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  1. M

    Vector Quotient Spaces

    I'm having a bit of trouble seeing Vector Quotient Spaces. Lets say I have a vector space $V$ and I want to quotient out by a linear subspace $N$. Then $V/N$ is the set of all equivalence classes $[N + v]$ where $v \in V$. For example, let me try to take $\mathbb{R}^{2} /$ x-axis. This...
  2. M

    Cal vs Chicago

    Hello, I am currently a high school student and I have been accepted into both UC Berkeley and the University of Chicago. I plan to major in math or physics, but I have no idea which college I want to go to. From what you know of their undergraduate mathematics or physics programs, which...
  3. M

    What the heck?

    This site was linked at the top of the physics forums website. www.relativitychallenge.com I guess everybody has freedom of speech. Do people just get a kick out of proving Einstein wrong?
  4. M

    Fields and new operations

    This problem comes from Halmos's Finite Dimensional Vector Spaces. Given that we can re-define addition or multiplication or both, is the set of all nonnegative integers a field? What about the integers? My thinking is that since the Rational numbers form a field, and they are countable, we...
  5. M

    Goursat and Analysis

    Has anyone read Cours d'analyse mathématique (Course in Mathematical Analysis)? If you have, do you find some the the problems just a might challenging? I've found that this is pretty characteristic of books of that time period. Why have problems in textbooks today gotten so much easier?
  6. M

    Medical Photographic Memory

    What is the Latest age at which one can acquire this ability? Are there people who can learn languages easily at older ages given that they had not been exposed at a younger age?
  7. M

    Probability and the Real Line

    Take the Interval [0,1] over the reals. Randomnly choosing a number, what is the probability that you will get an irrational number? A rational one?
  8. M

    Methods of Teaching Mathematics

    Where is mathematics learning going today? There are many books nowadays, that emphasis conciseness and rigour over all else. The Rudin "series" is a perfect example. There is hardly any motivation, and emphasis is put on rigour, rather than intuition. I am sure that this could be argued...
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    Bourbakist approach to writing textbooks

    After much much reading, I have come to the conclusion that the Bourbakist approach to writing textbooks is only useful for Algebra and Analysis, not for things like Dynamical Systems and Differential Geometry/Topology. Does anybody want to agree or disagree?
  10. M

    A simple integration question.

    Lets say we have two functions f and g that are riemann integrableon the interval [a,b] If (f(x)-q(g(x)))^2 is greater than zero for all real q, is the integral from a to b of (f(x)-q(g(x)))^2 greater than zero? Also, lets say we have a function h(x) that is defined from [0,1] as follows...
  11. M

    Topological Spaces

    Is it true that a perfect generalized ordered space can be embedded in a perfect linearly ordered space? It is true that a perfect generalized ordered space can be embedded as a closed subset in a perfect linearly ordered space.
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    Prove: A mapping f:S->T is bijective if and only if it has an inverse?

    Can a mapping from f:S->T associate an element of s into several elements of T? Also, how do you prove: A mapping f:S->T is bijective if and only if it has an inverse?
  13. M

    Better Analysis Book

    Which is the more thorough more rigorous book? Rudin's Principles of Mathematical Analysis, or Lang's Undergraduate Analysis?
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