Tootfinder

"A common Israeli taking point [is] 'How should America respond if rockets were launched from Tijuana at San Diego?' ... it highlights both Israeli ignorance and fears and Palestinian and Xicano/Migrant commonality. Just like how Tel Aviv and Gaza are part of Historic Palestine, San Diego and Tijuana were both Mexico but more importantly part of the homeland of the Kumeyaay People."

"As demographers who study fertility, family behaviors and childbearing intentions, we can say with certainty that population decline is not imminent, inevitable or necessarily catastrophic."
https://theconversation.com/fear…

Fears that falling birth rates in US could lead to population collapse are based on faulty assumptions
While the changes in population structure that accompany low birth rates are real, the impact of these changes has been dramatically overstated.

Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt
Lior Fishman, David Lambert, Keith Merrill, David Simmons
https://arxiv.org/abs/2506.19060

Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt
Following the work of Waldschmidt, we investigate problems in Diophantine approximation on abelian varieties. First we show that a conjecture of Waldschmidt for a given simple abelian variety is equivalent to a well-known Diophantine condition holding for a certain matrix related to that variety. We then posit a related but weaker conjecture, and establish the upper bound direction of that conjecture in full generality. For rank 1 elliptic curves defined over a number field $K \subset \mathbb{R…

Distinguishing Filling Invariants Associated to Conjugacy in Groups
Conan Gillis, Timothy Riley
https://arxiv.org/abs/2506.19264 https://

Distinguishing Filling Invariants Associated to Conjugacy in Groups
Brick and Corson introduced annular Dehn functions in 1998 to quantify the conjugacy problem for finitely generated groups and gave the fundamental relationships between it, the Dehn function, and the conjugator length function. We furnish the theory with diverse examples groups. In particular, we show that these three invariants are independent -- no two of the three functions determine the other.

Die chinesische Firma Gotion High-Tech hat mit der Produktion seiner 5MWh #Batteriespeicher in #Göttingen begonnen.
Ja, in Deutschland.🤪
Nach erfolgreicher #TÜV-Zertifizieru…

Gotion starts manufacturing 5MWh BESS in Germany following key certifications
Gotion High-Tech has launched manufacturing of its 5MWh BESS units at its manufacturing facility in Göttingen, Germany.

Poster No 857: Plastic changes in the occipital regions during pitch perception associate with blindness onset age; "Specifically, the primary visual cortex remains engaged in pitch processing tasks following late blindness." https://doi.org/10.5281/zenodo.15641972

On Gy\'arf\'as' Path-Colour Problem
Ben Cameron, Alexander Clow
https://arxiv.org/abs/2506.19100 https://arxiv.org/pdf/25…

On Gyárfás' Path-Colour Problem
In their 1997 paper titled ``Fruit Salad", Gyárfás posed the following conjecture: there exists a constant $k$ such that if each path of a graph spans a $3$-colourable subgraph, then the graph is $k$-colourable. It is noted that $k=4$ might suffice. Let $r(G)$ be the maximum chromatic number of any subgraph $H$ of $G$ where $H$ is spanned by a path. The only progress on this conjecture comes from Randerath and Schiermeyer in 2002, who proved that if $G$ is an $n$ vertex graph, then $χ(G) \le…

On the PI-exponent of matrix algebras and algebras with generalized actions
Thiago Castilho de Mello, Felipe Yukihide Yasumura
https://arxiv.org/abs/2506.21257

On the PI-exponent of matrix algebras and algebras with generalized actions
We compute the PI-exponent of the matrix ring with coefficients in an associative algebra. As a consequence, we prove the following. Let $\mathcal{R}$ be a PI-algebra with a positive PI-exponent. If $M_n(\mathcal{R})$ and $M_m(\mathcal{R})$ satisfy the same set of polynomial identities then $n=m$. We provide examples where this result fails if either $\mathcal{R}$ is not PI or has zero exponent. We obtain the same statement for certain finite-dimensional algebras with generalized action over an…

Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors
Jialai She
https://arxiv.org/abs/2507.16828 https://

Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors
The Erdős-Mollin-Walsh conjecture, asserting the nonexistence of three consecutive powerful integers, remains a celebrated open problem in number theory. A natural line of inquiry, following recent work by Chan (2025), is to investigate potential counterexamples centered around perfect cubes, which are themselves powerful. This paper establishes a new non-existence result for a family of such integer triplets with distinct structural constraints, combining techniques from modular arithmetic, $…

The infinite dimensional geometry of conjugation invariant generating sets
Sabine Chu, George Domat, Christine Gao, Ananya Prasanna, Alex Wright
https://arxiv.org/abs/2506.18618

The infinite dimensional geometry of conjugation invariant generating sets
We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all non-filling curves; mapping class groups and outer automorphism groups of free groups with the generating sets of all reducible elements; and groups with suitable actions on Gromov hyperbolic spaces with a generating set of elliptic elements. In these Cayley graphs we …