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  1. M

    WHat's wrong in my argument?

    I was asked to prove, every punctured open set in R^2 is path connected. My argument : take points x and y. let z be the point we've taken off from U (open). if x, y,z do not pass through a staright line, we have a segment between a and y. Now if the 3, i.e. x,y,z lie on a straight...
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    Generating Functions!

    We are about to begin this topic soon in class... I'd like to know all about generating functions and their application. I tried reading it up...One thing I'd like to know is, once you have a generating function...then what? You get some information...but what is it? I tried to find no. of...
  3. M

    Assume that f is non-negative on (0,1)

    Hello...I need help and I know this is a very simple problem...I don't know why I'm getting stuck( Maybe because it's past midnight here:frown: ) Assume that f is non-negative on (0,1) and the third derivative of f exists on (0,1).If f(x)=0 for at least 2 values of x in (0,1), show that there...
  4. M

    World Music

    The reason I decided to post my queries out here is simple:I believe that ideas of PF members,by and large,represent the ideas of the world. :smile: I've been learning indian classical music for about 12 years now...have lots of friends who're equally passionate about music.I guess...
  5. M

    Prove that the sequence converges and find its limit

    given a recurrence relation, a_1 =2^(1/2) and a_n = (2 +a_n-1)^1/2 ...prove that the sequence converges and find its limit.. are we supposed to begin by guessing the limit and the bounds ??
  6. M

    Integral problems

    i think i don't have adequate theory to solve these problems.... :frown: 1.given that f(x) =cos(x) sin^k(x) / (1+x). calculate integral of this function wrt x between limits 0 and pi/2 . then find the it's limit as k tends to infinity... 2.let f(x)={e^(-ax)-e^(-bx)}/x, 0< a< b .let I be...
  7. M

    Green's Theorem

    find the double integral of the function e^(x^2) over the region where y/2 <= x<= 1 and 0<=y <=2 USING GREEN's THEOREM. I can't imagine how we'd use green's theorem here...if F=(P,Q) is the function, are we supposed to find P and Q using green's theorem and then parametrize the boundry of...
  8. M

    More group theory!

    seeing lots of group theory here after a really long time... let G be a finite group of order n, where n is not divisible by 3. suppose (ab)^3 = a^3 b^3 ,for a, b in G . prove that G is abelian.
  9. M

    Irreducible elements

    Here’s an interesting question… Let R be a commutative ring and ‘a’ an element in R. If the principal ideal Ra is a maximal ideal of R then show that ‘a’ is an irreducible element. If a is prime, this is pretty obvious…if a is not prime, then we say a= bc for some b,c in R. Now we need to show...
  10. M

    Euclidean domains

    this seems to be a very fundamental problem...but i need help... prove or disprove : let D be a euclidean domain with size function d, then for a,b in D, b != 0, there exist unique q,r in D such that a= qb+r where r=0 or d(r) < d(b). first of all, what is size function? next...do we only...
  11. M

    Idempotent elements

    here's a real tough one ( at least for me) ....show that the ring Z/mnZ where m ,n are relatively prime has an idempotent element other than 0 and 1. i looked at examples and it works.... do we look for solutions of the equation a^2 -a = kmn , for some k in Z( that is, other than 0 and 1)? help!
  12. M

    Finite fields

    let F= { 0=a1, a2,a3,a4.....an} be a finite field. show that (1+a2)(1+a3).........(1+an) = 0. when n is odd, it's simple since 1 belongs to F. then odd number of elements are left( they're distinct from 0 and 1). at least one of them, say x,must have itself as its inverse. x^2 = 1...
  13. M

    Compactness contradiction physics

    Let A be a compact subset of a metric space (X,d). Show that there exist a,b in A such that d(A) = d(a,b) where d(A) denotes the diameter of A. I guess...we're supposed to use the fact that a compactness of A implies that it is closed and bounded or alternately...we could assume that...
  14. M

    Metric spaces

    Here’s a problem I’ve been struggling with, for a while…. If (X,d) is a metric space and f:X-->X is a continuous function, then show that A={ x in X : f(x)=x} is a closed set. One possible way that I can think of is defining a new function g(x) = f(x)-x .Then A={x in X : g(x) =0}. Now {0} is...
  15. M

    What a homeomorphism means

    hi! would like to know what a homeomorphism means ( how do you geometrically visualize it?) AND is the symbol 8 homeomorphic to the symbol X? Why or why not? ( from whatever little i know intuitively about homeomorphisms, i think it is not....)
  16. M

    Find the subgroup!

    let p and q be distinct primes. suppose that H is a proper subset of integers and H is a group under addition that contains exactly 3 elements of the set { p,p+q,pq, p^q , q^p}. Determine which of the foll are the 3 elements in H a. pq, p^q, q^p b. P+q, pq,p^q c. p, p+q, pq d. p...
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    2 functions f,g:X -> X that are discontinuous

    2 functions f,g:X --> X that are discontinuous looking for 2 functions f,g:X --> X that are discontinuous but their composition gof continuous...
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