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The classification and representations of positive definite ternary quadratic forms of level 4N
Yifan Luo, Haigang Zhou
https://arxiv.org/abs/2402.17443 ht…

The classification and representations of positive definite ternary quadratic forms of level 4N
Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer $N$, first we give the classification of positive definite ternary quadratic forms of level $4N$ explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level $4N$, which are involved with modified Hurwitz class number. …

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This https://arxiv.org/abs/2311.06144 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csRO_…

Multi-Agent Reinforcement Learning for the Low-Level Control of a Quadrotor UAV
By leveraging the underlying structures of the quadrotor dynamics, we propose multi-agent reinforcement learning frameworks to innovate the low-level control of a quadrotor, where independent agents operate cooperatively to achieve a common goal. While single-agent reinforcement learning has been successfully applied in quadrotor controls, training a large monolithic network is often data-intensive and time-consuming. Moreover, achieving agile yawing control remains a significant challenge due …

Inverse Optimal Control for Linear Quadratic Tracking with Unknown Target States
Yao Li, Chengpu Yu, Hao Fang, Jie Chen
https://arxiv.org/abs/2402.17247 ht…

Inverse Optimal Control for Linear Quadratic Tracking with Unknown Target States
This paper addresses the inverse optimal control for the linear quadratic tracking problem with a fixed but unknown target state, which aims to estimate the possible triplets comprising the target state, the state weight matrix, and the input weight matrix from observed optimal control input and the corresponding state trajectories. Sufficient conditions have been provided for the unique determination of both the linear quadratic cost function as well as the target state. A computationally effi…

Complexity of Minimizing Regularized Convex Quadratic Functions
Daniel Berg Thomsen, Nikita Doikov
https://arxiv.org/abs/2404.17543 https://

Complexity of Minimizing Regularized Convex Quadratic Functions
In this work, we study the iteration complexity of gradient methods minimizing the class of uniformly convex regularized quadratic functions. We prove lower bounds on the functional residual of the form $Ω(N^{-2p/(p-2)})$, where $p > 2$ is the power of the regularization term, and $N$ is the number of calls to a first-order oracle. A special case of our problem class is $p=3$, which is the minimization of cubically regularized convex quadratic functions. It naturally appears as a subproblem at…

On the Iterated Monodromy Groups of Quadratic Rational functions with periodic critical points
Özlem Ejder
https://arXiv.org/abs/2401.14862 https://…

On the Iterated Monodromy Groups of Quadratic Rational functions with periodic critical points
Let $f$ be a quadratic rational function defined over a field of characteristic not equal to $2$. We assume that the critical points of $f$ are all periodic. We first determine the geometric iterated monodromy group up to conjugacy and prove that it is of weakly branch type. This type of result was previously only known for the discrete iterated monodromy groups of PCF quadratic polynomials due to work of Bartholdi and Nekrashevyich. Furthermore, motivated by a question of Jones and Levy on s…

Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville
Barinder S. Banwait, Maarten Derickx
https://arXiv.org/abs/2401.14514

Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville
We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbović. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$ for all squarefree $d$ with $|d| < 800$, improving on the previously known range of $-5 < d < 26$…

This https://arxiv.org/abs/1103.5906 has been replaced.
link: https://scholar.google.com/scholar?q=a<…

Torsion groups of elliptic curves over quadratic fields
We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the smallest absolute value of it's discriminant such that there exists an elliptic curve with that torsion. We also examine the interplay of the torsion and rank over a fixed quadratic field and see that what happens is very different than over $\Q$. Finally we…

Sails for universal quadratic forms
V\'it\v{e}zslav Kala, Siu Hang Man
https://arxiv.org/abs/2403.18390 https://arxiv.org/pdf/240…

Sails for universal quadratic forms
We establish a new connection between sails, a key notion in the geometric theory of generalised continued fractions, and arithmetic of totally real number fields, specifically, universal quadratic forms and additively indecomposable integers. Our main application is to biquadratic fields, for which we show that if their signature rank is at least 3, then ranks of universal forms and numbers of indecomposables grow as a power of the discriminant. We also construct a family in which these number…

Joint distribution of $L$-values and orders of Sha groups of quadratic twists of elliptic curves
Peng-Jie Wong
https://arxiv.org/abs/2403.18382 https://

Joint distribution of $L$-values and orders of Sha groups of quadratic twists of elliptic curves
We study the joint distribution of central $L$-values and orders of Tate-Shafarevich groups of quadratic twists of elliptic curves. In particular, adapting Radziwill and Soundararajan's principles of establishing upper and lower bounds for the distribution of central values in families of $L$-functions, we obtain conditional upper and lower bounds for such a joint distribution for rank zero twists. These lead us to a conjecture on the full asymptotic for the joint distribution.