Tootfinder

The number of countable models of first-order theories
Anand Pillay, Predrag Tanovi\'c
https://arxiv.org/abs/2508.06854 https://arxiv.org/pdf/2508.0685…

The number of countable models of first-order theories
Throughout, $T$ denotes a complete first-order theory in a countable language $L$ that has infinite models and $I(\aleph_0,T)$ denotes the number of countable models of $T$, up to an isomorphism. To determine $I(\aleph_0,T)$, it suffices to consider only countable models of $T$ with domain $ω$; since there are at most continuum many $L$-structures with domain $ω$, $I(\aleph_0,T)\leqslant 2^{\aleph_0}$ holds. Theories with $I(\aleph_0,T)=1$ are the $\aleph_0$-categorical theories. These includ…

from my link log —
CandyFab: a DIY 3D sugar printer.
https://candyfab.org/
saved 2024-12-20 https://dotat.at/:/3D5SH.html

Some remarks on decay in countable groups and amalgamated free products
Srivatsav Kunnawalkam Elayavalli, Gregory Patchell, Lizzy Teryoshin
https://arxiv.org/abs/2509.08754 http…

Some remarks on decay in countable groups and amalgamated free products
In this note, we first study the notion of subexponential decay (SD) for countable groups with respect to a length function, which generalizes the well-known rapid decay (RD) property, first discovered by Haagerup in 1979. Several natural properties and examples are studied, especially including groups that have SD, but not RD. This consideration naturally has applications in $C^*$-algebras. We also consider in this setting a permanence theorem for decay in amalgamated free products (proved als…

Isomorphism of almost locally compact Polish metric structures
Maciej Malicki
https://arxiv.org/abs/2508.07765 https://arxiv.org/pdf/2508.07765

Isomorphism of almost locally compact Polish metric structures
A topological space is almost locally compact if it contains a dense locally compact subspace. We generalize a result from \cite{Ma}, showing that isomorphism on Borel classes of almost locally compact Polish metric structures is always classifiable by countable structures. This allows to remove a gap in the proof presented in \cite{Ma} of a positive answer to a question of Gao and Kechris, who asked whether isometry of locally compact Polish metric spaces is classifiable by countable structure…

Stabilisability and beta representations
Carl P. Dettmann (University of Bristol, UK)
https://arxiv.org/abs/2509.08532 https://arxiv.org/pdf/2509.08532

Stabilisability and beta representations
We consider a one dimensional affine switched system obtained from a formal limit of a two dimensional linear system. We show this is equivalent to minimising the average digit in beta representations with unrestricted digits. We give a countable set of $β$ for which the result is given by the usual (greedy) beta expansion, an interval of values for which it is strictly less, and a conditional lower bound for all $β$.

HUAC sucked. J. Edgar sucked. Half our elected government sucks.
"...movie that came under suspicion was “It’s a Wonderful Life.” The FBI complained that the 1946 blockbuster, which starred Jimmy Stewart as a broken man who learns the true value of his life, “deliberately maligned the upper classes” with its negative portrayal of Mr. Potter, the town’s richest man."
It DID malign Mr. Potter. He sucked.
Help Jane Fonda and the new Committee for the First Amendment!…

Committee for the First Amendment

Trump Translator #trumpimpression #comedyvideo #funnyvideo - YouTube

Metric criteria for fixed price of countable groups
Erin Bevilacqua, Lewis Bowen
https://arxiv.org/abs/2510.05459 https://arxiv.org/pdf/2510.05459

Metric criteria for fixed price of countable groups
We establish general criteria for a countable group $Γ$ to have fixed price 1 depending on a choice of left-invariant proper metric on $Γ$. We apply this criterion to show that if $Γ_1,Γ_2$ are two countable groups satisfying a certain growth condition then $Γ_1\times Γ_2$ has fixed price 1. For example, $Γ\times Γ$ has fixed price 1 for any countable group $Γ$.

Borel distinguishing number
Onur Bilge, Burak Kaya
https://arxiv.org/abs/2508.08201 https://arxiv.org/pdf/2508.08201

Borel distinguishing number
In this paper, we study definable variants of the notion of the distinguishing number of a graph in descriptive set theoretic setting. We introduce the notion of the Borel distinguishing number of a Borel graph and provide examples that separate distinguishing number and Borel distinguishing number at various levels. More specifically, we prove that there exist Borel graphs with countable distinguishing number but uncountable Borel distinguishing number and that, for every integer $n \geq 3$, t…

Products of Finitely-Generated Groups with a Certain Growth Condition Have Fixed Price One
Ali Khezeli
https://arxiv.org/abs/2509.08325 https://arxiv.org/p…

Products of Finitely-Generated Groups with a Certain Growth Condition Have Fixed Price One
An open problem posed by Gaboriau is whether the product of any two infinite countable groups has fixed price one. We provide an affirmative answer if the two groups are finitely generated and their growths satisfy a specific condition. The proof uses the propagation method to construct a Poisson horoball process as a weak factor of i.i.d., where each horoball is equipped with a marking that depends only on the first coordinate, in an i.i.d. manner. Then, a low-cost graphing of this process is …