Tootfinder

Quadratic units and cubic fields
Florian Breuer, James Punch
https://arxiv.org/abs/2507.06579 https://arxiv.org/pdf/2507.06579…

Quadratic units and cubic fields
We investigate Eisenstein discriminants, which are squarefree integers $d \equiv 5 \pmod{8}$ such that the fundamental unit $\varepsilon_d$ of the real quadratic field $K=\mathbb{Q}(\sqrt{d})$ satisfies $\varepsilon_d \equiv 1 \pmod{2\mathcal{O}_K}$. These discriminants are related to a classical question of Eisenstein and have connections to the class groups of orders in quadratic fields as well as to real cubic fields. We present numerical computations of Eisenstein discriminants up to $10^{1…

What's my phase again? Computing the vacuum-to-vacuum amplitude of quadratic bosonic evolution
Nicol\'as Quesada
https://arxiv.org/abs/2507.06154 h…

What's my phase again? Computing the vacuum-to-vacuum amplitude of quadratic bosonic evolution
Quadratic bosonic Hamiltonians and their associated unitary transformations form a fundamental class of operations in quantum optics, modeling key processes such as squeezing, displacement, and beam-splitting. Their Heisenberg-picture dynamics simplifies to linear (or possibly affine) transformations on quadrature operators, enabling efficient analysis and decomposition into optical gate sets using matrix operations. However, this formalism discards a phase, which, while often neglected, is ess…

Diverse Neural Sequences in QIF Networks: An Analytically Tractable Framework for Synfire Chains and Hippocampal Replay
Genki Shimizu, Taro Toyoizumi
https://arxiv.org/abs/2508.06085

Diverse Neural Sequences in QIF Networks: An Analytically Tractable Framework for Synfire Chains and Hippocampal Replay
Sequential neural activity is fundamental to cognition, yet how diverse sequences are recalled under biological constraints remains a key question. Existing models often struggle to balance biophysical realism and analytical tractability. We address this problem by proposing a parsimonious network of Quadratic Integrate-and-Fire (QIF) neurons with sequences embedded via a temporally asymmetric Hebbian (TAH) rule. Our findings demonstrate that this single framework robustly reproduces a spectrum…

Super-Quadratic Quantum Speed-ups and Guessing Many Likely Keys
Timo Glaser, Alexander May, Julian Nowakowski
https://arxiv.org/abs/2509.06549 https://arxi…

Super-Quadratic Quantum Speed-ups and Guessing Many Likely Keys
We study the fundamental problem of guessing cryptographic keys, drawn from some non-uniform probability distribution $D$, as e.g. in LPN, LWE or for passwords. The optimal classical algorithm enumerates keys in decreasing order of likelihood. The optimal quantum algorithm, due to Montanaro (2011), is a sophisticated Grover search. We give the first tight analysis for Montanaro's algorithm, showing that its runtime is $2^{H_{2/3}(D)/2}$, where $H_α(\cdot)$ denotes Renyi entropy with paramete…

Symplectic configurations: a homological and computer-aided approach
Weimin Chen
https://arxiv.org/abs/2509.07429 https://arxiv.org/pdf/2509.07429

Symplectic configurations: a homological and computer-aided approach
We formulated a homological and computer-aided approach to study certain unions of symplectic surfaces, called symplectic configurations, in a rational $4$-manifold $X=CP^2\# N\overline{CP^2}$. We addressed several fundamental theoretical questions, and also as a technical device, developed a symplectic analog of the so-called quadratic Cremona transformations in complex algebraic geometry. As an application, we gave a new proof that a certain line arrangement in $CP^2$, called Fano planes, doe…

The windowed quadratic phase Fourier transform: structure, convolution theorem and application
Sarga Varghese, Manab Kundu
https://arxiv.org/abs/2507.03911

The windowed quadratic phase Fourier transform: structure, convolution theorem and application
The windowed quadratic phase Fourier transform (WQPFT) combines the localization capabilities of windowed transforms with the phase modulation structure of the quadratic phase Fourier transform (QPFT). This paper investigates fundamental properties of the WQPFT, including linearity, shifting, modulation, conjugation, and symmetry. In addition, we derive the reproducing kernel, establish a reconstruction formula, and characterize the range of the transform. Convolution theorems in both the spect…

Cost Functions in Economic Complexity
Alessandro Bellina, Paolo Butt\`a, Vito D. P. Servedio
https://arxiv.org/abs/2507.04054 https://

Cost Functions in Economic Complexity
Economic complexity algorithms, including the Economic Complexity Index (ECI) and Non-Homogeneous Economic Fitness and Complexity (NHEFC), have proven effective at capturing the intricate dynamics of economic systems. Here, we present a fundamental reinterpretation of these algorithms by reformulating them as optimization problems that minimize specific cost functions. We show that ECI computation is equivalent to finding eigenvectors of the network's transition matrix by minimizing the quadrat…

Diophantine tuples and Integral Ideals of $\mathbb{Q}(\sqrt{d})$
Kalyan Chakraborty, Shubham Gupta, Krishnarjun Krishnamoorthy
https://arxiv.org/abs/2509.06522 https://

Diophantine tuples and Integral Ideals of $\mathbb{Q}(\sqrt{d})$
Suppose $n$ is the fundamental discriminant associated with a quadratic extension of $\mathbb{Q}$. We show that for every Diophantine $m$-tuple $ \{t_1, t_2, \ldots, t_m\} $ with the property $ D(n) $, there exists integral ideals $ \mathfrak{t}_1, \mathfrak{t}_2, \ldots, \mathfrak{t}_m $ of $ \mathbb{Q}(\sqrt{n}) $ and $c\in \{1,2\}$ such that $ t_i= c\mathcal{N}(\mathfrak{t}_i) $ for $ i=1,2, \ldots, m $. Here, $ \mathcal{N}(\cdot) $ denotes the norm map from $\mathbb{Q}(\sqrt{n})$ to $\mathb…

More regular formal moduli spaces and arithmetic transfer conjectures: the ramified quadratic case
Yu Luo, Michael Rapoport, Wei Zhang
https://arxiv.org/abs/2507.01395

More regular formal moduli spaces and arithmetic transfer conjectures: the ramified quadratic case
For unitary groups associated to a ramified quadratic extension of a $p$-adic field, we define various regular formal moduli spaces of $p$-divisible groups with parahoric levels, characterize exceptional special divisors on them, and construct correspondences between them. We formulate arithmetic transfer conjectures, which are variants of the arithmetic fundamental lemma conjecture in this context. We prove the conjectures in the lowest dimensional cases.

The shape of quadratic Gauss paths
Justine Dell, Djordje Mili\'cevi\'c
https://arxiv.org/abs/2508.21707 https://arxiv.org/pdf/2508.21707

The shape of quadratic Gauss paths
We consider the distribution of quadratic Gauss paths, polygonal paths joining partial sums of quadratic Gauss sums to square-free fundamental discriminant moduli in a dyadic range [Q,2Q]. We prove that this striking ensemble converges in law, as Q->\infty, to a random Fourier series we explicitly describe, and we prove a convergence in probability result and a classification result for the limiting shapes that explain the visually remarkable properties of these Gauss paths.