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On quadratic persistence and Pythagoras numbers of totally real projective varieties
Jong In Han, Jaewoo Jung, Euisung Park
https://arxiv.org/abs/2506.13247

On quadratic persistence and Pythagoras numbers of totally real projective varieties
In this paper, we study the relationship between quadratic persistence and the Pythagoras number of totally real projective varieties. Building upon the foundational work of Blekherman et al. in arXiv:1902.02754, we extend their characterizations of arithmetically Cohen-Macaulay varieties with next-to-maximal quadratic persistence to arbitrary case. Our main result classifies totally real non-aCM varieties of codimension $c$ and degree $d$ that exhibit next-to-maximal quadratic persistence in t…

Coxeter matrices and homological quadratic forms of $\boldsymbol{n}$-hereditary algebras
Raziyeh Diyanatnezhad, Alireza Nasr-Isfahani
https://arxiv.org/abs/2506.13136

Coxeter matrices and homological quadratic forms of $\boldsymbol{n}$-hereditary algebras
We study the Coxeter matrices and the homological quadratic forms of $n$-hereditary algebras within the framework of higher dimensional Auslander--Reiten theory. Let $Λ$ be a finite dimensional $n$-hereditary algebra with the Coxeter matrix $Φ$ and the homological quadratic form $χ$. We prove that if $Λ$ is $n$-representation finite, then there exists a positive integer $d$ such that $Φ^d=1$. In the case $n$ is an odd number, we show that if there exists a positive integer $d$ such that $Î…

From Complexity to Simplicity: A Polynomial Time Solution To The Subset Sum Problem And A Milestone In Proving P Equal NP
Thami Nkosi
https://arxiv.org/abs/2506.12019

From Complexity to Simplicity: A Polynomial Time Solution To The Subset Sum Problem And A Milestone In Proving P Equal NP
The Subset Sum Problem being one of the NP-complete problems, till present, has no polynomial time solution. The aim of this research is to present a way to solve all instances of the Subset Sum Problem in quadratic time worst case. With showing that the solution functions within polynomial time, it would hence prove that indeed P equals NP, theoretically that is. When it comes to the practical extension of the solution to the NP class, reductions would have to be tailored and standardized due …

A High-Order Quadrature Method for Implicitly Defined Hypersurfaces and Regions
Zibo Zhao
https://arxiv.org/abs/2506.13078 https://ar…

A High-Order Quadrature Method for Implicitly Defined Hypersurfaces and Regions
This paper presents a high-order accurate numerical quadrature algorithm for evaluating integrals over curved surfaces and regions defined implicitly via a level set of a given function restricted to a hyperrectangle. The domain is divided into small tetrahedrons, and by employing the change of variables formula, the approach yields an algorithm requiring only one-dimensional root finding and standard Gaussian quadrature. The resulting quadrature scheme guarantees strictly positive weights and …

Simple quintessence models in light of DESI-BAO observations
James M. Cline, Varun Muralidharan
https://arxiv.org/abs/2506.13047 https://

Simple quintessence models in light of DESI-BAO observations
Recent analyses from the DESI collaboration suggest that the dark energy density of the Universe may be decreasing with time, slowing the acceleration of the scale factor $a$. Typically these studies are performed assuming an ansatz for the equation of state $w(a)$. In this work, we instead consider simple models of a scalar quintessence potential with linear and quadratic behavior, which could be more representative of real models than particular parametrizations of $w(a)$. We observe a signif…

On the Robustness of Derivative-free Methods for Linear Quadratic Regulator
Weijian Li, Panagiotis Kounatidis, Zhong-Ping Jiang, Andreas A. Malikopoulos
https://arxiv.org/abs/2506.12596

On the Robustness of Derivative-free Methods for Linear Quadratic Regulator
Policy optimization has drawn increasing attention in reinforcement learning, particularly in the context of derivative-free methods for linear quadratic regulator (LQR) problems with unknown dynamics. This paper focuses on characterizing the robustness of derivative-free methods for solving an infinite-horizon LQR problem. To be specific, we estimate policy gradients by cost values, and study the effect of perturbations on the estimations, where the perturbations may arise from function approx…

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Isoperimetric Inequality on Manifolds with Quadratically Decaying Curvature
Dangyang He
https://arxiv.org/abs/2506.13365 https://arxi…

Isoperimetric Inequality on Manifolds with Quadratically Decaying Curvature
In this paper, we investigate the reverse improvement property of Sobolev inequalities on manifolds with quadratically decaying Ricci curvature. Specifically, we establish conditions under which the uniform decay of the heat kernel implies the validity of an isoperimetric inequality. As an application, we demonstrate the existence of isoperimetric sets in generalized Grushin spaces. Our approach is built on a weak-type Sobolev inequality, gradient estimates on remote balls, and a Hardy-type glu…

Gated Rotary-Enhanced Linear Attention for Long-term Sequential Recommendation
Juntao Hu, Wei Zhou, Huayi Shen, Xiao Du, Jie Liao, Junhao Wen, Min Gao
https://arxiv.org/abs/2506.13315

Gated Rotary-Enhanced Linear Attention for Long-term Sequential Recommendation
In Sequential Recommendation Systems (SRSs), Transformer models show remarkable performance but face computation cost challenges when modeling long-term user behavior sequences due to the quadratic complexity of the dot-product attention mechanism. By approximating the dot-product attention, linear attention provides an efficient option with linear complexity. However, existing linear attention methods face two limitations: 1) they often use learnable position encodings, which incur extra compu…