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0821-2500873 (ext: 556/557) 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: XY 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: XY 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