Tootfinder

Apple and Google partnering for an AI deal is beneficial for both, giving Apple a compelling AI story for investors and Google access to affluent users (Dave Lee/Bloomberg)
https://www.bloomberg.com/opinion/articles

For Apple-Google Frenemies, AI Partnership Makes Sense for iPhone and Gemini
A deal that would bring Gemini artificial intelligence to the iPhone can solve short-term headaches for both companies.

A note on weighted consecutive Davenport constant
A. Lemos, A. O. Moura, S. Ribas, A. T. Silva
https://arxiv.org/abs/2404.11312 https://

A note on weighted consecutive Davenport constant
Let $G$ be a group and $A\subseteq [1,\exp(G)-1]$. We define the constant ${\sf C}_A(G),$ which is the least positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has an $A$-weighted consecutive product-one subsequence. In this paper, among other things, we prove that ${\sf C}_A(C_n^2)=4$ with $A=[1,n-1],$ and ${\sf C}(H\times K)=|H||K|$, where $H$ is a finite abelian group and $K$ is a metacyclic group.

#NowPlaying - #NeilYoung will always be known as the first-take/best-take/preferably-under-a-full-moon guy, but the ongoing avalanche of mid-'70s lost albums & box sets & loosie outtakes on his streaming service all point to him actually being a totally ruthless self editor. anyway, &quot…

Peter Navarro is due to report to a federal prison on Tuesday after an appeals court ruled that his appeal wasn’t likely to overturn his conviction for refusing to cooperate with a congressional investigation into the #January6 attack that Trump supporters aimed at the US Capitol in 2021.
#Navarro has maintained th…

Jetzt also das Kind in #Offenburg abholen, weil die deutsche #Bahn nicht weiter fahren kann.
#regen

A Graded Schur Lemma and a graded-monoidal structure for induced modules over graded-commutative algebras
J\"urgen Fuchs, Tobias Gr{\o}sfjeld
https://arxiv.org/abs/2403.10366

A Graded Schur Lemma and a graded-monoidal structure for induced modules over graded-commutative algebras
We consider algebras and Frobenius algebras, internal to a monoidal category, that are graded over a finite abelian group. For the case that A is a twisted group algebra in a linear abelian monoidal category we obtain a graded generalization of the Schur Lemma for the category of induced A-modules. We further show that if the monoidal category is braided and A is commutative up to a bicharacter of the grading group, then the category of induced A-modules can be endowed with a graded-monoidal st…

Kuo: Apple to use higher energy density battery cells for iPhone 16 Pro Max and stainless steel battery cases for better replaceability, an EU requirement (William Gallagher/AppleInsider)
https://appleinsider.com/articles/24/0…

iPhone 16 Pro Max to get new battery that could last longer
Analyst Ming-Chi Kuo reports that the iPhone 16 Pro Max will get a new battery that has a greater energy density than before, which means a longer life if Apple keeps the battery to the same size.

On the number of subsequence sums of a sequence in finite abelian groups related to the support of the sequence
Rui Wang, Han Chao, Jiangtao Peng
https://arxiv.org/abs/2404.11307 …

On the number of subsequence sums of a sequence in finite abelian groups related to the support of the sequence
Let $G$ be a finite abelian group and $S$ a sequence with elements of $G$. Let $|S|$ denote the length of $S$ and $k$ an integer with $k\in [1, |S|]$. Let $Σ_{k}(S) \subset G$ denote the set of group elements which can be expressed as a sum of a subsequence of $S$ with length $k$. Let $Σ(S)=\cup_{k=1}^{|S|}Σ_{k}(S)$ and $Σ_{\geq k}(S)=\cup_{t=k}^{|S|}Σ_{t}(S)$. It is known that if $0\not\in Σ(S)$, then $|Σ(S)|\geq |S|+|\mat…

This https://arxiv.org/abs/2210.14039 has been replaced.
link: https://scholar.google.com/scholar?q=a

Stability, corners, and other 2-dimensional shapes
We introduce a relaxation of stability, called robust stability, which is insensitive to perturbations by subsets of Loeb measure $0$ in a pseudofinite group. We show that robust stability satisfies a stationarity principle for measure independent elements. We apply this principle to deduce the existence of squares in dense robustly stable subsets of Cartesian products of non-standard finite groups, possibly non-abelian. Our results imply qualitative asymptotic versions for Cartesian products o…