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  1. G

    Product of Two Levi-Civita Symbols in N-dimensions

    When N=2 we have for example \varepsilon_{12}=+1. Now, you didn't define \varepsilon^{12} for us. Also, is there some Einstein summation convention in effect here?
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    Concept Decomposing of Partial Fractions

    You could, instead, do this: \frac{A}{x-1} + \frac{Bx+C}{(x-2)^2}. The point is that the numerator can be anything with degree less than the degree of the denominator.
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    How to measure perimeter without pi?

    the fact that circumference/diameter is the same for all circles is in Euclid, so it predates Archimedes.
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    A weird equation to solve

    Can you do these two Taylor series (at 0): (ln(1-z))^2 and 1/(1-z)^{m+1} ? Then take the product of the two series.
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    Inverse function in one dimension

    If a function is not one-to-one, then it has no inverse. Whether yours is one-to-one probably depends on the coefficients.
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    P-adic analysis question

    Power series is probably not useful. Your integrand |x|_p has only countably many values, and integrals of that kind are best converted to sums.
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    Is this considered as a function?

    Actually, it is an equation, not a function. This equation can be used to define a function by someone who knows what he is doing.
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    P-adic analysis question

    When you say p-adic analysis, p is a prime, so p=0 is not used. |0|_p = 0. Sometimes the usual absolute value |x| is called the \infty-adic absolute value, and \infty is listed among the "primes". The p-adic absolute value is defined for the p-adic numbers, not the real numbers. Except the...
  9. G

    A Function to Integrate

    if I say to simplify, Maple it does
  10. G

    Are the definitions of vector in Linear Algebra and physics compatible?

    The covectors constitute the dual space of the vectors. This tells mathematicians how to treat them when you change variables. Physicists have these weird formulas to use for that... Why? You'll have to ask a physicist.
  11. G

    Diagonalization of complex symmetric matrices

    Woah, U^T in your formula and not U^* ... so in general U^T is not the inverse of U . Why did you choose that?
  12. G

    Number of points on the plane vs. number of points on the line

    Wait: there's a bijection between the integers and the even integers, even though the big one has TWO copies (evens and odds) of the small one. Cardinality does these strange things.
  13. G

    Give an example to show that if not assuming independence of

    What do you get if X_1 = X_2 = ..., which is sort of the EXTREME example of non-independence?
  14. G

    Convolutions of Lebesgue integrable functions

    He writes $\int_R f(x) dx$ to mean the Lebesgue integral.
  15. G

    Pi Paradox

    Nope. You have not proved the length of the circle (your limiting curve) is 4. You have proved, instead, that the length of the limit is not equal to the limit of the lengths. No need to use a circle to prove this: For example, you can use stairsteps converging to a sloping line segment.
  16. G

    How to evaluate this infinite series?

    Why do you think there is a simple way to write this sum? For most sums, there isn't.
  17. G

    Integral of e^{1/x}dx

    If you ask Maple, you get an answer in terms of \mathrm{Ei}_1(x)
  18. G

    Negative times negative

    In order to prove anything, you have to start with something. What are you starting with?
  19. G

    Showing two matrices are not unitarily similar

    The same thing said in another way: The columns of B are orthogonal, unitary transformation preserves orthogonality, but the columns of A are not orthogonal. Therefore A and B are not unitarily equivalent.
  20. G

    Generalization of Hyperoperations / fractional operations

    http://en.wikipedia.org/wiki/Tetration
  21. G

    Three-way majority

    Depends on the number of voters. For example, with 1 voter, the probability is 0. In general, look up "binomial distribution" I guess: you are asking for the probability that some party attains at least half the votes.
  22. G

    Borel Measuble function

    In more detail, for each fixed n the function f_n defined by f_n(x) = \frac{f(x+1/n)-f(x)}{1/n} is continuous, and f'(x) is the pointwise limit.
  23. G

    What is the limit of a^x when a tend to infinity and x tend to 0?

    Calculus textbooks say it this way: " \infty^0 is an indeterminate form ".
  24. G

    Place of Analytic geometry in modern undergraduate curriculum

    Long ago there were two separate courses, "Calculus" and "Analytic Geometry". Then (perhaps around the 1950s ... ???) they were generally combined into a single course "Calculus with Analytic Geometry". Those courses hope to intersperse the topics so that some analytic geometry can be used in...
  25. G

    Right handed system

    in general (not orthogonal, not unit length) you might say that A,B,C in that order is right-handed iff the triple scalar product (A x B) . C is positive.
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    How to relate P(F) with P(F') where F' is F's closure (P(F)\neq 0)

    In general, since F is a subset of its closure, the probability of F is less than or equal to the probability of the closure.
  27. G

    What is this?: SUM(<1M> X^N)

    \sum_{N=1}^M x^N is called a "geometric series" is that what you mean?
  28. G

    Algebraic independence

    Definitions needed. What are x,y,z? What does it mean for an ordered triple and and ordered pair to be algebraically dependent?
  29. G

    ZFC universe vs. ZF universe

    Take your example, a vector space without a basis. We can fix that problem in two ways: Making the universe smaller, throwing out that vector space. Making the universe bigger, adding a basis of the space. Now, for example: is there a basis of R over Q? Unfortunately, we cannot throw out R...
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