M.Sc Mathematics Question Paper

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Loyola College M.Sc. Mathematics April 2003 Number Theory Question Paper PDF Download

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LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – mathematics

FourTh SEMESTER  – APRIL 2003

MT  4950/ M  1055   number theory

23.04.2003

1.00 – 4.00                                                                                            Max: 100 Mark

 

 

  1. (a) (i)   If  (a,m)=1, prove that af(m) º1(modm).  Hence deduce Fermats theorem.

(OR)

(ii)   of n ³1 , prove that                                                                          (8)

  • (i)   State and prove Wilson’s theorem

(ii)   Salve the congruence x2 + x +7 º 0 (mod 189)

(OR)

  • (iii)  Solve x2 + x +7 º 0(mod 73)

(iv)  Reduce the congruence 4x2 +2x +1 º0 (mod5) to the form x2 º a (mod p)

hence find the solutions                                                                          (17)

 

  1. a) (i)    Let P be an odd prime with (a, p) = 1. Consider he least non-negative

residues  module p of the integers a, 2a, 3a, ….

If n denotes the number of these residues that exceed p/2, then

prove that the Legendre symbol

(OR)

(ii)    Find the  value of the Legendre symbol

(iii)   Find the highest power of  7 that divides 1000!                                        (8)

 

  1. b) (i)   If  p is an odd prime and (a,2p) = 1 then prove that the

Legendre symbol

(ii)   Define the Jacobi symbol

(OR)

(iii)  If  f(n) is a multiplicative function and if   then prove that

F(n) in multiplicative.

 

(iv)  Define the Moebius function m(n) and prove that inversion formula that

if  for every positive integer n

then F(n)  .                                                                  (17)

 

 

 

 

 

  1. (a) (i) Find all the integral solutions of the equation ax + by =c  if they

exist, where a, b, c and integers.

(OR)

(ii) Show that there exists at least one positive solution of ax + by = c if

g = (a, b) satisfies the condition g|c and gc >ab.                          (8)

 

(b)  (i)  Prove that all solutions of 3x +5y =1 can be written in the

form  x = 2 + 5t, y = -1 -3t.

 

(ii)  Define a primitive solution  of x2 + y2 = z2.  Prove that the positive

primitive solutions of x2 + y2 = z2 with  y even are give by x = r2 = s2,

y = 2rs, z = r2 + s2, where r and s are  arbitrary integers of opposite

parity  with  r > s >0 and (r,s) =1.

(OR)

(ii)   Prove that every positive integer is a sum of four squares of integers.   (17)

 

  1. (a) (i) If P(n) is the partition function, with the usual notation

prove that pm(n) = pm-1(n) + Pn(n-m) if  n ³ m>1.

 

(ii)    using the graph of a partition, prove that the  usual notation prove that

the number of partitions of n into m summands is the same as the

number of partitions of n having largest sum m and m.

(OR)

(iii)    State Euler’s formula and use it prove that Euler’s identify for

any positive integer  n.                                                                                (8)

 

(b)  (i)   If n ³ 0 then prove that

 

 

(OR)

 

(ii)    of

 

 

  • (iii) Prove that for o £ x<1, the series      (17)

 

 

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Loyola College M.Sc. Mathematics April 2003 Mechanies – II Question Paper PDF Download

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LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – mathematics

FourTh SEMESTER  – APRIL 2003

MT  4801/ M  1026   mechanies – II

16.04.2003

1.00 – 4.00                                                                                             Max: 100 Marks

 

Answer ALL the questions

 

  1. a)  Explain the term ‘ABERRATION’.  Also derive the relativistic formula

 

for aberration in the form      (8)

(OR)

 

  1. Show that the operator  is an invariant for

Lorentz  transformation.                                                                       (8)

 

  1. a)   State ‘ETHER’ Hypothesis.  Explain the  Michelson- Morley experiment

and give the conclusion.                                                                         (17)

 

(OR)

  1. b) Show that Lorentz  transformations forma group      .                            (17)

 

  1. a) Obtain the transformation formula for mass in the form (8)

(OR)

 

  1. If a body of mass m disintegrates while at rest in to two parts of rest masses m1 and m2, show that the energies E1 and E2   of the parts are

given by       (17)

 

  1. a) Derive the equation E = m C2, Deduce that p2 – is an invariant under

Lorentz  transformation.                                                                      (17)

 

 

 

 

 

 

 

 

 

 

 

 

 

(OR)

  1. Obtain the transformation formula  for force components in the

(17)

 

 

 

  1. a)    Explain ‘contravariant vectors’,   covariant vectors,

‘contravariant tensors’  and ‘covariant tensors’.                                   (8)

 

(OR)

  1. If a vector has components, on cartesion coordinates then

the components in polar coordinates are and if the components be

then the polar coordinates  components are         (8)

 

  1. a)      Define fundamental tensors and show that gmg is a Covariant  tensor of

rank two. Also transform ds2 = dx2 + dy2 + dz2 in polar and cylindrical

coordinates.                                                                                            (17)

 

(OR)

  1. b) Define Christoffel’s 3-index symbols of the first and second kind.  Also

calculate christoffel’s symbols corresponding to the metric

ds2 = dr2 + r2dq2 + r2sin2q df2.                                                                (17)

 

  1. a) Define ‘Energy Tensor’.  Show that the equationfor m = 4

gives the equation of continuity in Hydrodynamics.                                  (8)

 

 

 

 

 

 

 

 

 

(OR)

  1. Obtain isotropic polar coordinates and Cartesian coordinates.

Also Deduce that the velocity of light at distance r1 from the origin is

 

(8)

 

  1. a) Obtain the schwarzchild  line element in the neighbourhood of  an attracting particle

 

in the from       (17)

 

(OR)

  1. b)     Derive the differential equation to the planetary orbits in the

 

form  .                                    (17)

 

 

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Loyola College M.Sc. Mathematics April 2003 Graph Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.Sc. DEGREE EXAMINATION – Mathematics

FourTH SEMESTER – APRIL 2003

MT 4951 / M 1056  –  GRAPH THEORY

 

23.04.2003

1.00 – 4.00                                                                                                     Max : 100 Marks

      Answer ALL questions. Each question carries 25 marks.

 

  1. (a) Obtain a necessary and sufficient condition for a sequence (d1, d2, …,dn) to be graphic.

(OR)

(b) Prove that a graph is bipartite if and only if it contains no odd cycle.          (8 marks)

(c) (i) Prove, with usual  notation, that  .

(ii) Find the shortest distance from the vertex uo  to all other vertices in the following
graph.

 

 

 

 

 

 

(OR)

(d) (i) Prove that a vertex  v of a tree G is a cut vertex of G if and only if d (v)>1.   Deduce
that every nontrivial loopless connected graph has at least two vertices that are not cut
vertices.

(ii)State Cayley’s formula and find the number of distinct spanning trees of the following
graph.                                                                                                (8+9 marks)

 

 

 

  1. (a) Define connectivity and edge-connectivity of a graph.   Prove, with usual notation, that
    .

(OR)

  • If G is 2-connected, then prove that any two vertices of G lie on a common cycle.

(8 marks)

  • (i) Prove that a non empty connected graph is Eulerian if and only if it has no vertex of
    odd degree.

(ii) Obtain Chvatal’s sufficient condition for a simple graph to be Hamiltonian.

(OR)

  • (i) Let G be a simple graph and let u and v be non-adjacent vertices in G such that
  1. n. Prove that G is Hamiltonian  is Hamiltonian.

(ii) Describe Chinese postman problem. State Fleury’s algorithm. Find an optimal

four in the following graph.

 

 

 

 

 

 

 

 

 

 

 

 

  1. (a) Prove that a matching M in G is a maximum matching if and only if G contains no
    M-augmenting path.

(OR)

(b)  Prove , with usual notation, that n if                                      (8 marks)

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

(ii)  Find an optimal matching in the graph given by the following matrix.

 

(OR)

  • (i) Define an independent set. Prove with usual notation that n.

(ii) Prove that in a bipartite graph the number of edges in a maximum matching is the
same as the number of vertices in a minimum covering.                          (8+9 marks)

  1. (a) State and prove Brook’s theorem.

(OR)

(b)  Let G be a k-critical graph with a 2-vertex cut {u, v}. Prove that  .

(8 marks)

(c)  State and prove kuratowski’s theorem.

(OR)

  • (i) Obtain Euler’s formula and deduce that k5 is non-

(ii)  State and prove the five-colour theorem.                                                (9+8 marks)

 

 

 

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Loyola College M.Sc. Mathematics April 2003 Computer Algorithms Question Paper PDF Download

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LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – mathematics

FourTh SEMESTER  – APRIL 2003

MT  4952/ M  1057    computer Algorithms

26.04.2003

1.00 – 4.00                                                                                            Max: 100 Mark

 

Answer ALL questions. Each question carries 25 marks

 

I  a)   (i)   Give procedure SEARCH to search for an element x in an array A(1:n)

and to return k if a(k) = x and zero otherwise.

(ii)   Give a recursive produce to find GCD of 2 accepted numbers.        (OR)

  1. b) Give a procedure to create a leap of n elements, inserting one item, at a time.    (8)
  2. c) (i)   Discuss: Analyzing  algorithms in general

(ii)   Explain the conditional statements and loop structures in SPARKS. (OR)

  1. d) Give HEAPSORT to sort numbers in an array. Simulate it on

A(1:6)  = (14,17,25,12,13,7)                                                                       (17)

 

II a)  Give procedure BINSRCH and simulate it on

A(1:7) = (45,  70,  82,  90,  95, 100, 110)when x = 46.(OR)

  1. b) Give procedure MAXMIN and find its  best, worst and average lese number

of comparisons when n is a power of 2.                                                              (8)

  1. c) Give procedure MERGESORT

If the time for merging operation is proportional to n, then find the computing

time ד (n) for MERGESORT.  When n = 2k, prove that ד(n) =O(n log n)  (OR)

  1. Give procedure SELECT to find the kth  smallest element in an array simulate it on

A(1:8) = (14, 12, 61, 60, 17, 20, 6, 10 )  to find the 3rd  smallest element.  (17)

 

III a)  Explain the problem of optimal storage on rapes. With usual notation.

if l1 £ l2 £ .-.£ ln then prove that  the ordering ij = ji £ n minimizes

outing over all possible permutations of the ij .                                          (OR)

  1. State a greedy algorithm to generate shortest paths from a given vertex to

all other  vertices in a graph.                                                                                 (8)

  1. Give line procedure due to krushkal to find a minimum sparring tree of a graph.

Prove also that Kruskal’s algorithm generates a minimum cost sparring

tree for every corrected undirected graph G.                                            (OR)

  1. State procedure GREEDY- With used notation, if

p1|w1³ p2|w2 ³ …³ pn|wn , then prove that  GREEDY- KNAPSACK generates

an optimal solution to the given instance of the knapsack problem.                  (17)

 

IV  a)   Explain sum of subsets problem and give 2 different formulations for the same.

(OR)

  1. b) Explain in detail low backtracking works on the 4 -queen problem.                 (8)
  2. c) (i)  Give recursive backtracking algorithm for sum of subsets problem

(ii)  State procedure MCOLO RING to find all m-colorings in a graph.

(OR)

  1. d)   Define a hamiltonian cycle.Give an example of (i)  a hamiltonian graph.

(ii) Give algorithm HAMITONIANJ to generate all hamiltonian cycles in a graph   (17)

 

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Loyola College M.Sc. Mathematics April 2003 Complex Analysis – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.Sc. DEGREE EXAMINATION – mathematics

SECOND SEMESTER – APRIL 2003

MT 2802 / M 823  –  COMPLEX ANALYSIS – II

 

24.04.2003

1.00 – 4.00                                                                                                      Max : 100 Marks

Answer ALL questions.

 

  1. (a) Show that .                                                                              (8)

 

  • Let Re Zn >0 for all Prove  that  Converges to a non zero number if and only if the series                                                                             (8)

 

  • (i) Obtain the Gauss formula for the Gamma function and show that
    éé

(ii)  If | z |  and p ³ o then prove that                            (7+2+8)

 

OR

  • (i) Let f be a real valued function defined on  such that for all  x > 0.
    Suppose f(x) statistics the following properties.
  • log f (x) is convex,
  • f (x+1) = xf (x) for all x,
  • f (1) =1. Then show that f (x) = [(x) for all x.

 

  • State and prove Euler’s Theorem.

 

II  (a)  State and prove first version of Maximum Principle for Harmonic Functions.      (8)

OR

(b)  Let be a path from a to b and let be an analytic
continuation along There is a number Î>0 such that if  is any path
from a to b with  for all t, and if is any
contribution along  with [go]a = [fo]a : than prove that [g1]b = [f1]b .                   (8)

 

  • (i) Define Poisson kernal and prove the four properties of Poisson kernal.

(ii)  Stating the required conditions, solve the Dirichlet’s problem for the unit disk.                                                                                                                     (6+11)

OR

(d)  State the prove Harnack’s inequality and hence prove Hernack’s theorem.           (5+12)

 

 

 

 

III  a.  State and prove Poisson – Jenson formula.                                                                 (8)

OR

  1. State and prove Little Picard’s theorem.                                                                    (8)
  1. Define order and genus of an entire function and prove that if f is an entire function

of finite genus then f is of finite order l £ +1.                                             (5+12)

 

OR

  1. State and prove Bloch’s theorem.           (17)

 

IV  a. Prove that a discrete module consists of ether of Zero alone, of the integral multiples
nco of a single complex number ¹ o or  1 or of all  linear combinations  n1 w1 + h2 w2
with integral coefficient of two numbers w1, w2 with non real ration .                  (8)

OR

  1. Show that the zeroes a1, a2 ….. an and poles b1, b2 ….bn of an elliptic function satisfy (modm)

 

  1. (i)  Define weierslvass p function.  Derive the differential equation satisfied by the

weierslvass p –function.

(ii)   Show that p (z +u) + p(z) + p (u)  =                          (2+8+7)

 

OR

  1. Show that ℒ(z) is an odd function and prove that   ℒ(z) = -p(z), also derive the le gendre relation.

 

 

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Loyola College M.Sc. Mathematics Nov 2003 Mathematical Statistics – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2003

ST 3951 / S 972 – MATHEMATICAL STATISTICS – I

10.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions.                                                                                   (10×2=20 marks)

 

  1. Illustrate that pairwise independence does not imply mutual independence of random events.
  2. Prove that every distribution function is continuous atleast from the left.
  3. Show that if the probability of a random event equals zero, it does not follow that this event is impossible. Similarly prove that if the probability of a random event equals one, its does not follow that this event is sure.
  4. Define truncated distribution of a random variable X and given an example.
  5. Give two examples of random variables for which expected value does not exist.
  6. Define convergence in law of a sequence of random variables and give an example.
  7. Show that if the moment of order k of a random variable X exists, then

where a > 0.

  1. State the theorem of Bochner, giving necessary and sufficient conditions for a function to be a characteristic function.
  2. The characteristic function of the random variable X is given by = exp Find the density function of this random variable.
  3. State Lindeberg – Levy Central  Limit

 

SECTION-B

Answer any FIVE questions.                                                                           (5×8=40 marks)

 

  1. Let {An}, n =1, 2, ….., be a non increasing sequence of events and let A be their product. Then show that P (A) = .
  2. Show that the conditional probability satisfies the axioms of the theory of probability.
  3. a) State and prove Bayes
  4. b) Illustrate the application of Bayes
  5. State and prove a necessary and sufficient condition for the independence of the random variables X and Y of the discrete type.
  6. If a random variable has a symmetric distribution and its expected value exists, then show that this expected value equals the center of symmetry. Hence show that for a symmetric distribution the central moments of odd orders (if they exist) are equal to zero.
  7. a) If not all the moments exist, then show that those moments that do exists fail to determine the distribution function F (x).
  8. b) Define convergence in rth mean and given an example.
  9. The random variables X and Y have the joint density given by

f (x,y) = .   Compute the coefficient of correlation.

  1. Define the t, chi-square and F distributions.

 

SECTION-C

Answer any TWO questions.                                                                           (2×20=40 marks)

 

  1. a) State and prove Lapunov inequality concerning absolute moments.            (10)
  2. b) Show that the expected value of the product of an arbitrary finite number of

independent random variables, whose expected values exist, equals the product of the

expected values of these variables.                                                                              (4)

  1. c) Show that the covariance of two independent random variables equals zero. Is the

converse true?  Justify your answer.                                                                             (6)

  1. a) State and prove Levy Inversion Theorem concerning the determination of the

distribution function by the characteristic function.                                                   (14)

  1. b) Prove that the probability function of the Poisson distribution can be obtained as the

limit of a sequence of probability functions of the binomial distribution.                   (6)

  1. a) Show that for n ³ 2, the binomial distribution can be obtained from the zero-one

distribution.                                                                                                                   (4)

  1. b) Examine the additive property for Gamma random variables. (6)
  2. c) State and prove Bernoulli’s weak law of large numbers.                      (10)
  3. a) State and prove the Chebyshev Show that in the class of random variables

whose second order moment exists, one cannot obtain a better inequality than the

Chebyshev inequality.                                                                                                (10)

  1. b) If the th moment of a random variable exists, then show that can be expressed

in terms of the th derivative of the characteristic function of this random variable

at t = 0.                                                                                                                          (8)

  1. c) Define the strong law of large numbers. (2)

 

 

 

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Loyola College M.Sc. Mathematics Nov 2003 Functional Analysis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

  1. Sc. DEGREE EXAMINATION – MATHEMATICS

IV SEMESTER – APRIL 2004

MT 4800/M 1025 – FUNCTIONAL ANALYSIS

 

Date   :                                                                                                            Max. Marks  :  100 Marks

Hours           :   3 hours

 

Answer All questions:                                                                         

 

  1. a) Show that every vector space has a Hamel basis

(Or)

If  , prove that the hull space Z(f) has deficiency O or 1 in X.  Conversely, show that if Z is a subspace of X of deficiency O or 1,   then there is  an   such  that

Z = Z(f).                                                                                                          (8)

 

  1. b) (i) Let X and Y be normed linear spaces and let B(X, Y) denote the set of all bounded linear transformations from X into Y. Then prove that B(X, Y) is a normed linear space.

 

(ii) Let X and Y be normed linear spaces and let T : XY be a linear transformation.  Prove that T is bounded if and only if T is continuous. (9 + 8)

 

(Or)

State and prove the Hahn – Banach Theorem (real version)           (  17 )

 

  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, then T  is bounded.

(Or)

State and prove F-Riesz  Lemma                                                    (8)

 

  1. 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.

(Or)

If X and Y are Banach spaces and if T is a continuous linear transformation of X onto Y, then prove that T is an open mapping.                                    (17)

 

  1. a) State and prove Bessel’s  inequality

(Or)

If T is an operator on X, then show that (Tx, x) = 0

 

  1. b) i) If x‘ is a bound linear functional on a Hilbert space X, prove that there is a unique

such that  x‘(x) = (x,

 

  1. ii) If M and N are closed linear subspaces of a Hilbert space H and if P and Q are projections on M and N, then show that    PQ =  O QP = O.      (9 + 8)

(Or)

Prove that two Hilbert spaces are isomorphic iff they have the same dimension. (17)

 

 

4)  a)  Define a topological  divisor of Zero.  Let  S be the set of singular elements in a Banach   algebra .  Prove that the set of all topological divisors of Zero is a subset of S.

 

(Or)

Let A be a Banach algebra  and   Then prove that the spectrum of x, is non-empty.

 

  1. State and prove the Spectral Theorem.

(Or)

 

Define the spectral radius of an element x in a Banach Algebra   A.  In the usual notation, prove that .                                                                                      (17)

 

 

 

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Loyola College M.Sc. Mathematics April 2006 Topology Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 10

THIRD SEMESTER – APRIL 2006

                                                                  MT 3800 – TOPOLOGY

 

 

Date & Time : 28-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

Answer ALL questions.  All questions carry equal marks.

  1. a) i) Let X be a metric space with metric d.  Show that d1 defined by  is also a metric on X.  Give an example of a pseudo metric which is not a metric.

(or)

  1. ii) In any metric space X, show that each open sphere is an open set.  Prove that any union of open sets in X is open.                                                                                                                 (8)
  2. b) i) Let X be a complete metric space and let Y be a subspace of X.  Prove that Y is complete iff it is closed.
  3. ii) State and prove Cantor’s Intersection Theorem.

iii)  If  is a sequence of nowhere dense sets in a complete metric space X, prove that there exists a point in X which is not any of the s.                                                               (6+6+5)

  1. iv) Let X and Y be metric spaces and f be a mapping of X into Y.  Then prove that f is continuous iff is open in X whenever G is open in Y.
  2. v) Prove that the set C(X,R) of all bounded continuous real functions defined on a metric space X is a Banech space with respect to point wise addition and scalar multiplication and the norm defined by .                                                                                                                  (6+11)
  3. a) i) Show that every separable metric space is second countable.

(or)

  1. ii) Prove that the product of any non-empty class of compact spaces is compact.

(8)

  1. b) i) Show that any continuous image of a compact space is compact.
  2. 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+5)

(or)

  1. State and prove Lindelof’s Theorem.
  2. v) Let X be any non-empty set, and let S be an arbitrary class of subsets of X.  Show that S can serve as an open subbase for a topology on X.                                                            (6+11)

III. a) i)   Prove that a metric space is compact iff it is complete and totally bounded.

(or)

  1. ii) Prove that every compact Hausdorff space is normal.                                     (8)
  2. b) i) In a sequentially compact metric space, prove that every open cover has a Lebesque number.
  3. ii) Show that every sequentially compact metric space is totally bounded.

iii)  Prove that every sequentially compact metric space is compact.             (9+4+4)

(or)

  1. b) iv) In a Hausdorff space, show that any point and disjoint compact subspace can be separated by open sets.
  2. v) Show that every compact subspace of a Hausdorff space is closed.
  3. vi) Prove that a 1–1 mapping of a compact space on to a Hausdorff space is homeomorphism.         (7+5+5)
  4. a) i) Prove that any continuous image of a connected space is connected.

(or)

  1. ii) Let X be a T1  Prove that X is normal iff each neighbourhood of a closed set F contains the closure of some neighbourhood of F.                                                                           (8)
  2. b) i) State and prove the Urysohn Imbedding Therorem.

(or)

  1. ii) State and prove the Weierstrass Approximation Theorem.                            (17)

 

 

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Loyola College M.Sc. Mathematics April 2006 Probability Theory And Stochastic Processes Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2006

                       ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

 

 

Date & Time : 28-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer ALL the questions                                                                                                              (10 ´ 2 = 20)

  1. Define probability by classical method.
  2. Give an example for a discrete probability distribution.
  3. Define an induced probability space.
  4. State the properties of a distribution function.
  5. Define the distributed function of a continuous random variable.
  6. Write the formula to find the conditional mean and variance of Y given X = x.
  7. What do you mean by a Markov matrix? Give an example
  8. Write a note on one-dimensional random walk.
  9. Define (i) recurrence of a state           (ii) periodicity of a state
  1. Define renewal function.

PART – B

Answer any FIVE questions.                                                                                                             (5 ´ 8 = 40)

  1. State and prove Boole’s inequality.
  2. Explain multinomial distribution with an example.
  3. Given the dF

F(x) =       0     ,  x < – 1

 

=    ,  -1

=      1        ,  1

 

compute (a) P(-1/2 < X  1/2)         (b) P(X = 0)    (c) P(X = 1)       (d) P (2 < X  3).

 

  1. Let X have the pdf f(x) = 2x,  0 < x < 1, zero elsewhere. Find the dF and p.d.f. of Y = X2.

 

  1. (a) When is a Markov process called a Markov chain?

(b) Show that communication is an equivalence relation.                                                              (2 + 6)

 

  1. A Markov chain on states {0,1,2,3,4,5} has t.p.m.

 

Find the equivalence classes.

 

  1. Find the periodicity of the various states for a Markov chain with t.p.m.

 

 

 

 

  1. Derive the differential equations for a pure birth process clearly stating the postulates.

 

PART – C

Answer any TWO questions.                                                                                                           (2 ´ 20 = 40)

 

  1. (a) The probabilities that the independent events A,B and C will occur are ¼, ½ , ¼ respectively.
    What is the probability that at least one of the three events will occur?

 

  • Find the mean and variance of the distribution that has the dF

 

F(x)  =  0         ,  x < 0

=  x/8      ,  0  £  x < 2

=  x2/16   ,  2  £  x < 4

=  1         ,  4  £  x                                                                                                     (5 + 15)

 

  1. If X1 and X2 have the joint p.d.f.

 

f(x1,x2) =

 

find     (i) marginal pdf of X1 and X2.

                                                (ii) conditional pdf  of X2 given X1 = x1 and X1 given X2 = x2.

(iii) find the conditional mean and variance of X2 given X1 = x1 and

X1 given X2 = x2.                                                                                                                  (4 + 4 + 12)                      

 

  1. Derive a Poisson process clearly stating the postulates.

 

  1. Derive the backward and forward Kolmogorov differential equations for a

birth and death process clearly stating the postulates.

 

 

 

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Loyola College M.Sc. Mathematics April 2006 Mathematical Statistics – II Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

AC 50

FOURTH SEMESTER – APRIL 2006

                                              ST 4900 – MATHEMATICAL STATISTICS – II

(Also equivalent to ST 4953)

 

 

Date & Time : 03-05-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION-A    (10 ´ 2 = 20)

Answer ALL questions.  Each question carries 2 marks.

  1. Let T have a t-distribution with 10 degrees of freedom. Find P(|T|  > 2.228).
  2. Find the variance of S2 = (1/n) ∑ ( xi – x )2 , when X1, X2,…., Xn is a random sample      from N(µ , σ2 ).
  3. How do you obtain the joint p.d.f. of any two order statistics Yr and Ys when Yr < Ys ?
  4. What do you understand by a sufficient statistic for a parameter?
  5. Define: UMVUE.
  6. State Rao-Cramer Inequality.
  7. Distinguish between randomized and non-randomized tests.
  8. Illustrate graphically, the meaning of UMPT of level α test.
  9. Define a renewal process.
  10. When do you say that a stochastic matrix is regular?

SECTION-B   (8 x 5 = 40)

Answer any 5 questions.  Each question carries 8 marks.

 

  1. Let and S2 be the mean and the variance of a random sample of size 25 from a distribution N (3, 100).   Evaluate P (0 < < 6, 55.2 < S2 < 145.6).
  2. Derive the central F-distribution with (r1, r2) degrees of freedom.
  3. Let Y1 < Y2 < Y3 be the order statistics of a random sample of size 3 from the uniform distribution having p.d.f.
    f(x; θ ) = 1/θ, 0 < x < θ, 0 < θ < ∞, zero elsewhere.  Show that 4Y1, 2Y2 and (4/3)Y3 are all unbiased estimators of θ.  Find the variance of  (4/3)Y3.
  4. If az2 + bz + c = 0 for more than two values of z, then show that a = b = c = 0. Use this result to show that the family{ B(2, p): 0 < p < 1}  is complete.
  5. State and prove Lehmann-Scheffe’s theorem.
  6. Let X have a p.d.f. of the form f(x; θ) = θ xθ-1 , 0 <x < 1,  θ =1,2, zero elsewhere.  To test H0 : θ =1 against H1: θ =2, use a random sample X1, X2 of size n = 2 and define the critical region to be C = { (x1, x2) : ¾ ≤ x1­ x2 }.  Find the power function of the test.
  7. Prove or disprove: “UMPT of level α always exists for all types of testing problems”. Justify your answer.
  8. A certain genetic model suggests that the probabilities of a particular trinomial distribution are, respectively, p1 =p2, p2 = 2p (1-p), and p3 = (1-p)2 , where  0 < p < 1.   If X1, X2, X3 represent the respective frequencies in ‘n’ independent trials, explain how we could check on the adequacy of the genetic model.

 

SECTION-C       ( 20 ´ 2 = 40 )

Answer any 2 questions. Each question carries 20 marks.

  1. a) State and prove Factorization theorem. (12)
  2. b) Given the p.d. f.  f(x; θ) = 1 / ( π [1 + ( x – θ)2 ) , -∞ < x  < ∞ , -∞ < θ <  ∞. Show that the Rao-Cramer lower bound  is 2/n, where n is the size of a random sample from this Cauchy distribution. (8)

 

  1. a) State and prove the sufficiency part of Neyman-Pearson theorem. (12)
  2. b) Let X1, X2,…, Xn denote a random sample from a distribution having the p.d.f.
    f(x; p) = px (1-p)1-x , x = 0,1, zero, elsewhere.  Show that C =  { (x1, …,xn) :  Σ xi ≤ k }is a best critical region for testing H0: p = ½ against  H1: p = 1/3. Use the central limit theorem to find n and k so that approximately the level of the test is 0.05 and the power of the test is 0.9. (8)

 

  1. a) Derive the likelihood ratio test for testing H0: θ1=0, θ2 > 0 against
    H1: θ1 ≠ 0,  θ2 >0  when a random sample of size n is drawn from N(θ1 , θ2 ). (12)
  2. b) By giving suitable examples, distinguish between unpaired and paired t-tests. (8)

 

  1. a) Show that the Markov chain is completely determined by the transition matrix and the initial distribution. (8)
  2. b) Give an example of a random walk with an absorbing barrier.  (4)
  3. c) Explain in detail the properties of a Poisson process. (8)

 

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Loyola College M.Sc. Mathematics April 2006 Mathematical Statistics – I Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

 

THIRD SEMESTER – APRIL 2006

                                               ST 3951 – MATHEMATICAL STATISTICS – I

 

 

Date & Time : 27-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

     SECTION-A    (10 ´ 2 = 20)

Answer ALL questions.  Each question carries 2 marks.

  1. Give an example for a non-decreasing sequence of sets.
  1. Distinguish between experiment and random experiment.
  2. Let f (x) =    x/15,  x=1,2,3,4,5

0,  otherwise.

Find the median of the above distribution.

  1.  Let f(x) = (4-x) / 16, -2 < x < 2 ,zero elsewhere, be the p.d.f. of X.

If Y = ‌ X ‌ , compute P(Y ≤ 1).

  1. Give an example of a random variable in which mean doesn’t exist.
  1. Prove that E(E(X / Y)) = E(X).
  2. Define Hyper Geometric distribution.
  3. Define the characteristic function of a multidimensional random vector.

p                         p                                                 p

  1.  If    Xn → X    and    Yn →   Y, then show that Xn + Yn   →   X +Y.
  2. State Lindeberg-Feller theorem.

SECTION-B   (8 x 5 = 40)

Answer any 5 questions.  Each question carries 8 marks.

  1. Let f(x) = ½, -1 < x < 1, zero elsewhere, be the p.d.f. of X.  Find the distribution

function and the p.d.f. of Y = X2.

  1. State and prove Chebyshev’s inequality.
  2. If X1 and X2 are discrete random variables having the joint p.m.f.

f(x1,x2) = ( x1 + 2 x2 ) / 18,  (x1, x2­) = (1,1), (1,2), (2,1), (2,2), zero elsewhere, determine the conditional mean and variance of X2, given X1 =x1, for x1 = 1 or 2.

Also, compute E[ 3X1 – 2 X2 ].

  1. State and prove any two properties of MGF.
  2. Stating the conditions, show that binomial distribution tends to Poisson

distribution.

  1. Obtain the central moments of N (µ, σ2).
  2. Let X ~ G (n1, α) and Y ~ G (n2, α) be independent. Find the distribution of X/Y.
  3. Explain in detail the difference between WLLN and SLLN.

 

                                           SECTION-C ( 20 x 2 = 40 )

Answer any 2 questions. Each question carries 20 marks.

  1. a) State and prove Bayes’ theorem. (10)
  2. b) Bowl I contains 3 red chips and 7 blue chips. Bowl II contains 6 red chips and 4 blue chips. A bowl is selected at random and then 1 chip is drawn from this bowl.  Compute the probability that this chip is red.  Also, relative to the hypothesis that the chip is red, find the conditional probability that is drawn from bowl II.  (10)

 

  1. a) Find the mean and variance of the random variable X having the distribution function:

 

F(x)   =   0,        x < 0,

( x/4),   0≤x <1,

(x2 /4), 1≤x<2,

1 ,        x≥ 2.                                                  (10)

  1. b) Let X have the uniform distribution over the interval ( -π/2 , π/2). Find the

distribution of Y  = tan X.    (10)

 

  1. a) State and prove Kolmogorov’s strong law of large numbers. (12)
  2. b) State and prove Borel-Cantelli lemma. (8)

 

  1. a) Examine if central limit theorem (using Lyapounov’s condition) holds for the

following sequence of independent variates:

 

 

P     Xk   =   ±  2k      =   2-(2k + 1) ,             P       Xk  =  0    =   1 – 2–2k   (8)

 

  1. b) State and prove Lindeberg-Levy central limit theorem. (12)

 

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Loyola College M.Sc. Mathematics April 2006 Functional Analysis Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 5

FOURTH SEMESTER – APRIL 2006

                                                      MT 4800 – FUNCTIONAL ANALYSIS

 

 

Date & Time : 20-04-2006/FORENOON     Dept. No.                                                       Max. : 100 Marks

 

 

ANSWER ALL QUESTIONS

  1. a) i) Show that every vector space has a Hamel basis

(or)

  1. ii) If f Î X*, prove that the null space Z(f) has deficiency 0 or 1 in X.  Conversely, show that if Z is a subspace of X of deficiency 0 or 1, then there is an f Î X* such that Z=Z(f).
  2. b) iii)   Show that every element of X/Y contains exactly one element of Z, where Y and Z are complementary subspaces of X.
  3. iv) Let X and Y be normed linear spaces and let B(X, Y) denote the set of all bounded linear transformations from X into Y.  Then prove that B(X,Y) is a normed linear space.

(or)

  1. v) Let X be a real vector space, p be a real valued function on X such that P(x+y) £ p(x) + p(y) and p(ax) = a p(x) ” x,y Î X and a ³ 0, and let Y be a subspace of X.  If f is a linear functional on Y and f(x)  £  p(x) ” x Î Y, prove that there is a linear functional F on X such that F(x)=f(x)  ” x Î Y and F(x) £ p(x) ” x Î
  2. a) i) If x ¹ 0 is an element of a real normed linear space X, then show that there exists an x Î x¢ such that x¢(x) = ||x||  and ||x¢|| = 1.

(or)

  1. ii) 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, then T is bounded.                                                                   (8)

(or)

  1. b) iii)   State and prove the uniform boundedness theorem.
  2. iv) Give an example to show that uniform boundedness principle is not for every normed vector space.                                                                                                             (10+7)

(or)

  1. v) Let X and Y be Banach spaces and if T is a continuous linear transformation of X onto Y, then prove that T is an open mapping.                                                                               (17)
  2. a) i) State and prove the Riesz – Representation Theorem.

(or)

  1. 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 M ^N Û PQ = 0 Û QP=0                                               (8)
  2. b) iii)   If T is an operator on a Hilbert space X, then prove that T is normal iff its real and imaginary parts commute.
  3. iv) Prove that how Hilbert spaces are isomorphic iff they have the same dimension.  (7+10)

(or)

  1. v) If P is a projection on a closed linear space M of a Hilbert space X, prove that M is invariant under T Û TP =PTP
  2. vi) If P1, P2, … Pn are projections on closed linear subspaces M1, M2, … Mn on X, then prove that P= P1 + P2 + …+Pn is a projection iff the Pi are pairwise orthogonal and in the case P is a projection on M=M1+M2+…+Mn.                                                                              (5+12)
  3. a) i) Prove that every element x in a Banech algebra A for which ||x–1|| < 1, is regular, and the inverse of such an element is given by .

(or)

  1. ii) Let A be a Banech algebra and x Π Then prove that the spectrum of x, s(x), is non-empty.   (8)
  2. b) iii)   Let G be the set of regular elements in A and S be the set of singular elements in A.  Prove that G is an open set and therefore S is a closed set.
  3. iv) Show that the mapping x à x–1 of G into G is continuous and is therefore a homeomorphism.   (5+12)

(or)

  1. v) State and prove the Spectral Theorem.                                                    (17)

 

 

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Loyola College M.Sc. Mathematics Nov 2006 Topology Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 23

THIRD SEMESTER – NOV 2006

         MT 3803 – TOPOLOGY

(Also equivalent to MT 3800)

 

 

Date & Time : 25-10-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

            Answer all the questions.

 

01.(a)(i)   Let X be a metric space with metric d.  Show that d1. defined by

d1(x,y) =

is also a metric on X.

(OR)

(ii)   Let X and Y be metric spaces and f be a mapping of X into Y.

Show that f –1(G) is open in X whenever G is open in Y.

 

(b)(i)  Let X be a metric space.  Prove that any arbitrary union of open sets in X is
open and any finite intersection of open sets in X is open.

 

(ii)   Give an example to show that any arbitrary intersection of open sets in X
need not be open.

 

(iii)   In any metric space X, prove that each closed sphere is a closed set.(6+4+5)

(OR)

(iv)  If a convergent sequence in a metric space has infinitely many distinct
points, prove that its limit is a limit point of the set of points of the
sequence.

 

(v)   State and prove Cantor’s Intersection Theorem.

 

(vi)   If {An} is a sequence of nowhere dense sets in a complete metric space X,
show that there exists a point in X which is not in any of the An’s.    (4+6+5)

 

02.(a) (i)    Prove that every separable metric space is second countable.

(OR)

(ii)   Let X be a non–empty set, and let there be give a “closure” operation
which  assigns to each subset A of X a subset   of X in such a manner
that  (1)  = ,  (2)  A Í ,  (3)  , and  (4)   =.

If a “closed” set A is defined to be one for which A = , show that the
class of all complements of such sets is a topology on X whole closure
operation is precisely that initially given.

 

  • (i) Show that any closed subspace of a compact space is compact.

 

(ii)   Give an example to show that a proper subspace of a compact space need
not be closed.

 

(iii)  Prove that any continuous image of a compact space is compact.     (5+4+6)

(OR)

(iv)  Let C(X  đ) be the set of all bounded continuous real functions defined
on a topological space X.  Show that  (1) C (X  đ) is a real Banach space
with respect  to pointwise addition and multiplication and the  norm
defined by  = sup;    (2)  If multiplication is defined pointwise
C(X,  R) is a commutative real algebra with identity in which
£     and   = 1.

 

03.(a) (i)     State and prove Tychonoff’s Theorem.

(OR)

(ii)     Show that a metric space is compact Û it is complete and totally
bounded.

 

(b) (i)    Prove that in a sequentially compact space, every open cover has a
Lesbesgue number.

 

(ii)    Show that every sequentially compact metric space is totally bounded.(9+6)

(OR)

(iii)    State and prove Ascoli’s Theorem.

 

04.(a)(i)     Show that every subspace of Hausdorff is also a Hausdorff.

(OR)

(ii)     Prove that every compact Haurdolff space is normal.

 

(b)(i)     Let X be a T1 – space.

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

 

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

(OR)

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

 

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

(OR)

(ii)    Show that the components of a totally disconnected space are its points.

 

(b)(i)    Let X be a topological space and A be a connected subspace of X.  If B is a
subspace of X such that A Í B Í , show that B is connected.

 

(ii)   If X is an arbitrary topological space, then prove the following:

(1)  each point in X is contained in exactly one component of X;

  • each connected subspace of X is contained in a component of X;
  • a connected subspace of X which is both open and closed is a            component of X.                                                                        (6+9)

(OR)

 

(iii)   State and prove the Weierstrass Approximation Theorem.

 

 

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Loyola College M.Sc. Mathematics Nov 2006 Ordinary Differential Equations Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 20

FIRST SEMESTER – NOV 2006

         MT 1806 – ORDINARY DIFFERENTIAL EQUATIONS

 

 

Date & Time : 31-10-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

 

 

 

 

ANSWER ALL QUESTIONS

 

  1. (a) If the Wronskian of 2 functions x1(t) and x2(t) on I is non-zero for at

least one point of the interval I, show that x1(t) and x2(t) are linearly

independent on I.

OR

Consider the Differential Equation x” + λ2 x = 0, prove that

A cos λx + B sin λx is also a solution of the Differential equation.

(5 Marks)

(b) State and prove the method of variation of parameters.

OR

By the method of variation of parameters solve x”’ − x’ = t.   (15 Marks)

 

  1. (a) Obtain the indicial form of the equation

2x2 (d2y/dx2 ) +  (dy/dx)   + y = 0

OR

Obtain the indicial form of the Bessel’s differential equation. (5 Marks)

(b) Solve the differential equation using Frobenius Method ,

x2 (d2y/dx2)  + x q(x) (dy/dx)  + r(x) y = 0 and discuss about their

solutions when it’s  roots differ by an integer .

OR

Solve the  Legendre’s equation,

(1 – x2) (d2y/dx2)  – 2x (dy/dx)   + L(L+1)y = 0.                     (15 Marks)

 

 

III. (a) Prove that ∫+1-1 Pn(x) dx = 2 if n = 0 and

∫+1-1 Pn(x) dx = 0 if n ≥ 1

OR

Show that Hypergeometric function does not change if the parameter α and

β are interchanged, keeping γ fixed.                                                 (5 Marks)

(b) Obtain Rodrigue’s Formula and hence find P0(x), P1(x), P2(x) & P3(x).

OR

Show that Pn(x) = 2F1[-n, n+1; 1; (1-x)/2]                                    (15 Marks)

 

 

 

 

 

 

 

 

 

IV.(a) Considering an Initial Value Problem x’ =  -x, x(0) = 1, t ≥ 0, find xn(t).

OR

Find the eigen value and eigen function of x” + λ x = 0, 0 < t ≤   (5 Marks)

(b) State and prove Picard’s Boundary Value Problem.

OR

State Green’s Function. Show that x(t) is a solution of L(x) + f(t) = 0 if and

only if  x(t) = ∫ba G(t,s) f(s) ds.                                                      (15 Marks)

 

V.(a) Discuss the fundamental Theorem on the stability of the equilibrium of

the system x’ = f(t, x).

OR

Obtain the condition for the null solution of the system x’ = A(t) x is

asymptotically stable.                                                                     (5 Marks)

(b) Study the stability of a linear system by Lyapunov’s direct method.

OR

Study the stability of a non-linear system by Lyapunov’s direct method.

(15 Marks)

 

 

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Loyola College M.Sc. Mathematics Nov 2006 Mathematical Methods In Biology Question Paper PDF Download

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                        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 27

THIRD SEMESTER – NOV 2006

MT 3875 – MATHEMATICAL METHODS IN BIOLOGY

 

 

Date & Time : 06-11-2006/9.00-12.00   Dept. No.                                                       Max. : 100 Marks

 

 

I    a)  Draw the state diagram for  M = { (q0,q1,q2,q3), {0,1}, δ,q0,{q0} }

δ 0 1
q0

q1

q2

q3

 q0,q1

q3

  _

 q3

 q0, q2

   _

  q3

   q3

 

 

 

 

 

 

(or)

  1. b) Why do we need to install a program from web ?                                     (5)

 

  1. c) How do you generate Data base? Explain with an example .

(or)

  1. d) Comment on ‘ Internet is a powerful tool for bio informatics ’. (15)

 

II   a) Expand HTTP and explain  Motif.

(or)

  1. b) Define Edit graph and explain it for ANN and CAN. (5)

 

  1. c) Write notes on recurrence relation and about the correctness

of general relation

(or)

  1. d) Briefly describe on dynamic programming. (15)

 

III   a) Explain briefly on calculations of edit distance using tabulation method .

(or)

  1. b) Construct a deterministic finite automata accepting words over {0,1}

ending with ‘111’.                                                                                                     (5)

 

  1. c) When both i and j are strictly positive, prove that

D(i,j) = min [D(i-1,j)+1, D(i,j-1)+1, D(i-1,j-1)+t(i,j)]

(or)

d). What skills does a bioinformatician  should have ?                                                 (15)

 

IV   What do you mean by sequence alignment data ?

(or)

  1. Define Global alignment problem .                              (5)

 

  1. c) Describe the salient features of Human Genome project.

(or)

  1. d) Bio informatics is just a collection of Building Data bases- Explain. (15)

 

V    a) What type of questions does the bio informatics to be  answered in the field of

biomaths ?

(or)

  1. b) Define string alignment with an example. (5)

 

  1. c) What does informatics mean to biologists ?

(or)

  1. d)  Explain about the sequence matching of aniridia a human gene and

eyeless a fruit fly gene.

(15)

 

 

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Loyola College M.Sc. Mathematics Nov 2006 Real Analysis Question Paper PDF Download

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                   LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034      M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 19

FIRST SEMESTER – NOV 2006

         MT 1805 – REAL ANALYSIS

 

 

Date & Time : 28-10-2006/1.00-4.00         Dept. No.                                                       Max. : 100 Marks

 

 

  1. a)(1) When does the Riemann-Stieltjes integral reduce to Riemann integral. Explain with usual notations.

OR

(2) If a < s < b, f ÎÂ (a) on [a,b] and a (x) = I (x – s), the unit step function, then prove that = f (s).                                                                                                             (5)

b)(1) Let f be a bounded function on [a,b] having finitely many points of discontinuity on [a,b]. Let a be continuous at every point at which f is discontinuous. Prove that f ÎÂ(a).                                                                                                                                                (8)

(2) Suppose f is strictly increasing continuous function that maps an interval [A.B] onto [a,b]. Suppose a is monotonically increasing on [a,b] and f ÎÂ (a) on [a,b]. Define b and g on [A,B] by b (y) = a (f (y)), g (y) = f (f (y)). Then prove that g ÎÂ (b) and .                                                                                                     (7)

OR

(3) Let a be monotonically increasing function on [a,b] and let a¢ Î R on [a,b]. If f is a bounded real function on [a,b] then prove that f ÎÂ (a) on [a,b] Û f a¢ ÎÂ (a) on [a,b].(8)

(4) Let f ÎÂ (a) on [a,b]. For a £ x £ b, define F(x) = , then prove that F is continuous on [a,b]. Also, if f is continuous at some x o Î (a,b) then prove that F is differentiable at x o and F¢ ( x o ) = f (x o ).                                                                           (7)

  1. a) Let : [a,b] ® R m and let x Î (a,b). If the derivatives of exist at x then prove that it is unique.

OR

(2) Suppose that  maps a convex open set E Í Rn into Rm,  is differentiable on E and there exists a constant M such that  M, ” x Î E, then prove that

ú  (b) –  (a)ú £ M ú b – aú , ” a, b Î E.                                                                      (5)

 

  1. b) (1) Suppose E is an open set in R n ; maps R into R m ; is differentiable at x o Î E,  maps an open set containing    (E) into R k and  is differentiable at f (xo). Then the mapping of E into R k, defined by is differentiable at xo and .                                                                                                  (8)

(2) Suppose  maps an open set EÍ Â n into  m. Let   be differentiable at x Î E, then prove that the partial derivatives (Dj f i) (x) exist and , 1£ j £ m, where {e 1, e  2, e  3, …, e n} and {u 1, u 2, u 3, …, u m} are standard bases of R n and R m.  (7)

(3) If X is a complete metric space and if f is a contraction of X into X, then prove that there exists one and only one x ÎX such that f (x) = x.                                          (15)

 

III.  a) (1) Prove:  where {f n} converges uniformly to a function f on E and x is a limit point of a metric space E.

OR

(2) Suppose that {f n} is a sequence of functions defined on E and suppose that                  ½f n (x)½£ M n, x ÎE, n = 1,2,… Then prove that converges uniformly on E if converges.                                                                                                                (5)

  1. b) (1) Suppose that K is a compact set and

* {f n} is a sequence of continuous functions on K

** {f n} converges point wise to a continuous function f on K

*** f n (x) ³ f n+1 (x), ” n ÎK, n= 1,2,… then prove that f n ® f  uniformly on K. (7)

(2) State and prove Cauchy criterion for uniform convergence of complex functions defined on some set E.                                                                                                                         (8)

OR

(3) State and prove Stone-Weierstrass theorem.                                                            (15)

IV a) (1)Show that  converges if and only if n >0.

OR

(2) Prove that G  = .                                                                                         (5)

b)(1) Derive the relation between Beta and Gamma functions.                                       (7)

(2) State and prove Stirling’s formula.                                                                          (8)

OR

3) If f is a positive function on (0,¥) such that f (x+1) = x f (x);  f (1) =1 and log f is convex then prove that f (x) = G (x).                                                                                               (8)

(4) If x >0 and y >0  then                                        (7)

  1. a) (1)If f (x) has m continuous derivatives and no point occurs in the sequence x 0, x 1, ..,x n more than (m+1) times then prove that there exists exactly one polynomial Pn (x) of degree £ n which  agrees with f (x) at x 0, x 1, …, x n.

OR

2) Show that the error estimation for sine or cosine function f in linear interpolation is given by the formula ½f(x)-P(x)½£ .                                                                    (5)

b)(1) Let x0, x1, …, xn be n+1 distinct points in the domain of a function f and let P be the interpolation polynomial of degree £ n, that agrees with f at these points. Choose a point x in the domain of f and let [a,b] be any closed interval containing the points x 0, x 1, …, x n  and x. If f has a derivative of order n+1 in the interval [a,b], then prove that there is at least one point c in the open interval (a,b) such that  where A (x) = (x – x0) (x – x1)…(x – x n).            (7)

(2) Let P n+1 (x)= x n+1 +Q(x) where Q is a polynomial of degree £ n and let maximum of ½P n+1 (x)½, -1 £ x £ 1.  Then prove that we get the inequality . Moreover , prove that   if and only if , where T n+1 is the Chebyshev polynomial of degree n+1.                                          (8)

OR

3) Let f be a continuous function on [a,b] and assume that T is a polynomial of degree £ n that best approximates f on [a,b] relative to the maximum norm. Let R(x) = f (x) –T(x) denote the error in the approximation and let D = . Then prove that

(i) If D= 0 the function R is identically zero on [a,b].

(ii) If D >0, the function R has at least (n+1) changes of sign on [a,b].              (15).

 

 

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Loyola College M.Sc. Mathematics Nov 2006 Fluid Dynamics Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 29

THIRD SEMESTER – NOV 2006

         MT 3953 – FLUID DYNAMICS

 

 

Date & Time : 01-11-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Answer ALL Questions.

I    a) (i) Derive the equation of continuity in the form

[OR]

(ii)State and prove Euler’s equation of motion.                               (8)

 

  1. b) (i) The velocity of an incompressible fluid is given by .

Prove that the liquid motion possible and that the velocity potential is .

Also find the stream lines.

[OR]

(ii)State and prove Holemn Hortz  vorticity theorem                   (17)

 

 

II  a) (i)Show that the two dimensional flow described by the equation

is irrotational. Find the stream lines and equaipotentials.

[OR]

(ii)State and prove Milne Thomson circle theorem.                         (8)

 

  1. b) (i) In a  two  dimensional  fluid  motion  the  stream  lines  are

given by .Then show that  where A and B are constants. Also find the velocity.

[OR]

(ii) State and prove Blasius theorem.                                (17)

P.T.O.

 

 

III  a)(i)Write a note on Joukowskis transformation.

[OR]

(ii) State and prove Kutta and Joukowskis theorem.                        (8)

 

b)(i) Discuss the geometrical construction of an aerofoil.

[OR]

(ii) Discuss the liquid motion past a sphere.                      (17)

 

 

IV  a) (i) Find the exact solution of a liquid past a pipe of elliptical cross section.

[OR]

(ii) Discuss the flow between two parallel plates.                            (8)

 

  1. b) (i) Prove that .

[OR}                                                                                                                                                  (ii) Derive the Navier-Stokes equation of motion for viscous fluid.      (17)

 

 

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Loyola College M.Sc. Mathematics Nov 2006 Linear Algebra Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 18

FIRST SEMESTER – NOV 2006

         MT 1804 – LINEAR ALGEBRA

 

 

Date & Time : 26-10-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

 

 

Answer ALL Questions.

I ) a)    Let T be a linear operator on an n-dimensional vector space V. Then prove that the characteristic and the minimal polynomials for T have the same roots, except for multiplicities.

[OR]

Let W be an invariant subspace for T. Then prove that the characteristic polynomial for the restriction operator divides the characteristic polynomial for T. Also prove that the minimal polynomial for divides the minimal polynomial for T.                                                                                                    (5)

  1. b) State and prove Cayley-Hamilton theorem.

[OR]

Let V be a finite-dimensional vector space V over F and let T be a linear transform on V. Then prove that T is diagonalizable if and only if the minimal polynomial for T has the  form  where    are  distinct  elements of  F.                                                                                    (15)

II )a)    Let V be a finite-dimensional vector space. Let   be the subspaces of V and let  . Then prove the following are equivalent.

  1. i)  are independent.
  2. ii) For each we have  = {0}.

[OR]

Let  be a non-zero vector in V and let  be the T-annihilator of .Then        prove that

  1. i) If the degree ofis k, then the vectors form a   basis for.
  2. ii) If U is the linear operator on induced by T, then the minimal polynomial for U is.                                                               (5)

 

 

 

  1. b) State and prove the primary decomposition theorem.

[OR]

Let T be a linear operator on a finite-dimensional vector space V over F. If T is diagonalizable and if  are the distinct characteristic values of T, then prove that there exist linear operators  on V such that

(i)  T;

(ii)  I=;

(iii);

(iv)

(v) the range of  is the characteristic  space for T associated with

Conversely, if there exist k distinct scalars  and  k  non-zero linear operators  which satisfy conditions (i),(ii) and (iii), then show that T is diagonalizable, are the distinct characteristic values of T, and conditions (iv) and (v) are satisfied .                                                                         (15)

 

III  a) Write a note on the Jordon form.

[OR]

Let T be a linear operator on  which is represented in the standard basis by the matrix. Find the minimal polynomial for T.                                  (5)

  1. b) State and prove cyclic decomposition theorem.

[OR]

State and prove generalized Cayley-Hamilton theorem.                                   (15)

 

 

 

 

 

 

 

 

 

IV  a)   Prove that a form f  is Hermitian if and only if the corresponding linear operator T is self adjoint.

[OR]

If  , then prove that .                                                    (5)

  1. b) i) State and prove Principal Axis Theorem.
  2. ii) Let V be a complex vector space and f a form on V such that fis real for every .Then prove that f is Hermitian.                                       (9+6)

[OR]

Let T be a diagonalizable normal operator  with spectrum S  on a finite-dimensional inner product  space V .Suppose f is a function whose domain contains S. Then prove that  f(T) is a  diagonalizable normal operator  with spectrum f(S) .If U is a unitary map of V onto V’ and   T’=UTU, prove that S is the spectrum of T’ and  f(T)= Uf(T)U .                                                  (15)

V  a)    Find all bilinear forms of  F over F.

[OR]

Let f be a non-degenerate bilinear form on a finite-dimensional vector space V.

Then prove that the set of all linear operators on V which preserve f is a group under the operation of composition.                                                                    (5)

 

  1. Let V be a finite-dimensional vector space V over a field of characteristic zero, and let f be a symmetric bilinear form on V. Then prove that there is an ordered basis for V in which f is represented by a diagonal matrix.

[OR]

Let V be an n-dimensional vector space over a sub field of the complex numbers, and let f be a skew-symmetric bilinear form on V. Then prove that the rank r of f is even, and if r = 2k, then there is an ordered basis for V in which the matrix of f is the direct sum of the (n-r) x (n-r) zero matrix and k copies of the 2×2 matrix

.                                                                                                          (15)

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Loyola College M.Sc. Mathematics Nov 2006 Differential Geometry Question Paper PDF Download

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                        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 21

FIRST SEMESTER – NOV 2006

MT 1807 – DIFFERENTIAL GEOMETRY

 

 

Date & Time : 02-11-2006/1.00-4.00    Dept. No.                                                       Max. : 100 Marks

 

 

Answer ALL the questions

 

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

(or)

  1. b) Show that the necessary and sufficient condition for a curve to be a plane curve

is  = 0.                                                                                                     [5]

 

  1. c) Derive the equation of the osculating plane at a point on the curve of intersection of

two surfacesin terms of the  parameter u.                          [15]

(or)

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

 

II a) Define involute and find the curvature of it.

(or)

  1. b) Prove that a curve is of constant slope if and only if the ratio of curvature to torsion

is  constant .                                                                                                                [5]

 

  1. c) State and prove the fundamental theorem for space curve. [15]

(or)

  1. d) Find the intrinsic equations of the curve given by

 

III a) What is metric? Prove that the first fundamental form is invariant under the

transformation of parameters.

(or)

  1. b) Derive the condition for a proper transformation from regular point. [5]

 

  1. c) Show that a necessary and sufficient condition for a surface to be developable is

that the Gaussian curvature is zero.                                                                       [15]

(or)

  1. d) Define envelope and developable surface. Derive rectifying developable associated

with a space curve.

 

IV a) State and prove Meusnier  Theorem.

(or)

  1. b) Prove that the necessary and sufficient condition that the lines of curvature may be

parametric curve is that                                                             [5]

 

  1. c) Prove that on the general surface, a necessary and sufficient condition that the curve

be a geodesic is  for all values of the parameter .  [15]

(or)

  1. d) Find the principal curvature and principal direction at any point on a surface

 

 

V a)  Derive Weingarten equation.                                                                                   [5]

(or)

  1. b) Prove that in a region R of a surface of a constant positive Gaussian curvature

without umbilics, the principal curvature takes the extreme values at the boundaries.

 

  1. c) Derive Gauss equation. [15]

(or)

  1. d) State the fundamental theorem of Surface Theory and illustrate with an example

 

 

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