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Solving Fermat-type equations over quadratic fields
Begum Gulsah Cakti
https://arxiv.org/abs/2509.14280 https://arxiv.org/pdf/2509.14280

Solving Fermat-type equations over quadratic fields
This paper applies the modular approach to obtain effectively computable bounds for Fermat-type equations over number fields, while also discussing the differences and obstructions that arise when considering such equations over totally real versus totally complex number fields. We use these techniques to study the generalized Fermat equation $d^ra^p+b^p+c^p=0$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ of class number one. Extending the results of Freitas\&Siksek and Turcas, we show that whe…

A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning
Juan Diego Toscano, Daniel T. Chen, Vivek Oommen, George Em Karniadakis
https://arxiv.org/abs/2509.14198

A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning
Residual-based adaptive strategies are widely used in scientific machine learning but remain largely heuristic. We introduce a unifying variational framework that formalizes these methods by integrating convex transformations of the residual. Different transformations correspond to distinct objective functionals: exponential weights target the minimization of uniform error, while linear weights recover the minimization of quadratic error. Within this perspective, adaptive weighting is equivalen…

Characterization of Gradient Index Fibers
Robert Leonard, Matthew Marshall, Seth Hyra, Meagan Plummer, Jacob Williamson, Spencer Olson
https://arxiv.org/abs/2509.14390 https://

Characterization of Gradient Index Fibers
Gradient index (GRIN) fibers are used to improve the design of many fiber optic devices. However, the properties of the GRIN fiber must be determined to optimally engineer a device which incorporates GRIN fiber components. The index of refraction of most GRIN fibers varies quadratically in the radial direction, where the quadratic coefficient is characterized by the gradient index constant $g$. We measured $g$ for Thorlabs GIF50C GRIN fiber at both $780~\mathrm{nm}$ and $1550~\mathrm{nm}$ using…

Decay of Chebyshev coefficients and error estimates of associated quadrature on shrinking intervals
Krishna Yamanappa Poojara, Sabhrant Sachan, Ambuj Pandey
https://arxiv.org/abs/2509.14602

Decay of Chebyshev coefficients and error estimates of associated quadrature on shrinking intervals
We analyze decay of Chebyshev coefficients and local Chebyshev approximations for functions of finite regularity on finite intervals, focusing on the framework where the interval length tends to zero while the number of approximation nodes remains fixed. For all four families of Chebyshev polynomials, we derive error estimates that quantify the dependence on the interval length for (i) the decay of Chebyshev coefficients, (ii) the approximation error between continuous and discrete Chebyshev co…

Anti-pre-Poisson bialgebras and relative Rota-Baxter operators
Qinxiu Sun, Min Wu
https://arxiv.org/abs/2509.13963 https://arxiv.org/pdf/2509.13963

Anti-pre-Poisson bialgebras and relative Rota-Baxter operators
In this paper, we first introduce the notion of an anti-pre-Poisson bialgebra, which is shown to be equivalent to both quadratic anti-pre-Poisson algebras and matched pairs of Poisson algebras. The study of coboundary anti-pre-Poisson bialgebras leads to the anti-pre-Poisson Yang-Baxter equation (APP-YBE). Skew-symmetric solutions of this equation give rise to coboundary anti-pre-Poisson bialgebras. Furthermore, we investigate how solutions without skew-symmetry can also induce such bialgebras,…

Replaced article(s) found for cs.MA. https://arxiv.org/list/cs.MA/new
[1/1]:
- Smooth Games of Configuration in the Linear-Quadratic Setting
Jesse Milzman, Jeffrey Mao, Giuseppe Loianno
…

Dynamical degrees of affine-triangular automorphisms in dimension four
Enbo Shao, Xiaosong Sun
https://arxiv.org/abs/2509.14584 https://arxiv.org/pdf/2509.…

Dynamical degrees of affine-triangular automorphisms in dimension four
In this paper, let k be the affine n-space over an arbitrary field k. We show that dynamical degrees of affine-triangular automorphisms in dimension 4 are algebraic integers of degree not larger than 4. As a consequence, if char(k) is not equal to 2, then dynamical degrees of quadratic automorphisms in dimension 4 are algebraic integers of degree not larger than 4. We also explore some results in higher dimensions. These results partially give the affirmative answer to the Conjecture in DF21

Generalized splitting of algebras with application to a bialgebra structure of Leibniz algebras induced from averaging Lie bialgebras
Chengming Bai, Li Guo, Guilai Liu, Quan Zhao
https://arxiv.org/abs/2509.14137

Generalized splitting of algebras with application to a bialgebra structure of Leibniz algebras induced from averaging Lie bialgebras
The classical notion of splitting a binary quadratic operad $\mathcal{P}$ gives the notion of pre-$\mathcal{P}$-algebras characterized by $\mathcal{O}$-operators, with pre-Lie algebras as a well-known example. Pre-$\mathcal{P}$-algebras give a refinement of the structure of $\mathcal{P}$-algebras and is critical in the Manin triple approach to bialgebras for $\mathcal{P}$-algebras. Motivated by the new types of splitting appeared in recent studies, this paper aims to extend the classical notion…

$S$-units and period length of continued fractions of linear recursions
Veekesh Kumar, Vivek Singh, Johannes Sprang
https://arxiv.org/abs/2509.14599 https://

$S$-units and period length of continued fractions of linear recursions
Let $(A_n)_{n\in \mathbb{Z}}$ be a linear recurrence sequence with values in a real quadratic field. In this paper, we study the question whether the period length of the continued fraction of $A_n$ is bounded as $n$ varies. The case where $(A_n)_n$ is a linear recurrence of degree $1$ has previously been solved by Corvaja and Zannier. Their result settled a problem posed by Mendès France about the length of the periods of the continued fractions for $α^n$.

Improvement of Drmota-Verwee's effective Erdos-Wintner theorem for Zeckendorf expansions
Johann Verwee
https://arxiv.org/abs/2509.14974 https://arxiv.o…

Improvement of Drmota-Verwee's effective Erdos-Wintner theorem for Zeckendorf expansions
We revisit the effective Erdos-Wintner theorem for Zeckendorf expansions. Drmota and the author obtained a uniform Kolmogorov bound whose error involves $T\sum_{j>L-2h}|f(F_j)|$, which assumes absolute convergence of the linear tail $\sum_j f(F_j)$. We remove this assumption. Grouping the transfer matrices in pairs and working to second order on the logarithm of the product, after extracting the common linear phase along the dominant direction, yields a quadratic tail $T^2\sum_{j>L-2h…