# All Questions

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### Is the set of all multilinear forms on a vector space $V$ over a field $F$ itself a vector space? [closed]

If so, what is vector addition on this space? Is it the usual pointwise addition of functions?
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### Diophantine's Problem

Find all integer solutions for $$x^3+1=y^2.$$ Attempt: By guessing, I found five pairs of integer solutions for the equation: $(2, \pm 3)$, $(0, 1)$, $(-1, 0)$ and $(0, -1)$, but really I don't know ...
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### Let $f:\Omega\to\textbf{R}^{m}$ be a function. Then $f$ is measurable if and only if $f^{-1}(B)$ is measurable for every open box $B$.

Let $\Omega$ be a measurable subset of $\textbf{R}^{n}$, and let $f:\Omega\to\textbf{R}^{m}$ be a function. Then $f$ is measurable if and only if $f^{-1}(B)$ is measurable for every open box $B$. My ...
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### Decomposition group does not depend on the prime

Suppose you are working with an abelian Galois group $G=G(L/K)$ of the Galois extension $L/K$. You know the Decomposition group is: $D=D(Q,P) = \{ \sigma \in G : \sigma(Q) = Q \}$ where $Q$ (in $L$...
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### Integral for homework (It's supposed to solved with integration by parts) [closed]

There is this integral in my homework I solved it using some trigonometry and u substitution bu it's supposed to be solved by integration by parts. If anyone could help it would be fantastic. Problem:...
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### The intuition behind using the Euler-Lagrange equation to solve PDEs

I was looking at some videos on using the Euler-Lagrange equation and the Calculus of Variations. I understand that the Euler-Lagrange equation can be used to solve some PDEs analytically, or simplify ...
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### Which is the correct way of showing $D \otimes \bar{D}$ is a representation?

To show that $D \otimes D$ is a representation, I know that I must show that*: $$\tag{1} D \otimes D (U_1 U_2) \phi = D \otimes D (U_1) (D \otimes D(U_2) \phi)$$ I understand that this can be ...
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### What can you say about $T$ if dim$(V) =$ Rank$(T - \lambda I)$?

I stumbled across this condition and I wanted to know what you could say about this: Let $T:V \to V$ be a linear transformation, with $V$ having a finite dimension. What can you say about $T$ if ...
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### Is this proof that relative entropy is never negative correct?

I wish to prove that relative entropy(Kullback-Liebler divergence) is always non-negative. I.e. that $$I^{KL}(F;G)=E_F\left[\log\frac{f(X)}{g(X)}\right]\geq0$$ where F,G are two different probability ...
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### Create a table of values and an associated graph that are increasing, with a point of inflection.

X Y 3 .. 4 9 5 14 6 17 7 19 8 .. The question is how I can find the missing value. As I did the problem, I got 1 and 21 but they are the wrong answers in this case.
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### Estimation of the Kolmogorov test power [Rstudio].

Let $x$ be a 20-element sample from the T-Student distribution with 4 degree of freedom. I want to do a simulation to estimate the power of the Kolmogorov test for $H_0=F\sim N(0,1)$ if $\alpha=0.01$. ...
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### Representations of elements in polynomial rings

I am reading a paper by Banaschewski (Krull implies Zorn) and I am trying to prove everything I find in order to understand it fully. There's a few facts he states before getting into the actual proof,...
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### Finding a fixed polynomial under the multiplicative inversion automorphism

Can anyone find a polynomial $f ∈ ℚ\left(X+\frac{1}{1-X} + \frac{X-1}{X}\right) ⊆ ℚ(X)$ that is fixed under the automorphism $(X ↦ \frac{1}{X})$? $f = X+\frac{1}{X}$ would be nice, but I don't know ...
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### Limit $\lim_{x\to\infty} x(\arctan(a^2x)-\arctan(ax))$

I have a limit $\lim_{x\to\infty} x(\arctan(a^2x)-\arctan(ax))$ and I know the solution $\frac{a-1}{a^2}$, but I dont have any Idea, how to calculate this limit or at least how to start. Any idea?
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### How many functions does it take to solve polynomial equations?

It is well known that polynomial equations of degree $\ge 5$ cannot be solved by just using addition, multiplication, division and radicals (Abel–Ruffini theorem). So what, who cares? Let's just add ...
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### 'Locally' Convex Function

I have a continously differentiable function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ which I am trying to prove is globally convex. Computing the Hessian directly is very difficult as it is a somewhat ...
### Proof Without contradiction $x^2 = 2$ has no rational solutions
I am in the process of learning real analysis and I was wondering if there was a way to prove no $x \in \mathbb{Q}$ satisfies $x^2 = 2$ without a proof by contradiction.