Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Birkhoff's representation theorem wikipedia , lookup
Linear algebra wikipedia , lookup
Hilbert space wikipedia , lookup
Algebraic K-theory wikipedia , lookup
Vector space wikipedia , lookup
Group action wikipedia , lookup
Bra–ket notation wikipedia , lookup
Fundamental group wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Basis (linear algebra) wikipedia , lookup
DE–7677 11 DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE EXAMINATION, DECEMBER 2009. COMMUTATIVE ALGEBRA (Upto 2006 Batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. 1. (a) (b) Let R be a local ring. Prove that any finitely generated projective Rmodule is free. Prove that the nil radical of R is the intersection of all prime ideals of R. 2. State and prove Chinese Remainder theorem. 3. (a) (b) 4. (a) (b) 5. (a) (b) Prove that for any ideal I of R, the radical is the intersection of all prime ideals containing I. Prove that the ideal I and J are co maximal if and only if their radicals are co maximal. Prove that R is a local ring if and only if it has a unique maximal ideal. Prove that a primary ideal need not be a power of a prime ideal. Prove that if R is a Noetherian ring so is R [x]. Prove that in an Artinian ring every nil radical is nil potent. 6. State and prove first uniqueness’ theorem on primary decomposition. 7. Prove that an Artinian ring is uniquely isomorphic to a finite direct product of Artinian local rings. 8. Prove the following. (a) Every discrete Valuation ring is Noetherian. (b) Every discrete Valuation ring is a local ring. (c) Any non zero ideal is a power of the maximal ideal. ——————— 1 DE-7677 Ws8 DE–7678 12 Ws8 DISTANCE EDUCATION M.Phil. (Maths) DEGREE EXAMINATION, DECEMBER 2009. MEASURE THEORY (Upto 2006 Batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. Each question carries 20 marks. 1. (a) If u and v are real measurable functions on a measurable space x and is a continuous mapping of the plane into a topological space y , prove that the function given by is h( x ) (u( x ), v( x )) h: x y measurable. (b) State and prove Fatou’s lemma. 2. (a) (b) State and theorem. prove Lebesgue’s monotone convergence If f : X [0, ] is measurable and ( E ) fd where E M , prove that is a E measurable and on M gd gfd . X X 3. State and prove Riesz Representation theorem. 4. Let X be a locally compact Hausdorff space in which every openset is . Compact. If is a positive Borel measure on X such that (k) for every compact set k , prove that is regular. 5. (a) (b) Define convex function and prove that any convex function on (a, b) is continuous. State and prove Jensen’s inequality. 6. State and prove the Hahn Decomposition theorem. 2 DE-7677 7. If 1 p , is a – finite positive measure on X and is a bounded linear functional on Lp ( ) then prove that there is a unique g Lq ( ) where q is the Ws8 exponent conjugate to p such that ( f ) fgd . X 8. Let ( X , S , ) and (Y , , ) be a – finite measure spaces and Q S . If ( x ) (Qx ) and ( y) (Q y ) x X and yY , Prove that is S – measurable and for every is – measurable and d d . X Y ———————— DE–7679 13 DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE EXAMINATION, DECEMBER 2009. TOPOLOGICAL VECTOR SPACES (Upto 2006 Batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. (5 20 = 100) 1. (a) Show that if B is a local base for a topological vector space X then every member of B contains the closure of same member of B. (b) Show that every locally compact topological vector space X has finite dimension. 2. (a) Show that if d1 and d2 are invariant metrices on a vector space X which induce the same topology on X then (b) (i) d1 , d2 have the some Cauchy sequence. (ii) d1 is complete if and only if d2 is complete. Show that the following are equivalent on a topological vector space E. 3 DE-7677 (i) E is bounded (ii) If x n is a sequence in E and { n } is a sequence of scalars such that xn 0, n then n x n 0 as n . 3. (a) (b) State and prove Banach-Steinhans theorem. Show that if n is a sequence of continuous linear mapping from on F-space X into a topological vector space Y and if x lim n x exists for every n x X then is continuous. 4. State and prove the open mapping theorem. 5. (a) (b) State and prove Hahn-Banach separation theorem. Show that if f is a continuous linear functional on a subspace M of a locally convex space X then there exists X * such that f on M. 6. State and prove that Banach – Alaoglu theorem. 7. (a) Suppose X is a topological vector space with X * separate points show that if K is a compact convex set in X then K is the closed convex hull of the set of its extreme points. (b) Suppose H is the convex hull of a compact set K in a topological space X. Show that if X is a Frechet space then H is compact. 8. (a) (i) Suppose X, Y are Banach spaces. Show that T BX ,Y T * B Y * , X * that satisfies Tx , y * x ,T * y * for all x X , y* Y * . (ii) (b) T* T . Suppose X, Y are Banach spaces and T B X , Y . Show that T is compact if and only if T * is compact. 4 DE-7677 Ws8 ——————— DE-7680 14 A Ws8 DISTANCE EDUCATION M.Phil. (Mathematics) DEGREE EXAMINATION, DECEMBER 2009. FUNDAMENTALS OF DOMINATIONS IN GRAPHS (Upto 2006 batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. All questions carry equal marks. 1. (a) Prove that if G is a graph without isolated vertices and if S is a minimal dominating set of G then V-S is a dominating set S. (b) Show that if a graph has no isolated vertices then G n . 2 2. State and prove a necessary and sufficient condition for a tree to be a wounded spider. 3. Prove that if G 2 then m 1 n G n G 2 . 2 4. If S V is a maximal irredundant set in a graph G , prove that it is a minimal external redundant set. 5. If T is a tree of order n 2 , prove that R T 6. Prove that if has G Dir G G OIR G IR G . no 3n 1 and F T 8 n 2 4 . 2 isolated vertices then 7. Prove that a vertex V is in V if and only if P V , S V for some set S containing . 8. Prove that a graph G UEA iff V is empty. ——————— DE–7681 14 B 5 DE-7677 DISTANCE EDUCATION M.Phil. (Maths) DEGREE EXAMINATION, DECEMBER 2009. DATA STRUCTURE AND ALGORITHMS (Upto 2006 Batch) Time : Three hours Maximum : 100 marks Answer any FIVE questions. All questions carry equal marks. 1. (a) (b) Discuss : (i) Dynamic object (ii) Default constructor. 2. (a) (b) Write a program to check if two trees are isomorphic or not. Write efficient functions that take only a pointer to the root of a binary tree, T, and compute? The number of nodes in T. 7. (a) (b) How to construct an expansion tree? Write a program to construct a tree and to compute the depth and height of a tree. 6. (a) (b) What is stack? Explain the array implementation of stack. Write a program to evaluate a postfix expression. 5. (a) (b) Discuss the merits and demerits of representing a data sequentially and non-sequentially. Write a program to add two polynomial have M and N terms, represented by a linked list, and calculate the time complexity of program? 4. (a) (b) How to estimate and reduce the running time of a program? What is list? Explain the cursor implementation in linked list. 3. (a) (b) Compare friend function with inline function. What is Graph? Write a program to find the shortest weighted path from S to every other vertex in G, Where G is a weighted Graph G = (V,E) given as input to a program. Define Graph, Write a program to traverse the graph. 8. Write and analyze prim's and kruskal algorithms. ———————— 6 DE-7677 Ws8