# All Questions

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### Every euclidean space ($\mathbb {R}^n$) is complete.

To prove this, I would like to use induction. For $n=1$ it is easy to prove that $\mathbb{R}$ is complete. For $n=k$ we assume it is true. For $n=k+1$, we have to show that $\mathbb {R}^{k+1}$ is ...
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### Definite integral involving root of polynomial

While studying models for fluid dynamics I stumbled upon the following integral: $$\int_0^R2r (1-(\frac{r}{R})^2)^\frac{1}{n}\,dr=R^2\frac{n}{n+1}$$ I would like to prove this relation but am having ...
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### Reference Request-Essential Extension

Let $R$ be a commutative ring with unit. Assume $R$ is an essential extension of each of its non-zero ideals. I feel that there should be something in the literature about this, but I could not find ...
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### Explicit heat kernels

For quite general domains, the Dirichlet heat kernel has a representation via the eigenfunctions of the corresponding Dirichlet problem. This form is usually not easy to analyse so I was wondering - ...
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### Solving for $x$ in $\tan(3x) \tan (2x)= 1$

If $$\tan(3x) \tan(2x)= 1$$ Then $x$ is equal to Attempt: I used the '$\tan$' identity but it showed no results. The identity: $$\frac{\tan(2x)+\tan(3x)}{1-\tan(2x)\tan(3x)}$$
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### How to evaluate $\int_0^1\frac{\ln(1-2t+2t^2)}{t}dt$?

The question starts with: $$\int_0^1\frac{-2t^2+t}{-t^2+t}\ln(1-2t+2t^2)dt\text{ = ?}$$ My attempt is as follows: $$\int_0^1\frac{-2t^2+t}{-t^2+t}\ln(1-2t+2t^2)dt$$ ...
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### Can we somehow use the functor $\mathbf{Set}(\mathbb{N},-)$ to define $\mathbb{N}$?

Hom functors can be used define coproducts in terms of products. In particular: $$\mathbf{Set}(A \sqcup B,X) \cong \mathbf{Set}(A,X) \times \mathbf{Set}(B,X)$$ To oversimplify a little: "a function ...
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### Calculate minimal Variance

My task is to calculate the minimal variance. I got a result, but don't know for sure if it's correct. Maybe some of you could help me out here. Let $X$ be some real-valued random variable. We know ...
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### Curve sketching without a computer program

How to sketch the curve x^6 + y^6 = (x^4)*y without using a computer program ? Could someone give me the step by step ?
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### An m-dimensional space with each 'point' in the space having an n-dimensional value

Say I have an $m$-dimensional space (continuous or discrete) such that every point in that space has a value, and that value is an $n$-dimensional vector (continuous or discrete). My question is how ...
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### Convex set in a vector space gives a norm

Given an $\mathbb{R}$ or $\mathbb{C}$ vector space $X$ and a function $p:X\rightarrow[0,\infty)$ with $p(x)=0$ iff $x=0$ and $p(\alpha x)=|\alpha|p(x)$ for all $x,\alpha$, I want to show that $p$ is a ...
L'Hospital's Rule states that $$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$ can be applied when: (1) $f$, $g$ are differentiable; (2) $g'(x) \neq 0$ for $x$ near $a$ (except ...