Tootfinder

Millions of inequivalent quadratic APN functions in eight variables
Christof Beierle, Philippe Langevin, Gregor Leander, Alexandr Polujan, Shahram Rasoolzadeh
https://arxiv.org/abs/2508.04644

Millions of inequivalent quadratic APN functions in eight variables
The only known example of an almost perfect nonlinear (APN) permutation in even dimension was obtained by applying CCZ-equivalence to a specific quadratic APN function. Motivated by this result, there have been numerous recent attempts to construct new quadratic APN functions. Currently, 32,892 quadratic APN functions in dimension 8 are known and two recent conjectures address their possible total number. The first, proposed by Y. Yu and L. Perrin (Cryptogr. Commun. 14(6): 1359-1369, 2022), sug…

Quadratic Chabauty Experiments on Genus 2 Bielliptic Modular Curves in the LMFDB
Kate Finnerty
https://arxiv.org/abs/2507.03784 https://

Quadratic Chabauty Experiments on Genus 2 Bielliptic Modular Curves in the LMFDB
We present results of quadratic Chabauty experiments on genus 2 bielliptic modular curves of Jacobian rank 2 that have recently been added to the LMFDB. We apply quadratic Chabauty methods over both the rationals and quadratic imaginary fields. In a number of cases, the experiments yielded algebraic irrational points among the set of mock rational points. We highlight specific notable examples, including the non-split Cartan modular curve $X^+_{ns}(15)$. Lastly, we offer a conjecture relating t…

Turnpike Property of a Linear-Quadratic Optimal Control Problem in Large Horizons with Regime Switching II: Non-Homogeneous Cases
Hongwei Mei, Rui Wang, Jiongmin Yong
https://arxiv.org/abs/2508.04993

Turnpike Property of a Linear-Quadratic Optimal Control Problem in Large Horizons with Regime Switching II: Non-Homogeneous Cases
This paper is concerned with an optimal control problem for a nonhomogeneous linear stochastic differential equation having regime switching with a quadratic functional in the large time horizon. This is a continuation of the paper \cite{Mei-Wang-Yong-2025}, in which the strong turnpike property was established for homogeneous linear systems with purely quadratic cost functionals. We extend the results to the current situation. It turns out that some of the results are new even for the cases wi…

Joyce structures from quadratic differentials on the sphere
Timothy Moy
https://arxiv.org/abs/2509.05275 https://arxiv.org/pdf/2509.05275

Joyce structures from quadratic differentials on the sphere
Motivated by known examples of Joyce structures on spaces of meromorphic quadratic differentials, we consider the isomonodromic deformations of particular second-order linear ODEs with rational potential. We show the infinitesimal isomonodromic deformations are the kernel of a closed $2$-form arising from the intersection pairing of an algebraic curve defined by the potential. This observation enables us to construct Joyce structures on a class of moduli spaces of meromorphic quadratic differen…

Bilinear Quadratic Output Systems and Balanced Truncation
Heike Fa{\ss}bender (Institute for Numerical Analysis, TU Braunschweig), Serkan Gugercin (Department of Mathematics and Division of Computational Modeling and Data Analytics, Academy of Data Science, Virginia Tech), Till Peters (Institute for Numerical Analysis, TU Braunschweig)
https://

Bilinear Quadratic Output Systems and Balanced Truncation
Dynamical systems with quadratic outputs have recently attracted significant attention. In this paper, we consider bilinear dynamical systems, a special class of weakly nonlinear systems, with a quadratic output. We develop various primal-dual formulations for these systems and define the corresponding system Gramians. Conditions for the existence and uniqueness of these Gramians are established, and the generalized Lyapunov equations they satisfy are derived. Using these Gramians and their tru…

Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms
Stanley Yao Xiao
https://arxiv.org/abs/2508.03848 https://

Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms
In this paper we prove a correspondence between a canonical degree six covariant of binary quartic forms $F$ and a cubic covariant of a pair of ternary quadratic forms $(f_A, f_B)$. In the process we obtain a canonical way to diagonalize a pair of $n$-ary quadratic forms over any field $K$ of characteristic zero. As a corollary, we give a precise criterion to decide whether a pair of $n$-ary quadratic forms over $\mathbb{Q}$ is diagonalizable over $\mathbb{Q}$.

A Sequential Quadratic Programming Perspective on Optimal Control
Abhijeet, Suman Chakravorty
https://arxiv.org/abs/2510.03475 https://arxiv.org/pdf/2510.0…

A Sequential Quadratic Programming Perspective on Optimal Control
This paper offers a unified perspective on different approaches to the solution of optimal control problems through the lens of constrained sequential quadratic programming. In particular, it allows us to find the relationships between Newton's method, the iterative LQR (iLQR), and Differential Dynamic Programming (DDP) approaches to solve the problem. It is shown that the iLQR is a principled SQP approach, rather than simply an approximation of DDP by neglecting the Hessian terms, to solve opt…

Class groups of imaginary quadratic points on $X_1(16)$
Maarten Derickx
https://arxiv.org/abs/2507.04604 https://arxiv.org/pdf/2507.0…

Class groups of imaginary quadratic points on $X_1(16)$
The main result is to show that if $K \ncong \mathbb Q(\sqrt{-15})$ is an imaginary quadratic field and $E$ is an elliptic curve over $K$ with a torsion point of order 16, then the class number of $K$ is divisible by 10. This gives an affirmative answer to a 12 year old question by David Krumm. This is done by setting up a more general framework for studying divisibility of class groups of imaginary quadratic points on hyper-elliptic curves and applying it to $X_1(16)$.

Inverse Continuous-Time Linear Quadratic Regulator: From Control Cost Matrix to Entire Cost Reconstruction
Yuexin Cao, Yibei Li, Zhuo Zou, Xiaoming Hu
https://arxiv.org/abs/2510.04083

Inverse Continuous-Time Linear Quadratic Regulator: From Control Cost Matrix to Entire Cost Reconstruction
This paper studies the inverse optimal control problem for continuous-time linear quadratic regulators over finite-time horizon, aiming to reconstruct the control, state, and terminal cost matrices in the objective function from observed optimal inputs. Previous studies have mainly explored the recovery of state cost matrices under the assumptions that the system is controllable and the control cost matrix is given. Motivated by various applications in which the control cost matrix is unknown a…

Sums of the floor function related to class numbers of imaginary quadratic fields
Marc Chamberland, Karl Dilcher
https://arxiv.org/abs/2510.04387 https://a…

Sums of the floor function related to class numbers of imaginary quadratic fields
A curious identity of Bunyakovsky (1882), made more widely known by Pólya and Szeg{\H o} in their ``Problems and Theorems in Analysis", gives an evaluation of a sum of the floor function of square roots involving primes $p\equiv 1\pmod{4}$. We evaluate this sum also in the case $p\equiv 3\pmod{4}$, obtaining an identity in terms of the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-p})$. We also consider certain cases where the prime $p$ is replaced by a composite integer. C…