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  1. H

    Transitive subgroup of the symmetric group

    Hi, I need help in proving the following statement: An abelian,transitive subgroup of the symmetric group Sn is cyclic,generated by an n-cycle. Thank's in advance
  2. H

    Quotient space of the unit sphere

    prove that the quotient space obtained by identifying the points on the southern hemisphere, is homeomorphic to the whole sphere.I am trying to define a homeomorphism between the quotient space and the sphere,and i need help doing it. Thank's in advance.
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    The box topology on R^ω

    Hi, What are the convergent sequences in the box topology on R^ω?Are they the eventually constant only? Thank's in advance
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    Determinant of a unit columns matrix

    If all columns of a matrix are unit vectors, the determinant of the matrix is less or equal 1 I am trying to prove this assertion,which i guess to be true. can anybody help me? Thank's in advance
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    Matrix logarithm

    Homework Statement Hi, how can i show that the matrix logarithm log(I+A) is continuously differentiable on the set of matrices having operator norm less than 1. Homework Equations http://planetmath.org/matrixlogarithm The Attempt at a Solution i tried to compute the...
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    Hausdorff dimension of the cantor set

    Hi, Using the definition of Hausdorff measure: http://en.wikipedia.org/wiki/Hausdorff_measure I am looking for a simple proof that Hd(C) is greater than 0, where C is the Cantor set and d=log(2)/log(3) Thank's in advance
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    Fractal (Hausdorff) dimension

    Hi, I am trying to understand why do the two versions of Hausdorff (fractal) dimension are actually the same.I refer to the definition by coverings and the definition by ratio of two logarythms. http://en.wikipedia.org/wiki/Hausdorff_measure...
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    Fractal (Hausdorff) dimension

    Hi, Can someone give me a link to a clear and relatively basic and short matirial introducing the notion of fractal dimension (Hausdorff dimension)? Thank's in advance.
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    Imbedding of the ratiomals

    Hi' can the rational numbers be imbedded in a countable complete metric space X? If D is the set of isolated points of X,then D is dense in X\D is countable complete metric space so it is homeomorphic to Q.Where am i wrong?
  10. H

    Product space

    Is there atopological space X such that XxX (the product space) is homeomorphic to the unit circle in the plane
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    Locally path connected

    Hi, I am trying to prove that any compact metric space that is also locally connected,must be locally path connected. can someone help? thank's in advance.
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    Complete countable metric space

    Homework Statement It is clear that a countable complete metric space must have an isolated point,moreover,the set of isolated points is dense.what example is there of a countable complete metric space with points that are not isolated? Homework Equations The Attempt at a Solution
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    The quotient topology

    Homework Statement X is a compact metric space, X/≈ is the quotient space,where the equivalence classes are the connected components of X.Prove that X/ ≈ is metrizable and zero dimensional. Homework Equations Y is zero dimensional if it has a basis consisting of clopen (closed and open at...
  14. H

    Argument principle for a rectangle

    Homework Statement . I want to prove that there is one solution for e^z-z in every shifted copy of the fundamental strip by applying the argument principle to the boundary of a rectangle −M≤Rez≤M , 2kπi≤Imz≤2(k+1)πi for large M and integer k . I need help in using the...
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    Number of zero's of holomorphic function

    Homework Statement Let f(z) be holomorphic in the open unit disc taking real values only on the real axis.show that f has at most one zero in the open disc. Homework Equations Rouche's theorem,the argument principle. The Attempt at a Solution obviously,f does not have non-real zero's .I...
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    The Cartesian product theorem for dimension 0

    The cartesian product ∏X = Xi of a countable family {Xi} of regular spaces is zero-dimensional i f and only i f all spaces Xi , are zero-dimensional. I wonder if the countability assumption is just to ensure the regularity of the product space ,or it is crucial for the clopen basis. Thank's
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    Non homeomorphic spaces

    Why are the irrationals R-Q and the product space (R-Q)XQ not homeomorphic? The first space i Baire space.may be the second space is not?
  18. H

    Locally uniformly convergence

    Homework Statement Let f(z) be holomorphic in the unit disc B(0,1),such thaf f(0)=0.Prove that the series Ʃf(z^n) is locally uniformly convergence in B(0,1). Homework Equations locally uniformly convergence:if it is uniformly convergence in a neibourhood of each point of B(0,1). The Attempt...
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    Quotient topology

    Hi, I am trying to prove the following proposition: Let F be a closed subset of the Euclidean space Rn.Then the quotient space Rn/F is first countable if and only if the boundary of F is bounded in Rn. Any ideas?
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    Topological property of the Cantor set

    Let X be a metric separable metric and zero dimensional space.Then X is homeomorphic to a subset of Cantor set. How can it be proved? Thank's a lot, Hedi
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    The sequence space lp

    Homework Statement how to prove that the sequences space lp is subspace of lq for p smaller than q? Homework Equations The Attempt at a Solution I try to imply holder inequality but meanwile unsuccesfully
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    Measurable function

    Let f be a measurable nonegative function on a positive measure space,such that for every positive t, μ{x:f(x)≥t}≤M/(t^5) M is constant.prove that f is in the space L3(dμ)
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    The FKG inequality

    Hi, can someone give me a refference for the FKG inequality? that is, let f,g be two nondecreasing and nonegative measurable functions on a probability space.Then ∫fgdμ≥(∫fdμ)(∫gdμ)
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    Lower bound

    Homework Statement Let p(x,y) be a positive polynomial of degree n ,p(x,y)=0 only at the origin.Is it possible that the quotient p(x,y)/[absolute value(x)+absval(y)]^n will have a positive lower bound in the punctured rectangle [-1,1]x[-1,1]-{(0,0)}? Homework Equations The...
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    Singularities of two variables rational functions

    Homework Statement Let p(x,y) be a positive polynomial of degree n ,p(x,y)=0 only at the origin.Is it possible that the quotient p(x,y)/[absolute value(x)+absval(y)]^n will have a positive lower bound in the punctured rectangle [-1,1]x[-1,1]-{(0,0)}? Homework Equations The Attempt at a...
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    Two variables polynomial

    Hi Let p(x,y)≥0 be a polynomial of degree n such that p(x,y)=0 only for x=y=0.Does there exist a positive constant C such that the inequality p(x,y)≥C (IxI+IyI)^n (strong inequality!) holds for all -1≤x,y≤1? The simbol I I stands for absolute value.
  27. H

    Buffon's cross

    Homework Statement The problem is Q2 in the attached Homework Equations The Attempt at a Solution I am trying to determine the rigion of parameters for each number of crosses,where x and y are the distances of the center of cross from the closest line respectively,and θ is the acute...
  28. H

    Prove monotonicity

    Homework Statement prove that the sequence (1+1/n)^(n+1) is monotonic decreasing Homework Equations The Attempt at a Solution I tried to use the binom expantion and the identity( Ck-1,n)+(Ck,n)=(Ck,n+1)
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    Lebesgue integral

    Homework Statement use the Lebesgue monotone convergence theorem to prove the following equality (attached) Homework Equations Lebesgue monotone convergence theorem The Attempt at a Solution i tried to identify a suitable monotonic sequence,from power series.
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    Base 3 decimal expansion

    Homework Statement How to compute the base 3 decimal expansion of 1/4? Homework Equations The Attempt at a Solution I tried sums of geometric sequences,but i need a clue for the computation.
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