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I have four coupled first-order non-linear differential equations, denoted as: A1(x), A2(x), A3(x), A4(x) which are all functions of x. I have the following code which attempts to solve the equations using ParametricSolveND by varying one of the initial conditions of the parameter (namely, A4(0) which I have denoted as the parameter j).

However, it is not plotting and I’m not sure what I’ve done wrong. Furthermore, I intend to plot A4(x) as a function of A4(0) for a fixed x (x=2000). How should I go about fixing this? Thank you.

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Consider the linear diophantine equation $$cex_1+cfx_2+dex_3+dfx_4=a$$

where $a=c-d=e-f$ holds with $c,d$ coprime and $e,f$ coprime and $a,d,f$ are odd and $T<c,d,e,f<2T$ holds and $T^alpha<a<2T^alpha$ holds where $alphain(0,1)$.

We pick uniformly random $ain(T^alpha,2T^alpha)$ and uniformly random $c,ein(T+a,2T)$ and set $d,f$ accordingly.

What is the minimum of $$L=|(x_1,x_2,x_3,x_4)|_infty?$$

If the constraints $a=c-d=e-f$ were not there and $c,d,e,f$ were random and unrelated to $a$ then since $L^4>T^2L$ should hold we need $L>T^{2/3}$.

However the constraints should force it to a smaller solution. How small is it?

In two variable case the situation corresponds to $cx+dy=(c-d)$ with $T<c,d<2T$ and $c,d$ coprime. Here $x=-y=1$ suffices instead of norm $|(x,y)|_infty$ being roughly $T$ where $(c-d)$ is replaced by a random number in $(T^alpha,2T^alpha)$ at an $alphain(0,1)$. This is a huge reduction.

I am working on a project that will programmatically take a given row in the database, and aggregate all related tables and specific rows that have either a direct or indirect FK relationship back to that single row.

This seems like a non-trivial problem and one that may have interesting solutions. The problem goes beyond simply following FK references. If one were to build this to be fast, there would have to be some level of batch processing involved.

I’m wondering if anyone has literature, or existing solutions on the subject that could help inform my approach to the problem. Thanks in advance

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Let $p$ be a fixed prime number and $S$ is a finite set of prime numbers which does not contain $p$. A theorem of Siegel asserts that the number of solutions to the $S$-unit equations are finite; that is, there are only finitely many $S$-unit $u$ such that $1-u$ is also an $S$-unit. Therefor for each such $S$ there exist a lower bound on $|u_1-u_2|_p$ where $u_1$ and $u_2$ are solutions to $S$-unit equations.

My question is: does there exist such a lower bound uniformly? More precisely, does there exist a lower bound for the $p$-adic distance between solutions to the $S$-unit equations that only depends on the size of $S$(and perhaps on $p$)? Here we are assuming $S$ does not contain $p$.

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