# All Questions

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### For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$?

Inspired by this question I ask this. For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$? The original question concerned $a=e$, the usual ...
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### compactness and subspaces

Let $T$ be a topological space. Let $K \subset Y \subset T$. Let $Y$ have the subspace topology. Let $K$ be compact in $Y$. Is $K$ compact in $T$ ? Well, for one hand, i think that if we think about ...
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### Derive the solution to the Lagrangian $\mathcal L= y(x)\sqrt{1+y'(x)^2}$

I am supposed to derive the solution to the Lagrangian $$\mathcal L= y(x)\sqrt{1+y'(x)^2}$$ Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be ...
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### Is the Fractal Dimension of a Space-Filling Curve in a Plane Always 2?

I have been playing around with space-filling curves that completely fill the unit square. All of them that I have seen have a fractal dimensional of 2. Makes sense that it would be 2, but a Google ...
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### $15a ≡ ca \pmod{25}$, then $15 ≡ c \pmod{25}$

For which numbers $a$ is it true that if $15a ≡ ca \pmod{25}$, then $15 ≡ c \pmod{25}$? I know that this means that $a\frac{15-c}{25}=k_1\in \mathbb{Z}$ and $\frac{15-c}{25}=k_2\in \mathbb{Z}$, but ...
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### Chi square independence test

How to work out chi square independence in the following table? Below is the observed and expected data concerning 7 themes displayed in a newspaper over a period of 3 months. I understand how to ...
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### Which line is better fit with least squares matrices

Given two best fit curves determined by the least squares method, how can I determine which line is a better fit. Specifics $X\boldsymbol\beta=\bf{y}$ is the representation of the "perfect fit" ...
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### Compute the splitting field

I have to compute the splitting field of $x^6-1 \in Q[x]$. I know that $x^6-1=(x+1)(x-1)(x^2-x+1)(x^2+x+1)$ but I don't know what to do after that. Please help me.
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### A question on torsion sheaves

Im not sure if Ive got this right: Let X be an integral scheme and $\mathcal{F}$ a coherent sheaf. Then $\mathcal{F}$ is torsion if and only if it is not supported at the generic point. It is is easy ...
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### How to find the order of these groups?

I don't know why but I just cannot see how to find the orders of these groups: $YXY^{-1}=X^2$ $YXY^{-1}=X^4$ $YXY^{-1}=X^3$ With the property that $X^5 = 1$ and $Y^4 =1$ How would I go about ...
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### Vague definitions of ramified, split and inert in a quadratic field

Our lecturer defined the following: Let $K=\mathbb Q(\sqrt d)$ be a quadratic field and $p$ a prime number, then (1) $\ p$ is ramified in $K$ if $\mathcal O_K⁄(p)\cong \mathbb F_p [x]⁄(x^2)$ ...
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### Extreme Value Theorem proof help

Extreme Value Theorem: If $f$ is a continuous function on an interval [a,b], then $f$ attains its maximum and minimum values on [a,b]. Proof from my book: Since $f$ is continuous, then $f$ has the ...