematical modeling.
Explain the construction of spring and dashpot systems also explain when
motion is said to be under damped, over damped and critically damped.
Describe a model for detection of diabetes.
Write a brief note on circular motion and elliptic motion of satellites.
Briefly explain on primary and secondary air pollutants.
Computer Programming - Math 3.5
1)
2)
3)
i.
ii.
iii.
iv.
Write a short note on operating systems.
Explain the basic components of a C program.
Explain the following along with example;
Loop
For loop
While loop
Do while loop.
4) Describe the following:
i.
ii.
iii.
iv.
Dynamic memory allocation
Memory allocations process
Allocating a block of memory
Allocating multiple blocks of memory.
5) Explain the silent features of Secant method and write its algorithm and C
program.
Facing sheet
KARNATAKA STATE
OPEN UNIVERSITY
Department of Studies in Mathematics
Mukthagangotri, Mysuru – 570 006
THIRD SEMESTER M.Sc. MATHEMATICS
ASSIGNMENT
COURSE CODE:
COURSE TITLE:
Name of the Candidate
Roll Number
Date of Submission
For Office Use only
Marks Obtained
Signature of The
Evaluator