Name of the University : Alagappa University
Degree : M.Phil
Department : Mathematics
Subject Code/Name : Commutative Algebra
Document Type : Model Question Papers
Website : alagappauniversity.ac.in
Download Model/Sample Question Paper : May 2008 : www.pdfquestion.in/uploads/alagappauniversity.ac.in/4116.-M.Phil.(Maths)upto2006bat.docx
Commutative Algebra :
1. (a) Define nil radical. Prove that the nil radical n of A is the intersection of all the prime ideal of A.
(b) Explain Tensor and Flat modulus.
Related : Alagappa University Theory Of History And Methods Of Research M.Phil Model Question Papers : www.pdfquestion.in/4111.html
(b) State and prove that the uniqueness theorem.
2. (a) Define Length.
Let A be a ring in which the zero ideal is a product m1,m2,m3…mn of (not necessarily district) maximal ideals. Then A is Noetherian iff A is Artinian.
(b) State and prove Jordan Holder Theorem.
3. (a) State and prove Going-up Theorem.
(b) Let (B,g) be a maximal element of Summation. Then B is a valuation ring of the field .
4. State and prove Norther’s normalization theorem.
5. (a) Define Discrete valuation ring.
(b) Let A be a Noetherian local domain of dimension 1, m its maximal ideal and k=A/m its residue field. Then the following are equivalent.
(i) A is a discrete valuation ring.
(ii) A is integrally closed.
(iii) m is a principal ideal.
(iv) Every non zero ideal is a power of .
Distance Education :
M.Phil. (Mathematics) Degree Examination, May 2008 :
Measure Theory : (upto 2006 batch)
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
Each question carries 20 marks.
1. (a) State and prove Lebesque monotone convergence theorem.
2. Explain briefly integration of complex functions.
3. State and prove Jensen’s inequality.
4. (a) Define conjugate exponents.
(b) State and prove Schwarz’s inequalities.
5. (a) State and prove Radon-Niscodyin theorem.
(b) Explain product measures.
Distance Education :
M.Phil. Degree Examination, May 2008 :
Mathematics L
Topological Vector Spaces : (Upto 2006 Batch)
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
Each question carries 20 marks.
1. (b) Prove that every locally compact topological vector space X has finite dimension. (10 + 10)
2. (a) In a topological vector space X, prove the following
(i) every neighborhood of O contains a balanced neighborhood of O and
(ii) every convex neighborhood of O contains a balance convex neighborhood of O .
(b) Suppose Y is a subspace of a topological vector space X and Y is locally compact, in the topology inherited from X . Prove that Y is a closed subspace of X . (10 + 10)
(b) Prove that a topological vector space X is normable if and only if its origin has a convex bounded neighborhood. (12 + 8)
3. (a) State and prove the category theorem.
(b) State and prove the Banach–Steinhaus. (10 + 10)
4. (a) State and prove the closed graph theorem.
5. State and prove the Banach-Alaoglu theorem.
Distance Education :
M.Phil. (Mathematics) Degree Examination, May 2008 :
Fundamentals Of Domination In Graph : (Upto 2006 batch)
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions
All questions carry equal marks.
1. For any graph T show that the following are equivalent
(a) T is connected and acyclic
(b) T is connected and m=n-1
(c) T is acyclic and m=n-1
(d) For any two vertices u and v in V(T) there exists a unique u-v path.
2. (a) State and prove ORE theorem on dominating set.
(b) Show that every connected graph G of order n>=2 has a dominating set S whose complement V-S is also a dominating set.
Distance Education :
M.Phil. (Mathematics) Degree Examination, May 2008 :
Data Structures And Algorithms : (Upto 2006 Batch)
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a) Describe the concept of polymorphism with an example. (10)
(b) Explain the following concepts with suitable example.
(i) Dynamic binding
(ii) Message passing. (5 + 5)
2. Explain the various statements available in C++ with suitable examples. (20)
3. (a) Explain how a multi-dimensional array can be implemented in a computer memory. (10)
(b) (i) Explain the structure of singly linked list. (5)
(ii) Write an algorithm to insert a node at the front of the linked list. (5)
4. (a) Explain doubly linked list concepts and its application. (10)
(b) Write short notes on circularly linked list with example. (10)
5. (a) Write down the stack operations with example. Discuss the applications of stack. (10)
(b) How are stacks used in evaluating a given numerical expression? Illustrate with example. (10)
6. (a) Write algorithms to do the following
(i) delete an element from the queue. (5)
(ii) insert an element into the queue. (5)
(b) Explain the following
(i) Priority queue (5)
(ii) Circular queue. (5)
7. (a) Write an algorithm for converting general trees to binary trees. (10)
(b) Write an algorithm for postorder tree traversal with example. (10)
8. (a) Explain the travelling salesman problem using BFS technique. (10)
(b) Write and explain Kruskal’s algorithm to find a minimum cost spanning tree. (10)