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Given an identity covariance matrix $I$, I want to perturb it such that the covariance entries no longer remain zero while the variance entries stay the same. I am looking for a method that allows me to control the effect of the perturbation. How do I do it? My idea: change the number of all-one rows and columns in the matrix. This does not work as the matrix will lose it positive definite property.

Any example of three (continuous) probability distributions so that $P(X < Y) > 0.5, P(Y < Z) > 0.5$ and $P(Z < X) > 0.5$ i.e., intransitive probabilities where $X, Y,$ and $Z$ are random variables drawn from the example probability distributions? I know about Efron's dice but I am looking for an example of probability distributions. And can such distributions be unimodal (and still lead to intransitivity)?

Let $M$ be a module over the integral domain $R$. Suppose that $M$ has rank $n$ and that $x_{1}, x_{2}, \ldots, x_{n}$ is any maximal set of linearly independent elements of $M$. Let $N = Rx_{1} +\cdots + Rx_{n}$ be the submodule generated by $x_{1},x_{2}, \ldots, x_{n}$. Prove that $M/N$ is a torsion $R$-module. Here's my idea. Since $M$ is a free module of rank $n$ and $N$ is a submodule of $M$, then $N$is free of rank $m \leq n$. And there exists a basis $y_{1}, \ldots, y_{m}$ of $M$ so that $a_{1}y_{1}, a_{2}y_{2}, \ldots, a_{m}y_{m}$ is a basis of $N$, where $a_{1}, \ldots, a_{m}$ are nonzero elements of $R$ with the divisibility relations $a_{1}|a_{2} | \cdots | a_{n}$. Since $x_{1}, x_{2}, \ldots, x_{n}$ is a maximal set of linearly independent elements of $M$, it's a basis of $M$. We want to show that $rx_{1}, \ldots, rx_{n}$ is a basis of $N$. Since $x_{1}, \ldots, x_{n}$ are linearly independent in $N$, it must be linearly independent in $N$. And we know that every element in $N$ can be written as a linear combination of $x_{1}, \ldots, x_{n}$. Thus, $\{rx_{1}, \ldots, rx_{n}\}$ forms a basis of $N$. Thus, $N$ is isomorphic to $R^n$. Eventually, I want to show that for any $y \in M$, there is a nonzero element $r \in R$ such that $ry$ can be written as a linear combination $r_{1}x_{1} + \cdots +r_{n}x_{n}$ of the $x_{i}$. But I'm not sure how to prove this from what I have so far. Please help! Thanks!

I would greatly appreciate recommendations for a good modern text for learning about primitive recursive arithmetic, ideally something that walks through a few proofs and discusses what can and cannot be proved in PRA. I have read through a number of the references on Wikipedia, but they seem to be mostly original sources.

I've been exploring patterns in Pythagorean quadruples (positive integers $(a,b,c,d)$ such that $a^2+b^2+c^2=d^2$) and have identified several parameterization patterns that generate valid quadruples. I'd like to know if these patterns are already known in the literature or if they might represent novel contributions. Pattern 1: Consecutive Term Parameterization Theorem: For any positive integer $a \geq 1$ and any non-negative integer $k \geq 0$, the following integers form a Pythagorean quadruple: $$\left[a, a+2k+1, \frac{a^2+(a+2k+1)^2-1}{2}, \frac{a^2+(a+2k+1)^2+1}{2}\right]$$ Examples: For $k=0, a=1$: $[1,2,2,3]$ since $1^2+2^2+2^2=9=3^2$ For $k=0, a=3$: $[3,4,12,13]$ since $3^2+4^2+12^2=169=13^2$ For $k=1, a=2$: $[2,5,14,15]$ since $2^2+5^2+14^2=225=15^2$ Pattern 2: Equal Initial Terms Theorem: For any even number $a=2k$ where $k>1$, the following integers form a Pythagorean quadruple: $$\left[a, a, \frac{a^2}{2}-1, \frac{a^2}{2}+1\right]$$ Examples: For $a=4$: $[4,4,7,9]$ since $4^2+4^2+7^2=81=9^2$ For $a=6$: $[6,6,17,19]$ since $6^2+6^2+17^2=361=19^2$ For $a=8$: $[8,8,31,33]$ since $8^2+8^2+31^2=1089=33^2$ Pattern 2A: A Corollary for Equal Initial Terms Theorem: For any even number $a=2(2k+1)$ where $k \geq 1$, the following integers form a Pythagorean quadruple: $$\left[a, a, \frac{a^2}{4}-2, \frac{a^2}{4}+2\right]$$ Examples: For $a=6$: $[6,6,7,11]$ since $6^2+6^2+7^2=121=11^2$ For $a=10$: $[10,10,23,27]$ since $10^2+10^2+23^2=729=27^2$ For $a=14$: $[14,14,47,51]$ since $14^2+14^2+47^2=2601=51^2$ Pattern 3: Scaled First Terms Theorem: For $a=2$ and any integer $m$, the following integers form a Pythagorean quadruple: $$\left[2, 2m, \frac{m^2-3}{2}, \frac{m^2+5}{2}\right]$$ Examples: For $m=3$: $[2,6,3,7]$ since $2^2+6^2+3^2=49=7^2$ For $m=7$: $[2,14,23,27]$ since $2^2+14^2+23^2=729=27^2$ For $m=9$: $[2,18,39,43]$ since $2^2+18^2+39^2=1849=43^2$ Pattern 4: Derived from Pythagorean Triples Theorem: For any odd number $a \geq 3$, if $[a, \frac{a^2-1}{2}, \frac{a^2+1}{2}]$ is a Pythagorean triple, then $[a, \frac{a^2-1}{2}, \frac{(a^2+1)^2-4}{8}, \frac{(a^2+1)^2+4}{8}]$ is a Pythagorean quadruple. Examples: From triple $[3,4,5]$, we can expand the the triple of last digit as $[5,12,13]$ and replace the last digit to get quadruple $[3,4,12,13]$ From triple $[5,12,13]$, we get quadruple $[5,12,84,85]$ From triple $[7,24,25]$, we get quadruple $[7,24,312,313]$ My questions: Are any of these parameterization patterns novel, or are they already documented in the literature? Do these represent special cases of the standard parameterization of Pythagorean quadruples? Are my proofs correct? (I've verified each pattern with numerous examples) Could these patterns lead to interesting mathematical insights about the structure of Pythagorean quadruples? I'd appreciate any feedback, references to existing literature, or suggestions for further exploration. Thanks! Disclaimer: I'm an independent enthusiast without formal mathematical research training. My background is in IT, though I've always been fascinated by number theory. I did reach out to a mathematics professor about these patterns, who directed me to literature about finite generation of points on curves and surfaces. However, without formal training, I've found it challenging to determine if my observations are already covered by established theory. I'm posting here hoping for more accessible guidance.