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5-regular graphs and the 3-dimensional rigidity matroid
Rebecca Monks, Anthony Nixon
https://arxiv.org/abs/2506.22214 https://arxiv.o…

5-regular graphs and the 3-dimensional rigidity matroid
A bar-joint framework $(G,p)$ in Euclidean $d$-space is rigid if the only edge-length-preserving continuous motions arise from isometries of $\mathbb{R}^d$. In the generic case, rigidity is determined by the generic $d$-dimensional rigidity matroid of $G$. The combinatorial nature of this matroid is well understood when $d=1,2$ but open when $d\geq 3$. Jackson and Jordán 2005 characterised independence in this matroid for connected graphs with minimum degree at most $d+1$ and maximum degree at…

"Le bouddhisme est généralement associé Š la paix, Š la tolérance et Š la compassion. Mais comme toutes les autres grandes religions, il a un côté violent."
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Commanders to retire Art Monk's number as team continues to honor their 'Super Bowl era'

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On $q$-analogs of the $3x 1$ Dynamical System
Kenneth G. Monks
https://arxiv.org/abs/2508.10153 https://arxiv.org/pdf/2508.10153

On $q$-analogs of the $3x+1$ Dynamical System
The $3x+1$ Conjecture asserts that the $T$-orbit of every positive integer $x$ contains $1$, where $T$ maps $x$ to $x/2$ for $x$ even and to $(3x+1)/2$ for $x$ odd. Several authors have studied the analogous map, $T_q$, which maps $x\in F_2[q]$ to $x/q$ if $q$ divides $x$ and $((1+q)x+1)/q$ otherwise. In particular, they showed that the $T_q$-orbit of every polynomial contains $1$. This seems analogous to the $3x+1$ conjecture, but does not prove the conjecture itself, as the dynamical systems …