Tootfinder

Quadratic Embedding Constants of Cartesian Products and Joins of Graphs
Projesh Nath Choudhury, Raju Nandi
https://arxiv.org/abs/2508.10834 https://arxiv.o…

Quadratic Embedding Constants of Cartesian Products and Joins of Graphs
The quadratic embedding constant (QEC) of a finite, simple, connected graph originated from the classical work of Schoenberg [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938] on Euclidean distance geometry. In this article, we study the QEC of graphs in terms of two graph operations: the Cartesian product and the join of graphs. We derive a general formula for the QEC of the join of an arbitrary graph with a regular graph and with a complete multipartite graph. We then provide quadratic…

FLARE: Agile Flights for Quadrotor Cable-Suspended Payload System via Reinforcement Learning
Dongcheng Cao, Jin Zhou, Xian Wang, Shuo Li
https://arxiv.org/abs/2508.09797 https:/…

FLARE: Agile Flights for Quadrotor Cable-Suspended Payload System via Reinforcement Learning
Agile flight for the quadrotor cable-suspended payload system is a formidable challenge due to its underactuated, highly nonlinear, and hybrid dynamics. Traditional optimization-based methods often struggle with high computational costs and the complexities of cable mode transitions, limiting their real-time applicability and maneuverability exploitation. In this letter, we present FLARE, a reinforcement learning (RL) framework that directly learns agile navigation policy from high-fidelity sim…

A Classical Quadratic Speedup for Planted $k$XOR
Meghal Gupta, William He, Ryan O'Donnell, Noah G. Singer
https://arxiv.org/abs/2508.09422 https://arxi…

A Classical Quadratic Speedup for Planted $k$XOR
A recent work of Schmidhuber et al (QIP, SODA, & Phys. Rev. X 2025) exhibited a quantum algorithm for the noisy planted $k$XOR problem running quartically faster than all known classical algorithms. In this work, we design a new classical algorithm that is quadratically faster than the best previous one, in the case of large constant $k$. Thus for such $k$, the quantum speedup of Schmidhuber et al. becomes only quadratic (though it retains a space advantage). Our algorithm, which also works in …

The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes
Ashay Burungale, Shinichi Kobayashi, Kazuto Ota
https://arxiv.org/abs/2507.08221

The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes
We prove an asymptotic formula for the $p$-adic valuation of Hecke $L$-values of an imaginary quadratic field at an inert prime $p$ along the anticyclotomic $\mathbb{Z}_p$-tower. The key is determination of the $p$-adic valuation of generalized Gauss sums defined using Coates-Wiles homomorphism, and of local resolvents in $\mathbb{Z}_p$-extensions. This answers a question of Rubin.

Infra-red enhanced loops in quadratic gravity
Alberto Salvio, Alessandro Strumia, Marco Vitti
https://arxiv.org/abs/2507.08803 https://

Infra-red enhanced loops in quadratic gravity
It has been suggested that logarithmically enhanced infra-red loop corrections arising in theories with four derivatives correspond to a physical running of couplings, rendering quadratic gravity asymptotically free. We find that these effects depend on the gauge and on the field parameterisation. We compute physical on-shell amplitudes and find genuine infra-red log-enhanced loop corrections, that are process-dependent and cannot be absorbed into running couplings. As a byproduct, we derive th…

Natively Trainable Sparse Attention for Hierarchical Point Cloud Datasets
Nicolas Lapautre, Maria Marchenko, Carlos Miguel Pati\~no, Xin Zhou
https://arxiv.org/abs/2508.10758 ht…

Natively Trainable Sparse Attention for Hierarchical Point Cloud Datasets
Unlocking the potential of transformers on datasets of large physical systems depends on overcoming the quadratic scaling of the attention mechanism. This work explores combining the Erwin architecture with the Native Sparse Attention (NSA) mechanism to improve the efficiency and receptive field of transformer models for large-scale physical systems, addressing the challenge of quadratic attention complexity. We adapt the NSA mechanism for non-sequential data, implement the Erwin NSA model, and…

Composition laws of binary quadratic forms and isolations of quadratic forms
Jangwon Ju, Daejun Kim, Kyoungmin Kim, Mingyu Kim, Byeong-Kweon Oh
https://arxiv.org/abs/2508.07916 …

Composition laws of binary quadratic forms and isolations of quadratic forms
A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an isolation of $f$. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binar…

Arboreal Galois groups of postcritically finite quadratic polynomials: The strictly preperiodic case
Robert L. Benedetto, Dragos Ghioca, Jamie Juul, Thomas J. Tucker
https://arxiv.org/abs/2507.08347

Arboreal Galois groups of postcritically finite quadratic polynomials: The strictly preperiodic case
In a previous paper, we provided an explicit description of the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 +c$ in the special case when the critical point $0$ is periodic under the action of $f(z)$. In the current paper, we complete the picture for all postcritically finite polynomials by addressing the cases when $0$ is strictly preperiodic for the polynomial $f(z)$.

Upper tail distributions of central $L$-values of quadratic twists of elliptic curves at the variance scale
Mr Nathan Creighton
https://arxiv.org/abs/2507.08640

Upper tail distributions of central $L$-values of quadratic twists of elliptic curves at the variance scale
We consider the large deviations at the order of the variance for the central value of a family of $L$-functions among the members with bounded discriminant. When there is an upper bound on an integer moment of the central value twisted by a short Dirichlet polynomial, we can establish upper bounds on the density of members exhibiting a large central value. We adapt the techniques from Arguin and Bailey for large deviations of the Riemann zeta function to prove results on the degree two family …

Counting points on a family of degree one del Pezzo surfaces
Katharine Woo
https://arxiv.org/abs/2508.09391 https://arxiv.org/pdf/2508.09391

Counting points on a family of degree one del Pezzo surfaces
We study rational points on the elliptic surface given by the equation: $$y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,$$ where $A,B\in \mathbb{Z}$ satisfy that $4A^3-27B^2\neq 0$ and $Q(u,v)$ is a positive-definite quadratic form. We prove asymptotics for a special subset of the rational points, specifically those that are integral with respect to the singularity. This method utilizes Mordell's parameterization of integral points on quadratic twists on elliptic curves, which is based on a syzygy for…