Tootfinder

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 $\mathbb{Q}$.

Learning Feasible Quantum States for Quadratic Constrained Binary Optimization Problems
Anthony Wilkie, Alexander DeLise, Andrew Del Real, Rebekah Herrman, James Ostrowski
https://arxiv.org/abs/2508.02590

Learning Feasible Quantum States for Quadratic Constrained Binary Optimization Problems
Quantum computing approaches excel at solving quadratic unconstrained binary optimization (QUBO) problems, however solving quadratic constrained binary optimization problems (QCBOs) is more challenging. In this work, we develop a variational approach that creates an equal superposition of quantum states that satisfy constraints in a QCBO. The method relies on flag qubits, one per constraint, to identify when a constraint is violated or not. The resulting equal superposition can be used as an in…

Quadrature Domains and the Faber Transform
Andrew J. Graven, Nikolai G. Makarov
https://arxiv.org/abs/2509.03777 https://arxiv.org/pdf/2509.03777

Quadrature Domains and the Faber Transform
We present a framework for reconstructing any simply connected bounded or unbounded, quadrature domain $Ω$ from its quadrature function $h$. Using the Faber transform, we derive formulae directly relating $h$ to the Riemann map for $Ω$. Through this approach, we obtain a complete classification of one point quadrature domains with complex charge. We proceed to develop a theory of weighted quadrature domains with respect to weights of the form $ρ_a(w)=|w|^{2(a-1)}$ when $a > 0$ ("po…

A Frank-Wolfe-based primal heuristic for quadratic mixed-integer optimization
Gioni Mexi, Deborah Hendrych, S\'ebastien Designolle, Mathieu Besan\c{c}on, Sebastian Pokutta
https://arxiv.org/abs/2508.01299

A Frank-Wolfe-based primal heuristic for quadratic mixed-integer optimization
We propose a primal heuristic for quadratic mixed-integer problems. Our method extends the Boscia framework -- originally a mixed-integer convex solver leveraging a Frank-Wolfe-based branch-and-bound approach -- to address nonconvex quadratic objective and constraints. We reformulate nonlinear constraints, introduce preprocessing steps, and a suite of heuristics including rounding strategies, gradient-guided selection, and large neighborhood search techniques that exploit integer-feasible verti…

Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws
G\'erard Ben Arous, Murat A. Erdogdu, N. Mert Vural, Denny Wu
https://arxiv.org/abs/2508.03688

Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws
We study the optimization and sample complexity of gradient-based training of a two-layer neural network with quadratic activation function in the high-dimensional regime, where the data is generated as $y \propto \sum_{j=1}^{r}λ_j σ\left(\langle \boldsymbol{θ_j}, \boldsymbol{x}\rangle\right), \boldsymbol{x} \sim N(0,\boldsymbol{I}_d)$, $σ$ is the 2nd Hermite polynomial, and $\lbrace\boldsymbolθ_j \rbrace_{j=1}^{r} \subset \mathbb{R}^d$ are orthonormal signal directions. We consider the ex…

Complexity of Quadratic Quantum Chaos
Pallab Basu, Suman Das, Pratik Nandy
https://arxiv.org/abs/2509.04075 https://arxiv.org/pdf/2509.04075

Complexity of Quadratic Quantum Chaos
We investigate minimal two-body Hamiltonians with random interactions that generate spectra resembling those of Gaussian random matrices, a phenomenon we term quadratic quantum chaos. Unlike integrable two-body fermionic systems, the corresponding hard-core boson models exhibit genuinely chaotic dynamics, closely paralleling the Sachdev-Ye-Kitaev (SYK) model in its spin representation. This chaotic behavior is diagnosed through spectral statistics and measures of operator growth, including Kryl…

Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, II
Fei Hou, Huicheng Yin
https://arxiv.org/abs/2508.03348 https…

Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, II
In the paper [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597-618], S. Alinhac established the global existence of small data smooth solutions to the Cauchy problem of 2-D quadratically quasilinear wave equations with null conditions. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. When the 2-D quadratic nonlinearity admits a s…

Redefining Euler-Rabinowitsch Polynomials with Heegner Number Based Quadratic Formulation
Sudarshan Kumaresan, Shipra Kumari, Neha Mishra
https://arxiv.org/abs/2508.02821 https:…

Redefining Euler-Rabinowitsch Polynomials with Heegner Number Based Quadratic Formulation
This paper introduces a novel class of prime-generating quadratic polynomials defined by $f_{Z,k,H}(n) = n^2 - (2Zk - 1)n + \frac{(2Zk - 1)^2 + H}{4}$, where $Zk \in \mathbb{Z}_{\geq 0}$ and $H$ belongs to the set of Heegner numbers. This form is closely related to the Euler-Rabinowitsch polynomials through specific substitutions. The structure enables algebraic tuning for prime-rich outputs and provides deeper insight into the impact of Heegner numbers on prime distribution. Using tools such a…

Love beyond Einstein: Metric reconstruction and Love number in quadratic gravity using WEFT
Arpan Bhattacharyya, Saptaswa Ghosh, Naman Kumar, Shailesh Kumar, Sounak Pal
https://arxiv.org/abs/2508.02785

Love beyond Einstein: Metric reconstruction and Love number in quadratic gravity using WEFT
We study tidal Love numbers of static black holes in four-dimensional quadratic theory of gravity, extending the result of GR. We use worldline effective field theory (WEFT) methods to compute metric perturbations from one-point functions, treating the higher-derivative terms perturbatively. We show that insertions of scalar fields on the worldline induce non-zero tidal tails, and the corresponding Love number displays no RG running. The same conclusion holds for the insertions of tensor fields…

Indefinite Linear-Quadratic Partially Observed Mean-Field Game
Tian Chen, Tianyang Nie, Zhen Wu
https://arxiv.org/abs/2508.01568 https://arxiv.org/pdf/2508…

Indefinite Linear-Quadratic Partially Observed Mean-Field Game
This paper investigates an indefinite linear-quadratic partially observed mean-field game with common noise, incorporating both state-average and control-average effects. In our model, each agent's state is observed through both individual and public observations, which are modeled as general stochastic processes rather than Brownian motions. {It is noteworthy that} the weighting matrices in the cost functional are allowed to be indefinite. We derive the optimal decentralized strategies using t…