# All Questions

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### About joint probability divided by the product of the probabilities

Let $X$ and $Y$ be two events. So $P(X)$ is the probability of $X$ happens, and $P(Y)$ is the probability of $Y$ happens. So $P(X,Y)$ is probability of both $X$ and $Y$ happen. So what is the ...
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### Automorphisms of a certain variety

Which is the group of automorphisms of the variety $(\mathbb{A}^1\setminus\{0,1\})^{\times 3}$ ? ($0,1$ are two points of $\mathbb{A}^1$, say the points corresponding to $0,1$ of the base field ...
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### Different standards for writing down expressions in a formal way

What are standard ways to write mathematical expressions in a (semi)formal way ? In different posts of mine concerning similar question I have encountered for a generic expression of the type "for all ...
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### The field of fractions of a field $F$ is isomorphic to $F$

Let $F$ be a field and let $\newcommand{\Fract}{\operatorname{Fract}}$ $\Fract(F)$ be the field of fractions of $F$; that is, $\Fract(F)= \{ {a \over b } \mid a \in F , b \in F \setminus \{ 0 \} \}$. ...
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### At most finitely many (Hamel) coordinate functionals are continuous - different proof

If $X$ is a vector space over $\mathbb R$ and $B=\{x_i; i\in I\}$ is a Hamel basis for $X$, then for each $i\in I$ we have a linear functional $a_i(x)$ which assigns to $x$ the $i$-th coordinate, ...
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### Decompose boolean function of multiple variables into multiple functions of one variable

say I have a function $$f(x, y) : bool$$ of two variables x and y - whose type can be anything - returning either true or false. I would like to create two functions of one variable each $g(x)$, ...
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### What does this quantifier evaluates to?

I need help solving the following question, Suppose U=Z. Simplify the quantifer, and say that if it is true or false with the help of and example, ~$\exists$x $(x| x|5 \Rightarrow x|15)$ Note ...
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### What does $\mathbb{Z}_{7429}$ mean?

What exactly does $\mathbb{Z}_{7429}$ mean? Is it the set of all integers up to and including 7429?
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### Convergence of a bounded sequence with bounded L2 variation

I have the following question, that a friend of mine asked me yesterday: Let $H$ be a Hilbert space, with norm $\|\cdot\|$. Let $(x_k)_{k \ge 0}$ be some sequence in $H$. Assume that $x_k$ is bounded ...
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### Elementary matrices

Is there a way to visualize the action of elementary matrices? (Or perhaps matrices in general). Perhaps someone could give an intuitive view of the effects of elementary row operations. Actually I ...
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### How formal or informal should math texts (written for different purposes) be?

When writing math articles (or just math text), do you write down mathematical expression in a formal way or describe it in words, e. g. "Let $X$ be a normed vector space. Then $X$ is called a ...
I'd like to know if I have got the following ideas right: 1) $f(r,\theta,t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)\exp[in\theta-j^2_{nk}t]$ subjected to initial ...