All Questions

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For $n\geq 2$, any continuous map $f:\mathbb{C}P^n\rightarrow S^2$ induces the zero map on $H_2(*)$

I am working through an old qualifying exam from another university. My course did not cover as much material as what is on this test (e.g. we did not cover cohomology). So I am just working through ...
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Sketch $f(x) =\frac{ 2x}{x^2-5x+4}$

I have already found the domain, intercepts and critical number along with max and min points. Now when finding the intervals of concavity I understand you must take the second derivative but I can't ...
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For $\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$, show there is a choice of signs such that it is irrational [on hold]

Considering $$\pm\sqrt 1\pm\sqrt 2 \pm\sqrt 3 \pm\cdots\pm\sqrt {2009}$$ where you can replace each $\pm$ with $+$ or $-$. Prove that there is at least one choice of signs such that the number is ...
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Material to learn some basic combinatorics?

I realize that I'm pretty weak when It comes to basic combinatorics, even with simple things like n choose k I don't feel confident. Furthermore, I've viewed some combinatorics books and the reasoning ...
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Find the exact critical numbers for $f(x) = 3x - \sqrt{x}$

I found the derivative of the function which I believe is $3-\frac{1}{2\sqrt{x}}$ but I am not sure how to find the $x$ value for the critical number.
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Compute Fourier coefficients of spline fit to data

Suppose you have data $$\{(x_i, 1)\}_{1\dots N}, \quad x_1=0<x_i<x_{i+1}<x_N=2\pi \ \forall i \in\{2,\dots N-1\}$$ In other words, we have a sequence of $y=1$ values at distinct random ...
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Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
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Given matrix $A$ such that $\forall x : |Ax| > |x|$, and eigenvalue $\lambda$ of $A$. Show $|\lambda |\geq 1$.

Say matrix $A$ has the property that for any non-zero vector $x$, left-multiplication of $x$ by $A$ increases the magnitude. That is, $\forall x$ $$|x| > 0 \implies |Ax| > |x|$$ Is it true ...
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In which topologies do open sets maintain open under countable or arbitrary intersection?

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology. I am curious if there is well known classes of ...
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Proof related to Harmonic Progression

The question is as follows: Let $m_1<m_2<m_3<\cdots<m_k$ be postive integers such that $\frac{1}{m_1}$, $\frac{1}{m_2}$, $\frac{1}{m_3}$, $\cdots$, $\frac{1}{m_k}$ are in arithmetic ...