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ws 2 DE–931 11 DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE EXAMINATION, DECEMBER 2010. COMMUTATIVE ALGEBRA (Upto 2006 Batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. (5 20 = 100) 1. (a) f g M M 0 is an exact sequence of R-modules If 0 M which splits, prove that there exists an R-homomorphism s : M M such that sf I M . (b) If 0 N P M 0 and 0 N P M 0 are two exact sequence of R-modules with P , P projective, prove that P N P N . (c) Prove that P P is projective P is projective for each . (6 + 6 + 8 = 20) 2. (a) If M n j 1 (b) m N M i and N j M prove that M N i N j . i, j j 1 f g N N 0 is an exact sequence of R-modules, If 0 N prove that for any R-module M, the sequence f g M N M N M N 0 is exact, where f and g are defined by f x y x f y and g x z x g z . (c) Prove that M is faithfully flat if and only if M is flat and for each maximal ideal m of R, mM M . (6 + 6 + 8 = 20) 3. (a) (b) Prove that an element a J R if and only if 1 ab is a unit, for all b R . State and prove Chinese remainder theorem. 1 DE–9020 ws 2 (c) If R is local ring, prove that any finitely generated projective R-module is free. (6 + 8 + 6 = 20) 4. (a) (b) (c) 5. (b) 6. If 0 M M M 0 is an exact sequence of R-modules prove that M is Noetherian if and only both M and M are Neotherian. If M is a Noetherian module prove that any irreducible submodule N of M is primary. If R is an Artinian ring prove that the nil radical N(R) is nilpotent. (8 + 6 + 6 = 20) Let R S be domains and S integral over R. Prove that R is a field if and only if S is a field. Let R be an integrally closed domain with quotient field K and S a normal extension of R with Galois group G = G(L/K). Prove that (i) G is the group of R-automorphisms of S (a) (ii) Two prime ideals P and Q of S lie over the same prime ideals of R if and only if there exists some G with P Q . (10 + 10 = 20) (a) If k is a field and R a finitely generated k-algebra prove that there exist y1 , y2 ,... yr R algebraically independent over k such that R is integral over ky1 , y2 ...., yr . (b) If R is a finitely generated k-algebra, k field and I an ideal of R, prove the intersection of all maximal ideals of R containing I. T is (c) If R is a Noetherian integrally closed domain with quotient field K and L is a finite separable extension of K, prove that the integral closure S of R in L Noetherian. (6 + 6 + 8 = 20) 7. (a) (b) If R is a subring a field K prove that the integral closure R of R in K is the intersection of all valuation rings V of K containing R. (c) Prove that the ring of integers in an algebraic number field is a Dedekind domain. (7 + 7 + 6 = 20) 8. (a) Let M be a fractionary ideal of R. Prove that the following are equivalent : (i) M is invertiable. Prove that the ideals of a valuation ring are totally ordered by inclusion and prove that if the ideals of a domain V with quotient field K are totally ordered by inclusion then V is a valuation ring of K. 2 DE–9020 ws 2 (ii) M is finitely generated and M p is invertible in R p for every prime ideal P of R. (iii) M is finitely generated and M m is invertible in Rm for every maximal ideal m of R. (b) If R is a domain, prove that R is a Dedekind domain if and only if every nonzero fractionary ideal of R is invertible. (10 + 10 = 20) ––––––––––––––– DE–932 12 DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE EXAMINATION, DECEMBER 2010. MEASURE THEORY (Upto 2006 Batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. Each question carries 20 marks. 1. (a) (b) State and prove Lebesque’s Monotone convergence theorem. Show that if F is any collection of subsets of X there exists a smallest – algebra m in X such that F ℳ*. 2. State and prove Lebesque’s dominated convergence theorem. 3. Let X be a locally compact Hausdorff space in which every open set is – compact. Let the any positive Borel measure on X such that (k ) for every compact set k. Then show that is regular. 4. (a) Define conjugate exponents. (b) Let p and q be conjugate exponents 1 p . Let X be a measure space with measure . Let f and g be measurable functions on X with range in [0, ] . Then show that 1 x 1 p q fgd fpd g q d and x x 3 DE–9020 ws 2 1/ p p ( f g ) d x 5. (a) 1/ p f x d x 1/ p g p d x Suppose and are measures on a – algebra ℳ is positive, and is complex. Then show that the following two conditions are equivalent. (i) (ii) To every 0 corresponds a 0 , such that | ( E ) | for all E ℳ with (E ) . (b) If is a positive -finite measure on a algebra ℳ in a set X , then show that there is a function W L ( ) such that 0 W ( x ) 1 for every xX. 6. Suppose 1 p , is a finite positive measure on X 1 and is a bounded linear functional on L p ( ) . Then show that there is a unique g Lq ( ) ( f ) where fgd( f L is q p the exponent conjugate to p such that ( )) . x 7. Let ( X , P , ) and (Y , P , ) be a -finite measure spaces. Suppose Q P J . If ( x ) (Qx ) , (Y ) (Q y ) for every x X and y Y then show that Q is P measurable, is J measurable and Qd d . x y 8. Suppose f L (R ) , g L (R ) . Then show that f ( x y)g( y) dy for almost all x . For these x , define h( x ) f ( x y)g( y)dy . Then show that h L (R ) and h 1 f 1 g 1 where f 1 f ( x ) dx . ––––––––––––––– DE–933 13 4 DE–9020 ws 2 DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE EXAMINATION, DECEMBER 2010. TOPOLOGICAL VECTOR SPACES (Upto 2006 Batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. (5 × 20 = 100) 1. (a) If Y is a subspace of a topological space X and Y is locally compact in the topology inherited from X, prove that Y is a closed subspace of X. 2. (b) Suppose Y is a subspace of a topological vector space X and Y is an Fspace in the topology inherited from X, prove that Y is a closed subspace of X. (a) Suppose X and Y are topological vector spaces and : X Y is linear. Consider the following statements : (i) is continuous. (ii) is bounded. (iii) If X n 0 then { X n : n = 1,2,3,...} is bounded. (iv) If X n 0 then X n 0 . Prove that (i) (ii) (iii) and if X is metrizable prove that (iii) (iv) (i). 3. (b) Prove that a topological vector space X is normable if and only if its origin has a convex bounded neighborhood. (a) If X and Y are topological vector spaces, K is a compact convex set in X, is a collection of continuous linear mappings of X into Y, and the orbits x x : are bounded subsets of Y, for every x K prove that there is a bounded set B Y such that K B for every . 4. (b) State and prove the closed graph theorem. (a) If X and Y are topological vector spaces and n is a sequence of continuous linear mappings of X into Y, prove the following : 5 DE–9020 ws 2 (i) If C is the set of all x X for which n x is a Cauchy sequence in Y and if C is of the second category in X then C = X. (ii) If L is the set of all x X at which x lim n x exists, if L is of n the second category in X and if Y is an F-space then L = X and : X Y is continuous. (b) If B : X Y Z is bilinear and separately continuous, X is an F-space and Y and Z are topological vector spaces, prove that Bx n , yn Bx 0 , y0 in Z when ever x n x 0 in X and yn y0 in Y. If Y is metrizable prove that B is continuous. 5. (a) Suppose (i) M is a subspace of a real vector space X. (ii) p:X R satisfies px y px p y and ptx tpx if x X , y X ,t 0 . (iii) f M R is linear and f x px on M. Prove that there exists a linear : X R such that x f x for x M and p x x px for x X . (b) If f is a continuous linear functional on a subspace M of a locally convex space X, prove that there exists X * such that f on M. 6. (a) Prove that in a locally convex space X, every weakly bounded set is originally bounded and vice versa. (b) Let H be the convex hull of a compact set K in a topological vector space X. Prove that (i) If X is a Frechet space, then H is compact and (ii) If X R n then H is compact. 7. (a) Suppose X and Y are normed space. Associate to each BX ,Y the number sup x : x X , x 1. Prove that if Y is a Banach space, so is B (X, Y). (b) If U and v are the open unit balls in the Banach spaces X and Y T BX ,Y and respectively and c 0 , prove that if the closure of T(U) contains cV then T(U) cV and if c y * T * y * for every y* Y * then T U cV . 6 DE–9020 ws 2 8. (a) Prove that on every compact group G there exists a unique regular Boral probability measure m which is left-invariant and right-invariant and it satisfies the relation f x dmx f x dmx for f C G . 1 G (b) G Suppose X is a Banach space, A and B are closed subspace of X and X = A + B. Prove that there exists a constant such that every x X has a representation x a b , where a A , b B and a b x . ——————— DE–934 14 (a) DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE EXAMINATION, DECEMBER 2010. FUNDAMENTALS OF DOMINATIONS IN GRAPH (Upto 2006 Batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. All questions carry equal marks. (5 20 = 100) 1. (a) Show that S V is a minimal dominating set if and only if every vertex in S has atleast one private neighbour. 2. (b) Show that for every connected graph G, rad (G) diam (G) 2 rad (G). (a) Show that if a graph G has no isolated vertices then (G ) (b) 3. n . 2 n Show that if G is a connected graph and (G ) then there is at most 2 one end vertex adjacent to each v V except for possible one vertex which may be adjacent to exactly two end vertices when n is odd. (a) Show that if a connected graph G is clear-force and net-free then n (G ) 4 (G ) 6 . 3 (b) Show that for any tree T, (T ) n (T ) iff T is a wounded spider. 7 DE–9020 ws 2 4. 5. 6. 7. 8. (a) Show that if a graph G n 1 ln ( (G ) 1)). (G ) (G ) 1 has no isolated vertices (b) Show that for any graph G and hereditary property ( P ) ( P ) n . (a) Show that for any graph G, (G ) (G ) . (b) Show that each knockout-with-replacement sequence on a graph of order n size m terminates in atmost m steps. Show that efficient dominating is NP complete (a) for bipartite graphs (b) for chordal graphs (a) Show that if G is an r-regular graph then F1 (G ) W1 (G ) W (G ) n . (b) State and prove automorphism class theorem. State and prove necessary and sufficient conditions for a graph G to be a member of UVR. ———————— DE–935 14 (B) DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE EXAMINATION, DECEMBER 2010. DATA STRUCTURES AND ALGORITHMS (Upto 2006 Batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. All questions carry equal marks. 1. (a) 2. then Explain OOP Paradigm. (15) (b) What are the benefits and applications of OOP? (a) With an example C++ program explain operator overloading. (10) 8 (5) DE–9020 ws 2 3. 4. 5. 6. 7. 8. (b) Explain different types of inheritance and use of access specifiers in inheritance. (10) (a) What is meant by complexity of an algorithm? Explain anyone notation. (10) (b) Write (a) How arrays are represented in memory? Explain address mechanism. (10) (b) Explain the concept of linked list. (10) (a) Explain array implementation of stack. (10) (b) Explain linked list representation of queue. (10) (a) Explain the binary tree representation of two-way linked list. (10) (b) What is circular list? Explain its structure. (10) (a) Explain Prim's algorithm. (10) (b) Explain and the explain way the of algorithm finding for (10) minimum (10) Write short notes on the following (a) Binary tree search. (6) (b) Depth first search. (8) (c) Merging. (6) ———————— 9 DE–9020 linear spanning search. tree.