# All Questions

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### In a lattice, $x \leq y$ and $x \leq z \iff x \leq y ∧ z$

Added later: Proof now in progress. Let $(L, \leq)$ be a lattice, $x, y, z \in L$. Prove that $x ≤ y$ and $x ≤ z \iff x ≤ y ∧ z$. I proceed to prove as follows: The statement can be split into ...
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### In a Lattice, x ∧ y ≤ x and x ∧ y ≤ y

Proof in progress.
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### What are good books/other readings for elementary set theory?

I am looking to expand my knowledge on set theory (which is pretty poor right now -- basic understanding of sets, power sets, and different (infinite) cardinalities). Are there any books that come to ...
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### random variable with binomial distribution

I have a probability space $\omega = 2^{\{1,\ldots,n\}}$ $\sigma$-algebra $2^\omega$ and $P(\{s\})=(p^{|s|})*(1-p)^{(n-|s|)}$ I assume that $n=2k$,$k$ natural number I need to find a random ...
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### One dimensional quotients of formal power series rings

Suppose $R=k[[x_1,...,x_n]]$ is a formal power series ring over the field $k$, what can we say about the structure of $R/p$ if $p$ is a prime ideal of $R$ such that dim$(R/p)=1$. In particular, are ...
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### Find the inverse of the following function

$g(x) =\frac{3-5x}{5-x}$ I know that I am retarded for not being able to do this on my own. Algebraically you just have to solve for x. Even so, I can't do it.
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### Is $L^2(\mathbb{R})$ with convolution a Banach Algebra?

Is $L^2(\mathbb{R})$ a Banach algebra, with convolution? I am pretty sure the answer is no, because I think that $f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I ...
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### Patterns in Prime numbers, and the null hypothesis

I've read about many attempts to find patterns in prime numbers. First, is there a mathematical way to prove there is not a pattern to prime numbers? Since there are ways to check if a number is ...
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### can I get a bound on the probability of deviation, similar to Markov inequality?

I have two random variables $X$ and $Y$, both receiving values between 0 and 1. I know that $E[X - Y] \ge 0$. Can I get any inequality of the form: $P(X - Y \ge \delta) \le F(\delta,X,Y)$ where ...
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### Lights out game on hexagonal grid

I greatly enjoyed the Lights Out game described here (I am sorry I had to link to an older page because some wikidiot keeps deleting most of the page). Its mathematical analysis is here (it's just ...
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### Maximize normal density function over a subset

For a 2D Normal distribution $N(0, \left[ \begin{array}{cc} 1 & -1/4 \\ -1/4 & 1 \end{array} \right])$, I am now trying to maximize its density function over $\{ x\geq 10, y \geq 10 \}$. My ...
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### Elementary problem in integration theory

If $f \in L(0,1)$ show that $x^{n} f(x) \in L(0,1)$ for $n \in \mathbb{N}$ and that $\displaystyle \int_{0}^{1} x^{n} f(x) dx \rightarrow 0$ as $n \to \infty$. Is the following attempt correct? ...
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Let ${f_{n}}$ be a sequence in $L^{2}(X,\mu)$ such that $||f_{n}||_{2} \rightarrow 0$ as $n \rightarrow \infty$. How to show that: $\displaystyle \lim_{n \to \infty} \int_{X} |f_n(x)| ... 2answers 445 views ### Prime factors of$n^2+1$I know it is unknown if there are infinitely many primes of the form$n^2+1$. Is it known if there is a positive integer$k$such that$|\{n\in\mathbb{Z}:n^2+1 \text{ has at most k prime ...
Assume $M$ is a (finitely generated) $A$-module such that $\wedge^n M$ is free of rank $1$ for some $n \geq 1$. Does it follow that $M$ is free of rank $n$? Or at least locally free of rank $n$? In ...