1. S

    Partition proof

    Hello All, I am stuck getting out of the blocks for this proof. Let [a,b] be a subset of the real numbers, and let [a,b] be a non-degenerate, closed, bounded interval. Let epsilon > 0. Prove that there is a partition R of [a,b] such that ||R||<epsilon. I have n>(b-a)/Epsilon. My professor...
  2. J

    Riemann Sum

    Find the Riemann sum Rp for the given function f on the indicated partition P by choosing on each subinterval of P (a) the left-hand endpoint (b) the right-hand endpoint, and (c) the midpoint. f(x) = 2x + 3; P = {1,3,4,5}, n = 3
  3. C

    Properties of Relations on X (Equivalence relation)

    Hello, first post here and I have a few questions. First question: Suppose X' is a partition of X. Define the relation: R = {(x, y) ∈ X × X : ∃Y ∈ X' , x ∈ Y ∩ y ∈ Y } Show that R is reflexive, symmetric and transitive. Second question: If f : X → Y and g : Y → X are functions, then g ◦ f...
  4. J

    smooth functions and partition unity

    Take U \subset \mathbb{R}^{n} where U is bounded. Assume we have an open covering U \subset \cup_{i=0}^{N}V_{i} where V_{i} \subset U for each i. For each V_{i} we have the mappings v_{i} \in C^{\infty}(\bar{V_{i}}). If we let \{ \zeta_{i} \}_{i}^{N} be a smooth partition of unity subordinate...
  5. K

    partition law

    Two basketball players throw a ball into the basket till the first hit. The probability for the first player tol hit the target is 5%, but for the second - 15%. What is the partition law for the number of attempts both player made. thanks for help.
  6. D

    Cosets form a partition of X

    Hello guys, I hope I'm posting in the right place. I'm having problems solve the following problem: Let Y be a subspace of a vector space X. Show that the distinct cosets x + Y (x in X) form a partition of X. I don't quite understand how these cosets work so I couldn't think of any way to...
  7. W

    Partition of a number with no 1s

    Hi. I have problems solving this problem: How many partitions of a number n are there, if none of the components of the partition equals 1? What I've got so far is that for n=1 we have 0, n=2 and n=3 we have 1 n=4 and n=5 we have 2 n=6 and n=7 we have 3 but then for n=8 we get 7 and for...
  8. D

    Partition of a set

    The question is to determine whether \mathcal{A} is a partition of the set A. A = \mathbb{R} and \mathcal{A} = \{S_y : y \epsilon \mathbb{R}\}, where S_y = \{x \epsilon \mathbb{R} : |x| = |y|\}. I understand what it means to be a partition, but I'm not sure what \mathcal{A}, or particularly...
  9. M

    The injectivity of a function defined on a set of partitions

    Hello. My name is Michael, I'm a freshman CS student and I'm stuck with this problem. I could really use your help: Let T be a non empty set, and let A and B two sets belonging to P(T), the set of partitions of T ( whatever x belongs to P(T), the complement of x = T - x ) Now, let f be a...
  10. S

    Combinatoric problem partition a set please help

    Let S be a set of 2n elements and let pn be the number of partitions of S into n parts, with two elements in each part. Then p1=1 and p2=3. Explain why pn = (2n-1)pn-1 for n>=2. Thanks!
  11. J

    Proving partition indentities

    How would i show the following: The number of partitions of the integer n into 3 parts is equal to the number of partitions of 2n into 3 parts of size less than n. I've tried drawing the Ferrers diagram, but I can't seem to extract any useful information. I am pretty sure the generating...
  12. A

    another special case of combinatorics (method of fictious partition)

    numbe of ways in which n identical things may be distributed among p persons if each may receive none,one or more things is n+p-1Cn` actually how does it happen?just the expression is given and nothing about the deduction. i am not able to use anything unless i know how to get please...
  13. M

    Partition Theory

    In how many ways can a positive integer n be broken into a sum of k positive integers ? ( While representing the number as a sum the order in which the addends are arranged for each of the ways is not taken into consideration) I know that we denote the number of ways to do this as p(n,k) and...
  14. A

    Partition under certain conditions

    Prove that if R is an equivalence relation on A \times A, then the set \{g(x): x \in A\} is a partition of A. My partner and I honestly have no idea where to begin with this.(Headbang)
  15. J

    Can a partition be a permutation OR combination?

    I'm so confused--I know what each of these is, but when using a partition (for groups of non-distinct objects), can that be applied to a permutation OR combination, or JUST combination?
  16. H

    homeomorphism from a partition of R^2 to (S^1) x (S^1)

    (Recall that S_1:=\{x\in\mathbb{R}^2:|x|=1\} is the unit circle in \mathbb{R}^2.) This is my first time dealing with quotient topologies and unit spheres and such, so I'm a little unsure. I was thinking of defining \sim as the relation induced by the partition P\times P of \mathbb{R}^2, where...
  17. wonderboy1953

    Partition relationship to fractals news

    From "New Math Theories Reveal the Nature of Numbers ScienceDaily (Jan. 20, 2011) — For centuries, some of the greatest names in math have tried to make sense of partition numbers, the basis for adding and counting...
  18. S

    Special Partition of unity

    Hello, I try to construct an example of a partition of unity on a manifold M, s.t. at least one function \psi_j \neq 0 and supp \psi_j is not compact. First I try to construct two functions, s.t. their sum is equal 1. But i couldn't find a example, s.t. the support is not compact. We know it...
  19. R

    For b>0, deduce that f(x)=x^3 is integrable on [0,b]

    This question is a peculiar one, to say the least. For b>0, let P be the partition of [0,b] into n equal subintervals. Calculate both S_P and s_P for f(x)=x^3 and prove that J(=glb(S_P))\leq 1/4b^4(1+1/n)^2 and that I(=lub(s_P))\geq 1/4b^4(1-1/n)^2 for any positive n. Deduce that f(x) is...
  20. R

    Peculiar means of proving a theorem on integrability

    This theorem has probably been seen a dozen times here, and I could find other formats of proving this theorem online, but the issue I'm having right now is a context/format issue. The question, listed below, is very specific on how we're supposed to answer it, and I'm not entirely sure how I'm...