Tootfinder

Composition laws of binary quadratic forms and isolations of quadratic forms
Jangwon Ju, Daejun Kim, Kyoungmin Kim, Mingyu Kim, Byeong-Kweon Oh
https://arxiv.org/abs/2508.07916 …

Composition laws of binary quadratic forms and isolations of quadratic forms
A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an isolation of $f$. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binar…

Performance of the Extended Ising Machine for the Quadratic Knapsack Problem
Haruka Akishima, Hirotaka Tamura, Kazue Kudo
https://arxiv.org/abs/2508.06909 https://

Performance of the Extended Ising Machine for the Quadratic Knapsack Problem
The extended Ising machine (EIM) enhances conventional Ising models, which handle only binary quadratic forms by allowing constraints through real-valued dependent variables. We address the quadratic knapsack problem (QKP), hard to solve using Ising machines when formulated as a quadratic unconstrained binary optimization (QUBO). We demonstrated the EIM's superiority by comparing it with the conventional Ising model-based approach, a commercial exact solver, and a state-of-the-art heuristic sol…

Rank 3 Quadratic Generators of Veronese Embeddings: The Characteristic 3 Case
Donghyeop Lee, Euisung Park, Saerom Sim
https://arxiv.org/abs/2509.09383 https://

Rank 3 Quadratic Generators of Veronese Embeddings: The Characteristic 3 Case
This paper investigates property QR(3) for Veronese embeddings over an algebraically closed field of characteristic $3$. We determine the rank index of $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (d))$ for all $n \geq 2$, $d \geq 3$, proving that it equals $3$ in these cases. Our approach adapts the inductive framework of [HLMP 2021], re-proving key lemmas for characteristic $3$ to establish quadratic generation by rank $3$ forms. We further compute the codimension of the span of rank $3$ quadr…

Linear-Quadratic Mean Field Games with Common Noise: A Direct Approach
Wenyu Cong, Jingtao Shi, Bingchang Wang
https://arxiv.org/abs/2508.07271 https://arx…

Linear-Quadratic Mean Field Games with Common Noise: A Direct Approach
This paper investigates a linear-quadratic mean field games problem with common noise, where the drift term and diffusion term of individual state equations are coupled with both the state, control, and mean field terms of the state, and we adopt the direct approach to tackle this problem. Compared with addressing the corresponding mean field teams problem, the mean field games problem with state coupling presents greater challenges. This is not only reflected in the explosive increase in the n…

Gauges for quadratic gravity: the extended transverse gauge and the energy-momentum tensor of the massive spin-2 field
Matheus F. S. Alves, L. G. Medeiros, Davi C. Rodrigues
https://arxiv.org/abs/2506.10793

Gauges for quadratic gravity: the extended transverse gauge and the energy-momentum tensor of the massive spin-2 field
We study the 4D Einstein-Hilbert action extension based on the square of the curvature tensors. Analyses of gauge and perturbation modes are often done considering the Teyssandier gauge condition. Although this approach can be useful for other higher-order extensions of quadratic gravity, a generalized transverse (or Lorentz) gauge is clear and sufficient for the present case, as explained here. We provide a detailed analysis of the generalized transverse gauge condition, its residual gauge sym…

Modified Cubic B-spline Based Differential Quadrature Methods for Time-fractional Black-Scholes Equation
Nizamudheen V, Riyasudheen TK, Noufal Asharaf, Shefeeq T
https://arxiv.org/abs/2508.06780

Modified Cubic B-spline Based Differential Quadrature Methods for Time-fractional Black-Scholes Equation
The time-fractional Black-Scholes equation (TFBSE) is intended to price the options for which the underlying price fluctuates within a correlated fractal transmission system. Although the TFBSE is an influential approach for grasping the long-term memory traits of financial markets, the non-local nature of fractional derivatives makes significant challenges in finding an accurate solution. We perform an efficient use of the differential quadrature method (DQM) based on modified cubic B-splines …

Fast attention mechanisms: a tale of parallelism
Jingwen Liu, Hantao Yu, Clayton Sanford, Alexandr Andoni, Daniel Hsu
https://arxiv.org/abs/2509.09001 https://

Fast attention mechanisms: a tale of parallelism
Transformers have the representational capacity to simulate Massively Parallel Computation (MPC) algorithms, but they suffer from quadratic time complexity, which severely limits their scalability. We introduce an efficient attention mechanism called Approximate Nearest Neighbor Attention (ANNA) with sub-quadratic time complexity. We prove that ANNA-transformers (1) retain the expressive power previously established for standard attention in terms of matching the capabilities of MPC algorithms,…

Parity of the partition function in quadratic progressions
Ken Ono
https://arxiv.org/abs/2509.09553 https://arxiv.org/pdf/2509.09553

Parity of the partition function in quadratic progressions
The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If $1

Turnpike properties for zero-sum stochastic linear quadratic differential games of Markovian regime switching system
Xun Li, Fan Wu, Xin Zhang
https://arxiv.org/abs/2509.09358 h…

Turnpike properties for zero-sum stochastic linear quadratic differential games of Markovian regime switching system
This paper investigates the long-time behavior of zero-sum stochastic linear-quadratic (SLQ) differential games within Markov regime-switching diffusion systems and establishes the turnpike property of the optimal triple. By verifying the convergence of the associated coupled differential Riccati equations (CDREs) along with their convergence rate, we show that, for a sufficiently large time horizon, the equilibrium strategy in the finite-horizon problem can be closely approximated by that of t…

Some bi-quadratic P\'{o}lya fields and large P\'{o}lya groups of compositum of simplest cubic and quintic fields
Md. Imdadul Islam, Debopam Chakraborty, Jaitra Chattopadhyay
https://arxiv.org/abs/2508.07894

Some bi-quadratic Pólya fields and large Pólya groups of compositum of simplest cubic and quintic fields
The Pólya group $Po(K)$ of an algebraic number field $K$ is the subgroup of the ideal class group $Cl_{K}$ generated by the ideal classes of the products of prime ideals of the same norm. If $Po(K)$ is trivial, then the number field $K$ is said to be a Pólya field. In this article, we furnish three families $\mathbb{Q}(\sqrt{p},\sqrt{qrs})$, $\mathbb{Q}(\sqrt{2p},\sqrt{qrs})$ and $\mathbb{Q}(\sqrt{2p},\sqrt{2qrs})$ of bi-quadratic Pólya fields $K$ involving prime numbers $p,q,r$ and $s$ that…