M.Sc Mathematics Question Paper

Loyola College M.Sc. Mathematics April 2012 Operations Research Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2012

MT 4811 – OPERATIONS RESEARCH

 

 

Date : 18-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL the questions

            All questions carry equal marks

 

I a) Explain sensitivity analysis. Is it really useful to a company?

(or)

  1. b) Explain branch and bound method.                             (5)

 

  1. c) Find the optimum integer solution to the following LPP.

 

 

(15)

(or)

 

  1. Solve the problem

Discuss the effect of changing the requirement vector from  on the optimum solution.

II a) What is goal programming? How is it useful for a manufacturing company?

(or)

  1. b) Mention the differences between LP and GP approach.                               (5)

 

  1. c) A firm produces two products A and B. Each product must be processed through two departments. Department I has 30 hours of production capacity per day, and department II has 60 hours. Each unit of Product A requires 2 hours in department I and 6 hours in department II. Each unit of product B requires 3 hours in department I and 4 hours in department II. Management has rank ordered the following goals it would like to achieve in determining the daily product mix.

P1 : Minimize the underachievement of joint total production of 10 units.

P2 : Minimize the underachievement of producing 7 units of product B.

P3 : Minimize the underachievement of producing 8 units of product A. Formulate

this problem as a GP problem and illustrate with graph.                                (15)

(or)

  1. d) A factory can manufacture two products A and B. The profit on a unit of A is `.80 and of B is `.40. The maximum demand of A is 6 units per day and of B is 8 units. The manufacturer has set up a goal of achieving a profit of `.640 per day. Formulate the problem as goal programming and solve it. (15)

 

III a) Explain the following terms in inventory: setup cost, holding cost, lead time,  optimal cost and stock out cost.

(or)

  1. b) Explain the term Price Break. Is it advisable to accept it always? (5)

 

  1. c) Group the items given below into an ABC classification.

 

Item No. Units Unit cost in Rs.
1 7000 5
2 2400 3
3 1500 10
4 600 22
5 3800 1.50
6 40000 0.50
7 60000 0.20
8 3000 3.50
9 300 8.00
10 29000 0.40
11 11500 7.10
12 4100 6.20

Explain by graphical representation.

(or)

  1. d) (i) A company operating 50 weeks in a year is concerned about its stock of

copper cables. One meter costs `.240 and there is a demand for 8000 meters per

week. The setup cost is `.2,700 and the holding cost is  25 % of  one meter cost.

Assuming no shortages are allowed, find the optimal inventory policy. Also find

the number of orders and total inventory cost.

(ii) Plastic drums are produced at the rate of 50 items per day. The demand

occurs at the rate of 25 items per day. If the setup cost is `.1000 per setup and

holding cost is `.1.00 per unit of item per day find the economic lot size for one

run, assuming that the shortages are not allowed. Also find the time of cycle

and the minimum total cost of one production run.                                   (8+7)

 

  1. a) Explain optimistic time and pessimistic time in network model.

(or)

  1. b) Explain Kendall’s notation for representing queuing models.       (5)

 

 

 

 

 

  1. c) Use Branch and Bound technique to solve the following:

 

(15)

(or)

 

  1. d) With usual notation show that the probability distribution of queue length is

given by  where .

V a) Write Kuhn-Tucker conditions for a quadratic programming problem.

(or)

  1. b) State Wolfe’s algorithm. (5)

 

  1. c) Using Kuhn-Tucker conditions

(15)

(or)

  1. d) State the special features of dynamic programming technique. Find the shortest

route for traveling from city 1 to 10 using dynamic programming technique.

 

 

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Loyola College M.Sc. Mathematics April 2012 Measure Theory And Integration Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

MT 2811 – MEASURE THEORY AND INTEGRATION

 

 

Date : 19-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

ANSWER ALL QUESTIONS:-

  1. (a) State and prove countable sub additive theorem for outer measures. (5)

(OR)

(b) Prove that every interval is measurable.                                                                 (5)

 

(c) Prove that there exists a non measurable set.                                                        (15)

(OR)

(d) Show that Lebesgue measure is regular.                                                               (15)

 

 

  1. (a) Let f and g be non negative measurable functions. Then prove ò f dx +  ò g dx = ò (f  + g) dx .                                                                                     (5)

(OR)

(b) Prove that if the sequence  is a sequence of non-negative measurable function

then .                                                                                   (5)

 

(c)  State and prove Lebesgue Dominated Convergence theorem.                            (15)

(OR)

(d) If f is Riemann integrable and bounded over the finite interval [a,b] then prove that f

is integrable and .                                                                          (15)

 

  • (a) Show that with a usual notations the outer measure m* on H(Â),and the         (5)

outer measure defined   by  on S( Â)  and on contains   are the same.

(OR)

 

(b) Prove that if m* is an outer measure on H(Â), defined by m on H(Â) then contains

, the  -ring generated by  Â.                                                                        (5)

 

(c) Show that if  is a measure on a -ring  then the class of sets of the form

for any sets E,N such that While N  is contained in some set in of zero

measure is a -ring and the set function defined by is a

complete measure on .                                                                                              (15)

(OR)

 

(d) Prove that if  is an outer measure on H(Â),. Let  denote  the class of

Measurable sets then Prove that  is a – ring and  restricted to is a complete

measure.                                                                                                                     (15)                                                                                                                                                                                    

 

  1. (a) State and prove Holder’s inequality. (5)

(OR)

(b)   Define the following terms: convergence in measure, almost uniform convergence and uniform convergence almost everywhere.                                                                                                           (5)

(c)  Let [X, S, ] be a measure space with . If  is convex on (a, b) where  and f is a measurable function such that , for all x, prove that . When does equality occur?                    (15)

(OR)

(d) State and prove completeness theorem for convergence in measure. Show that if  almost uniform then  in measure and almost everywhere.                                                                 (15)

 

  1. (a) Define a positive set and show that a countable union of positive sets with respect to a

signed measure v is a positive set.                                                                                                             (5)

(OR)

(b)  Let v be a signed measure and let  be measure on [X, S] such that  are – finite, «, « then prove that .                                                                                                                    (5)

 

 

(c)  Let v be a signed measure on [X, S]. (i) Let  S and . Can you construct a positive set A with respect to v, such that  and ? Justify your answer. (ii) Construct a positive set A and a negative set B such that .        (15)

(OR)

(d) State and prove Lebesgue decomposition theorem.                                                   (15)

 

 

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Loyola College M.Sc. Mathematics April 2012 Mathematical Methods Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

MT 2901 – MATHEMATICAL METHODS

 

 

Date : 28-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION A

Answer any five questions                                                                                   (5 x 4 = 20)

  1. Find the equation of the line through the point and parallel to the line through the points  and .
  2. Suppose the fixed cost of production for a commodity is $20000; the variable cost is $15 per unit and the commodity sells for $20 per unit. What is the break-even quantity?
  3. If the marginal revenue function is , determine the revenue and demand functions.
  4. Evaluate if  when .
  5. Find the general solution of the differential equation .
  6. If , find and .
  7. If and , show that .

SECTION B

Answer any four questions                                                                               (4 x 10 = 40)

  1. Find the equation of the line that passes through the intersection of the lines and  and is perpendicular to the line .
  2. (i) Find the equation of the curve which has a slope of zero at the point , has a point of inflection at  and for which .

(ii) Evaluate      .                                                                                (6+4)

  1. Evaluate  using partial fractions.
  2. (i) Show that is a solution of  where c is an arbitrary constant and find a particular solution that satisfies the condition  when .

(ii) Solve the equation .                                                                (6+4)

  1. Solve the difference equation , if and also calculate , ,  and .
  2. If , . Find the least squares estimates for the regression equation .

SECTION C

Answer any two questions                                                                                 (2 x 20 = 40)

  1. (i) What relation (parallel, perpendicular, coincident or intersecting) does the line have to the following lines?
  • (b)  (c)

(d)  (e)  (f) .

(ii) Find the equation of the line which is parallel to the line through the points (5,6) and (7,8) and also passes through the intersection of the line having slope -2 through the point (-4,-6) and the line having slope 3 through the point (2,2).

(12+8)

  1. (i) The quantity demanded and the corresponding price, under pure competition, are determined by the demand and supply functions and , respectively. Determine the corresponding consumer’s surplus and producer’s surplus. (ii) Evaluate (a)  (b)                                                                                                                                                                  (10+10)
  2. (i) Solve the equation .

(ii) State the type, order and degree of the following equations

  • ; (b) .

(iii) Show that  is a solution of  and find a particular solution if ,,

(6+4+10)

  1. (i) Write the difference equation in terms of lagged values of .

(ii) If , find .

(iii) If ,   and  then show that  .                                                                                                         (8+4+8)


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Loyola College M.Sc. Mathematics April 2012 Graph Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2012

MT 4802 – GRAPH THEORY

 

 

Date : 21-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

 

Answer all the questions. Each question carries 25 marks.

 

  1. (a) State Cayley’s formula and determine the number of spanning trees of the following graph.

 

(8)

(OR)

(b) Define a graphic sequence. Obtain a necessary and sufficient condition for a sequence d = (d1, d2 … dn) to be graphic.                                                                         (8)

 

(c) (i) Obtain a characterization for bipartite graphs.

(ii) Prove that an edge of a graph G is a cut edge if and only if it is contained in no cycle of G.                                                                                                 (8 + 9)

(OR)

(d) Determine the shortest path / distance between u0 and all other vertices of the following graph.

(17)

 

  1. (a) Prove with usual notation that . (8)

(OR)

(b) Prove that a simple graph with  is Hamiltonian if and only if its closure is Hamiltonian.                                                                                               (8)

 

(c) (i) If G is a non-Hamiltonian simple graph with n ≥ 3, then prove that G is degree-majorised by some Cm, n .

(ii) Prove that a nonempty connected graph is Eulerian if and only if it has no vertex of odd degree.                                                                                        (7 + 10)

(OR)

(d) Describe the Chinese Postman problem. Obtain an optimal tour in the following weighted graph.

(17)

 

  1. (a) State and prove Berge theorem on maximum matching. (8)

(OR)

(b) State Kuhn-Munkres algorithm.                                                                            (8)

 

(c) (i) State and prove Hall’s theorem.

(ii) Prove that in a bipartite graph, the number of edges in a maximum matching is equal to the number of vertices in a minimum covering.                                 (9 + 8)

(OR)

(d) Prove, with usual notation, that a + b = n = a ¢ + b ¢, if d > 0.                           (17)

 

  1. (a) Let G be a k-critical graph with a 2-vertex cut {u, v}. Then prove that G = G1 G2 where Gi is a {u, v} – component of type i ( i = 1, 2) and that G1 + uv is k – critical.                                                                                                                                   (8)

(OR)

(b) Obtain Euler’s formula and deduce that K5 are non-planar.                                 (8)

 

(c) Prove that a graph is planar if and only if it contains no subdivision of K5 or K3, 3.

(17)

(OR)

(d) (i) State and prove Brook’s theorem.

(ii) State and prove five color theorem.                                                                (8 + 9)

 

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Loyola College M.Sc. Mathematics April 2012 Functional Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2012

MT 4810 – FUNCTIONAL ANALYSIS

 

 

Date : 16-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

Answer ALL questions:                                                                                           (5 x 20 = 100 Marks)

  1. a) Show that every element of X/Y contains exactly one element of Z where Y and Z are

complementary subspaces of a vector space X.

(OR)

If , prove that the null space has deficiency 0 or 1 in a vector space X.

Conversely, if Z is a subspace of X of deficiency 0 or 1, show that there is an

such that .                                                                                              (5)

  1. b)  Prove that every vector space X has a Hamel basis and all Hamel bases on X have the

same cardinal number.                                                                               (6+9)

(OR)

Let X be a real vector space, let Y be a subspace of X and   be a real valued function

on X such that and  for  If f

is a linear functional on Y and prove that there is a linear

functional F on X such that  and     (15)

  1. a) Let X and Y be normed linear spaces and let T be a linear transformation of X onto
  2. Prove that T is bounded if and only if T is continuous.

(OR)

If is an element of a normed linear space X, then prove that there exists an

such that  and .                                                           (5)

  1. b) State and prove Hahn Banach Theorem for a Complex normed linear space.

(OR)

State and prove the uniform Boundedness Theorem. Give an example to show that the

uniform Boundedness Principle is not true for every normed vector space.         (9+6)

 

 

 

 

  1. a)  Let X and Y be Banach spaces and let T be a linear transformation of X into Y. Prove that if the graph of T is closed, than T is bounded.

(OR)

If x1 is a bounded linear functional on a Hilbert space X, prove that there is a unique

such that .                                                         (5)

  1. b) If M is a closed subspace of a Hilbert space X, then prove that every x in X has a

unique representation  where .

(OR)

State and prove Open Mapping Theorem.                                                    (15)

  1. a) If T is an operator on a Hilbert space X, show that T is normalits real and imaginary parts commute.

(OR)

If and  are normal operators on a Hilbert space X with the property that either

commute with adjoint of the other, prove that and are normal.

  1. b) (i) If T is an operator on a Hilbert space X, prove that

(ii)  If M and N are closed linear subspaces of a Hilbert space X and if P and Q are projections

on M and N, then show that       (6+9)

(OR)

State and prove Riesz-Fischer Theorem.                                                        (15)

 

 

  1. a) Prove that the spectrum of  is non-empty.

(OR)

Show that given by  is continuous, where G is the set of regular

elements in a Banach Algebra.                                                                        (5)

  1. b) State and prove the Spectral Theorem.

(OR)

Define spectral radius and derive the formula for the same.                          (15)

 

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Loyola College M.Sc. Mathematics April 2012 Formal Languages And Automata Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

MT 2960 – FORMAL LANGUAGES AND AUTOMATA

 

 

Date : 26-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

 

ANSWER ALL QUESTIONS

 

I a)  Construct a finite automation that accepts exactly those input strings of 0’s and1’s

that end in 111.

[OR]

  1. b) Construct a DFA accepting all strings in (0 + 1)* having odd number of zeros.    (5)

 

 

  1. c) i)Let L be a set accepted by a nondeterministic finite automation. Then prove

that there exists a deterministic finite automation that accepts L.

ii)Write a note on Epsilon-Closure and give an example.                                  (10+5)

[OR]

  1. d) i)Let r be a regular expression. Then prove that there exists an NFA with

– transitions that accepts L(r).

  1. ii) An NFA has moves.

Find an  equivalent DFA.                                                                                 (8+7)

 

 

II a)   Prove that L = { / n is an integer, n  1} is not regular.

[OR]

  1. b) State and prove pumping lemma.                                                                         (5)

 

 

  1. Minimize the following automation.

 

0 1
A B C
     B D E
     C F G
  D D E
     E F G
    *F E D
   *G G F

[OR]

 

 

 

 

d)i) State and prove any three closure properties of regular languages.

  1. ii) Construct an equivalent DFA for the following NFA

 

a b
 
 
 
 

(15)

 

III a)   Construct a grammar generating all palindromes over {0, 1}.

[OR]

  1. b) Construct a grammar to generate L = { / n is an integer, n  1}        (5)

 

  1. Write GNF grammar for for the following set of production rules

,

[OR]

  1. d) Consider a grammar with production  rules

. The terminals

are Derive an equivalent grammar in CNF                         (15)

 

IV a)   Define a PDA and give an example.

[OR]

  1. b) Construct a PDA that accepts {/ n  1} by empty stack                         (5)

 

  1. c) If a PDA  A  accepts L by empty store then prove that there exists another PDA

B accepting L by final state.

[OR]

  1. d) Let M be a PDA with    as  (q0, 0,Z0) = {( q0, XZ0)}, (q0, 0,X) = {( q0, XX)}

(q0, 1,X) = {( q1, )},  (q1, 1,X) = {( q1, )},  (q1, ,X) = {( q1, )},

(q1,,Z0) = {( q1, )}. Construct a CFG generating  N(M).                      (15)

 

 

V a)  Design a  Turing Machine to add two positive integers.                      .

[OR]

  1. b) Design a Turing Machine to compute .                               (5)

 

  1. c)  Design a TM to accept the language L = { }

[OR]

  1. d) Design a TM to perform proper subtraction. (15)

 

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Loyola College M.Sc. Mathematics April 2012 Differential Geometry Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2012

MT 1813 – DIFFERENTIAL GEOMETRY

 

 

Date : 03-05-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer all the questions:

All questions carry equal marks:

 

 

I a) Obtain the equation of the tangent at any point on the circular helix.

(or)

  1. b) Derive the equation of osculating plane at a point on the circular helix.                   [5]

 

  1. c) Derive the formula for torsion of a curve in terms of the parameter u and hence

calculate the torsion and curvature of the curve.

(or)

  1. d) Derive the Serret-Frenet formulae and  deduce them in terms of  Darboux vector.[15]

 

 

II a) Find the plane that has three point of contact at origin with the curve

 

(or)

  1. b) Prove that the necessary and sufficient condition that a space curve may be helix is

that the ratio of its curvature to torsion is always a constant.                                     [5]

 

  1. c) Define evolute and involute. Also find their equations.

(or)

  1. d) State and prove the fundamental theorem of space curves. [15]

 

 

III a) Derive the equation satisfying the principal curvature at a point on the space curve.

(or)

  1. b) Prove that the first fundamental form is positive definite. [5]

 

  1. c) Prove the necessary and sufficient condition for a surface to be developable.

(or)

  1. d) Derive any two developables associated with a space curve. [15]

 

 

IV a) State the duality between  space curve and developable.

(or)

  1. b) Derive the geometrical interpretation of second fundamental form. [5]

 

  1. c) Find the first and second fundamental form of the curve

.

(or)

  1. d) (1) How can you find whether the given equation represents a curve or a surface?

(2) State and prove Euler’s Theorem.

(3) Define oblique, normal, principal sections of a surface.                            [3+6+6]

 

 

V a) Derive Weingarton equation.

(or)

  1. b) Show that sphere is the only surface in which all points are umbilics. [5]

 

  1. c) Derive Gauss equation.

(or)

(d) State the fundamental theorem of Surface Theory and demonstrate it in the case

of unit sphere .                                                                                                      [15]

 

 

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Loyola College M.Sc. Mathematics April 2012 Complex Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2012

MT 3811 – COMPLEX ANALYSIS

 

 

Date : 24-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

Answer all the questions.

 

  1. a) Prove that  if using Leibniz’s rule

OR

  1. b) Let be a non-constant polynomial. Prove that there is a complex number  such that.                                                                                                       (5)
  2. c) Let be an analytic function. Prove that  for  where  Hence prove that if f is analytic in an open disk  then prove that  for  where.

OR

  1. d) State and prove homotopic version of Cauchy’s theorem.      (15)

 

  1. a) State and prove Morera’s theorem.

OR

  1. b) Prove that a differentiable function  on  is convex if and only if  is

increasing.                                                                                                             (5)

  1. c) State and prove the Arzela-Ascoli theorem.

OR

  1. d) State and prove the Riemann mapping theorem.

(15)

 

  1. a) Show that  in the usual notation.

OR

  1. b) If and  then prove that .

(5)

 

 

 

 

 

 

 

  1. c) (i) Let be a compact metric space and let  be a sequence of continuous functions from X into  such that  converges absolutely and uniformly for x in X. Then prove that the product  converges absolutely and uniformly for x in X. Also prove that there is an integer  such that  if and only if  for some n, .

(ii) State and prove Weierstrass factorization theorem.                                     (7+8)

OR

  1. d) Let , then prove that converges absolutely if and only if  converges absolutely.
  2. e) State and prove Bohr-Mollerup theorem.    (7+8)

 

  1. a) State and prove Jensen’s formula.

OR

  1. b) If  is a metric space, then prove that  is also a metric on                                                                                                                                            (5)
  2. c) State and prove Rung’s theorem.                                                                        (15)

OR

  1. d) State and prove Hadamard’s factorization theorem.                                           (15)

 

  1. a) Prove that any two bases of a same module are connected by a unimodular transformation.

OR

  1. b) Prove that an elliptic function without poles is a constant.

(5)

  1. c) (i) Prove that the zeros  and poles  of an elliptic function satisfy .

(ii) Derive Legendre’s relation                                                                           (7+8)

OR

  1. d) (i) State and prove the addition theorem for the Weierstrass

(ii) Show that

(8+7)

 

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Loyola College M.Sc. Mathematics April 2012 Algorithmic Graph Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

MT 2813 – ALGORITHMIC GRAPH THEORY

 

 

Date : 24-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer all questions. Each question carries 20 marks.

 

  1. (a) Prove that every nontrivial loopless connected graph has at least two vertices that are

not cut vertices.                                                                                                    (5)

(OR)

 

  • Define walk, trail, path, cycle and tree with examples. (5)

 

  • (i) Prove that a connected graph is a tree if and only if every edge is a cut edge.

(ii)  State Dijkstra’s algorithm and use it to find the shortest distance between the vertices A and H in the weighted graph given below.                                      (5+10)

(OR)

  • (i) Find a Hamiltonian path or a Hamiltonian cycle if it exists in each of the graphs given below. If it does not exist, explain why?
  • Prove that an edge e of G is a cut edge of G if and only if e is contained in no cycle of G.                                                                         (6+9)
  1. (a) Define the connectivity and edge connectivity of a graph G and give an example of a graph G in which k(G) < k¢ (G) < d (G).                                                                   (5)

 

(OR)

 

(b)  Prove that K5 is non-planar.                                                                                  (5)

 

(c)  Define Eulerian graph. State and prove a necessary and sufficient condition for a graph to be Eulerian.                                                                                          (15)

(OR)

 

  • Explain Chinese Postman problem. Use Fleury’s algorithm to find the Euler tour for the following graph.

(15)

 

  1. (a) Define and obtain a minimal vertex separator for b and g.

 

(5)

(OR)

(b)  State the Lexicographic breadth first search algorithm.                                       (5)

 

(c)  Let G be an undirected graph. Then prove that the following statements are equivalent.

(i) G is triangulated.

(ii) G is the intersection graph of a family of subtrees of a tree.

(iii) There is a tree T whose vertex set is the set of maximal cliques of a graph G such that each induced subgraph: v ε V is connected, where Kv consists of those maximal cliques which contain v.                                       .           (15)

(OR)

  • (i) Prove that a family of subtrees of a tree satisfies the Helly property.

(ii) Let G be an undirected graph. Then prove that the following statements are equivalent.

(1) G is triangulated.

(2) Every minimal vertex separator induces a complete subgraph of G.                                                                                                                                                        (6+9)

 

  1. (a) Let G be a split graph with the vertex set partitioned into a stable set S and a clique K. If |S| = α(G) and |K| = ω(G) – 1, then prove that there exists an x ε S such that K +{x} is a clique.                                                                                                          (5)

(OR)

(b) Define a permutation graph. Draw the permutation graph corresponding to the permutation [9, 7, 1, 5, 2, 6, 3, 4, 8].                                                              (5)

 

(c)      Let G be an undirected graph with degree sequence d1 ≥ d2 ≥ … ≥ dn and let m = max {i : di ≥ i – 1}. Then prove that G is a split graph if and only if                           .                                                         (15)

(OR)

(d)      (i) Prove that an undirected graph G is a permutation graph if and only if G and  are comparability graphs.

(ii) Obtain the permutation from the following transitive orientations of G and.

(8+7)

 

  1. (a) State the depth-first search algorithm.                                                                  (5)

(OR)

(b)  Obtain an interval representation for the following interval graph.

(5)

 

(c)  State the breadth-first search algorithm and simulate it on the following graph by

selecting the vertex a.

(OR)

(d)      Let G be an undirected graph. Then prove that the following statements are

equivalent.

(1) G is an interval graph

(2) G contains no chordless 4-cycle and its complement  is comparability

graph.

(3) The maximal cliques can be linearly ordered such that for every vertex v of

G the maximal cliques containing v occurs consecutively.                       (15)

 

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Loyola College M.Sc. Mathematics April 2012 Algebra Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc., DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2012

MT 2810/MT 2804 – ALGEBRA

 

 

Date :17-04-2012              Dept. No.                                                 Max. : 100 Marks

Time:  9.00-12.00

 

Answer ALL the Questions

  1. a) If G is a finite group, then prove that in other words,  show that the number of

elements conjugate  to a in G is the index of the normalizer of a in G.

(OR)

  1. b) If p is a prime number and , then show that G has an element of order p.     (5)

 

  1. c) Prove that G has a subgroup of order where p is a prime number and divides O (G).

(OR)

  1. d) Prove that every finite abelian group is the direct product of cyclic groups.   (15)

 

 

  1. a) Given two polynomials and in then prove that there exist two polynomials

and  in such that where or .

(OR)

  1. If the primitive polynomial can be factored as the product of two polynomials having rational coefficients prove that it can be factored as the product of two polynomials having integer coefficients.      (5)

 

  1. (i) State and prove the Eisenstein criterion.

(ii)  State and prove Gauss lemma.

(OR)

  1. Let R be a Euclidean ring, then show that any finitely-generated R-module M is the direct sum of a finite number of cyclic submodules.                                                                        (15)

 

  1. a) If L is an algebraic extension of K and K is an algebraic extension of F, prove that L is an

algebraic extension of F.

(OR)

  1. If is a polynomial in F[x] of degree and is irreducible over F, then prove that there is an extension E of F, such that [E : F] = n in which  has a root.                                           (5)

 

  1. The element a Î K is said to be algebraic over F iff F(a) is a finite extension of F.

(OR)

  1. (i) If a, b in K are algebraic over F then show that a ± b, ab and are algebraic over F.

(ii) If F is of characteristic 0 and a, b are algebraic over F, then show that there exists an

element c Î F(a,b) such that F (a,b) = F(c).                                                                      (15)

 

 

  1. a) Find the fixed field of G (K, F) where F is the field of real numbers and K is the field of

complex numbers.

(OR)

  1. b) Prove that S4 is solvable.      (5)

 

  1. c) State and prove the fundamental theorem of Galois theory.

(OR)

  1.        d)  K is the normal extension of F iff K is the splitting field of some polynomial over          (15)

 

  1. a) Let G be a finite abelian group such that the relation satisfied by atmost n elements of G for every positive integer n then prove that G is a cyclic group.

(OR)

  1. b) For every prime number p and for every positive integer m, prove that there is a unique field

having pm elements.                                                                                                                  (5)

 

  1. c) State and prove Wedderburn’s theorem on finite division rings.

 

(OR)

 

  1. d) Let K be the normal extension of F and H G (K, F), is the

fixed field of H then prove that

(i) [K : KH]= O (H)

(ii) H = G (K, KH).   In particular,  H = G (K,F) and [K : F] = O .                                                                                                                                                                  (15)

 

 

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Loyola College M.Sc. Mathematics Nov 2012 Topology Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3810 / 3803 – TOPOLOGY

 

 

Date : 01/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer all questions. All questions carry equal marks:                      5 x 20 = 100

01. (a) (i) Let X be a metric space with metric . Show that defined by is also a metric on X.

(OR)                                                                                                                              (OR)
    (ii) Define a Pseudo metric space on a non-empty set X. Give an example of a pseudo metric which is not a metric.               (5)

 
  (b) (i) Let X be a complete metric space, and let Y be a subspace of X. Prove that Y is complete iff Y is closed.
    (ii) State and prove Cantor Intersection Theorem.   (8+7)
    (OR)
    (iii) Prove that f is continuous at .
    (iv) Show that  f is continuous is open in X whenever G is open in Y.
02. (a) (i) Prove that every second countable space is separable.

(OR)                                                                                                                              (OR)
    (ii) Define a topology on a non-empty set  with an example. Let  be a topological space and  be an arbitrary subset of . Show that each neighbourhood of intersects .                                                                                                 (5)

 
  (b) (i) Show that any continuous image of a compact space is compact.
    (ii) Prove that any closed subspace of a compact space is compact.
    (iii) Give an example to show that a compact subspace of a compact space need not be closed.     (6+6+3)

(OR)

(OR)
    (iv)  Show that a topological space is compact, if every subbasic open cover has a finite subcover. (15)

 
03. (a) (i) Show that every compact metric space has the Bolzano-Weirstrass property.

(OR)                                                                                                                              (OR)
    (ii) State and prove Tychanoff’s Theorem.         (5)
  (b) (i) Prove that In a sequentially compact metric space every open cover has a Lebesgue number.

 
    (ii) Show that every sequentially compact metric space is compact. (10+5)

(OR)                                                                                                                              (OR)
    (iii) State and prove Ascoli’s Theorem               (15)
04 (a) (i) Show that the product of any non-empty class of Hausdorff spaces is a Hausdorff spaces.

(OR)                                                                                                                              (OR)
    (ii) Prove that every compact Haurdorff space is normal.                           (5)                                                                                        (5)
  (b) (i) Let X be a T1 – space.

Show that X is a normal  each neighbourhood of a closed set F contains the closure of some neighbourhood of F.

 
    (ii) State and prove Urysohn’s Lemma.                (6+9)                                                                                                                                                                                                                                                                                                                       (6+9)

                                                                    (OR)                                                                                                                              (OR)
    (iii) If  X is a second countable normal space, show that there exists a homeomorphism f of X onto a subspace of .                    (15)                                                                                                                                (15)

 
05. (a) (i) Show that any continuous image of a connected space is connected.

(OR)                                                                                                                              (OR)
    Prove that if a subspace of a real line is connected, then it is an internal.(5)
  (b) (i) Show that the product of any non-empty class of connected spaces is connected.
    (ii) Let X be a Compact Hausdorff Space. Show that X is totally disconnected iff it has open base whose sets are also closed.                                   (6+9)                                                                                                                                                      (6+9)

(OR)                                                                                                                              (OR)
    (iii) State and prove Weierstrass Approximation Theorem.           (15)                        (15)

 

 

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Loyola College M.Sc. Mathematics Nov 2012 Real Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

MT 1816 – REAL ANALYSIS

(12 BATCH STUDENTS ONLY)

 

Date : 05/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Answer all the questions. Each question carries 20 marks.

I.a) 1) Prove that refinement over an interval increases the lower sum and decreases the   upper sum.

OR

a)2. Using the notion of Upper sums and Lower sums of a bounded function, when do you say that  on [a,b]. When does the Riemann-Stieltjes integral reduces to Riemann integral?                                                                                      (5)

 

b)1) Suppose f is bounded on [a,b]. f has finitely many points of discontinuity on [a,b] and is continuous at every point at which f is discontinuous the prove that  .

b)2) Prove: Let  be a monotonically increasing function on [a,b] and let on [a,b]. If f is a bounded real function on [a,b] then prove that on [a,b] on [a,b]. In this case                                                                                   (5 +10)

OR

c)1) Prove: on [a,b]

c)2) Suppose c Î (a,b) and two of the three integrals ,  and  exist. Then prove that the third also exists and

c)3) If f is monotonic on [a,b] and if   is continuous on [a,b], then  prove that .                                   (4+4+7)

  1. a)1) State and prove the theorem on Cauchy criterion for uniform convergence.

OR

    a)2) Using a suitable example show that the limit of the integral need not be equal to the integral of the limit.                                                                                           (5)

b)1) If X is a metric space and denote the set of all complex valued, continuous, bounded functions with domain X. Choosing a suitable distance function between the elements of   prove that is a metric space  and also a complete metric space.

b)2) Suppose that is a sequence of differentiable functions on [a,b]. Suppose that converges at some point . If converges uniformly on [a,b] then prove that converges uniformly to some function f and                                            (5+10)

OR

c)1) Let . Verify whether .

c)2) Suppose  converges uniformly to a function f on E, where E is a set as a metric space. Let x be a limit point of  E and suppose that . Then prove that {An}conveges and                                                               (5+10)

III. a)1) State and prove Parseval’s formula.

OR

      a)2) State and prove Dini’s theorem.                                                                          (5)

      b)1) State and prove Jordan’s theorem.

      b)2)State and prove Riemann- Lebesgue Lemma.                                                   (6+9)

OR

c)1) State and prove Riesz – Fischer’s theorem.

c)2) State and prove Riemann Localization theorem.                                                   (6+9)

 

  1. a)1) Suppose E is an open set in Rn and f maps E into Rm and x is an element in E such that when , . Then prove that A is unique.

OR

a)2) Let be the set of all invertible linear operators on Rn. If then prove that .                                                                                                                                                                           (5)

  1. b) State and prove Inverse function theorem.                           (15)

OR

c)1) If  and c is a scalarthen prove that  . And with the distance between A and B defined as   , prove that is a metric space.

c)2) Define Contraction principle and prove the following theorem: Let X is a complete metric space and if  is a contraction of  X into X then  there exists only one xX  such that (x)=x.                                                                                                                        (5+10)

 

Va)1). Derive the rectilinear co ordinates.

OR

 a)2) Derive the sum of powers of .                                                                         (5)

 

 b)1) Explain how the product and quotient rule are derived for functions f(x) and g(x)?

OR

b)2) Derive the expression for D’ Alembert’s wave equation for a vibrating string.

 

 

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Loyola College M.Sc. Mathematics Nov 2012 Number Theory And Cryptography Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3960 – NUMBER THEORY AND CRYPTOGRAPHY

 

 

Date : 08/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

ANSWER ALL QUESTIONS

 

I    a ) Describe subtraction bit operations.

[OR]

  1. b) Prove that .                             (5)

 

  1. c) i) If eggs are removed from a basket 3, 5 and  7 at a time there remain respectively

1, 2 and 3 eggs . Using Chinese Remainder Theorem, find the least number of

eggs that could have been in the basket.

  1. ii) If

[OR]

  1. d) i) Find the upper bound for the number of bit operations required to compute
  2. ii) Prove that the Euclidean algorithm always gives the greatest common divisor in

a finite number of steps.                                                                                 (8 + 7)

 

 

II   a) State and prove any two properties of Legendre’s symbols.

[OR]

  1. b) Find the value of the Legendre’s symbol . (5)

 

 

  1. c) Prove that where is a prime number.                                        (15)

[OR]

  1. d) (i) Prove that the order of any  .

(ii) Find the Gaussian sum                                         (8 + 7)

 

 

III   a) Decipher the message  DWWDF   NDWGD  ZQ, which was enciphered

using Ceasar cipher.

[OR]

 

 

b)Decipher  FWMDIQ , which has been enciphered using the matrix

(5)

 

  1. c) i) Discuss about Knapsack ciphers.
  2. ii) Solve the Knapsack problem . (8 + 7)

[OR]

  1. A person is using 2 x 2 enciphering matrix with a 29 letter alphabet, where  A – Z

have the usual numerical equivalents , blank = 26,  ? = 27,  ! = 28. He receives the message “GFPYJP  X?UYXSTLADPLW”. He knows that  the last five letters of  the plaintext are KARLA . Find the deciphering matrix and read the message. (15)

 

 

IV   a) Find all bases for which 21 is a pseudo prime.

[OR]

  1. b) If n is an Euler pseudo prime to the base b , then prove that it is a pseudo prime

to  the base b .Also discuss about the converse.                                                      (5)

 

  1. c) Let n be an odd composite integer. Then prove that

(i) n is a pseudoprime to the base b,  where g.c.d.(b, n) = 1, then n is a pseudoprime

to the base

(ii) n is a pseudoprime to the base b,  where g.c.d.(b, n) = 1, if and only if the order

of b in ( Z/nZ )* divides n –1

(iii) n is a pseudoprime to the bases  ,  then n is a pseudoprime to the base

and also to the base  .

(iv) If n is not a pseudoprime to a single base b( Z/nZ )*, then n is not a

pseudoprime to atleast half of the possible bases b( Z/nZ )*

 

[OR]

  1. d) Discuss about any two primality tests                                                                 (15)

 

V    a) Check whether 141467  is a prime number.

[OR]

  1. b) Write a note on Fermat factorization method           (5)

 

  1. c) Let E be the elliptic curve  defined over . Then
  2. i) List the points on E
  3. ii) Compute P + Q if P = (2, 3)  and Q = (4, 0)

iii)  Compute 2P if P = (2, 3)                                                                        (7 + 4 + 4)

[OR]

  1. d) Write about elliptic curve discrete log problem. (15)

 

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Loyola College M.Sc. Mathematics Nov 2012 Measure Theory And Integration Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034    M.Sc. DEGREE EXAMINATION – MATHEMATICSSECOND SEMESTER – NOVEMBER 2012MT 2811 – MEASURE THEORY AND INTEGRATION
Date : 06/11/2012 Dept. No.   Max. : 100 Marks    Time : 1:00 – 4:00
ANSWER ALL QUESTIONS EACH QUESTION CARRIES 20 MARKS :        5 x 20 = 100
I. (a) Define outer measure and show that it is translation invariant (5)(OR)(b) Prove that B is the  algebra generated by each of the following classes: the open    intervals open sets, the  sets, and the   sets.   (5)

(c) Prove that the outer measure of an interval equals its length   (15)(OR)(d) Prove that Not every measurable set is a Borel set.      (15)
II. (a) State and prove Lebesque Monotone Convergence theorem.    (5)(OR)            (b) Prove that if f  is a non negative measurable function then there exists a sequence (5)      of  measurable monotonically increasing simple function such that .(c) State and prove Fatou’s Lemma  for measurable functions.   (15)(OR)(d) State and prove Lebesgue Dominated Convergence theorem.    (15)
III. (a) Show that if    is a sequence in a ring Âthen there is a sequence   of disjoint      sets of  such that   for each i and   for each N so that                                   (5)(OR) (b) Prove that with a usual notations the outer measure   on H(Â),and the outer measure outer measure defined   by   on S( Â)  and  on    are the same.    (5)

 

 

(c) Show that if   is a measure on a  -ring   then the class  of sets of the form                            for any sets E,N such that  While N  is contained in some set in  of zero                            measure is a  -ring and the set function  defined by  is a                            complete measure on  .   (15)(OR) (d) Prove that if   is an outer measure on H(Â),. Let   denote  the class of                    Measurable sets then Prove that   is a  – ring and   restricted to is a complete         measure.    (15)
IV. (a) Prove that   space is a vector space for  . (5)(OR)(b) State and prove Minkowski’s inequality. (5)

(c) State and prove Jensen’s inequality. Also prove that every function convex on an open interval is continuous. (15)(OR)(d) Prove that   where   is convex on (a, b) and  . Also prove that a differentiable function   is convex on (a, b) if and only if  is a monotone increasing function. (15)V. (a) Define the following terms: total variation, absolutely continuous, and mutually singular      with respect to signed measure. (5)(OR)(b) Let v be a signed measure on [X, S]. Construct the measures v+ and v- on [X, S] such that v = v+ – v- and v+ ┴ v-. (5)

(c) If  , ,  and   are  – finite signed measure on [X, S] and  « , «  then prove that  . Also prove that a countable union of positive sets with respect to a signed measure v is a positive set. (15)(OR)(d) State and prove Hahn decomposition theorem. (15)
 

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Loyola College M.Sc. Mathematics Nov 2012 Complex Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3811 – COMPLEX ANALYSIS

 

 

Date : 03/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer all the questions:

 

  1. a) Prove that  if  using Leibniz’s rule.

OR

  1. b) State and prove Liouville’s theorem.        (5)
  2. c) State and prove first version of Cauchy’s integral formula.

OR

  1. d) State and prove the homotopic version of Cauchy’s theorem                  (15)

 

  1. a) State and prove Hadamard’s three circles theorem.

OR

  1. b) Define a convex function and prove that a function  is convex if and

only if the set  is a convex set.                           (5)

  1. c) State and prove Goursat’s theorem.

OR

  1. d) State and prove Arzela Ascoli theorem.      (15)

 

  1. a) Let , for all . Then prove that  converges to a complex number different from zero if and only if  converges.

OR

  1. b) Show that in the usual notation.                                                                                                                                            (5)

 

 

  1. c) (i) If and  then prove that .

(ii) Prove that .

(iii) State and prove Gauss’s Formula.                                                                                                                                                                                              (5+5+5)

OR

  1. d) (i) State and prove Bohr-Mollerup theorem.

(ii) Prove that (a)  converges to  in  and (b) if  then  for all .                                                                                                                                                                                          (8+7)

  1. a) State and prove Jensen’s formula.

OR

  1. b) Let  be a rectifiable curve and let K be a compact set such that .  If f is a continuous function on  and  then prove that there is a rational function  having all its poles on  and  such that  for all z in K.

(5)

  1. c) State and prove Mittag-Leffler’s theorem.                                                          (15)

OR

  1. d) State and prove Hadamard’s Factorization theorem.

(15)

  1. a) Prove that any two bases of a same module are connected by a unimodular transformation.

OR

  1. b) Show that and it is an odd function.                                                                                                                                                        (5)

 

  1. c) (i) Prove that the zeros and poles  of an elliptic function satisfy .

(ii) Prove that   .                                  (7+8)

OR

  1. d) (i) Show that

(ii) State and prove the addition theorem for the Weierstrass -function.                                                                                                                                                 (7+8)

 

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Loyola College M.Sc. Mathematics Nov 2012 Classical Mechanics Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3812 – CLASSICAL MECHANICS

 

 

Date : 06/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer ALL the questions:

 

  1. a. i. The quantity exerted by the outside agent that causes the change of position is called ———
  2.   I get up in the morning and go to work – denotes ————–type of motion.

iii. I get up in the morning and go to work but evening I’m back at home  – denotes ————–type

of motion.

  1. The generalized displacement is —————————-
  2. Holo means —————-in Greek.

OR

  1. Derive the equation of motion of Atwood’s machine  [ 5 ]

 

  1. .i. State and prove the principle of Virtual Work and deduce D’Alembert’s principle
  2. An inextensible string of negligible mass hanging over a smooth peg at A connects another

mass m1on a frictionless inclined plane of angle q to another mass m2 . Use  D’Alembert’s

principle to prove that the mass will be in equilibrium if  m2  = m1sinq.

OR

  1. Classify constraints and state the principles involved in choosing the generalised coordinates and

classify the constraints with reasons for any one of  the following cases

  1. A bead moving on a circular wire.
  2. A sphere rolling down a rough inclined plane without slipping.

iii. The molecules moving inside a gas container.                                                                      [15]

 

  1. a. i. An ignorable coordinate is one which ———————-
  2.  In a conservative system,  H = ——– + V

iii. In  variation , energy is ————————

  1. In – variation process, System point is speeded up or slowed down in order to make the total

travel time along every path ——————

OR

  1. Find the Routh’s function for the motion for the motion of a projectile.  Hence deduce equation of

motion.                                                                                                                                            [5]

 

  1. State Hamilton’s principle and deduce Lagrange’s equation from Hamilton’s principle and hence

find the equation of motion of  one dimension Harmonic oscillator.                                              [7+8]

OR

  1. i. Derive the Hamiltonian function
  2. Derive Hamilton canonical equation of motion.

iii.Give the physical significance of Hamilton’s function                                                           5+5+5]

 

3.a. i.

ii.

iii. The time taken by a light ray to travel between two points is —————.

  1. If the dynamical system have one degree freedom, then pdq – PdQ = —————
  2. generates an ————– transformation.  

OR

  1. b. Show that is a canonical transformation

                                                                                                                  [5]

  1. State and prove Integral Invariant theorem of Poincare

.

OR

  1. Discuss about the motion of a top

[7+8]

 04.a. i. The solution of H (q1, q2 q3,…. qn , ) + = 0 is known as ———————-

  1. In any dynamical system, the collection of points is called a—————–

iii.  =

  1. If qi is cyclic, then pi a ———————–

v.

OR

  1. State and prove Liouvilli’s theorem. [5] 

                                                     

  1. Derive the conservation theorem of angular momentum using Infinite decimal

contact transformation                  

OR

d.Derive the Hamilton – Jacobi equation for the Hamilton’s principle function S. and deduce that

[15+5]                                                                                                      

05.a.i.  The  Complete integral W of Hamilton –Jacobi equation is called  ——-

  1. Separation of variables in Hamilton Jacobi’s equation is possible only if ——————

iii. For a conservative dynamical system in which the generalized coordinates are q , f  cyclic, then

the solution is given by ————-

  1. Action integral denoted by A is defined to be —————–
  2. If Wk denotes characteristic function, then  Jk = is known as —————

OR

  1. Discuss the motion of a particle moving in a plane under the action of central

force using Hamilton – Jacobi equation.                                                                                                [5]

  1. Derive the Hamilton – Jacobi equation for the Hamilton’s characteristic function

OR

  1. Discuss Kepler’s problem using action angle variable.

[15]

 

 

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