# All Questions

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### Real Analysis book with pictures and ideas of proofs

I am taking real analysis course in my graduate class of Maths. My classes will start in 3 months. I have studied real analysis but not very rigorously. Whenever I see theorem I have no idea on how ...
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### Nature of an improper integral

I want to study the convergence of this integral at 0: $$\int_0^{1}\frac{e^{\frac 1 t}}{\sqrt{t(1+t^2)}}\;dt.$$
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### Finding the derivative of an inverse function.

Let $f(x) = (-x^2)/(x^2+1)$. If g(x) is the inverse function of f(x) and f(1)=-1/2, what us g'(-1/2)? Can someone explain how to do the above problem as I am not even sure where to start. Would I ...
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### Can someone help me why this equation equals zero?

I played around with some numbers and stuff and made this weird equation: $$\huge x^{- \frac{n}{n^{-x}}}$$ So the thing is, with every number I tried typing this into a calculator, I got 0. Can ...
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### Finite dimensional algebra

Let $A$ be a finite dimensional algebra. Prove that an element of $A$ is invertible iff it is not a zero divisor. Let $a$ be an invertible element, then there exists an element $b$ such that $ab=1$ ...
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### Multiplying logarithms of different bases [on hold]

How do you multiply the following logs... $$\log_5(n) * \log_2(n)$$
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### Need help solving a system of iterative differential equations

Here I have a system of differential equations: $u_{0}''=-1$ $u_{0}u_{0}''+u_{1}''=-1$ $u_{2}''+u_{1}''u_{0}+u_{0}''u_{1}=-1$ $u_{3}''+u_{2}''u_{0}+u_{1}''u_{1}+u_{2}u_{0}''=-1$ ...
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### Solve $2y^{(5)}-7y^{(4)}+12y'''+8y''=0$ [on hold]

Find the general solution of higher order linear differential equation? Find the general solution of Differential equation using auxiliary equation? $$2y^{(5)}-7y^{(4)}+12y'''+8y''=0$$
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### Holder continuous functions embedded in Sobolev

For simplicity, I will consider $u\in W^{1,p}(\Omega)$, where $1<p<\infty$ and $\Omega\subset\mathbb{R}$ is open and bounded. I am able to show that |u(x)-u(y)| \leq ...
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### Lower bound for the chromatic number of $\mathbb{R}^n$

I'm going through a proof that of the following lower bound for the chromatic number of $\mathbb{R}^n$: $$\chi(\mathbb{R}^n) \geq (1.2 + o(1))^n$$ At some point in the proof we get that ...
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### Closed Form Summation Example

$$\sum_{i=1}^n (ai +b)$$ Let $n \geq 1$ be an integer, and let $a,b > 0$ be positive real numbers. Find a closed form for the following expression. In other words you are to eliminate the ...
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### Finding the frequency of a fan using a slow motion camera

This problem has me a little stumped. I'm not sure if my answer is correct and would just like to check: The camera shoots at 187 frames per second. The fan takes 33 frames to complete one revolution ...
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### A Sequence That has No Upper Bound But Does Not Tend To Infinity

Let $a_n$ be a sequence which has no upper bound. Give an counterexample sequence for the statement $$\lim_{n\to\infty} a_n=\infty$$ Any hint?
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### Spaces where distance between a point and a hyperplane is always reached

In this article I give an example of a Banach space where the distance between a point and a hyperplane is not reached. In the same article, I mention that for finite dimensional vector spaces or ...
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### Partition on a Closed Set A= [2,3]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?
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### Two questions concerning ideal factorization and norm

$\bullet$ In $\mathbb Z[\sqrt{-5}]$ why is $(2)=(2,1+\sqrt{-5})(2,1-\sqrt{-5})$ Actually both ideals on the RHS contain $(2)$, but also their product ? Can we just multiply RHS in the normal sense; ...
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### Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$ Can I multiply and distribute the $\ x^{7/2}\$ and $\ sec^2 \$ together. What is the strategy to solve this problem.
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### Help to prove that a group is cyclic

As part of my study of Abstract Algebra I'm trying to prove that $U_p$ si cyclic for $p$ a prime number. It's a classical result, but I'm trying to prove it following 4 steps stated as problems in my ...
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