# All Questions

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### Complex structure of HyperKaehler manifold

Let $X$ be a hyperKaehler manifold with complex structure $I$ and a metric $g$. It is known that there are two other complex structures $J,K$ on $X$ that generate $S^2$ of possible complex structures ...
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### Prove $\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$

I am trying to prove $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$ This problem is a classic, but I seem to be missing one step or the understanding of ...
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### When is the set statement: (A⊕B) = (A ∪ B) true?

"When is the set statement: (A⊕B) = (A ∪ B) a true statement? Is it true sometimes, never, or always? If it is sometimes, state the cases where it is." How would you go about finding the answer ...
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### I calculated $\lim_{x\to 0} {\sin x\over x}$ as 0.01745 instead of 1

When using a graphing utility to generate a table, I got 0.01745 as the limit rather than 1. Where did I go wrong?
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### Can standard Linear Programming algorithms return all valid solutions without losing their efficiency?

I have a (generalized) Linear Programming problem to solve. I anticipate exactly two equally valid optimizations of my objective function. I would be happy if I could receive both these points; it ...
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### Calculating mean after removing smallest items

I'm having a tough time wrapping my head around this question. Lets say that I am doing an experiment where I roll 10 dice. Each time I roll the dice, I record the average value and repeat the ...
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### Contracting an angle (using straightedge and compass)

In my field theory lecture notes I have it that a regular polygon with $n$ sides is constructable iff $\zeta_{n}=\frac{2\pi}{n}$ is constructable. Shouldn't this be $\frac{\pi}{n}$ instead of ...
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### Express the vector $\vec{v}$ as the sum of a vectors parallel to $\vec{b}$ and a vector component of $\vec{v}$ orthogonal to $\vec{b}$.

$$\vec{v}=2i-4j,\qquad \vec{b}=i+j$$ I have no idea where to start on this.
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### Paracompactness and Filters

If one use ultrafilters to describe a compact space, one gets Tychonoff Theorem as a trivial result. So, im just asking if there is such a "useful" equivalence, but concerning paracompact spaces ( ...
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### Degree of the extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$

Following my previous question the book then asks Use this to determine when the field extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$ is biquadratic (where $a,b\in\mathbb{Q}$) ...
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### what is the fourier series of $x(at)$ when $a = 0, a>0$ or $a<0$? How do I go about solving this?

Given Fourier series co-efficient of $x(t)$ is $X$; how do I go about solving fourier series of $x(at)$ when $a = 0$ ? I can easily get to the step where I insert $x(0)$ into the Fourier coefficient ...
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### Showing that the Poisson process is characterized by five properties

Klenke defines the Poisson process as a family of non-negative integer valued random variables with independent increments, whose increments are distributed according to the Poisson distribution ...
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### “$f$ is differentiable at $x_0$” implies…

Just making sure I understood: $$\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}=f'(x_0)$$ At a first glance I didn't understand why the above is true. It's because (in the case above) we can say that ...
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### Prove that $\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6$

Prove that $$\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6$$ I'm trying induction, this is what I have so far: Basecase $(n=6): 64<720<729$ is true. Inductive Case: Assume ...
### Inequality. $(n+1)\left(a^{n+1}+b^{n+1}\right) \geq (a+b)\left(a^n+a^{n-1}b+\ldots+b^{n}\right).$
Let $a$ and $b$ be positive numbers, and $n \in \mathbb{N}$. Prove that (using Rearrangement Inequality) (n+1)\left(a^{n+1}+b^{n+1}\right) \geq (a+b)\left(a^n+a^{n-1}b+\ldots+b^{n}\right). ...
I have some problems with my homework. We have two integers $a$ and $b$ which satisfy the equation $a^b - b^a = 1008$. Show that $a$ and $b$ have the same remainder when divided by $1008$.