Loyola College M.Sc. Mathematics Nov 2006 Analytic Number Theory Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 25

THIRD SEMESTER – NOV 2006

MT 3805 – ANALYTIC NUMBER THEORY

Date & Time : 30-10-2006/9.00-12.00 Dept. No. Max. : 100 Marks

Answer ALL questions.

a) i) Define Mobius function and Euler function
ii) Prove that for n≥1. (2+3)

iii) Prove that log n = and. (5)

b) i) Prove that the set of all arithmetical functions f with f(1)≠0 forms an abelian group

with respect to Dirichlet product , the identity element being the function I.

ii) Let f be multiplicative. Then prove that f is completely multiplicative if and only if

f for all n1.

iii) If f is multiplicative then prove that. (10+5)

a) i) State and prove Euler’s summation formula.

ii) Prove that where C is Euler’s constant. (5)
b) i) State and prove weak and strong versions of Dirichlet asymptotic formulae for

the partial sums of the divisor function d(n).

ii) ) State and prove Asymptotic formulae for the partial sums of divisor functions

and (15)

III. a) i) An integer n>0 is divisible by 9 if and only if the sum of its digits in its decimal

expansion is divisible by 9. Prove this using congruences.

ii) If acand if d= (m,c), then prove that a≡b. (5)
b) i) State and prove Lagrange’s theorem.
ii) For any prime p prove that all the coefficients of the polynomial

f(x)=(x-1)(x-2)(x-3)…………(x-p+1)-x+1 are divisible by p. (10+5)

iii) If (a,m)=1, prove that the solution of the linear congruence ax≡b (mod m) is

is given by x≡ba (mod m).

iv) State and prove Chinese remainder theorem. (6+9)

a) i) Let p be an odd prime. Then for all n prove that.

ii) Prove that Legendre’s symbol () is a completely multiplicative

function of n. (5)

b) i) For every odd prime p, Prove that and

ii) State and prove Gauss’ Lemma. (7+8)

iii) State and prove Quadratic reciprocity law. Use it to determine those odd

primes p for which 3 is a quadratic residue and those for which it is a

nonresidue (15)

a) i) Evaluate where P is an odd positive integer.

ii) Determine whether 888 is a quadratic residue or nonresidue of the prime 1999.
b) i) Prove that for <1 ,,where p(0)=1 and

p(n) is the partition function.

ii) State and prove Euler’s pentagonal-number theorem. (15)

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Loyola College M.Sc. Mathematics Nov 2006 Algorithmic Graph Theory Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 26

THIRD SEMESTER – NOV 2006

MT 3806 – ALGORITHMIC GRAPH THEORY

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

Answer all questions.

1(a)(i). Define a graph and reversal of a graph with examples. What do you mean by

symmetric closure?

(OR)

(ii). Define a graphic sequence. Check whether the sequence (8, 8, 6, 5, 5, 4, 3, 3, 2) is

graphic.

(5 marks)

(b)(i). Prove that the complement of an interval graph satisfies the transitive orientation

property.

(ii). State the depth-first search algorithm and simulate it on the following graph by

selecting the vertex a.

(OR)

(iii). Prove that an interval graph satisfies the triangulated graph property.

(iv). Obtain a necessary and sufficient condition for a sequence to be

graphic. (3+12 marks)

2(a)(i). Define a triangulated graph, a simplicial vertex and a vertex separator with

examples.

(OR)

(ii). What is a perfect vertex elimination scheme? Obtain the same for the following

graph.

(5 marks)

(b)(i).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 in G.

(ii). Prove that every triangulated graph has a simplicial vertex.

(10+5 marks)

(OR)

(iii). Prove that an undirected graph is triangulated if and only if the ordering produced

by the Lexicographic breadth first search is a perfect vertex elimination scheme.

(iv). Apply the above algorithm for the following graph.

(5+10 marks)

3(a)(i). Define a split graph. Give an example of two non isomorphic split graphs with the

same degree sequence.

(OR)

(ii). 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.

(b)(i). Let G be an undirected graph. Prove that the following statements are equivalent.

(1). G is a split graph

(2). G and are triangulated graphs

(3). G contains no induced subgraph isomorphic to 2K₂, C₄ or C₅.

(OR)

(ii). Let G be an undirected graph with degree sequence d₁≥ d₂ ≥ … ≥ d_n and let m =

max {i : d_i ≥ i – 1}. Then prove that G is a split graph if and only if

(15 marks)

4(a)(i). Define a permutation graph. Draw the permutation graph corresponding to the

permutation [5,6,1,2,4,3,7].

(OR)

(ii). What is a permutation labeling? Illustrate with an example. (5 marks)

(b)(i). Prove that an undirected graph G is permutation graph if and only if G and

are comparability graphs.

(OR)

(ii). Let G be an undirected graph. Prove with usual notations that a bijection L

from V to {1, 2, 3 … n} is a permutation labeling if and only if the mapping

, is an injection.

(15 marks)

5(a)(i). Define a circular arc graph.

(OR)

(ii). Obtain an interval representation of the interval graph given below.

(5 marks)

(b)(i). 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 comparability graph.

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

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

(OR)

(ii). Define circular 1’s property. Prove that a matrix M has circular 1’s property if and

only if M’ has consecutive 1’s property.

(iii). Prove that an m x n (0, 1) with nonzero entries can be tested for the circulars 1’s

property in O(m+ n +f) steps. (8+7 marks)

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Loyola College M.Sc. Mathematics April 2007 Real Analysis Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 28

FIRST SEMESTER – APRIL 2007

MT 1805 – REAL ANALYSIS

Date & Time: 27/04/2007 / 1:00 – 4:00Dept. No. Max. : 100 Marks

Answer all the questions. Each question carries 20 marks.

(a). (i). Prove that refinement of partitions decreases the upper Riemann Stieltjes sum.

(OR)

(ii). If f is monotonic on [a, b], and if is continuous on [a, b], then prove that on [a, b]. (5)

(b). (i). Suppose c_n ≥ 0, for n = 1, 2, 3 …, converges, and { s_n} is a sequence of

distinct points in [a, b]. If and f is continuous on [a, b], then prove that .

(ii). Suppose that on [a, b], m ≤ f ≤ M, is continuous on [m,M],and on [a, b]. Then prove that on [a, b]. (7+8)

(OR)

(iii). Assume that increases monotonically and on [a, b]. Let f be a

bounded real function on [a, b]. Then prove that if and only if and in

that case.

(iv). State and prove the fundamental theorem of Calculus. (8+7)

(a). (i). Prove that a linear operator A on a finite dimensional vector space X is

one-to-one if and only if the range of A is all of X.

(OR)

(ii). If then prove that and . (5)

(b). (i). Let be the set of all invertible linear operators on R^k. If

, and then prove that .

(ii). Obtain the chain rule of differentiation for the composition of two

functions. (7+8)

(OR)

(iii). Suppose maps an open set E Rⁿinto R^m. Then prove that

if and only if the partial derivatives exist and are continuous on E for , .

(iv). If X is a complete metric space and if is a contraction of X into X,

then prove that there exists one and only one x in X such that . (8+7)

(a). (i).Show by means of an example that a convergent series of continuous functions

may have a discontinuous sum.

(OR)

(ii). State and prove the Cauchy criterion for uniform convergence. (5)

(b). (i). Suppose on a set E in a metric space. Let x be a limit point of E

and suppose that . Then prove that converges and that .

(ii). Let be monotonically increasing on [a, b]. Suppose on [a, b],

for n = 1, 2, …, and suppose that uniformly on [a, b]. Then prove that on [a, b] and that. (8+7)

(OR)

(iii). If f is a continuous complex function on [a, b], then prove that there

exists a sequence of polynomials P_n such that uniformly on [a, b]. (15)

(a). (i). Define the exponential function and obtain the addition formula.

(OR)

(ii). If , prove with usual notation that E(it) 1. (5)

(b). (i). Given a double sequence, i = 1,2,…, j = 1,2,…, suppose that

and converges. Then prove that .

(ii). Suppose that the series and converges in the segment

S = (–R, R). Let E be the set of all x in S at which = . If E has a limit point in S, then prove that for all n. (7+8)

(OR)

(iii). State and prove the Parseval’s theorem. (15)

(a). (i). If f has a derivative of order n at a point x₀, then prove that the Taylor

polynomial is the unique polynomial such that

for any polynomial Q of degree ≤ n.

(OR)

(ii). Define the Chebychev polynomial T_n and prove that it is of degree n and that

the coefficient of xⁿ is 2ⁿ^–1. (5)

(b). (i). State and prove the construction theorem.

(ii). Let where is a polynomial of degree ≤ n, and let

. Then prove that , with equality if and

only if where is the Chebychev polynomial of degree n+1. (8+7)

(OR)

(iii). Let x₀, x₁, …, x_n be n+1 distinct points in the domain of a function f and let P

be the interpolating 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 x₀, x₁, …, x_n and x. If f has derivative of order n+1 in [a, b] then prove that there is a point c in (a, b) such that , where .

(iv). If f(x) has m continuous derivatives and no point occurs in the sequence x₀,

x₁, …, x_n more than m+1 times, then prove that there exists one polynomial P_n(x) of degree ≤ n which agrees with f(x) at x₀, x₁, …, x_n. (8+7)

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Loyola College M.Sc. Mathematics April 2007 Probability & Stochastic Processes Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 36

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2007

ST 2902 – PROBABILITY & STOCHASTIC PROCESSES

Date & Time: 26/04/2007 / 1:00 – 4:00 Dept. No. Max. : 100 Marks

PART-A

Answer all the questions. 10×2=20 marks

Define sample space and give an example.
If A is a subset of B, show that P(A)≤P(B).
Define conditional probability.
When three events are said to be mutually independent?
Define normal distribution.
Provide two properties of a distribution function.
Define Markov process.
When a state of a Markov chain is called recurrent and transient?
Define excess life and current life of a renewal process.

10.Write any two applications of stochastic processes.

PART-B

Answer any two questions. 5×8=40 marks

11.If 12 fair coins are tossed simultaneously, find the probability of getting

(a) exactly 6 heads (b) atleast 3 heads (c) almost 10 heads (d) not more than 5 heads.

12.State and prove Boole’s inequality.

13.Find the mean and variance of the distribution that has the distribution function

F(x) = 0, x<0,

= x/8, 0≤x<2,

=x²/16, 2 ≤x<4,

. =1, 4 ≤x.

14.Let X and Y have the joint p.d.f.

f(x,y)= 6x²y, 0<x<1,0<y<1,

= 0 , elsewhere.

Find (i) the marginals of X and Y.

(ii) P(0<X<3/4, 1/3<Y<2).

15.Let the random variables X and Y have the joint pdf

f(x,y) = x+y , 0<x<1,0<y<1,

= 0, elsewhere.

Find the correlation coefficient of X and Y.

Communication is an equivalence relation-Prove.

17.Determine the classes and the periodicity of the various states for a Markov chain with

transition probability matrix

18.Derive the differential equation for a pure birth process clearly stating

the assumptions.

PART-C

Answer any two questions. 2×20=40 marks.

(a) Derive the Poisson process clearly stating the postulates.

(b)Explain different types of stochastic processes.

Let f(x₁,x₂)=21 x₁² x₂³, 0<x₁<x₂<1,zero elsewhere,be the joint pdf of X₁ and

X₂.Find the conditional mean and variance of X₁ given X₂=x₂, 0<x₂<1

21.A Markov chain on states {0,1,2,3,4,5} has transition probability matrix

(a) Find all classes.

(b) Find periodicity of various states.

(d) Obtain mean recurrence time of states.

22.(a) Explain renewal process in detail.

(b) Derive forward and backward Kolmogorov differential equations for a birth and

death, process, clearly stating the postulates.

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Loyola College M.Sc. Mathematics April 2007 Mathematical Statistics – II Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 56

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2007

ST 4900 – MATHEMATICAL STATISTICS – II

Date & Time: 25/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

SECTION-A (10 x 2 = 20)

Answer ALL the questions. Each carries 2 marks.

Distinguish between point and interval estimation.
Examine whether the sample variance is a biased estimator of s² when a random sample of size ‘n’ is drawn from N(m, s² ).
When do you say that a statistic is a consistent estimator of a parameter?
Distinguish between power function and power of the test.
What is degree of freedom?
Illustrate graphically the meaning of UMPT.
What do you understand by likelihood ratio tests?
Define F statistic.

Define: Markov Chain.

Classify the stochastic processes with respect to time and state space.

SECTION-B (5 x 8 = 40)

Answer any FIVE questions. Each carries 8 marks.

Let Y₁ and Y₂ be two independent unbiased estimators of θ. Let the variance of Y₁ be twice the variance of Y₂. Find the constants k₁ and k₂ so that k₁ Y₁ + k₂ Y₂ is an unbiased estimator with smallest possible variance for such a linear combination.
State and prove Rao-Cramer Inequality.

Let X₁, …, X_n be i.i.d., each with distribution having p.d.f.

f(x; θ₁, θ₂ ) = (1 / θ₂) exp{ – ( x- θ₁ ) /θ₂}, -θ₁ ≤ x< ∞, -∞ < θ₁< ∞, 0 <θ₂< ∞,

0; elsewhere.

Find the maximum likelihood estimators of θ₁ and θ₂.

Let X ≥ 1 be the critical region for testing H₀: θ = 1 against H₁: θ = 2 on the basis of a single observation from the population with pdf

f(x ,θ) = θ exp{ – θ x }, 0 < x <∞; 0 otherwise.

Obtain the size and power of the test.

Let X₁, X₂…X_n be iid U(0,q), q>0. Show that the family of distributions has MLR in X_(n).

A random sample of size 14 drawn from a normal population provides a sample mean of 3.22mm with an unbiased standard deviation of 0.34mm. Can you conclude at 5% level of significance that it meets the company’s specification of 2.7mm against more than 2.7mm? Construct 95% confidence limits for the population mean.
Let X_1,X_2,…,X_nbe iid N(m,s²). Derive LRT for testing H₀: m = m ₀against

H₁: m ¹ m_0,s² is unknown.

Describe Poisson process.

SECTION-C (2 x 20 =40)

Answer any TWO questions. Each carries 20 marks.

a) State and prove factorization criterion for determining sufficient statistics. (12)
b) Show that the first order statistic Y₁ of a random sample of size n from the

distribution having p.d.f.

f(x: θ) = e^{–(x – θ)}, -¥ < x < ¥, – ¥ < θ < ¥, zero elsewhere,

is a complete sufficient statistic for θ.

Find the unique function of this statistic, which is the unbiased minimum variance

estimator of θ. (8)

State and prove the necessary and sufficient conditions of Neyman-Pearson

Fundamental Lemma. (10+10)

a) Let X_1,X₂and X₃ be a sample of size 3 from Poisson distribution with mean θ.

Consider the problem of testing H₀: θ = 2 against H₁: θ = 3. Find the randomized

MPT of level α =0.05. (12)

b) Prove or disprove:

“UMPT of level α always exists for all types of testing problems”.

Justify your answer. (8)

a) A number is to be selected from the interval (x : 0 < x < 2) by a random process.

Let A_i = {(x : (i-1)/2 < x < i/2}, i = 1,2,3, and let A₄ = {x : 3/2 < x < 2}. A certain

hypothesis assigns probabilities p_io to these sets in accordance with

p_io = ò_Ai (1/2) (2-x) dx, i = 1,2,3,4. If the observed frequencies of the

sets A_i, i = 1,2,3,4, are respectively, 30, 30, 10, 10, would H_o be accepted at the

(approximate) 5 percent level of significance ? (10)

b) Consider a Markov chain having state space S = {0,1,2} and transition probability

matrix

1/3 1/3 1/3

¼ ½ ¼

1/6 1/3 ½

Show that this chain has a unique stationary distribution p and find p. (10)

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Loyola College M.Sc. Mathematics April 2007 Functional Analysis Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 58

FOURTH SEMESTER – APRIL 2007

MT 4804 – FUNCTIONAL ANALYSIS

Date & Time: 16/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

Answer all questions.

01.(a) Show that every vector space has a Hamel basis.

(OR)

Prove that a subset S of a vector space X is linearly independent Û for every
subset {x₁, x₂, …, x_n} of S, åa_ix_i = 0 Þ a_i = 0, for all i. (5)

(b)(i) Show that every element of X/Y contains exactly one element of z, where Y
and z are complementary subspaces of X.

(ii) If Z is a subspace of a vector space X of deficiency 0 or 1, show that there is
an f Î X^* such Z = Z(f). (7 + 8)

(OR)

(iii) Let X be a real vector space. Let Y be a subspace of X and p be a real
valued function on X such that p(x) ³ 0, p(x + y) = p(x) + p(y) and
p(ax) = a(Px) for a ³ 0. If f is a linear functional on Y such that
£ p(x) for every x Î Y, show that there is a linear functional F on X
such that F(x) = f(x) on Y and £ p(x) on X. (15)

(a) State and prove F-Riesz Lemma.

(OR)

Let X and Y be normed linear spaces and let T be a linear transformation
of X into Y. Prove that T is bounded if and only if T is continuous. (5)

(b) State and prove the Hahn Banach Theorem for a complex normed linear
space.

(OR)

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. Show that B(X, Y) is a
normed linear space and B(X, Y) is a Banach space, if Y is a Banach space.

(15)

(a) State and prove Riesz Representation Theorem.

(OR)

Prove that a real Banach space is a Hilbert space iff the parallelogram law
holds in it. (5)

(b) State and prove the Projection Theorem.

(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. (15)

(a) State and prove Bessel’s inequality.

(OR)

If T is an operator on a Hilbert space X, show that T is a normal Û its real
and imaginary parts commute. (5)

(b)(i) If T is an operator in a Hilbert space X, then show that

(Tx, x) = 0 Þ T = 0.

(ii) If N₁ and N₂ are normal operators on a Hilbert space X with the property
that either commute with adjoint of the other, prove that N₁ + N₂ and N₁N₂
are normal. (7 + 8)

(OR)

(iii) State and prove Riesz-Fischer Theorem. (15)

(a) Prove that the spectrum of x, , is non-empty.

(OR)

Define a Banach algebra A, set of regular elements G, set of singular
elements S, and prove that G is open and S is closed. (5)

(b) State and prove the Spectral theorem.

(OR)

Let G be a set of regular elements in a Banach algebra A. (5)

Prove that f : G ® G given by f(x) = x^-1 is a homeomorphism.

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Loyola College M.Sc. Mathematics April 2007 Functional Analysis Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 54

FOURTH SEMESTER – APRIL 2007

MT 4800 – FUNCTIONAL ANALYSIS

Date & Time: 16/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

Answer ALL questions.

01.(a)(i) 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)

(ii) Prove that a subset S of a vector space X is linearly independent Û for
every subset {x₁, x₂, …, x_n} of S, i. (8)

(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. Show that B(X, Y) is a
normed linear space and B(X, Y) is a Banach space, if Y is a Banach space.

(OR)

(ii) Let X be a real vector space. Let Y be a subspace of X and p be a real
valued function on X such that p(x) ³ 0, p(x + y) = p(x) + p(y) and
p(ax) = a(Px) for a ³ 0. If f is a linear functional on Y such that

£ p(x) for every x Î Y, show that there is a linear functional F on X
such that F(x) = f(x) on Y and £ p(x) on X. (17)

02.(a)(i) Show that a normed vector space is finite dimensional iff the closed and
bounded sets are compact.

(OR)

(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)

(b)(i) State and prove the Uniform Boundeness Theorem. Give an example to
show that the uniform boundedness principle is not true for every normed
vector space. (9 + 8)

(OR)

(ii) 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)

03.(a)(i) Let X be a Hilbert space and S = {x_a} _a_{Î A} be an orthonormal set in X.

Prove that S is a basis iff it is complete in X.

(OR)

(ii) If T is an operator on a Hilbert space X, then show that

(Tx, x) = 0 Þ T = 0. (8)

(b)(i) State and prove Riesz Representation Theorem

(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 M ^ N Û PQ = O Û QP = 0.

(OR) (9 + 8)

(iii) State and prove Riesz – Fischer Theorem. (17)

04.(a) (i) Prove that the spectrum of x, s(x), is non-empty.

(OR)

(ii) Define a Banach Algebra A, set of regular elements G, set of singular
elements S, and prove that G is open and S is closed. (8)

(b)(i) Define spectral radius and derive a formula for the same.

(OR)

(ii) State and prove the Spectral theorem. (17)

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Loyola College M.Sc. Mathematics April 2007 Algorithmic Graph Theory Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

CV 52

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2007

MT 3806 – ALGORITHMIC GRAPH THEORY

Date & Time: 28/04/2007 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

Answer all questions.

1(a)(i). Define clique cover, stability number and chromatic number. Illustrate with

examples.

(OR)

(ii). What is an intersection graph? Explain with an example.

(5 marks)

(b)(i). Prove that the complement of an interval graph satisfies the transitive orientation

property.

(ii). Let and . Then

prove that d is graphic if and only if is graphic.

(OR)

(iv). Prove that an interval graph satisfies the triangulated graph property.

(v). State the breadth-first search algorithm. Simulate it on the following graph

by selecting the vertex p.

(3+12 marks)

2(a)(i). Define a simplicial vertex, a vertex separator and a perfect vertex elimination

scheme. Illustrate with examples.

(OR)

(ii). Obtain a clique tree representation of the following graph.

(b)(i).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 in G.

(ii). Prove that every triangulated graph has a simplicial vertex. (10+5 marks)

(OR)

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

(iv). Prove that an undirected graph is triangulated if and only if the ordering produced

by the Lexicographic breadth first search is a perfect vertex elimination scheme. (5+10 marks)

3(a)(i). Define a split graph and give an example.

(OR)

(ii). Let G be a split graph with the vertex set partitioned into a stable set S and a clique

K. If |S| = α(G) – 1, and |K| = ω(G) then prove that there exists an y ε K such that

S +{y} is stable.

(b)(i). Let G be an undirected graph. Prove that the following statements are equivalent.

(1). G is a split graph

(2). G and are triangulated graphs

(3). G contains no induced subgraph isomorphic to 2K₂, C₄ or C₅.

(OR)

(ii). Let G be an undirected graph with degree sequence d₁≥ d₂ ≥ … ≥ d_n and let m =

max {i : d_i ≥ i – 1}. Then prove that G is a split graph if and only if

(15 marks)

4(a)(i). Define and determine the permutation graph corresponding to the permutation

[8,4,5,2,1,3,7,6].

(OR)

(ii). What is a permutation labeling? Illustrate with an example.

(b)(i). Prove that an undirected graph G is permutation graph if and only if G and

are comparability graphs.

(OR)

(ii). Let G be an undirected graph. Prove with usual notations that a bijection L

from V to {1, 2, 3 … n} is a permutation labeling if and only if the mapping

, is an injection. (15 marks)

5(a)(i). Define an interval graph with an example.

(OR)

(ii). Obtain an interval representation of the interval graph given below.

(b)(i). 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 comparability graph.

(3). The maximal cliques can be linearly ordered such that, for every vertex x of G

the maximal cliques containing v occurs consecutively.

(OR)

(ii). An undirected graph G is a circular arc-graph if and only if its vertices can be circularly indexed v₁, v₂,…,v_n so that for all i and j

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 42

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2008

MT 3803 / 3800 – TOPOLOGY

Date : 26/04/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

Answer ALL questions.: (5 X 20 = 100)

(a) (i) Let X be a metric space with metric d.

Show that d₁ defined by

is also a metric on X.

(OR)

(ii) In any metric space X, prove that each open sphere is an open set.

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

Prove that f is continuous at x₀ iff and f is

continuous iff f^-1(G) is open in X whenever G is open in Y. (15)

(a) (i) Prove that every second countable space is reparable.

(OR)

(ii) Prove that every separable metric space is second (5)

(b) (i) Show that every continuous image of a compact space is compact.

(ii) Prove that any closed subspace of a compact space is compact. (8+7)

(OR)

space X, then show that f+g and are also continuous.

(ii) Let C(X,R) be the set of all bounded continuous real functions defined on

metric space, show that C(X,R) is a closed subset of the metric space. (8+7)

(a) (i) State and prove Tychnoff’s Theorem.

(OR)

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

(b) (i) Show that a metric space is sequentially compact it has the Bolzano

Weierstrass property.

(ii) Prove that every Compact metric space has the Bolzano Weierstrass

Property. (10+5)

(OR)

(iii) State and prove Lebesgue Coverity Lemme.

(iv) Prove that a metric space is compact it is complete and totally

bounded. (9+6)

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

(OR)

(ii) Prove that every compact Hausdorff Space is normed. (5)

(b) (i) Prove that the product of any non-empty class of Hausdorff Spaces is a

Hausdorff Space.

(ii) Prove that every compact subspace of a Hausdorff space is closed.

(iii) Show that a one-to-one continuous mapping of a compact space onto a

Hausdorff Space is a homeomorphism. (6+4+5)

(OR)

(iv) State and prove Uryshon Imbedding Theorem. (15)

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

(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)

(OR)

(iii) State and prove Weierstrass Approximation Theorem.

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 26

FIRST SEMESTER – APRIL 2008

MT 1805 – REAL ANALYSIS

Date : 30-04-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

a) 1) Let and on [a,b] then prove that

(i) on [a,b] and (ii)

2) Define step function and prove: If a < s < b, on [a,b] and , the unit step

function, then prove that (5)

b) 1) Let , n = 1,2,3,… . Suppose that is convergent and {s_n} is a sequence of distinct

numbers in (a,b). Let . Let f be continuous on [a,b] then prove that

2) Let be monotonically increasing function on [a,b] and let on [a,b]. If f is a bounded real

function on [a,b] then prove that

3) Let on [a,b] for . Define , then prove that F is continuous on [a,b].

If F is continuous at some point , then prove that F is differentiable at x_oand .

4) State and prove the fundamental theorem of Calculus and deduce the following result:

Suppose F and G are differentiable functions on [a,b], then prove that (6 + 9)

a) 1) Let exists then prove that it is unique.

2) Define a convex set and prove: Suppose that maps a convex set ; is

differentiable on E and there exists a constant M such that then prove that

(5)

b) 1) When do you say a function is continously differentiable? Letmaps an open set

show that if and only if the partial derivatives D_jf_i exists and are

continuous on E for (15)

2) a) State and prove the Contraction principle.

b) Let C(X) denote the set of all continuous, complex valued, bounded functions onX. Prove that C(X)

is a complete metric space. (5+10)

III. a)1) Prove that every converging sequence is a Cauchy’s sequence. Is the converse true?

b) 1) State and prove the Cauchy criterion for uniform convergence.

2) Suppose {f_n} is a sequence of differentiable functions on [a,b]. Suppose that {f_n(x₀)} converges uniformly on [a,b] then prove that {f_n} converges uniformly on [a,b] to some function f and (5 + 10)

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

a)1) Is the trignometric series a Fourier series? Justify your answer.

2) Define a Gamma function and state the three properties that characterize Gamma function completely. (5)

b)1) State and prove the Parseval’s theorem.

2) If f is continuous (with period ) and if then prove that there is a trignometric polynomial P such that for all real x. (10 + 5)

3) State and prove the Dirichlet’s necessary and sufficient condition for a Fourier series to converge to a sum s. (15)

a)1) Write a note on Lagrange’s polynomial.

2) Write a note on Chebyshev polynomial. (5)

b)1) 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 this approximation and let . Then prove

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)

2) If f(x) has m continuous derivatives and no point occurs in the sequence x_o, x₁, x₂, …, x_n more than (m + 1) times then prove that there exists exactly one polynomial P_n(x) of degree n which agrees with f(x) at x_o, x₁, x₂, …, x_n.

3) Let P _n+1 (x) = x ⁿ⁺¹+ Q(x), where Q(x) is a polynomial of degree n, and let . Then prove . (10+5)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 27

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2008

MT 1806 – ORDINARY DIFFERENTIAL EQUATIONS

Date : 03/05/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

ANSWER ALL QUESTIONS

a) Suppose x₁ (t) and x₂ (t) satisfy a x”(t) + b x'(t) + c x(t) = 0,

where ‘a’ is not zero then show that A x₁ (t) + B x₂ (t) satisfy the Differential Equation.

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. (5 Marks)

b) i) State and prove the Abel’s Formulae. (8 Marks)

(ii) Solve x” – x’ – 2x = 4t² using the method of variation of parameters.

(7 Marks)

(iii) If λ is a root of the quadratic equation a λ² + b λ + c = 0,

prove that e^λt is a solution of a x” + bx’ + c x = 0. (15 Marks)

a) Prove that exp[ x/2( t – t^{– 1})] = .

Prove that (1 – 2tx + t² )^{– ½} = if │t│< 1 & │x│≤ 1. (5 Marks)

b) Solve the Legendre’s Equation ( 1 – x² ) y´´– 2xy´ + ny = 0

Solve x ( 1 – x ) y´´ + ( 1 – x ) y´ – y =0 (15Marks)

III. (a) Prove that = 2 if n = 0 and

= 0 if n ≥ 1

Find (d/dx) F (α; β; γ; x ) (5 Marks)

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

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

a) Considering the Differential Equation of the Sturm-Liouville,

show that the eigen values λ_m and λ_n corresponding to eigen functions

x_m(t) and x_n(t) are orthogonal with respect to weight function r(t).

Solve the initial value problem x´ = t + x, x(0) = 1 (5 Marks)

b) State Green’s Function. x(t) is a solution of L(x) + f(t) = 0

if and only if x(t) = .

State and prove Picard’s Initial value Problem. (15 Marks)

a) Define Lyapunov’s Stability Statements.

Prove that the null solution of x’ = A (t) x is stable if and only if there

exists a positive constant k such that | Φ | ≤ k, t ≥ t₀ . (5 Marks)

b) State and prove the Fundamental Theorem on the stability of the

equilibrium of a system x’ = f (t, x).

Explain the stability of Quasi-linear system x’ = A(t) x. (15 Marks)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 25

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2008

MT 1804 – LINEAR ALGEBRA

Date : 28/04/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL Questions.

a) i) Let T be the linear operator on which is represented in the standard ordered

basis by the matrix . Find a basis of, each vector of which is a

characteristic vector of T.

ii) Let T be a linear operator on a finite dimensional vector space V. Let

be the distinct characteristic values of T and let be the null space of

(T-I). If W=then prove that

dim W=. (5)

b) i) State and prove Cayley-Hamilton theorem

ii) Let V be a finite dimensional vector space over F and T be a linear operator on V

then prove that T is triangulable if and only if the minimal polynomial for T is a

product of linear polynomials over F. (15)

a) i) Let V be a finite dimensional vector space. Let be subspaces of V

let Then prove that the following are equivalent :

1) are independent.

2) For each j,,= {0}

iii) Let T be a linear operator on a finite dimensional vector space V and let

are linear operators on V such that 1) each is a projection

2) 3) and let is the range of. If each

is invariant under T then prove that T=T, i=1,2,..k. (5)

b) i) Let T be a linear operator on a finite dimensional vector space V. Suppose that

the minimal polynomial for T decomposes over F into a product of linear

polynomials. Then prove that there is a diagonalizable operator D on V and

nilpotent operator N on V such that 1) T= D+ N

2) DN=ND

ii) Let T be a linear operator on a finite dimensional vector space V. Then

prove that T has a cyclic vector if only if the minimal and characteristic

polynomial for T are identical. (15)

III. a) i) Let T be a linear operator on which is represented in standard ordered basis

by the matrix . Prove that T has no cyclic vector. What is the

T-cyclic subspace generated by the vector (1,-1,3)?

ii) If U is a linear operator on the finite dimensional vector space W and if U has a

cyclic vector then prove that there is an ordered basis for W in which U is

represented by the companion matrix of the minimal polynomial for U. (5)

b) i) State and prove Cyclic Decomposition theorem.

ii) If T is a nilpotent operator on a vector space V of dimension n then prove that

characteristic polynomial for T is (15)

a) i) Let V be a finite dimensional complex inner product space and f a form on V.

Then prove that there is an orthonormal basis for V in which the matrix of f is

upper-triangular.

ii) Let T be a linear operator on a complex finite dimensional inner product space
Then prove that T is self-adjoint if and only if is real for every in V. (5)

b) i) Let f be the form on defined by f=.Find the

matrix of f with respect to the basis {(1,-1),(1,1)}.

ii) State and prove the spectral theorem. (6+9)

iii) Let f be a form on a real or complex vector space V and a basis

for the finite dimensional subspace W of V. Let M be the rxr matrix with

entries and W the set of all vectors in V such that

f ()=0 for all W. Then prove that W is a subspace of V and

={0} if and only if M is invertible and when this is the case V=W+W.

(15)

a) i) Let F be a field. Find all bilinear forms on the space .

ii) State and prove polarization identity for symmetric bilinear form f. (5)

b) i) Let V be a finite dimensional vector space over the field of complex numbers.

Let f be a symmetric bilinear form on V which has rank r. Then prove that there

is an ordered basis for V such that the matrix of f in the

ordered basis B is diagonal and f () =

ii) If f is a non-zero skew-symmetric bilinear form on a finite dimensional vector

space V then prove that there exist a finite sequence of pairs of

vectors,with the following properties:

1) f ()=1, j=1,2,,…,k.

2) f ()=f ()=f ()=0, ij.

3) If is the two dimensional subspace spanned by and , then

V=where is orthogonal to all and and the restriction

of f to is the zero form. (15)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 48

FOURTH SEMESTER – APRIL 2008

MT 4804 – FUNCTIONAL ANALYSIS

Date : 16/04/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

Answer ALL questions: (5 x 20 = 100)

a) (i) 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)

(ii) Prove that every vector space has a Hamel Basis. (5)

(i) Prove that a subset S of a vector space X is linearly independent for every subset of S, for all i.

(ii) If prove that the null space has deficiency 0 or 1 in X.

Conversely, if Z is a subspace of X of deficiency 0 or 1, show that there is an such that . (7+8)

(OR)

(iii) Let X be a real vector space, let Y be a subspace of X and p be a real valued function on X such that and p(ax)=a p(x) x,yX, for . If is a linear functional on Y and prove that there is a linear function F on X such that and . (15)

a) Let X and Y be normal linear spaces and let T be a linear transformation

of X into Y. Prove that T is bounded if and only if T is continuous.

(OR)

State and prove F-Rierz Lemma. (5)

State and prove Hahn Banach Theorem for a complex normal 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 normal

vector space. (15)

a) Prove that a real Banach space is a Hilbert space iff the parallelogram law

holds in it. (5)

(OR)

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.

State and prove Projection Theorem.

(OR)

State and prove Open Mapping Theorem. (15)

a) If T is an operator on a Hilbert Space X, show that T is normal its real

and imaginary parts commute

(OR)

If T is an operator on a Hilbert space X, prove that ( (5)

(i) If N₁ and N₂ are normal operators on a Hilbert space X with the

property that either commute with adjoint of the other, prove that N₁+N₂ and N₁N₂ are normal.

(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 (8+7)

(OR)

(iii) State and prove Rierz – Fischer Theorem.

a) Prove that the spectrum of is non-emply.

(OR)

Show that given is continuous. (5)

State and prove the Spectral Theorem.

(OR)

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

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 28

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2008

MT 1807 – DIFFERENTIAL GEOMETRY

Date : 05-05-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer all the questions

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

(or)

b) Prove that the curvature is the rate of change of angle of contingency with respect to

arc length. [5]

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)

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

II a) If the curve has three point contact with origin

withthen prove that .

(or)

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]

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

(or)

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)

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

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

(or)

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

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

(or)

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

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

(or)

d) Find the principal curvature and direction of the surface

. [15]

V a) Derive Weingarton equation.

(or)

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

c) Derive Gauss equation in terms of Christoffel’s symbol.

(or)

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

of unit sphere . [15]

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XZ 29

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2008

MT 1808 – COMPUTER ALGORITHMS

Date : 06/05/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL questions. Each question carries 20 marks

1 a. Define a queue. Give an algorithm to delete an element from a circular queue.

Give algorithm INSERT used to form a heap on n elements. (5)

(i) Discuss in detail how to analyze algorithms.

(ii) Define a binary tree. Give the parent, left child and the right child of a node of a binary tree labeled j. (9+6)

State algorithm HEAPSORT. Simulate it on a(1:5) = ( 67,23,12,78,90). (15)

2 a. Draw the tree of calls of MERGESORT when n =13.

State algorithm PARTITION to partition the array a[m:p-1] about a[m]. (5)

State algorithm BINSRCH. Simulate it on a(1:10) = (12, 23, 34, 45, 56, 67, 78, 89, 90, 98 ) to search for (i) x = 34 (ii) x = 80 (iii) x = 89. Draw the binary decision tree when n = 10.

State algorithm MERGESORT. Show that the computing time for MERGESORT is O(n log n ) where n is the number of inputs. (15)

3 a. Give the control abstraction for greedy method.

Explain the ‘optimal storage on tapes’ problem with an example. (5)

Explain the problem ‘job sequencing with deadlines’. Describe a greedy method to solve it and prove that the method always obtains an optimal solution to the job sequencing problem.

State algorithm GREEDY KNAPSACK and prove that it generates an optimal solution to the given instance of the knapsack problem. (15)

4 a. Explain two different formulations of ‘sum of subsets’ problem.

Define inorder traversal of a tree with an example. (5)

Explain how backtracking works on 4-queens problem. State algorithm NQUEENS.

State an algorithm for breadth first graph traversal. Draw a graph on 8 vertices and explain the concept. Also explain depth first graph traversal showing how it differs from breadth first graph traversal. (15)

5 a. Write note on ‘Nondetermininistic Algorithms’.

What is a formula in propositional calculus? When is it said to be in (i) conjunctive normal form and (ii) disjunctive normal form. (5)

Explain the clique decision problem Prove that CNF-satisfiability α clique decision problem

Define a node cover for a graph with an example. Prove that the clique decision problem α the node cover decision problem. (15)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 45

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2008

MT 3806 – ALGORITHMIC GRAPH THEORY

Date : 03/05/2008 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

Answer all questions.

1(a)(i). Obtain clique covers of size 2 and 3 for the following graph.

(OR)

(ii). Prove that an interval graph satisfies the triangulated graph property.

(5 marks)

(b)(i). State the breadth-first search algorithm. Simulate it on the following graph by selecting the vertex a.

(ii). State the depth-first search algorithm and simulate it on the following graph by selecting the vertex a.

(15 marks)

2(a)(i). Define a simplicial vertex. Obtain a perfect elimination scheme for the following graph

(OR)

(ii). Define a vertex separator and a minimum vertex separator. Obtain a minimum

vertex separator for the following graph.

(5 marks)

(b)(i). 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 in G.

(ii). Prove that every triangulated graph has a simplicial vertex.

(OR)

(iii). Prove that an undirected graph is triangulated if and only if the ordering produced

by the Lexicographic breadth first search is a perfect vertex elimination scheme.

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

(10+5 marks)

3(a)(i). Define a split graph; prove that an undirected graph is a split graph if and only if its complement is a split graph.

(OR)

(ii). 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 marks)

(b)(i). Let G be an undirected graph. Prove that the following statements are equivalent.

(1). G is a split graph

(2). G and are triangulated graphs

(3). G contains no induced subgraph isomorphic to 2K₂, C₄ or C₅.

(OR)

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

(15 marks)

4(a)(i). Define a permutation graph. Draw the permutation graph corresponding to the

permutation [7,2,5,1,6,8,3,4].

(OR)

(ii). What is a permutation labeling? Illustrate with an example.

(5 marks)

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

(OR)

(ii). Let G be an undirected graph. Prove with usual notations that a bijection L from V to {1, 2, 3 … n} is a permutation labeling if and only if the mapping

, is an injection.

(15 marks)

5(a)(i). Define a circular arc graph. Illustrate with an example.

(OR)

(ii). Obtain an interval representation of the interval graph given below.

(5 marks)

(b)(i). 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 a 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.

(OR)

(iii). Prove that an undirected graph G is a circular arc graph if and only if its

vertices can be circularly indexed v₁.v₂ …v_n so that for all i and j

(15 marks)

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AB 32

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – November 2008

MT 3803 – TOPOLOGY

Date : 03-11-08 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

Answer ALL questions. All questions carry equal marks.

(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 be a metric space. Prove that a subset F of X is closed Û its complement
F’ is open. (5)

(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.

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

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

(6 + 5 + 4)

(OR)

(iv) Prove that the set C(X, R) of all bounded continuous real functions defined
on a metric space X is a Banach space with respect to pointwise addition and
scalar multiplication and the norm defined by (15)

II.(a) (i) If X is a second countable space X, prove that X is separable.

(OR)

(ii) Let X be a topological space, and let {f_n} be a sequence of real functions
defined on X which converges uniformly to a function f defined on X. If all
the f_n’s are continuous, show that f is continuous. (5)

(i) Show that the 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 + 5 + 4)

(OR)

(iv) Prove that a topological space is compact, if every subbasic open cover has a
finite sub cover. (15)

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

(OR)

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

(5)

(b)(i) In a sequentially compact metric space, prove that every open cover has a
Lebesgue number.

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

(OR)

(iii) State and prove Ascoli’s Theorem (15)

IV.(a) (i) Prove that a 1 – 1 mapping of a compact space onto a Haurdorff space is a
homeomorphism.

(OR)

(ii) Show that every compact space is normal. (5)

(b)(i) State and prove the Tietze Extension Theorem.

(15)

(OR)

(ii) Let X be a T₁ – space.

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

(iii) State and prove URYSOHN’s Lemma

(6 + 9)

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

(OR)

(ii) Let X be a topological space and A be a connected subspace of X. (5)

If B is a subspace of X such that A Í B Í , then show that B is connected.

(b)(i) Show that a topological space X is disconnected Û there exists a continuous
mapping of X onto the discrete two-point space {0, 1}.

(ii) Prove that the product of any non-empty class of connected spaces is
connected. (6 + 9)

(OR)

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

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Loyola College M.Sc. Mathematics Nov 2008 Real Analysis Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 27

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

MT 1805 – REAL ANALYSIS

Date : 06-11-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL the questions

I a)1) If (with the usual notations) holds for some P and some prove that the same holds for every refinement of P.

2) If f is continuous on [a,b] then prove that on [a,b]. (5)

b) Suppose on [a,b], , is continuous on [m,M], and h(x) = (f(x)) on [a,b]. Then prove that on [a,b]
c) State and prove the fundamental theorem of Calculus for a function on [a,b]. (9 + 6)

d) 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
e) If f maps [a,b] into R^k and if for some monotonically increasing function on [a,b] then prove that and (8+7)
II. a) 1) Let be the set of all invertible operators on .Then prove that is open and the mapping A A^-1 is continuous on .

2) Let f be a differentiable function from E into R^m where E is an open set contained in Rⁿ. Then prove that the linear transformation from Rⁿ to R^m is unique. (5)

b) Define Convex set and prove: Suppose that maps a convex set E into ; is differentiable on E and there exists a constant M such that then . Also prove that if f’(x) = 0 for all x in E then f is constant.
c) State and prove the chain rule on the differentiability of a function. (7+8)

d) Suppose that maps a convex set E into . Let is differentiable at x Then prove that the partial derivatives D_jf_i (x) exists and , where {e₁, e₂, …, e_n} and {u₁, u₂, …, u_m} are the standard basis of and respectively. (15)

III.a) 1) Let denote the set of all continuous, complex valued, bounded functions on X. prove that is a complete metric space.

2) If is a sequence of continuous functions on E, and if uniformly on E, then prove that f is continuous on E. Is the converse true? Justify your answers.

b) State and prove the Weierstrass approximation theorem. (15)

c) Let be monotonically increasing on [a,b]. Suppose on [a,b], for n = 1,2,3,… , and suppose uniformly on [a,b], Then prove that on [a,b] and
d) Define equicontinuity of a function and prove: If K is compact, if for n = 1,2,3,… and if {f_n} is pointwise bounded and equicontinuous on K, then

(i) {f_n} is uniformly bounded on K,

(ii) {f_n} contains a uniformly convergent subsequence. (6+9)

a)1) Prove that G = .

2) If then prove that where E is a periodic function with period 2. (5)

b) Define Gamma function and derive a simple approximate expression for when x takes on very large values.
c) Derive the relationship between Beta and Gamma function. (10+5)

d) Explain with usual notations: Fourier series, orthogonal and orthonormal system. And prove the following theorem: Let {f_n } be orthonormal on [a,b]. Let S _n(x) = be the n^th partial sum of the Fourier series of f and suppose that t_n(x) = . Then prove that and equality holds if and only if g_m = c _m, m = 1,2, …,n. (15)
V) a) 1) If f has a derivative of order n at a point x₀, then prove that the Taylor Polynomial is the unique polynomial such that whatever Q may be in P ^{( n )}.

2) Define Chebyshev polynomial and list down its properties. (5)

b) Given n+1 distinct points x ₀,x ₁, …, x _n and n+1 real numbers f (x₀), f (x₁), …, f (x _n) not necessarily distinct, then prove that there exists one and only one polynomial P of degree £ n such that P (x _j) = f (x _j) for each j = 0,1,2,…,n. and the polynomial is given by the formula where .
c) 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 £ 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. (7 + 8)

c) 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

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

(b) 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 2008 Ordinary Differential Equations Question Paper PDF Download

November 15, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 28

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

MT 1806 – ORDINARY DIFFERENTIAL EQUATIONS

Date : 08-11-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

ANSWER ALL QUESTIONS

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

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

independent on I. Hence show that sin x, sin 2x, sin 3x are

linearly independent on [ 0, 2 ].

(ii) Suppose x₁ (t) and x₂ (t) satisfy a x”(t) + b x'(t) + c x(t) = 0,

where a is not zero, Show that A x₁ (t) + B x₂ (t) satisfies the

differential equation. Verify the same in x” + λ² x = 0. (5 Marks)

b) (i) State and prove the Abel’s Formulae. (8 Marks)

(ii) Solve x” – x’ – 2x = 4t² using the method of variation of

parameters. (7 Marks)

(iii) If λ is a root of the quadratic equation a λ² + b λ + c = 0,

prove that e^λt is a solution of a y” + by’ + c y = 0. (15 Marks)

a) (i) Whenever n is a positive or negative integer,

show that .

(ii) Obtain the linearly independent solution of the Legendre’s

differential equation. (5 Marks)

b) (i) For the differential equation

Obtain the indicial equation by the method of Frobenius. (8 Marks)

(ii) Prove that (7 Marks)

(iii) Solve the Bessel’s equation . (15 Marks)

III. a) (i) Express x4 using Legendre’s polynomial.

(ii) Show that F ( 1; p; p; x ) = 1/ (1 – x ) (5 Marks)

b) (i) State and prove Rodriguez’s Formula and find the value of

{8 P₄ (x) + 20 P₂ (x) + 7 P₀ (x)}

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

a) (i) Considering the differential equation of the Sturm-Liouville

problem, prove that all the eigen values are real.

(ii) Considering an Initial Value Problem x’ = 2x, x(0) = 1, t ≥ 0, find x_n(t).

(5 Marks)

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

and only if .

(ii) State and prove Picard’s initial value problem. (15 Marks)

a) (i) Write down Lyapunov’s stability statements.

(ii) Prove that the null solution of x’ = A (t) x is stable if and only if there

exists a positive constant k such that | φ | ≤ k, t ≥ t₀ . (5 marks)

b) (i) State and prove the Fundamental Theorem on the stability of the

equilibrium of a system x’ = f (t, x).

(ii) Discuss the stability of a linear system x’ = A x by

Lyapunov’s Direct Method.

(15 Marks)

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