Tootfinder

On Triple Quadratic Residue Symbols in Real Quadratic Fields
Atsuki Kuramoto
https://arxiv.org/abs/2509.00667 https://arxiv.org/pdf/2509.00667

On Triple Quadratic Residue Symbols in Real Quadratic Fields
We introduce triple quadratic residue symbols $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3]$ for certain finite primes $\mathfrak{p}_i$'s of a real quadratic field $k$ with trivial narrow class group. For this, we determine a presentation of the Galois group of the maximal pro-2 Galois extension over $k$ unramified outside $\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3$ and infinite primes, from which we derive mod 2 arithmetic triple Milnor invariants $μ_2(123)$ yielding the triple symbo…

Maximizing the Promptness of Metaverse Systems using Edge Computing by Deep Reinforcement Learning
Tam Ninh Thi-Thanh, Trinh Van Chien, Hung Tran, Nguyen Hoai Son, Van Nhan Vo
https://arxiv.org/abs/2506.02657

Maximizing the Promptness of Metaverse Systems using Edge Computing by Deep Reinforcement Learning
Metaverse and Digital Twin (DT) have attracted much academic and industrial attraction to approach the future digital world. This paper introduces the advantages of deep reinforcement learning (DRL) in assisting Metaverse system-based Digital Twin. In this system, we assume that it includes several Metaverse User devices collecting data from the real world to transfer it into the virtual world, a Metaverse Virtual Access Point (MVAP) undertaking the processing of data, and an edge computing ser…

🇺🇦 Auf radioeins läuft...
Matt Berninger:
🎵 Bonnet Of Pins (Radio-Edit)
#NowPlaying #MattBerninger
https://mattberninger.bandcamp.com/track/bonnet-of-pins
https://open.spotify.com/track/0hLFRJetaM0qj09lyyJ9a5

Cyclic $\mathcal{U}_\xi(\mathfrak{sl}_2)$-modules and invariants of knots with flat $\mathfrak{sl}_2$ connections in the complement
Calvin McPhail-Snyder, Nicolai Reshetikhin
https://arxiv.org/abs/2509.02354

Cyclic $\mathcal{U}_ξ(\mathfrak{sl}_2)$-modules and invariants of knots with flat $\mathfrak{sl}_2$ connections in the complement
The main result of this paper is the factorization of the holonomy $R$-matrix for quantum $\mathfrak{sl}_2$ at a root of unity into a product of four quantum holonomy dilogarithms. This factorization extends previously known results in this direction. We collect many existing results needed for the factorization. We use the holonomy $R$-matrices to define representations of a groupoid of braids with flat $\mathfrak{sl}_{2}$ connections, which also define invariants of knots with such connection…

The metaverse is going great!

Moving as a metaphor for software management.
https://www.garfieldtech.com/blog/moving-metaphor

Finite-dimensional modules over associative equivariant map algebras
Alexandru Chirvasitu
https://arxiv.org/abs/2509.01386 https://arxiv.org/pdf/2509.01386…

Finite-dimensional modules over associative equivariant map algebras
Let $X$ and $\mathfrak{a}$ be an affine scheme and (respectively) a finite-dimensional associative algebra over an algebraically-closed field $\Bbbk$, both equipped with actions by a linearly-reductive linear algebraic group $G$. We describe the simple finite-dimensional modules over the algebra of $G$-equivariant maps $X\to \mathfrak{a}$ in terms of the representation theory of the fixed-point subalgebras $\mathfrak{a}^x:=\mathfrak{a}^{G_x}\le \mathfrak{a}$, $G_x$ being the respective isotropy…

Forcing Axioms and construction schemes
Jorge Antonio Cruz Chapital, Osvaldo Guzman, Stevo Todorcevic
https://arxiv.org/abs/2509.01712 https://arxiv.org/pd…

Forcing Axioms and construction schemes
We continue the development of the theory of construction schemes over $ω_1$ as introduced by the third author by studying their relation with forcing axioms. Formally, we introduce the cardinals $\mathfrak{m}^n_{\mathcal{F}}$ and use the consistency of $\mathfrak{m}^2_\mathcal{F}>ω_1$ to prove a fundamental result relating gaps and almost disjoint families over $ω$. The cardinals $\mathfrak{m}_\mathcal{F}$ are also used to prove some limiting results for contstruction schemes, some of which…

The $\mu-$invariant of fine Selmer groups associated to general Drinfeld modules
Hang Chen
https://arxiv.org/abs/2507.02688 https://a…

The $μ-$invariant of fine Selmer groups associated to general Drinfeld modules
Let $F$ be a global function field over the finite field $\mathbb{F}_q$ where $q$ is a prime power and $A$ be the ring of elements in $F$ regular outside $\infty$. Let $ϕ$ be an arbitrary Drinfeld module over $F$ For a fixed non-zero prime ideal $\mathfrak{p}$ of $A$, we show that on the constant $\mathbb{Z}_{\textit{p}}-$extension $\mathfrak{F}$ of $F$, the Pontryagin dual of the fine Selmer group associated to the $\mathfrak{p}-$primary torsion of $ϕ$ over $\mathfrak{F}$ is a finitely gener…

Polynomials associated to Lie algebras
Mat\'ias Bruna, Alex Capu\~nay, Eduardo Friedman
https://arxiv.org/abs/2507.00326 https://…

Polynomials associated to Lie algebras
We associate to a semisimple complex Lie algebra $\mathfrak{g}$ a sequence of polynomials $P_{\ell,\mathfrak{g}}(x)\in\mathbb{Q}[x]$ in $r$ variables, where $r$ is the rank of $\mathfrak{g}$ and $\ell=0,1,2,\ldots $. The polynomials $P_{\ell,\mathfrak{g}}(x)$ are uniquely associated to the isomorphism class of $\mathfrak{g}$, up to re-numbering the variables, and are defined as special values of a variant of Witten's zeta function. Another set of polynomials associated to $\mathfrak{g}$ were de…