Tootfinder

Hilbert spaces admit no finitary discrete imaginaries
Ruiyuan Chen, Isabel Trindade
https://arxiv.org/abs/2509.11321 https://arxiv.org/pdf/2509.11321

Hilbert spaces admit no finitary discrete imaginaries
We prove that every functor from the category of Hilbert spaces and linear isometric embeddings to the category of sets which preserves directed colimits must be essentially constant on all infinite-dimensional spaces. In other words, every finitary set-valued imaginary over the theory of Hilbert spaces, in a broad signature-independent sense, must be essentially trivial. This extends a result and answers a question by Lieberman--Rosický--Vasey, who showed that no such functor on the supercate…

Nef cones of Hilbert schemes of points on some K3 surfaces
Uttaran Dutta, Sean Edwards, Neelarnab Raha
https://arxiv.org/abs/2509.10724 https://arxiv.org/p…

Nef cones of Hilbert schemes of points on some K3 surfaces
Let $X$ be a Mori dream K3 surface of Picard rank $2$. We compute the nef cone of the Hilbert scheme $X^{[n]}$ of $n$ points on $X$ for large $n$ using Bridgeland stability methods. We also find bounds on the nef cone for small $n$. Studying the geometry of a $2$-to-$1$ cover $X\rightarrow Q$ of a smooth quadric $Q$ in $\mathbb{P}^3$, and its induced rational map $X^{[2]}\dashrightarrow Q^{[2]}$ on the Hilbert schemes, we calculate the nef cone of $X^{[2]}$ in a particular case. Finally, we als…

Flow Straight and Fast in Hilbert Space: Functional Rectified Flow
Jianxin Zhang, Clayton Scott
https://arxiv.org/abs/2509.10384 https://arxiv.org/pdf/2509…

Flow Straight and Fast in Hilbert Space: Functional Rectified Flow
Many generative models originally developed in finite-dimensional Euclidean space have functional generalizations in infinite-dimensional settings. However, the extension of rectified flow to infinite-dimensional spaces remains unexplored. In this work, we establish a rigorous functional formulation of rectified flow in an infinite-dimensional Hilbert space. Our approach builds upon the superposition principle for continuity equations in an infinite-dimensional space. We further show that this …

Absolutely Abelian Hilbert Class Fields and $\ell$-torsion conjecture
Mahesh Kumar Ram, Prem Prakash Pandey, Nimish Kumar Mahapatra
https://arxiv.org/abs/2510.10725 https://

Absolutely Abelian Hilbert Class Fields and $\ell$-torsion conjecture
There are several recent works where authors have shown that number fields $K$ with `sufficiently many' units and cyclic class group contain a Euclidean ideal class provided the Hilbert class field $H(K)$ is an abelian extension of $\mathbb{Q}$. In this article, we explore the latter hypothesis: how often a number field $K$ satisfies that its Hilbert class field $H(K)$ is an abelian extension of $\mathbb{Q}$? We also noticed that for such number fields, when the Hilbert class field is an abelia…

C*-submodule preserving module mappings on Hilbert C*-modules
Michael Frank
https://arxiv.org/abs/2507.11206 https://arxiv.org/pdf/25…

C*-submodule preserving module mappings on Hilbert C*-modules
Let $A$ be a (non-unital, in general) C*-algebra with center $Z(M(A))$ of its multiplier algebra, and let $X$ be a full Hilbert $A$-module. Then any injective bounded module morphism $T$, for which every norm-closed $A$-submodule of $X$ is invariant, is of the form $T=c \cdot {\rm id}_X$ where $c \in Z(M(A))$. The same is true if the restriction to be norm-closed is dropped. At the other extrem, for two given strongly Morita equivalent C*-algebras $A$ and $B$ and an $A$-$B$ equivalence bimodule…

Triangular solutions to a Riemann-Hilbert problem from superelliptic curves
Anwar Al Ghabra, Benjamin Pich\'e, Vasilisa Shramchenko
https://arxiv.org/abs/2507.10692

Triangular solutions to a Riemann-Hilbert problem from superelliptic curves
We give fundamental solutions of arbitrarily sized matrix Fuchsian linear systems, in the case where the coefficients $B^{(i)}$ of the systems are matrix solutions of the Schlesinger system that are upper triangular, and whose eigenvalues follow an arithmetic progression of a rational difference. The values on the superdiagonals of the matrices $B^{(i)}$ are given by contour integrals of meromorphic differentials defined on Riemann surfaces obtained by compactification of superelliptic curves.

Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies
H. Aratyn, C. P. Constantinidis, J. F. Gomes, T. C. Santiago, A. H. Zimerman
https://arxiv.org/abs/2507.10744

Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies
We propose a generalized Riemann-Hilbert-Birkhoff decomposition that expands the standard integrable hierarchy formalism in two fundamental ways: it allows for integer powers of Lax matrix components in the flow equations to be increased as compared to conventional models, and it incorporates constant non-zero vacuum (background) solutions. Two additional parameters control these features. The first one defines the grade of a semisimple element that underpins the algebraic construction of the…

Etudes in the inverse spectral problem, II
Nikolai Makarov, Alexei Poltoratski
https://arxiv.org/abs/2509.10699 https://arxiv.org/pdf/2509.10699

Etudes in the inverse spectral problem, II
We apply the approach developed in our previous papers to obtain examples of solutions to the inverse spectral problem (ISP) for the canonical Hamiltonian system. One of our goals is to illustrate connections of ISP with classical tools of analysis, such as the Hilbert transform and solutions to the Riemann-Hilbert problem. A key role in our study is played by the systems with homogeneous and quasi-homogeneous spectral measures. We show how some of such systems give rise to families of Bessel f…

Approximating the operator norm of local Hamiltonians via few quantum states
Lars Becker, Joseph Slote, Alexander Volberg, Haonan Zhang
https://arxiv.org/abs/2509.11979 https://…

Approximating the operator norm of local Hamiltonians via few quantum states
Consider a Hermitian operator $A$ acting on a complex Hilbert space of dimension $2^n$. We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian, its operator norm can be approximated independently of $n$ by maximizing $|\braket{ψ|A|ψ}|$ over a small collection $\mathbf{X}_n$ of product states $\ketψ\in (\mathbf{C}^{2})^{\otimes n}$. More precisely, we show that whenever $A$ is $d$-local, \textit{i.e.,} $°(A)\…

On Absolutely norm (minimum) attaining $2\times 2$ block operator matrix
Puspendu Nag, Ramesh Golla
https://arxiv.org/abs/2507.11148 https://

On Absolutely norm (minimum) attaining $2\times 2$ block operator matrix
In this article, we study absolutely norm attaining operators ($\mathcal{AN}$-operators, in short), that is, operators that attain their norm on every non-zero closed subspace of a Hilbert space. Our focus is primarily on positive $2\times2$ block operator matrices in Hilbert spaces. Subsequently, we examine the analogous problem for operators that attain their minimum modulus on every nonzero closed subspace; these are referred to as absolutely minimum attaining operators (or $\mathcal{AM}$-op…

Active Subspaces in Infinite Dimension
Poorbita Kundu, Nathan Wycoff
https://arxiv.org/abs/2510.11871 https://arxiv.org/pdf/2510.11871

Active Subspaces in Infinite Dimension
Active subspace analysis uses the leading eigenspace of the gradient's second moment to conduct supervised dimension reduction. In this article, we extend this methodology to real-valued functionals on Hilbert space. We define an operator which coincides with the active subspace matrix when applied to a Euclidean space. We show that many of the desirable properties of Active Subspace analysis extend directly to the infinite dimensional setting. We also propose a Monte Carlo procedure and discus…

Orbit method for Quantum Corner Symmetries
Giulio Neri, Ludovic Varrin
https://arxiv.org/abs/2507.10683 https://arxiv.org/pdf/2507.10…

Orbit method for Quantum Corner Symmetries
The classification of the unitary irreducible representations of symmetry groups is a cornerstone of modern quantum physics, as it provides the fundamental building blocks for constructing the Hilbert spaces of theories admitting these symmetries. In the context of gravitational theories, several arguments point towards the existence of a universal symmetry group associated with corners, whose structure is the same for every diffeomorphism-invariant theory in any dimension. Recently, the repr…

Quantum spread complexity as a probe of NSI, $CP$ Violation, and mass ordering in neutrino oscillations in matter
Abhishek Kumar Jha, Govind Krishna G, Anandbhai Pravinbhai Prajapati, Subhashish Banerjee
https://arxiv.org/abs/2509.09854

Quantum spread complexity as a probe of NSI, $CP$ Violation, and mass ordering in neutrino oscillations in matter
Quantum spread complexity characterizes how a quantum state evolves and becomes distributed over the Hilbert space under unitary dynamics. In this work, we employ a cost function as a quantitative measure of spread complexity. We investigate this cost function within the framework of three-flavor neutrino oscillations in vacuum and matter, incorporating the $CP$-violation phase and Non-Standard Interaction (NSI) effects, under both normal and inverted mass ordering scenarios. The cost function …

Second Quantization for the Kepler Problem
John C. Baez
https://arxiv.org/abs/2507.11428 https://arxiv.org/pdf/2507.11428

Second Quantization for the Kepler Problem
The Kepler problem concerns a point particle in an attractive inverse square force. After a brief review of the classical and quantum versions of this problem, focused on their hidden $\text{SU}(2) \times \text{SU}(2)$ symmetry, we discuss the quantum Kepler problem for a spin-$\frac{1}{2}$ particle. We show that the Hilbert space $\mathcal{H}$ of bound states for this problem is unitarily equivalent, as a representation of $\text{SU}(2) \times \text{SU}(2)$, to the Hilbert space of solutions o…

Moduli spaces of generalised tautological bundles on Hilbert schemes
Andreas Krug, Fabian Reede, Ziyu Zhang
https://arxiv.org/abs/2510.11298 https://arxiv.…

Moduli spaces of generalised tautological bundles on Hilbert schemes
We construct new stable vector bundles on Hilbert schemes of points on algebraic surfaces, which are parametrised by connected components of their moduli spaces. This work generalises aspects of our previous work on tautological bundles and of recent work of O'Grady.

Kernel Learning for Mean-Variance Trading Strategies
Owen Futter, Nicola Muca Cirone, Blanka Horvath
https://arxiv.org/abs/2507.10701 https://

Kernel Learning for Mean-Variance Trading Strategies
In this article, we develop a kernel-based framework for constructing dynamic, pathdependent trading strategies under a mean-variance optimisation criterion. Building on the theoretical results of (Muca Cirone and Salvi, 2025), we parameterise trading strategies as functions in a reproducing kernel Hilbert space (RKHS), enabling a flexible and non-Markovian approach to optimal portfolio problems. We compare this with the signature-based framework of (Futter, Horvath, Wiese, 2023) and demonstrat…

Partial Dirac Structures and Dynamical Systems
Fernand Pelletier, Patrick Cabau
https://arxiv.org/abs/2508.10799 https://arxiv.org/pdf/2508.10799

Partial Dirac Structures and Dynamical Systems
In a previous paper (PeCa24), the notion of Dirac structure in finite dimension was extended to the convenient setting. In particular, we introduce the notion of \emph{partial Dirac structure on a convenient manifold} and look for which all geometrical results in finite dimension which are still true in this infinite dimensional framework. Note that this context is justified by many mechanical infinite dimensional examples which recover all the classical ones, such as Hilbert, Banach, Fréchet …

From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma
Mircea Lazar
https://arxiv.org/abs/2508.07494 https://a…

From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma
The generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma are two open problems that play a key role in the development of data-driven control methods for nonlinear systems. Both problems hinge on the construction of observable or basis functions and their corresponding Hilbert space that enable an infinite-dimensional, linear system representation. In this paper we derive a novel solution to these problems based on orthonormal …

On the semi-infinite cohomology of graded-unitary vertex algebras
Christopher Beem, Niklas Garner
https://arxiv.org/abs/2509.10364 https://arxiv.org/pdf/25…

On the semi-infinite cohomology of graded-unitary vertex algebras
Recently, the first author with A. Ardehali, M. Lemos, and L. Rastelli introduced the notion of graded unitarity for vertex algebras. This generalization of unitarity is motivated by the SCFT/VOA correspondence and introduces a novel Hilbert space structure on the state space of a large class of vertex algebras that are not unitary in the conventional sense. In this paper, we study the relative semi-infinite cohomology of graded-unitary vertex algebras that admit a chiral quantum moment map for…

Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE
Dominic Blanco, Matthieu Cadiot
https://arxiv.org/abs/2509.10375 h…

Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE
In this article, we extend the framework developed previously to allow for rigorous proofs of existence of smooth, localized solutions in semi-linear partial differential equations possessing both space and non-space group symmetries. We demonstrate our approach on the Swift-Hohenberg model. In particular, for a given symmetry group $\mathcal{G}$, we construct a natural Hilbert space $H^l_{\mathcal{G}}$ containing only functions with $\mathcal{G}$-symmetry. In this space, products and different…

Bound on entanglement in neural quantum states
Nisarga Paul
https://arxiv.org/abs/2510.11797 https://arxiv.org/pdf/2510.11797

Bound on entanglement in neural quantum states
Variational wavefunctions offer a practical route around the exponential complexity of many-body Hilbert spaces, but their expressive power is often sharply constrained. Matrix product states, for instance, are efficient but limited to area law entangled states. Neural quantum states (NQS) are widely believed to overcome such limitations, yet little is known about their fundamental constraints. Here we prove that feed-forward neural quantum states acting on $n$ spins with $k$ scalar nonlinearit…

Rigged Hilbert Space Formulation of Quantum Thermo Field Dynamics and Mapping to Rigged Liouville Space
Junichi Takahashi, Shousuke Ohmori
https://arxiv.org/abs/2508.06923 https…

Rigged Hilbert Space Formulation of Quantum Thermo Field Dynamics and Mapping to Rigged Liouville Space
The rigged Hilbert space, a triplet extension of the Hilbert space, provides a mathematically rigorous foundation for quantum mechanics by extending the Hilbert space to accommodate generalized eigenstates. In this paper, we construct a triplet structure for the thermal space arising in Thermo Field Dynamics with the aid of the tensor product formulation of rigged Hilbert spaces, a formalism that reformulates thermal averages as pure-state expectation values in a doubled Hilbert space. We then …

Hilbert schemes of elliptic surfaces: group actions and derived categories
David Zhiyuan Bai
https://arxiv.org/abs/2508.09065 https://arxiv.org/pdf/2508.09…

Hilbert schemes of elliptic surfaces: group actions and derived categories
Let $X\to C$ be an elliptic surface with integral fibers and a section. The Hilbert scheme $X^{[n]}$ fibers over $C^{[n]}$. We construct a commutative group scheme over the entire base $C^{[n]}$ that embeds as an open subscheme of the Hilbert scheme, such that its action on itself extends to the entirety of $X^{[n]}$. We show that the action is $δ$-regular in the sense of Ngô. Using the derived McKay correspondence, we construct an exact autoequivalence of $D^b\operatorname{Coh}(X^{[n]})$ who…

The Baby Universe is Fine and the CFT Knows It: On Holography for Closed Universes
Stefano Antonini, Pratik Rath, Martin Sasieta, Brian Swingle, Alejandro Vilar L\'opez
https://arxiv.org/abs/2507.10649

The Baby Universe is Fine and the CFT Knows It: On Holography for Closed Universes
Big bang/big crunch closed universes can be realized in AdS/CFT, even though they lack asymptotically AdS boundaries. With enough bulk entanglement, the bulk Hilbert space of a closed universe can be holographically encoded in the CFT. We clarify the relation of this encoding to observer-clone proposals and refute recent arguments about the breakdown of semiclassical physics in such spaces. In the limit of no bulk entanglement, the holographic encoding breaks down. The oft-cited one-dimensional…

A Schauder Basis for Multiparameter Persistence
Peter Bubenik, Zachariah Ross
https://arxiv.org/abs/2510.10347 https://arxiv.org/pdf/2510.10347

A Schauder Basis for Multiparameter Persistence
Certain classes of multiparameter persistence modules may be encoded as signed barcodes, represented as points in a polyhedral subset of Euclidean space, we refer to as signed persistence diagrams. These signed persistence diagrams exist in the dual space of compactly supported, Lipschitz functionals on a polyhedral pair. In the interest of statistics and machine learning on multiparameter persistence modules, we aim to embed these signed persistence diagrams into Banach or Hilbert space. We us…

Characterizations of dual of K-frames in quaternionic Hilbert spaces
Chander Shekhar
https://arxiv.org/abs/2508.08263 https://arxiv.org/pdf/2508.08263

Characterizations of dual of K-frames in quaternionic Hilbert spaces
Dual of K-frames in a right quaternionic Hilbert space has been recently introduced and studied by Ellouz[1]. In this paper, we study duals of K-frames and prove a characterization of a K-dual in terms of the canonical K-dual of a K-frame in the case when the range of K is not necessarily closed. Moreover, we defined and studied the Approximate K- Dual of a K-frame in a right quaternionic Hilbert space and proved various results for their existence.

QPOs in a highly magnetized ultra-compact X-ray binary 4U 1626-67
Zi-Yi Zhou, Long Ji, Ling-Da Kong, Sergey S. Tsygankov, Qing-Cang Shui, Lian Tao
https://arxiv.org/abs/2510.09986

QPOs in a highly magnetized ultra-compact X-ray binary 4U 1626-67
We report the detection of mHz quasi-periodic oscillations (QPOs) in four NuSTAR observations of 4U 1626-67 during its recent spin-down episode. By using a novel method based on the Hilbert-Huang Transform (HHT), we present the first QPO-phase-resolved timing and spectral analysis of accreting X-ray pulsars in low mass X-ray binaries. Broadband QPO waveforms have been reconstructed and exhibit approximately sinusoidal shapes, with fractional amplitudes that vary with energy. In addition, we fin…

Convergence Analysis of Galerkin Approximations for the Lindblad Master Equation
R\'emi Robin, Pierre Rouchon
https://arxiv.org/abs/2510.11416 https://…

Convergence Analysis of Galerkin Approximations for the Lindblad Master Equation
This paper analyzes the numerical approximation of the Lindblad master equation on infinite-dimensional Hilbert spaces. We employ a classical Galerkin approach for spatial discretization and investigate the convergence of the discretized solution to the exact solution. Using \textit{a priori} estimates, we derive explicit convergence rates and demonstrate the effectiveness of our method through examples motivated by autonomous quantum error correction.

On S-Split p-Hilbert Class Field Towers with Prescribed Galois Groups
Christian Maire (FEMTO-ST, UMLP), Karim Sankara (UNB)
https://arxiv.org/abs/2508.07946 https://

On S-Split p-Hilbert Class Field Towers with Prescribed Galois Groups
In this work, we show that given a finite p-group G, a number field K having a trivial p-class group Cl K , and a finite set of primes S of K, there exists a finite extension F/K such that the S-split p-Hilbert class field tower L S p (F ) of F has G as its Galois group. This extends results by Ozaki and Hajir-Maire-Ramakrishna.

When are two algorithms the same? Towards addressing Hilbert's 24th problem
Konstantin Doubrovinski
https://arxiv.org/abs/2508.02764 https://arxiv.org/…

When are two algorithms the same? Towards addressing Hilbert's 24th problem
The informal question of when two theorem proofs are "essentially the same" goes back to David Hilbert, who considered adding it (or something largely equivalent) to his famous list of open problems, but eventually decided to leave it out. Given that the notion of a formal proof is closely related to that of a (computer) program, i.e. a recursive function, it may be useful to ask the same question with regard to programs instead. Here we propose a minimalistic approach to this question within R…

Efficient Preparation of Decoherence Free Subspace Basis States
Zi-Ming Li, Yu-xi Liu
https://arxiv.org/abs/2509.11544 https://arxiv.org/pdf/2509.11544

Efficient Preparation of Decoherence Free Subspace Basis States
Decoherence-free subspace (DFS) provides a crucial mechanism for passive error mitigation in quantum computation by encoding information within symmetry-protected subspaces of the Hilbert space, which are immune from collective decoherence. Constructing a complete set of orthogonal basis states for the DFS is essential to realize fault-tolerant quantum computation by using the DFS codes. However, existing methods for preparing these basis states are often non-scalable, platform-specific, or yie…

A Rigorous Floquet Approach to Parabolic PDEs, with Applications to Walking Droplet Statistics
Skyler Mao, David Darrow
https://arxiv.org/abs/2508.09506 https://

A Rigorous Floquet Approach to Parabolic PDEs, with Applications to Walking Droplet Statistics
Floquet theory provides a powerful tool for studying time-periodic dynamics, allowing the stability and long-time behavior of any periodic, linear ODE to be recast in terms of a linear time-invariant system. However, except when dealing with Hilbert spaces or certain well-behaved classes of operators, Floquet theory cannot be rigorously applied to PDEs without dimensional reduction. We describe a simple, rigorous extension of Floquet theory to a wide class of parabolic PDEs on Banach spaces, en…

Limit cycles and invariant algebraic curves
Armengol Gasull, Paulo Santana
https://arxiv.org/abs/2510.11705 https://arxiv.org/pdf/2510.11705

Limit cycles and invariant algebraic curves
We give lower bounds in terms of~$n,$ for the number of limit cycles of polynomial vector fields of degree~$n,$ having any prescribed invariant algebraic curve. By applying them when the ovals of this curve are also algebraic limit cycles we obtain a new recurrent property for the Hilbert numbers. Finally, we apply our results to two important families of models: Kolmogorov systems and a general family of systems appearing in Game Theory.

Actions of Fell bundles
Erik B\'edos, Roberto COnti
https://arxiv.org/abs/2509.10822 https://arxiv.org/pdf/2509.10822

Actions of Fell bundles
We introduce and study actions of Fell bundles over discrete groups on Hilbert bundles. Many examples of such actions are presented. We discuss the connection with positive definite bundle maps between Fell bundles, culminating in the unital case in a Gelfand-Raikov type theorem. We also use these actions to construct C*-correspondences over cross-sectional C*-algebras of Fell bundles.

Hadamard-L\'{e}vy theorems for maps taking values in a finite dimensional space
Yacine Chitour, Zhengping Ji, Emmanuel Tr\'elat
https://arxiv.org/abs/2510.11206 https://…

Hadamard-Lévy theorems for maps taking values in a finite dimensional space
We propose global surjectivity theorems of differentiable maps based on second order conditions. Using the homotopy continuation method, we demonstrate that, for a $C^2$ differentiable map from a Hilbert space to a finite-dimensional Euclidean space, when its second-order differential has uniform upper and lower bounds, it has a global path-lifting property in the presence of singularities. This is then applied to the nonlinear motion planning problem, establishing in some cases the well-posedn…

Compositional Concept Generalization with Variational Quantum Circuits
Hala Hawashin, Mina Abbaszadeh, Nicholas Joseph, Beth Pearson, Martha Lewis, Mehrnoosh sadrzadeh
https://arxiv.org/abs/2509.09541 …

Compositional Concept Generalization with Variational Quantum Circuits
Compositional generalization is a key facet of human cognition, but lacking in current AI tools such as vision-language models. Previous work examined whether a compositional tensor-based sentence semantics can overcome the challenge, but led to negative results. We conjecture that the increased training efficiency of quantum models will improve performance in these tasks. We interpret the representations of compositional tensor-based models in Hilbert spaces and train Variational Quantum Circu…

Kernel Two-Sample Testing via Directional Components Analysis
Rui Cui, Yuhao Li, Xiaojun Song
https://arxiv.org/abs/2508.08564 https://arxiv.org/pdf/2508.0…

Kernel Two-Sample Testing via Directional Components Analysis
We propose a novel kernel-based two-sample test that leverages the spectral decomposition of the maximum mean discrepancy (MMD) statistic to identify and utilize well-estimated directional components in reproducing kernel Hilbert space (RKHS). Our approach is motivated by the observation that the estimation quality of these components varies significantly, with leading eigen-directions being more reliably estimated in finite samples. By focusing on these directions and aggregating information a…

Topological Vibration Analysis of Elastic Lattices via Bloch Sphere Mapping
Kazi Tahsin Mahmood, M. Arif Hasan
https://arxiv.org/abs/2510.11930 https://arx…

Topological Vibration Analysis of Elastic Lattices via Bloch Sphere Mapping
Mechanical lattices support topological wave phenomena governed by geometric phases. We develop a compact Hilbert space description for one-dimensional elastic chains, expressing intra-cell motion as a normalized superposition of orthogonal eigenstates and tracking complex amplitudes as trajectories on a Bloch sphere. For diatomic lattices, this framework makes inversion symmetry protection explicit: the relative phase between in-phase and out-of-phase modes is piecewise locked, and the Zak pha…

A Hilbert Series for Generalized Toric Polygons
Ignacio Carre\~no Bolla, Sebasti\'an Franco, Diego Rodr\'iguez-G\'omez
https://arxiv.org/abs/2509.08876 https://

A Hilbert Series for Generalized Toric Polygons
We study the Hilbert series for $5d$ Superconformal Field Theories (SCFTs) engineered by Generalized Toric Polygons (GTPs), which extend the geometric realization of these theories from toric Calabi-Yau 3-folds to theories associated to general webs of 5- and 7-branes. Smoothed T-cones provide fundamental building blocks of GTP tessellations, generalizing the role of minimal triangles in toric diagrams. Building on this construction, we propose an extension of the Martelli-Sparks-Yau algorithm …

Quantum Functional Information through the Evolution of Random Circuits
Rodrigo Pasti, Jonas Krause
https://arxiv.org/abs/2509.11409 https://arxiv.org/pdf/…

Quantum Functional Information through the Evolution of Random Circuits
We introduce Quantum Functional Information (QFI), a new metric to quantify the rarity and utility of quantum states and circuits. Unlike standard measures such as fidelity or entropy, QFI captures the balance between functionality and rarity within the Hilbert space. We validate QFI through two approaches: random circuit sampling and evolutionary algorithms guided by fidelity or QFI objectives. Random sampling shows that states with near-perfect fidelity are less informational than s…

The Levelwise Finite Generation of Free Tambara Functors
Emory Sun
https://arxiv.org/abs/2508.08340 https://arxiv.org/pdf/2508.08340

The Levelwise Finite Generation of Free Tambara Functors
We prove the levelwise finite generation of free polynomial $G$-Tambara functors in a collection of cases, most notably when $G$ is a finite Dedekind group or when $G \cong C_p \rtimes C_q$, $p > q$ primes. In the process, we establish the permanence of various finiteness conditions under box products and norms $n_H^G$ of Tambara functors, including a weak Hilbert Basis Theorem.

More birational involutions
Pietro Beri, Laurent Manivel
https://arxiv.org/abs/2509.10130 https://arxiv.org/pdf/2509.10130…

More birational involutions
For $S$ a very general polarized K3 surface of degree $8n-6$, we describe in geometrical terms a birational involution of the Hilbert scheme $S^{[n]}$ of $n$ points on the surface, whose existence was established from lattice theoretical considerations. In a previous work we studied this involution for $n=3$, with the help of the exceptional Lie group $G_2$, since the Mukai model of $S$ is embedded in its projectivized Lie algebra. Here we use different, more general arguments to show that some…

Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant
Malte Gerhold, Marcel Scherer, Orr Shalit
https://arxiv.org/abs/2510.12540

Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant
For a tuple $T$ of Hilbert space operators, the 'commuting dilation constant' is the smallest number $c$ such that the operators of $T$ are a simultaneous compression of commuting normal operators of norm at most $c$. We present numerical experiments giving a strong indication that the commuting dilation constant of a pair of independent random $N{\times}N$ unitary matrices converges to $\sqrt2$ as $N \to \infty$ almost surely. Under the assumption that this is the case, we prove that the commu…

The index of a pair of pure states and the interacting integer quantum Hall effect
Sven Bachmann, Jacob Shapiro, Cl\'ement Tauber
https://arxiv.org/abs/2507.10807

The index of a pair of pure states and the interacting integer quantum Hall effect
We introduce the index $\mathcal{N}(ω_1,ω_2)$ of a pair of pure states on a unital C*-algebra, which is a generalization of the notion of the index of a pair of projections on a Hilbert space. We then show that the Hall conductance associated with an invertible state $ω$ of a two-dimensional interacting electronic system which is symmetric under $U(1)$ charge transformation may be written as the index $\mathcal{N}(ω,ω_D)$, where $ω_D$ is obtained from $ω$ by inserting a unit of magnetic …

Replaced article(s) found for math.OA. https://arxiv.org/list/math.OA/new
[1/1]:
- The matched projections of idempotents on Hilbert $C^*$-modules
Xiaoyi Tian, Qingxiang Xu, Chunhong Fu

Deciding Whether a C-Q Channel Preserves a Bit is QCMA-Complete
Kiera Hutton, Arthur Mehta, Andrej Vukovic
https://arxiv.org/abs/2508.10664 https://arxiv.o…

Deciding Whether a C-Q Channel Preserves a Bit is QCMA-Complete
We prove that deciding whether a classical-quantum (C-Q) channel can exactly preserve a single classical bit is QCMA-complete. This "bit-preservation" problem is a special case of orthogonality-constrained optimization tasks over C-Q channels, in which one seeks orthogonal input states whose outputs have small or large Hilbert-Schmidt overlap after passing through the channel. Both problems can be cast as biquadratic optimization with orthogonality constraints. Our main technical contribution u…

Mean-square and linear convergence of a stochastic proximal point algorithm in metric spaces of nonpositive curvature
Nicholas Pischke
https://arxiv.org/abs/2510.10697 https://

Mean-square and linear convergence of a stochastic proximal point algorithm in metric spaces of nonpositive curvature
We define a stochastic variant of the proximal point algorithm in the general setting of nonlinear (separable) Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field and prove its convergence under a suitable strong monotonicity assumption, together with a probabilistic independence assumption and a separability assumption on the tangent spaces. As a particular case, our results transfer previous work by P. Bianchi on that method in Hilbert space…

Introduction to quantitative De Giorgi methods
Giovanni Brigati, Cl\'ement Mouhot
https://arxiv.org/abs/2510.11481 https://arxiv.org/pdf/2510.11481

Introduction to quantitative De Giorgi methods
The theory of De Giorgi (1958) and Nash (1959) solves Hilbert's 19th problem and constitutes a major advance in the analysis of PDEs in the 20th century. This theory concerns the Hölder regularity of solutions to elliptic and parabolic equations with non-regular coefficients, and it was extended by Moser (1960) to include the Harnack inequality. This course reviews the classical De Giorgi method in the elliptic and parabolic cases and introduces its recent extension to hypoelliptic equations w…

Kernel VICReg for Self-Supervised Learning in Reproducing Kernel Hilbert Space
M. Hadi Sepanj, Benyamin Ghojogh, Paul Fieguth
https://arxiv.org/abs/2509.07289 https://

Kernel VICReg for Self-Supervised Learning in Reproducing Kernel Hilbert Space
Self-supervised learning (SSL) has emerged as a powerful paradigm for representation learning by optimizing geometric objectives--such as invariance to augmentations, variance preservation, and feature decorrelation--without requiring labels. However, most existing methods operate in Euclidean space, limiting their ability to capture nonlinear dependencies and geometric structures. In this work, we propose Kernel VICReg, a novel self-supervised learning framework that lifts the VICReg objective…

On Mori dreamness of blowups along space curves
Tiago Duarte Guerreiro, Sokratis Zikas
https://arxiv.org/abs/2509.10148 https://arxiv.org/pdf/2509.10148

On Mori dreamness of blowups along space curves
We study the problem of determining when the blowup $X \to \mathbb{P}^3$ along a smooth space curve $C$ is a Mori Dream Space. We obtain sufficient conditions, as well obstructions to the Mori dreamness of $X$ based on the external geometry of $C$. We furthermore find infinitely many pairs $(g,d)$ such that the corresponding Hilbert schemes $H_{g,d}^S$ admit components whose general element has these obstructions. As a consequence we show that Mori dreamness is not an open property in flat fami…

Towards Fock Spaces in Hypercomplex Analysis
Kamal Diki
https://arxiv.org/abs/2510.10688 https://arxiv.org/pdf/2510.10688

Towards Fock Spaces in Hypercomplex Analysis
The Bargmann-Fock space(or Fock space for short) is a fundamental example of reproducing kernel Hilbert spaces that has found fascinating applications across multiple fields of current interest, including quantum mechanics, time-frequency analysis, mathematical analysis, and stochastic processes. In recent years, there has been increased interest in studying counterparts of the Fock space and related topics in hypercomplex analysis. This chapter presents a survey exploring various aspects of th…

Algorithm for constructing optimal explicit finite-difference formulas in the Hilbert space
R. S. Karimov, D. D. Atoev
https://arxiv.org/abs/2510.06643 https://

Algorithm for constructing optimal explicit finite-difference formulas in the Hilbert space
This work presents problems of constructing finite-difference formulas in the Hilbert space, i.e., setting problems of constructing finite-difference formulas using functional methods. The work presents a functional statement of the problem of optimizing finite-difference formulas in the space $W_{2}^{\left(m,m-1\right)} \left(0,1\right)$. Here, representations of optimal coefficients of explicit finite-difference formulas of the Adams type on classes $W_{2}^{\left(m,m-1\right)} \left(0,1\right…

Observation of quantum-field-theory dynamics on a spin-phonon quantum computer
Anton T. Than, Saurabh V. Kadam, Vinay Vikramaditya, Nhung H. Nguyen, Xingxin Liu, Zohreh Davoudi, Alaina M. Green, Norbert M. Linke
https://arxiv.org/abs/2509.11477

Observation of quantum-field-theory dynamics on a spin-phonon quantum computer
Simulating out-of-equilibrium dynamics of quantum field theories in nature is challenging with classical methods, but is a promising application for quantum computers. Unfortunately, simulating interacting bosonic fields involves a high boson-to-qubit encoding overhead. Furthermore, when mapping to qubits, the infinite-dimensional Hilbert space of bosons is necessarily truncated, with truncation errors that grow with energy and time. A qubit-based quantum computer, augmented with an active boso…

Anchored Implication & Event-Indexed Fixed Points in Hilbert Spaces: Uniqueness and Quantitative Rates
Faruk Alpay, Bugra Kilictas, Taylan Alpay
https://arxiv.org/abs/2508.08397

Anchored Implication & Event-Indexed Fixed Points in Hilbert Spaces: Uniqueness and Quantitative Rates
We develop a synthesis of orthomodular logic (projections as propositions) with operator fixed-point theory in Hilbert spaces. First, we introduce an anchored implication connective $A \Rightarrow^{\mathrm{comm}}_{P} B$, defined semantically so that it is true only when either $A$ is false or else $A$ is true and $B$ is true in a ''commuting'' context specified by a fixed nonzero projection $P$. This connective refines material implication by adding a side condition $[E_B,P]=0$ (commutation of …

Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions
Jeanine Van Order
https://arxiv.org/abs/2510.10303 https://

Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions
We give two distinct proofs of the Gross-Zagier formula in terms of sums of automorphic Green's functions realized as regularized theta lifts, including one involving arithmetic Hirzebruch-Zagier divisors on the Hilbert modular surface $X_0(N) \times X_0(N)$. We then describe applications to the refined conjecture of Birch and Swinnerton-Dyer. Through these calculations, we also describe known and conjectural relations of the central derivative values of Rankin-Selberg $L$-functions that appear…

STRIDE: Scalable and Interpretable XAI via Subset-Free Functional Decomposition
Chaeyun Ko
https://arxiv.org/abs/2509.09070 https://arxiv.org/pdf/2509.0907…

STRIDE: Scalable and Interpretable XAI via Subset-Free Functional Decomposition
Most explainable AI (XAI) frameworks face two practical limitations: the exponential cost of reasoning over feature subsets and the reduced expressiveness of summarizing effects as single scalar values. We present STRIDE, a scalable framework that aims to mitigate both issues by framing explanation as a subset-enumeration-free, orthogonal functional decomposition in a Reproducing Kernel Hilbert Space (RKHS). Rather than focusing only on scalar attributions, STRIDE computes functional components…

Gravitational Hilbert spaces: invariant and co-invariant states, inner products, gauge-fixing, and BRST
Jesse Held, Henry Maxfield
https://arxiv.org/abs/2509.05412 https://

Gravitational Hilbert spaces: invariant and co-invariant states, inner products, gauge-fixing, and BRST
Hilbert spaces in theories of gravity are notoriously subtle due to the Hamiltonian constraints, particularly regarding the inner product. To demystify this subject, we review and extend a collection of ideas in canonical gravity, and connect to the sum-over-histories approach by clarifying the Hilbert space interpretation of various gravitational path integrals. We use one-dimensional (or mini-superspace) models as the simplest context to exemplify the conceptual ideas. We emphasise that a phy…

HilbertA: Hilbert Attention for Image Generation with Diffusion Models
Shaoyi Zheng, Wenbo Lu, Yuxuan Xia, Haomin Liu, Shengjie Wang
https://arxiv.org/abs/2509.26538 https://

HilbertA: Hilbert Attention for Image Generation with Diffusion Models
Designing sparse attention for diffusion transformers requires reconciling two-dimensional spatial locality with GPU efficiency, a trade-off that current methods struggle to achieve. Existing approaches enforce two-dimensional spatial locality but often incur uncoalesced memory access. We present HilbertA, a 2D-aware and GPU-efficient sparse attention mechanism. HilbertA reorders image tokens along Hilbert curves to achieve a contiguous memory layout while preserving spatial neighborhoods, and …

Dichotomies for triangular systems on Hilbert spaces
Davor Dragicevic, Kenneth J. Palmer, Boris Petkovic
https://arxiv.org/abs/2508.04320 https://arxiv.org…

Dichotomies for triangular systems on Hilbert spaces
In this article, we study the relationship between the exponential dichotomy properties of a triangular system of linear difference equations and its associated diagonal system on Hilbert spaces. We stress that all previous results in this direction were restricted to the finite-dimensional case. As in the previous work of the first two authors, we rely on the relationship between exponential dichotomies and the so-called admissibility properties. However, this approach requires nontrivial chan…

The tensor product of p-adic Hilbert spaces
Paolo Aniello, Lorenzo Guglielmi, Stefano Mancini, Vincenzo Parisi
https://arxiv.org/abs/2510.07504 https://arx…

The tensor product of p-adic Hilbert spaces
In the framework of quantum mechanics over a quadratic extension of the ultrametric field of p-adic numbers, we introduce a notion of tensor product of p-adic Hilbert spaces. To this end, following a standard approach, we first consider the algebraic tensor product of p-adic Hilbert spaces. We next define a suitable norm on this linear space. It turns out that, in the p-adic framework, this norm is the analogue of the projective norm associated with the tensor product of real or complex normed …

Random Nonlinear Fusion Frames from Averaged Operator Iterations
James Tian
https://arxiv.org/abs/2509.09927 https://arxiv.org/pdf/2509.09927

Random Nonlinear Fusion Frames from Averaged Operator Iterations
We study random iterations of averaged operators in Hilbert spaces and prove that the associated residuals converge exponentially fast, both in expectation and almost surely. Our results provide quantitative bounds in terms of a single geometric parameter, giving sharp control of convergence rates under minimal assumptions. As an application, we introduce the concept of random nonlinear fusion frames. Here the atoms are generated dynamically from the residuals of the iteration and yield exact s…

A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization
Hideaki Kim, Tomoharu Iwata
https://arxiv.org/abs/2510.08916 https://ar…

A Representer Theorem for Hawkes Processes via Penalized Least Squares Minimization
The representer theorem is a cornerstone of kernel methods, which aim to estimate latent functions in reproducing kernel Hilbert spaces (RKHSs) in a nonparametric manner. Its significance lies in converting inherently infinite-dimensional optimization problems into finite-dimensional ones over dual coefficients, thereby enabling practical and computationally tractable algorithms. In this paper, we address the problem of estimating the latent triggering kernels--functions that encode the interac…

On the geometry of punctual Hilbert schemes on singular curves
Mounir Hajli, Hussein Mourtada, Wenhao Zhu
https://arxiv.org/abs/2509.06761 https://arxiv.or…

On the geometry of punctual Hilbert schemes on singular curves
Inspired by the work of Soma and Watari, we define a tree structure on the sub-semimodule of a semi-group $Γ$ associated an irreducible curve singularity $(C,O).$ Building on the results of Oblomkov, Rasmussen and Shende, we show that for certain singularities, this tree encodes some aspects of the geometric structure of the punctual Hilbert schemes of $(C,O).$ As an application, we compute the motivic Hilbert zeta function for some singular curves. A point of the Hilbert scheme corresponds to…

$L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings
Jeanine Van Order
https://arxiv.org/abs/2510.10277 https://

$L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings
We derive new integral presentations for central derivative values of $L$-functions of elliptic curves defined over the rationals, basechanged to a real quadratic field $K$, twisted by ring class characters of $K$ in terms of sums along ``geodesics" corresponding to the class group of $K$ of automorphic Green's functions for certain Hirzebruch-Zagier-like arithmetic divisors on Hilbert modular surfaces. We also relate these sums to Birch-Swinnerton-Dyer constants and periods.

Frame redundancy and Beurling density
Marcin Bownik, Jordy Timo van Velthoven
https://arxiv.org/abs/2509.11887 https://arxiv.org/pdf/2509.11887

Frame redundancy and Beurling density
We show that the frame measure function of a frame in certain reproducing kernel Hilbert spaces on metric measure spaces is given by the reciprocal of the Beurling density of its index set. In addition, we show that each such frame with Beurling density greater than one contains a subframe with Beurling density arbitrary close to one. This confirms that the concept of frame measure function as introduced by Balan and Landau is a meaningful quantitative definition for the redundancy of a large c…

Newton-Okounkov bodies for nested Hilbert schemes
Ian Cavey, Eugene Gorsky, Alexei Oblomkov, Joshua P. Turner
https://arxiv.org/abs/2510.07420 https://arxi…

Newton-Okounkov bodies for nested Hilbert schemes
We study sections of line bundles on the nested Hilbert scheme of points on the affine plane. We describe the spaces of sections in terms of certain ideals introduced by Haiman, and find explicit bases for them by analyzing the trailing terms in some monomial order. As a consequence, we compute the Newton-Okounkov bodies for nested Hilbert schemes.

B-RNS-GSS formalism and $L_{\infty}$-actions
Andrei Mikhailov
https://arxiv.org/abs/2510.10400 https://arxiv.org/pdf/2510.10400

B-RNS-GSS formalism and $L_{\infty}$-actions
Pure spinor formalism and RNS formalism are related by a chain of equivalences constructed by introducing and integrating-out BRST quartets. This is known as B-RNS-GSS formalism. One of the steps can be understood as adding auxiliary fields to lift a strong homotopy action of the SUSY Lie superalgebra in the large Hilbert space to a strict action. We develop a general prescription for this ``strictification'' procedure, which can be applied for any strong homotopy action of a Lie superalgebra. …

On $\alpha$-monotone operators and their resolvent in Banach spaces
Changchi Huang, Jigen Peng, Yuchao Tang
https://arxiv.org/abs/2510.12538 https://arxiv.…

On $α$-monotone operators and their resolvent in Banach spaces
This paper introduces a new definition of $α$-monotone operators in real 2-uniformly convex and smooth Banach spaces. Based on this new definition, we establish several novel structural and analytical properties of such operators, which not only extend classical results from Hilbert spaces but also reveal new insights into the geometry of Banach spaces. In particular, we examine the resolvent of $α$-maximal monotone operators and demonstrate how its behavior is consistent with, and generalize…

Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces
Septimus Boshoff, Sebastian Peitz, Stefan Klus
https://arxiv.org/abs/2509.22482 https://ar…

Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces
The Koopman operator, as a linear representation of a nonlinear dynamical system, has been attracting attention in many fields of science. Recently, Koopman operator theory has been combined with another concept that is popular in data science: reproducing kernel Hilbert spaces. We follow this thread into Gaussian process methods, and illustrate how these methods can alleviate two pervasive problems with kernel-based Koopman algorithms. The first being sparsity: most kernel methods do not scale…

Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$
Nobu Kishimoto, Kiyeon Lee
https://arxiv.org/abs/2508.08647 https://a…

Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$
In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schrödinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with derivative terms and a non-local term arising from the Hilbert transform, modeling interactions in plasma physics. We establish global existence for small initial data in the weighted Sobolev space $H^2 \cap H^{1,1}$ and optimal time decay effect. Using energy …

Generalized probabilistic approximation characteristic based on Birkhoff orthogonality and related conclusions in $S_{\infty}$-norm
Weiye Zhang, Chong Wang, Huan Li
https://arxiv.org/abs/2509.10458

Generalized probabilistic approximation characteristic based on Birkhoff orthogonality and related conclusions in $S_{\infty}$-norm
In this article, we generalize the definition of the probabilistic Gel'fand width from the Hilbert space to the strictly convex reflexive space by giving Birkhoff left orthogonal decomposition theorem. Meanwhile, a more natural definition of Gel'fand width in the classical setting is selected to make sure probabilistic and average Gel'fand widths will not lose their meaning, so that we can give the equality relation between the probabilistic Gel'fand width and the probabilistic linear width of …

Local Equations for Hilbert Schemes of Points
Nathan Ilten, Francesco Meazzini, Andrea Petracci
https://arxiv.org/abs/2510.07066 https://arxiv.org/pdf/2510…

Local Equations for Hilbert Schemes of Points
We compute the completion of the local ring of the Hilbert scheme of degree $n+1$ subschemes of $\mathbb{A}^n$ at the point corresponding to the ideal $\langle x_1,\ldots,x_n\rangle^2$, and describe the completion of the universal family. For the purposes of comparison, we do this computation with both classical and DGLA methods. We use our explicit equations to produce high dimensional linear subspaces of the Hilbert scheme, and compare our equations with those coming from deformations of base…

The phi-Process: Operator-Algebraic Embeddings of Possibilities, Transfinite Stabilization, and a Quantitative Application to Sensory Depletion
Bugra Kilictas, Faruk Alpay
https://arxiv.org/abs/2508.10650

The phi-Process: Operator-Algebraic Embeddings of Possibilities, Transfinite Stabilization, and a Quantitative Application to Sensory Depletion
We formalize a transfinite Phi process that treats all possibility embeddings as operators on structured state spaces including complete lattices, Banach and Hilbert spaces, and orthomodular lattices. We prove a determinization lemma showing that lifting to sets or distributions yields a deterministic global dynamic, an ordinal stabilization theorem sending operator transforms to the fixed subspace by stage omega under normal spectral contraction, and a product of Riesz projections theorem for …

Explicit Hecke eigenform product identities for Hilbert modular forms
Zeping Hao, Chao Qin, Yang Zhou
https://arxiv.org/abs/2508.03071 https://arxiv.org/pd…

Explicit Hecke eigenform product identities for Hilbert modular forms
Let $F$ be a totally real number field, and $g,f,h$ be Hilbert modular forms over $F$ that are Hecke eigenforms satisfying $g=f\cdot h$. We characterize such product identities among all real quadratic fields of narrow class number one, proving they occur only for $F=\mathbb Q(\sqrt{5})$, with precisely two such identities. We also shed some light on the general totally real case by showing that no such identity exists when both $f$ and $h$ are Eisenstein series of distinct weights.

$C^1$ perturbations of a continuum of critical points
Rafael Ortega, Antonio J. Urena
https://arxiv.org/abs/2509.10457 https://arxiv.org/pdf/2509.10457

$C^1$ perturbations of a continuum of critical points
Given a real valued function having a nondegenerate compact manifold of critical points, some of these points survive under small $C^2$ perturbations. This is a well-known result in critical point theory. In 1986 Weinstein obtained the analogous conclusions when the perturbation is only $C^2$ and the ambient space is a finite dimensional manifold. In this work we present a complete proof for $C^1$ perturbations in infinite dimensional Hilbert spaces.

Continuous Linear Series
Eduardo Esteves, Antonio Nigro, Pedro Rizzo
https://arxiv.org/abs/2510.11623 https://arxiv.org/pdf/2510.11623

Continuous Linear Series
We parameterize by a fine moduli space all degenerations of linear series to a singular curve which is the union of two smooth components meeting transversally at a single point. For this we introduce a novel object in the study of degenerations of linear series, which is the continuous linear series. Our moduli space can be regarded as a Hilbert quotient, in the terminology introduced by Kapranov, and is a new compactification of Osserman moduli space of exact limit linear series, and conseque…

Effective approach to open systems with probability currents and the Grothendieck formalism
A. Vourdas
https://arxiv.org/abs/2509.07882 https://arxiv.org/p…

Effective approach to open systems with probability currents and the Grothendieck formalism
An effective approach to open systems and irreversible phenomena is presented, where an open system $Σ(d)$ with $d$-dimensional Hilbert space, is a subsystem of a larger isolated system $Σ(2d)$ (the `full universe') with $2d$-dimensional Hilbert space. A family of Bargmann-like representations (called $z$-Bargmann representations) introduces naturally the larger space. The $z$-Bargmann representations are defined through semi-unitary matrices (which are a coherent states formalism in disguise…

Random Operator-Valued Kernels and Moment Dilations
James Tian
https://arxiv.org/abs/2508.10321 https://arxiv.org/pdf/2508.10321

Random Operator-Valued Kernels and Moment Dilations
This paper studies random operator-valued positive definite (p.d.) kernels and their connection to moment dilations. A class of random p.d. kernels is introduced in which the positivity requirement is imposed only in expectation, extending the deterministic case. Basic properties are established, including convexity, sampling stability, and the relationship between mean kernels and deterministic ones. A Gaussian factorization result is proved under a Hilbert-Schmidt/trace condition, clarifying …

Replaced article(s) found for math.FA. https://arxiv.org/list/math.FA/new
[1/1]:
- On the Probabilistic Approximation in Reproducing Kernel Hilbert Spaces
Dongwei Chen, Kai-Hsiang Wang

Multiscale Hilbert expansion and hydrodynamics beyond Navier-Stokes
O. M. Chekmareva, I. B. Chekmarev
https://arxiv.org/abs/2509.02580 https://arxiv.org/pd…

Multiscale Hilbert expansion and hydrodynamics beyond Navier-Stokes
A truncated Hilbert expansion with multiple scales is used to construct asymptotic solution of the linearized one-dimensional Boltzmann equation with small Knudsen number. The freedom ganed by introducing new independent time variables is exploited to eliminate the sources of secular behavior as the time interval increases. The regular gas-dynamic equations then appear as necessary conditions for expansion to be uniformly valid on a long time intervals.

Testing Quantum Mechanics with Quantum Computers: Qubit Information Capacity
Tim Palmer
https://arxiv.org/abs/2510.02877 https://arxiv.org/pdf/2510.02877…

Testing Quantum Mechanics with Quantum Computers: Qubit Information Capacity
Motivated by John Wheeler's assertion that the continuum nature of Hilbert Space conceals the information-theoretic nature of the quantum wavefunction, a specific discretisation of complex Hilbert Space is proposed, leading to the notion of qubit information capacity $N_{\mathrm{max}}$: for any $N \ge N_{\mathrm{max}}$-qubit state, there is insufficient information in the $N$ qubits to allocate even one bit to each of the $2^{N+1}-2$ degrees of freedom demanded by complex Hilbert Space and henc…

Strongly continuous fields of operators over varying Hilbert spaces
Ali BenAmor, Batu G\"uneysu, Thomas Kalmes, Peter Stollmann
https://arxiv.org/abs/2509.07479 https://

Strongly continuous fields of operators over varying Hilbert spaces
After introducing a natural notion of continuous fields of locally convex spaces, we establish a new theory of strongly continuous families of possibly unbounded self-adjoint operators over varying Hilbert spaces. This setting allows to treat operator families defined on bundles of Hilbert spaces that are not locally trivial (such as e.g.~the tangent bundle of Wasserstein space), without referring to identification operators at all.

A penalty-free quantum algorithm to find energy eigenstates
Nannan Ma, Heng Dai, Jiangbin Gong
https://arxiv.org/abs/2509.09148 https://arxiv.org/pdf/2509.…

A penalty-free quantum algorithm to find energy eigenstates
Finding eigenstates of a given many-body Hamiltonian is a long-standing challenge due to the perceived computational complexity. Leveraging on the hardware of a quantum computer accommodating the exponential growth of the Hilbert space size with the number of qubits, more quantum algorithms to find the eigenstates of many-body Hamiltonians will be of wide interest with profound implications and applications. In this work, we advocate a quantum algorithm to find the ground state and excited stat…

Direct integral of locally Hilbert spaces
Chaitanya J. Kulkarni, Santhosh Kumar Pamula
https://arxiv.org/abs/2508.03407 https://arxiv.org/pdf/2508.03407

Direct integral of locally Hilbert spaces
In this work, we introduce the concept of the direct integral of locally Hilbert spaces. This notion is formulated such that the direct integral of locally Hilbert spaces forms a locally Hilbert space. We then define two classes of locally bounded operators on the direct integral of locally Hilbert spaces namely the class of decomposable locally bounded operators and the class of diagonalizable locally bounded operators. We prove that the set of all decomposable locally bounded operators forms …

Hilbert scheme and Hilbert functions of smooth curves of degrees at most $15$ in $\mathbb{P}^5$
Edoardo Ballico, Changho Keem
https://arxiv.org/abs/2507.16547

Hilbert scheme and Hilbert functions of smooth curves of degrees at most $15$ in $\mathbb{P}^5$
Denoting $\mathcal{H}_{d,g,5}$ by the Hilbert scheme of smooth curves of degree $d$ and genus $g$ in $\mathbb{P}^5$, let $\mathcal{H}$ be an irreducible component of $\mathcal{H}_{d,g,5}$. We study the Hilbert function $h_X:\mathbb{N}\longrightarrow\mathbb{N}$, $h_X(t):= h^0(\mathcal{I}_X(t))$ of a general member $X\in\mathcal{H}$, especially when the degree of the curve is low; $d\le 15$. We also determine the irreducibility of $\mathcal{H}_{d,g,5}$ for $d\le 14$ and study the natural functori…

Bargmann invariants of Gaussian states
Jianwei Xu
https://arxiv.org/abs/2508.07155 https://arxiv.org/pdf/2508.07155…

Bargmann invariants of Gaussian states
Given a set of ordered quantum states, described by density operators $% \{ρ_{j}\}_{j=1}^{n}$, the Bargmann invariant of $\{ρ_{j}\}_{j=1}^{n}$ is defined as tr($ρ_{1}ρ_{2}...ρ_{n}$). Bargmann invariant serves as a fundamental concept for quantum mechanics and has diverse applications in quantum information science. Bosonic Gaussian states are a class of quantum states on infinite-dimensional Hilbert space, widely used in quantum optics and quantum information science. Bosonic Gaussian stat…

Logarithmic Quot spaces, boundedness, and K-tropicalizations
Patrick Kennedy-Hunt, Dhruv Ranganathan
https://arxiv.org/abs/2508.08175 https://arxiv.org/pdf…

Logarithmic Quot spaces, boundedness, and K-tropicalizations
Logarithmic Hilbert and Quot spaces are generalizations of their traditional versions adapted to study pairs and degenerations. The logarithmic Quot spaces of $(X,D)$ parameterize "algebraically transverse" (logarithmically flat) quotient sheaves on degenerations of $X$. We prove boundedness and deduce properness of logarithmic Quot spaces. The results complete the basic foundations of logarithmic Quot spaces and specialize to work of Li-Wu and Maulik and the second author in special cases. Bou…

Direct integral of locally Hilbert spaces over a locally measure space
Chaitanya J. Kulkarni, Santhosh Kumar Pamula
https://arxiv.org/abs/2508.03779 https://

Direct integral of locally Hilbert spaces over a locally measure space
In this work, we introduce the concept of the direct integral of locally Hilbert spaces by generalizing the classical notion of a measure space to that of a locally measure space. We establish that the direct integral of a family of locally Hilbert spaces over a locally measure space forms a locally Hilbert space. We then define two important subclasses of locally bounded operators on such direct integrals, namely decomposable locally bounded operators and diagonalizable locally bounded operato…

Exploring Photon Blockade in Multimode Jaynes-Cummings Models with Two-Photon Dissipation
Caden McCollum, Imran M. Mirza
https://arxiv.org/abs/2509.09084 https://

Exploring Photon Blockade in Multimode Jaynes-Cummings Models with Two-Photon Dissipation
The photon blockade phenomenon, a promising tool for realizing efficient single-photon sources, is the central focus of our work. We study this phenomenon within the context of the multimode extension of the Jaynes-Cummings model, incorporating two-photon dissipation and external coherent driving. Operating in the weak-driving regime, we confine our analysis to the two-excitation sector of the Hilbert space, initially exploring the single-mode case and then focusing on the corresponding multimo…

On the spectrum of the symmetric tensor products of certain Hilbert-space operators
Yuchi Yang, Yuanhang Zhang
https://arxiv.org/abs/2509.07374 https://arx…

On the spectrum of the symmetric tensor products of certain Hilbert-space operators
This paper primarily investigates spectral properties of symmetric tensor products of Hilbert-space operators. For a unilateral weighted shift operator $S_w$, we present an algorithm to compute the point spectrum of its symmetric and antisymmetric tensor products with the adjoint $S_w^*$. Additionally, we analyze the symmetric tensor product of an injective unilateral weighted shift $S_α$ and a diagonal operator $M$ on $l^2$, demonstrating that its point spectrum must be contained in $\{0\}$.

Projective models for Hilbert squares of $K3$ surfaces
\'Angel David R\'ios Ortiz, Andr\'es Rojas, Jieao Song
https://arxiv.org/abs/2510.02065 https://

Projective models for Hilbert squares of $K3$ surfaces
For a very general polarized $K3$ surface $S\subset \mathbb{P}^g$ of genus $g\ge 5$, we study the linear system on the Hilbert square $S^{[2]}$ parametrizing quadrics in $\mathbb{P}^g$ that contain $S$. We prove its very ampleness for $g\geq 7$. In the cases of genus 7 or 8, we describe in detail the projective geometry of the corresponding embedding by making use of the Mukai model for $S$. In both cases, it can be realized as a degeneracy locus on an ambient homogeneous space, in a strikingly…

Transfinite Operator Fixed Points on Hilbert Spaces: An Alpay Algebra Approach
Faruk Alpay, Hamdi Alakkad, Taylan Alpay
https://arxiv.org/abs/2508.04890 https://

Transfinite Operator Fixed Points on Hilbert Spaces: An Alpay Algebra Approach
This work develops a functional-analytic framework based on the transfinite iteration of a self-adjoint operator. Beginning with a densely defined self-adjoint operator $A$ on a Hilbert space $H$, a spectral-transform functor $Φ$ is applied iteratively. This process generates a transfinite sequence of operators, $\{Φ^α(A)\}_{α<Ω}$, by progressively enlarging the ambient Hilbert space at each ordinal stage. Under suitable continuity and monotonicity conditions on $Φ$, it is established via…

Optimal detection of quantum states via projective measurements
Giuseppe Del Vecchio Del Vecchio, Satya N. Majumdar
https://arxiv.org/abs/2509.08556 https://

Optimal detection of quantum states via projective measurements
We consider the quantum dynamical evolution of a fully-connected quantum system subjected to random projective measurements and study the first detection time of an extended subspace of the Hilbert space. Exact analytical expressions are obtained for the mean first detection time and the full first detection probability distribution as functionals of the initial state $\ket{ψ_0}$ and the measurement rate $r$. Exact enumeration agrees perfectly with our analytical predictions.

Commutators of the Unilateral Shift and Adjoint for Reproducing Kernel Hilbert Spaces on the Disk
Nathan Parker
https://arxiv.org/abs/2508.02896 https://ar…

Commutators of the Unilateral Shift and Adjoint for Reproducing Kernel Hilbert Spaces on the Disk
We generalise the result of Berger and Shaw the trace formula for Hardy Hilbert space to a larger class of rotation invariant Hilbert function spaces on the unit disk. We also demonstrate many meaningful examples of these Hilbert spaces by computing the inner products. We also extend to a wider class than the unilateral shift, that is, weighted shifts under certain restrictions.

Orthocomplemented subspaces and partial projections on a Hilbert space
Iosif Petrakis
https://arxiv.org/abs/2508.15906 https://arxiv.org/pdf/2508.15906

Orthocomplemented subspaces and partial projections on a Hilbert space
We introduce the notion of an orthocomplemented subspace of a Hilbert space H, that is, a pair of orthogonal closed subspaces of H, as a two-dimensional counterpart to the one-dimensional notion of a closed subspace of H. Orthocomplemented subspaces are the Hilbert space-analogue to Bishop's complemented subsets. To complemented subsets correspond their characteristic functions, which are partial, Boolean-valued functions. Similarly, to orthocomplemented subspaces of H correspond partial projec…

Descriptive set theory of separable Fr\'echet spaces
Bruno de Mendon\c{c}a Braga, Willian Hans Goes Corr\^ea, Valentin Ferenczi
https://arxiv.org/abs/2508.09368 https://

Descriptive set theory of separable Fréchet spaces
In the past few decades, much has been done regarding the descriptive set theory of separable Banach spaces. However, the descriptive properties of separable Fréchet spaces have not yet been investigated. In these notes, we look at this problem, its relation with the (now standard) theory for separable Banach spaces, and we compute/estimate the descriptive complexity of some classical classes of separable Fréchet spaces such as Fréchet-Hilbert, Schwartz, nuclear, and Montel spaces. Our main …

The Born rule as a natural transformation of functors
Boyu Yang, James Fullwood
https://arxiv.org/abs/2509.08323 https://arxiv.org/pdf/2509.08323

The Born rule as a natural transformation of functors
In this work, we show that the quantum mechanical notions of density operator, positive operator-valued measure (POVM), and the Born rule, are all simultaneously encoded in the categorical notion of a natural transformation of functors. In particular, we show that given a fixed quantum system, there exists an explicit bijection from the set of density operators on the associated Hilbert space to the set of natural transformations between the canonical measurement and probability functors associ…

Composition operators on some semi-Hilbert spaces
Y. Estaremi, M. S. Al Ghafri
https://arxiv.org/abs/2508.05042 https://arxiv.org/pdf/2508.05042

Composition operators on some semi-Hilbert spaces
This paper investigates composition operators and weighted composition operators on semi-Hilbert spaces induced by positive multiplication operators on \( L^2(μ) \). Within the framework of \( A \)-adjoint operators, we characterize conditions under which a composition operator \( C_φ\) is \( A \)-selfadjoint, \( A \)-normal, \( A \)-quasinormal, or \( A \)-unitary, where \( A = M_u \) denotes a positive multiplication operator. Additionally, we provide necessary and sufficient conditions for…

The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology
S. V. Dzhenzher
https://arxiv.org/abs/2510.04077 https://ar…

The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology
We consider random linear continuous operators $Ω\to \mathcal{L}(\mathcal{H}, \mathcal{H})$ on a Hilbert space $\mathcal{H}$. For example, such random operators may be random quantum channels. The Central Limit Theorem is known for the sums of i.i.d. random operators. Instead of the sum, there may be considered the composition of random exponents $e^{A_i/n}$. We obtain the Central Limit Theorem in the Weak Operator Topology for centralized and normalized random exponents of i.i.d. linear conti…

Construction of linearly independent and orthogonal functions in Hilbert function spaces via Wronski determinants
Athanasios Micheas
https://arxiv.org/abs/2508.04639 https://

Construction of linearly independent and orthogonal functions in Hilbert function spaces via Wronski determinants
Based on the Wronski determinant, we propose a construction of linearly independent and orthogonal functions in any Hilbert function space. The method requires only an initial function from the space of functions under consideration, that satisfies minimal properties.