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Quadratic forms and their duals
Hans Havlicek
https://arxiv.org/abs/2506.23613 https://arxiv.org/pdf/2506.23613

Quadratic forms and their duals
There are many specific results, spread over the literature, regarding the dualisation of quadrics in projective spaces and quadratic forms on vector spaces. In the present work we aim at generalising and unifying some of these. We start with a quadratic form $Q$ that is defined on a subspace $S$ of a finite-dimensional vector space $V$ over a field $F$. Whenever $Q$ satisfies a certain condition, which comes into effect only when $F$ is of characteristic two, $Q$ gives rise to a dual quadratic…

A general approach to permutation polynomials from quadratic forms
Ruikai Chen
https://arxiv.org/abs/2506.24012 https://arxiv.org/pdf…

A general approach to permutation polynomials from quadratic forms
We investigate a family of permutation polynomials of finite fields of characteristic 2. Through a connection between permutation polynomials and quadratic forms, a general treatment is presented to characterize these permutation polynomials. By determining some character sums associated with quadratic forms, we explicitly describe several classes of permutation polynomials.

Quadratic Bureau-Guillot systems with the first and second Painlev\'e transcendents in the coefficients. Part I: geometric approach and birational equivalence
Marta Dell'Atti, Galina Filipuk
https://arxiv.org/abs/2506.22896

Quadratic Bureau-Guillot systems with the first and second Painlevé transcendents in the coefficients. Part I: geometric approach and birational equivalence
Bureau proposed a classification of systems of quadratic differential equations in two variables which are free of movable critical points, which was recently revised by Guillot. We revisit the quadratic Bureau-Guillot systems with the first and second Painlevé transcendent in the coefficients. We explain their birational equivalence by using the geometric approach of Okamoto's spaces of initial conditions and the method of iterative polynomial regularisation, solving the Painlevé equivalence…

Efficient Numerical Integration for Finite Element Trunk Spaces in 2D and 3D using Machine Learning: A new Optimisation Paradigm to Construct Application-Specific Quadrature Rules
Tomas Teijeiro, Pouria Behnoudfar, Jamie M. Taylor, David Pardo, Victor M. Calo
https://arxiv.org/abs/2506.23741…

Efficient Numerical Integration for Finite Element Trunk Spaces in 2D and 3D using Machine Learning: A new Optimisation Paradigm to Construct Application-Specific Quadrature Rules
Finite element methods usually construct basis functions and quadrature rules for multidimensional domains via tensor products of one-dimensional counterparts. While straightforward, this approach results in integration spaces larger than necessary, especially as the polynomial degree $p$ or the spatial dimension increases, leading to considerable computational overhead. This work starts from the hypothesis that reducing the dimensionality of the polynomial space can lead to quadrature rules wi…

Wasserstein projections in the convex order: regularity and characterization in the quadratic Gaussian case
Aur\'elien Alfonsi, Benjamin Jourdain
https://arxiv.org/abs/2506.23981

Wasserstein projections in the convex order: regularity and characterization in the quadratic Gaussian case
In this paper, we first show continuity of both Wasserstein projections in the convex order when they are unique. We also check that, in arbitrary dimension $d$, the quadratic Wasserstein projection of a probability measure $μ$ on the set of probability measures dominated by $ν$ in the convex order is non-expansive in $μ$ and Hölder continuous with exponent $1/2$ in $ν$. When $μ$ and $ν$ are Gaussian, we check that this projection is Gaussian and also consider the quadratic Wasserstein p…

Linear-Quadratic Discrete-Time Dynamic Games with Unknown Dynamics
Shengyuan Huang, Xiaoguang Yang, Zhigang Cao, Wenjun Mei
https://arxiv.org/abs/2506.22073

Linear-Quadratic Discrete-Time Dynamic Games with Unknown Dynamics
Considering linear-quadratic discrete-time games with unknown input/output/state (i/o/s) dynamics and state, we provide necessary and sufficient conditions for the existence and uniqueness of feedback Nash equilibria (FNE) in the finite-horizon game, based entirely on offline input/output data. We prove that the finite-horizon unknown-dynamics game and its corresponding known-dynamics game have the same FNEs, and provide detailed relationships between their respective FNE matrices. To simplify …

A Comparison of Relativistic Coupled Cluster and Equation of Motion Coupled Cluster Quadratic Response Theory
Xiang Yuan, Lo\"ic Halbert, Lucas Visscher, Andr\'e Severo Pereira Gomes
https://arxiv.org/abs/2506.22905

A Comparison of Relativistic Coupled Cluster and Equation of Motion Coupled Cluster Quadratic Response Theory
We present the implementation of relativistic coupled cluster quadratic response theory (QR-CC), following our development of relativistic equation of motion coupled cluster quadratic response theory (QR-EOMCC) [X. Yuan et al., J. Chem. Theory Comput. 2023, 19, 9248]. These codes, which can be used in combination with relativistic (2- and 4-component based) as well as non-relativistic Hamiltonians, are capable of treating both static and dynamic perturbations for electric and magnetic operators…

An improved large sieve for quadratic characters via Hooley neutralisers and its applications
Cameron Wilson
https://arxiv.org/abs/2506.22667 https://

An improved large sieve for quadratic characters via Hooley neutralisers and its applications
We combine Hooley neutralisers and the large sieve for quadratic characters. We give applications to character sums with a hyperbolic height condition.

Quadratic Wasserstein distance between Gaussian laws revisited with correlation
Aur\'elien Alfonsi, Benjamin Jourdain
https://arxiv.org/abs/2506.23742 …

Quadratic Wasserstein distance between Gaussian laws revisited with correlation
In this note, we give a simple derivation of the formula obtained in Dowson and Landau (1982), Olkin and Pukelsheim (1982) and Givens and Shortt (1984) for the quadratic Wasserstein distance between two Gaussian distributions on $\R^d$ with respective covariance matrices $Σ_μ$ and $Σ_ν$. This derivation relies on the existence of an orthogonal matrix $O$ such that $O^*Σ_μO$ and $O^*Σ_νO$ share the same correlation matrix and on the simplicity of optimal couplings in the case with the sa…