Tootfinder

Titan submarine: Also in the same ntsb set of documents are a) The interview with the Softee who worked on some of the software, (describing Jira and Git to marine investigators..), and replugging the Logitech joystick dongle, and who had also got into screwing the end cap of the sub on and torquing the bolts up (!).
and b) The details of the recovery of the images off the camera - which turn out to be useless - but involved some fun with Luks, and a Qualcomm SoM.

Constructing crowded Hausdorff $P$-spaces in set theory without the axiom of choice
Eleftherios Tachtsis, Eliza Wajch
https://arxiv.org/abs/2510.11935 https://

Constructing crowded Hausdorff $P$-spaces in set theory without the axiom of choice
For an infinite set $X$, a closed under finite unions family $\mathcal{Z}$ with $[X]^{<ω}\subseteq\mathcal{Z}\subseteq\mathcal{P}(X)$, and any $\mathcal{A}\subseteq\mathcal{P}(X)$, the topology $τ_{\mathcal{A}}[\mathcal{Z}]=\{V\in\mathcal{A}: (\forall x\in V)(\exists z\in \mathcal{Z})(x\cap z=\emptyset \wedge \{y\in\mathcal{A}: x\subseteq y\subseteq X\setminus z\}\subseteq V)\}$ on $\mathcal{A}$ is investigated to give answers to the following open problem in various models of $\mathbf{ZF}$ o…

I am these cost 25¢ per song years old.
I wonder how much the artists got paid being played on these compared to Spotify or any of the other streaming services? Assuming they did in fact get paid.
Think how annoying it is when some AH plays their phone on speaker. Now imagine a diner full of tables all playing their own tunes on speaker (the only way you could).

The maximum product of sizes of cross-\(t\)-intersecting families
Jingjun Bao, Lijun Ji
https://arxiv.org/abs/2510.11724 https://arxiv.org/pdf/2510.11724…

The maximum product of sizes of cross-\(t\)-intersecting families
Two families of sets \(\mathcal{A}\) and \(\mathcal{B}\) are called \emph{cross-\(t\)-intersecting} if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\). Determining the maximum product of sizes for such cross-\(t\)-intersecting families is an active problem in extremal set theory. In this paper, we verify the following cross-\(t\)-intersecting version of the Erdős-Ko-Rado theorem: For \(k\geq l \geq t \geq 3\) and \(\min\{m,n\} \geq (t+1)(k-t+1)\), the ma…

I got my kids each a set of dice and my son’s life is now ruled by a D20!
(e.g number of minutes he’s going to do homework)

The odd independence number of graphs, II: Finite and infinite grids and chessboard graphs
Yair Caro, Mirko Petru\v{s}evski, Riste \v{S}krekovski, Zsolt Tuza
https://arxiv.org/abs/2510.01897

The odd independence number of graphs, II: Finite and infinite grids and chessboard graphs
An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $α_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. …

A Bouquet of Results on Maximum Range Sum: General Techniques and Hardness Reductions
Rachana Gusain, Saladi Rahul, Aditya Subramanian
https://arxiv.org/abs/2509.16008 https://

A Bouquet of Results on Maximum Range Sum: General Techniques and Hardness Reductions
We revisit the maximum range sum (MaxRS) problem: given a set $P$ of $n$ weighted points in $\mathbb{R}^d$ and a range $Q$ (typically axis-aligned $d$-box or $d$-ball), the goal is to place $Q$ to maximize the total weight of points in $P\cap Q$. We study three natural variations: (1) Dynamic MaxRS: The goal is to update the placement of a $d$-ball under point insertions and deletions. We give a randomized $(\frac{1}{2}-ε)$-approximation with update time $O_ε(\log n)$. The approximation fac…

The odd independence number of graphs, I: Foundations and classical classes
Yair Caro, Mirko Petru\v{s}evski, Riste \v{S}krekovski, Zsolt Tuza
https://arxiv.org/abs/2509.20763 h…

The odd independence number of graphs, I: Foundations and classical classes
An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $α_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. …

Lower estimates for the norm and the Kuratowski measure of moncompactness of Wiener-Hopf type operators
Oleksiy Karlovych, Eugene Shargorodsky
https://arxiv.org/abs/2509.20296 h…

Lower estimates for the norm and the Kuratowski measure of moncompactness of Wiener-Hopf type operators
Let $X(\mathbb{R}^n)$ be a Banach function space and $Ω\subseteq\mathbb{R}^n$ be a measurable set of positive measure. For a Fourier mutliplier $a$ on $X(\mathbb{R}^n)$, consider the Wiener-Hopf type operator $W_Ω(a):=r_ΩF^{-1}aF e_Ω$, where $F^{\pm 1}$ are the Fourier transforms, $r_Ω$ is the operator of restriction from $\mathbb{R}^n$ to $Ω$ and $e_Ω$ is the operator of extension by zero from $Ω$ to $\mathbb{R}^n$. Let $X_2(Ω)$ be the closure of $L^2(Ω)\cap X(Ω)$ in $X(Ω)$. We sho…