Tootfinder

On Binary Codes That Are Maximal Totally Isotropic Subspaces with Respect to an Alternating Form
Patrick King, Mikhail Kotchetov
https://arxiv.org/abs/2510.05464 https://…

On Binary Codes That Are Maximal Totally Isotropic Subspaces with Respect to an Alternating Form
Self-dual binary linear codes have been extensively studied and classified for length n <= 40. However, little attention has been paid to linear codes that coincide with their orthogonal complement when the underlying inner product is not the dot product. In this paper, we introduce an alternating form defined on F_2^n and study codes that are maximal totally isotropic with repsect to this form. We classify such codes for n <= 24 and present a MacWilliams-type identity which relates the weight …

Kernel ridge regression under power-law data: spectrum and generalization
Arie Wortsman, Bruno Loureiro
https://arxiv.org/abs/2510.04780 https://arxiv.org/…

Kernel ridge regression under power-law data: spectrum and generalization
In this work, we investigate high-dimensional kernel ridge regression (KRR) on i.i.d. Gaussian data with anisotropic power-law covariance. This setting differs fundamentally from the classical source & capacity conditions for KRR, where power-law assumptions are typically imposed on the kernel eigen-spectrum itself. Our contributions are twofold. First, we derive an explicit characterization of the kernel spectrum for polynomial inner-product kernels, giving a precise description of how the ker…

A Dwyer-Rezk classification for polynomial functors in Weiss calculus
David Barnes, Magdalena K\k{e}dziorek, Niall Taggart
https://arxiv.org/abs/2508.03808 https://

A Dwyer-Rezk classification for polynomial functors in Weiss calculus
In Goodwillie calculus, unpublished work of Dwyer and Rezk provides a classification of reduced filtered colimit preserving $d$-excisive functors from pointed spaces to spectra as spectrum-valued functors on the category of finite sets of cardinality at most $d$ and epimorphisms. We prove through different methods the analogous result in Weiss calculus: $d$-polynomial functors are equivalent to spectrum-valued functors on the category of finite-dimensional inner product spaces of dimension at m…

Weak containment of representation on topological groupoids
K. N. Sridharan, N. Shravan Kumar
https://arxiv.org/abs/2510.05898 https://arxiv.org/pdf/2510.0…

Weak containment of representation on topological groupoids
Let $G$ be a second-countable, locally compact Hausdorff groupoid equipped with a Haar system. This paper investigates the weak containment of continuous unitary representations of groupoids. We show that both induction and inner tensor product of representations preserve weak containment. Additionally, we introduce the notion of a topological invariant mean on $G/H$ and explore its connection to amenability. With that, we establish a groupoid analogue of Greenleaf's theorem. Finally, we provid…

$\mathbb{F}_{2}\mathbb{F}_{4}$-Additive Complementary Dual Codes
S. Ouagagui, N. Benbelkacem, A. Batoul, T. Abualrub
https://arxiv.org/abs/2508.05317 https://

$\mathbb{F}_{2}\mathbb{F}_{4}$-Additive Complementary Dual Codes
In this paper, we investigate the structure and properties of additive complementary dual (ACD) codes over the mixed alphabet $\mathbb{F}_2\mathbb{F}_4$ relative to a certain inner product defined over $\mathbb{F}_2\mathbb{F}_4$. We establish sufficient conditions under which such codes are additive complementary dual (ACD) codes. We also show that ACD codes over $\mathbb{F}_{2}\mathbb{F}_{4}$ can be applied to construct binary linear complementary dual codes as their images under the linear ma…

Uncertainty Relation for Pseudo-Hermitian Quantum Systems
Boubakeur Khantoul, Bilel Hamil, Amar Benchikha
https://arxiv.org/abs/2508.00648 https://arxiv.or…

Uncertainty Relation for Pseudo-Hermitian Quantum Systems
This study investigates pseudo-Hermitian quantum mechanics, where the Hamiltonian satisfies a modified Hermiticity condition. We extend the uncertainty relation for such systems, demonstrating its equivalence to the standard Hermitian case within a pseudo-Hermitian inner product. Analytical solutions to the time-dependent Schrödinger equation with a linearly evolving potential are derived. Furthermore, we show that the uncertainty relation for position and momentum remains real and greater tha…

Multiplier-free In-Memory Vector-Matrix Multiplication Using Distributed Arithmetic
Felix Zeller, John Reuben, Dietmar Fey
https://arxiv.org/abs/2510.02099 https://

Multiplier-free In-Memory Vector-Matrix Multiplication Using Distributed Arithmetic
Vector-Matrix Multiplication (VMM) is the fundamental and frequently required computation in inference of Neural Networks (NN). Due to the large data movement required during inference, VMM can benefit greatly from in-memory computing. However, ADC/DACs required for in-memory VMM consume significant power and area. `Distributed Arithmetic (DA)', a technique in computer architecture prevalent in 1980s was used to achieve inner product or dot product of two vectors without using a hard-wired mult…

A way to treat dual Hahn polynomials as Racah polynomials via the theory of Leonard pairs
Hau-Wen Huang
https://arxiv.org/abs/2508.02032 https://arxiv.org/…

A way to treat dual Hahn polynomials as Racah polynomials via the theory of Leonard pairs
The dual Hahn polynomials $\{u_i(x)\}_{i=0}^d$ are a family of discrete orthogonal polynomials involving two real parameters $r$ and $s$. Let $L,L^*$ denote the corresponding Leonard pair. Assume that $r\not=0$ and $r+s=0$. We show that $L,(L^*+\frac{r-d}{2})^{2}$ is a Leonard pair. According to the theory of Leonard pairs, the polynomials $\{u_i(x)\}_{i=0}^d$ are not only the dual Hahn polynomials but also the Racah polynomials with respect to the same inner product.

Funnel control for passive infinite-dimensional systems
Thavamani Govindaraj, Anthony Hastir, Lassi Paunonen, Timo Reis
https://arxiv.org/abs/2510.01027 https://

Funnel control for passive infinite-dimensional systems
We consider funnel control for linear infinite-dimensional systems that are impedance passive, meaning that they satisfy an energy balance in which the stored energy equals the squared norm of the state and the supplied power is the inner product of input and output. For the analysis we employ the system node approach, which offers a unified framework for infinite-dimensional systems with boundary and distributed control and observation. The resulting closed-loop dynamics are governed by a nonl…

SINDI: an Efficient Index for Approximate Maximum Inner Product Search on Sparse Vectors
Ruoxuan Li, Xiaoyao Zhong, Jiabao Jin, Peng Cheng, Wangze Ni, Lei Chen, Zhitao Shen, Wei Jia, Xiangyu Wang, Xuemin Lin, Heng Tao Shen, Jingkuan Song
https://arxiv.org/abs/2509.08395

SINDI: an Efficient Index for Approximate Maximum Inner Product Search on Sparse Vectors
Sparse vector Maximum Inner Product Search (MIPS) is crucial in multi-path retrieval for Retrieval-Augmented Generation (RAG). Recent inverted index-based and graph-based algorithms have achieved high search accuracy with practical efficiency. However, their performance in production environments is often limited by redundant distance computations and frequent random memory accesses. Furthermore, the compressed storage format of sparse vectors hinders the use of SIMD acceleration. In this paper…

Massey products for homotopy inner products
Kate Poirier, Thomas Tradler, Scott O. Wilson
https://arxiv.org/abs/2507.15494 https://ar…

Massey products for homotopy inner products
This paper defines Massey-type products for a homotopy inner product on an $A_\infty$ algebra, called Massey inner products. We include an explicit description of ordinary Massey products for $A_\infty$ algebras, and for $A_\infty$ modules, and show that Massey product sets can have interesting internal structure. We give non-trivial applications from lens spaces, links, and low dimensional manifolds, showing Massey inner products contain information beyond ordinary Massey products.

Inner-product Functional Encryption with Fine-grained Revocation for Flexible EHR Sharing
Yue Han, Jinguang Han, Liqun Chen, Chao Sun
https://arxiv.org/abs/2509.07804 https://…

Inner-product Functional Encryption with Fine-grained Revocation for Flexible EHR Sharing
E-health record (EHR) contains a vast amount of continuously growing medical data and enables medical institutions to access patient health data conveniently.This provides opportunities for medical data mining which has important applications in identifying high-risk patients and improving disease diagnosis, etc.Since EHR contains sensitive patient information, how to protect patient privacy and enable mining on EHR data is important and challenging.Traditional public key encryption (PKE) can p…

Symmetrized operators or modified integration measure in Generalized Uncertainty Principle Models
Michael Bishop, Daniel Hooker, Doug Singleton
https://arxiv.org/abs/2509.20466 …

Symmetrized operators or modified integration measure in Generalized Uncertainty Principle Models
Many Generalized Uncertainty Principle (GUP) models modify the inner-product measure to ensure symmetric position or momentum operators. We show that an alternate approach to these GUPs is to symmetrize the operators rather than modifying the inner product. This preserves the standard momentum space allowing the eigenstates and maximally localized states of the modified position operator to have a standard position representation. We compare both approaches and highlight their merits.

Transients in black hole perturbation theory
J\'er\'emy Besson, Javier Carballo, Christiana Pantelidou, Benjamin Withers
https://arxiv.org/abs/2507.16493

Transients in black hole perturbation theory
Black hole quasinormal modes arise as eigenmodes of a non-normal Hamiltonian and consequently they do not obey orthogonality relations with respect to commonly used inner products, for example, the energy inner product. A direct consequence of this is the appearance of transient phenomena. This review summarises current developments on the topic, both in frequency- and time-domain. In particular, we discuss the appearance of i) transient plateaus: arbitrarily long-lived sums of quasinormal mode…

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…

Best approximation from the positive cone of an inner product lattice
Marouen Abdouli, Karim Boulabiar
https://arxiv.org/abs/2508.13722 https://arxiv.org/p…

Best approximation from the positive cone of an inner product lattice
Let $E$ be a directed (i.e., positively generated) ordered vector space endowed with an inner product. In this note, we prove that the following statements are equivalent: i) $E$ is a vector lattice and its norm induced by its inner product is a lattice norm. ii) The metric projection onto the positive cone $E^+$ of $E$ exists and it is both isotone and subadditive. Moreover, in this case, the best approximation to any $x \in E$ from $E^+$ coincides with its positive part $x^+$. This result ext…

NLAFormer: Transformers Learn Numerical Linear Algebra Operations
Zhantao Ma, Yihang Gao, Michael K. Ng
https://arxiv.org/abs/2508.19557 https://arxiv.org/…

NLAFormer: Transformers Learn Numerical Linear Algebra Operations
Transformers are effective and efficient at modeling complex relationships and learning patterns from structured data in many applications. The main aim of this paper is to propose and design NLAFormer, which is a transformer-based architecture for learning numerical linear algebra operations: pointwise computation, shifting, transposition, inner product, matrix multiplication, and matrix-vector multiplication. Using a linear algebra argument, we demonstrate that transformers can express such o…

A novel generalized inner product-based wave scattering from an underwater source in a compressible ocean
R. Pethiyagoda S. Das, B. Wilks, M. H. Meylan
https://arxiv.org/abs/2509.19196

A Universal Banach--Bregman Framework for Stochastic Iterations: Unifying Stochastic Mirror Descent, Learning and LLM Training
Johnny R. Zhang (Independent Researcher), Xiaomei Mi (University of Manchester), Gaoyuan Du (Amazon), Qianyi Sun (Microsoft), Shiqi Wang (Meta), Jiaxuan Li (Amazon), Wenhua Zhou (Independent Researcher)
https://arx…

A Universal Banach--Bregman Framework for Stochastic Iterations: Unifying Stochastic Mirror Descent, Learning and LLM Training
Stochastic optimization powers the scalability of modern artificial intelligence, spanning machine learning, deep learning, reinforcement learning, and large language model training. Yet, existing theory remains largely confined to Hilbert spaces, relying on inner-product frameworks and orthogonality. This paradigm fails to capture non-Euclidean settings, such as mirror descent on simplices, Bregman proximal methods for sparse learning, natural gradient descent in information geometry, or Kullb…

Equivalence of mutually unbiased bases via orbits: general theory and a $d=4$ case study
Amit Te'eni, Eliahu Cohen
https://arxiv.org/abs/2508.15412 https://

Equivalence of mutually unbiased bases via orbits: general theory and a $d=4$ case study
In quantum mechanics, mutually unbiased bases (MUBs) represent orthonormal bases that are as "far apart" as possible, and their classification reveals rich underlying geometric structure. Given a complex inner product space, we construct the space of its orthonormal bases as a discrete quotient of the complete flag manifold. We introduce a metric on this space, which corresponds to the "MUBness" distance. This allows us to describe equivalence between sets of mutually unbiased bases in terms of…

Symplectic Quasi Self-dual, Self-dual, and LCD Codes over Non-unital Rings of Order Six
Altaf Alshuhail
https://arxiv.org/abs/2508.16344 https://arxiv.org/…

Symplectic Quasi Self-dual, Self-dual, and LCD Codes over Non-unital Rings of Order Six
We consider codes over the two semi-local non-unital rings of order six, \[ H_{23} = \langle a,b \mid 2a=0, 3b = 0, a^2=a, b^2 = 0, ab = 0 = ba \rangle,\] and \[H_{32} = \langle a,b \mid 2a=0, 3b = 0, a^2=0, b^2 = b, ab = 0 = ba \rangle. \] with respect to a symplectic inner product. Via the decomposition $\mathcal{C} = a\,\mathcal{C}_a\oplus b\,\mathcal{C}_b$, any $H_z$-code splits into a binary component $\mathcal{C}_a$ and a ternary component $\mathcal{C}_b$; this yields characterizations of…

Smoothness in the space of bounded linear operators on semi-Hilbert space
Somdatta Barik, Souvik Ghosh, Kallol Paul, Debmalya Sain
https://arxiv.org/abs/2509.00335 https://

Smoothness in the space of bounded linear operators on semi-Hilbert space
Given a nonzero positive operator $A$ on a Hilbert space $\mathbb{H}$, a semi-inner product is naturally induced on $\mathbb{H}$. In this work, we introduce the notion of \emph{$A$-smoothness} for bounded linear operators on the resulting semi-Hilbert space and investigate its various properties. We provide a comprehensive characterization of the $A$-smoothness for $A$-bounded operators and further analyze the $A$-smoothness of $A$-compact operators in terms of their $A$-norm attainment sets. U…

Wave-Based Semantic Memory with Resonance-Based Retrieval: A Phase-Aware Alternative to Vector Embedding Stores
Aleksandr Listopad
https://arxiv.org/abs/2509.09691 https://

Wave-Based Semantic Memory with Resonance-Based Retrieval: A Phase-Aware Alternative to Vector Embedding Stores
Conventional vector-based memory systems rely on cosine or inner product similarity within real-valued embedding spaces. While computationally efficient, such approaches are inherently phase-insensitive and limited in their ability to capture resonance phenomena crucial for meaning representation. We propose Wave-Based Semantic Memory, a novel framework that models knowledge as wave patterns $ψ(x) = A(x) e^{iϕ(x)}$ and retrieves it through resonance-based interference. This approach preserves…

Higher Structures on Boundary Conformal Manifolds: Higher Berry Phase and Boundary Conformal Field Theory
Yichul Choi, Hyunsoo Ha, Dongyeob Kim, Yuya Kusuki, Shuhei Ohyama, Shinsei Ryu
https://arxiv.org/abs/2507.12525

Higher Structures on Boundary Conformal Manifolds: Higher Berry Phase and Boundary Conformal Field Theory
We introduce the notion of higher Berry connection and curvature in the space of conformal boundary conditions in (1+1)d conformal field theories (CFT), related to each other by exactly marginal boundary deformations, forming a "boundary conformal manifold." Our definition builds upon previous works on tensor networks, such as matrix product states (MPS), where the triple inner product or multi-wavefunction overlap plays the key geometric role. On the one hand, our boundary conformal field theo…

Energy Estimates for Fractional Evolution Equations
Paulo M. Carvalho-Neto, Cicero L. Frota, Juan C. Oyola Ballesteros, Pedro G. P. Torelli
https://arxiv.org/abs/2508.05780 http…

Energy Estimates for Fractional Evolution Equations
This work presents a more broadly applicable version of an energy inequality for weak solutions of evolution equations involving fractional time derivatives. Unlike the classical identity that relates the time derivative of the squared norm to the inner product of the derivative and the function itself, its fractional analogue typically requires a regularity condition that is difficult to verify in practice. We revisit this inequality in the setting of square integrable functions over an open d…

Generalised state space geometry in Hermitian and non-Hermitian quantum systems
Kunal Pal
https://arxiv.org/abs/2507.18486 https://arxiv.org/pdf/2507.18486…

Generalised state space geometry in Hermitian and non-Hermitian quantum systems
One of the key features of information geometry in the classical setting is the existence of a metric structure and a family of connections on the space of probability distributions. The uniqueness of the Fisher--Rao metric and the duality of these connections is at the heart of classical information geometry. However, these features do not carry over straightforwardly to quantum systems, where a Hermitian inner product structure on the Hilbert space induces a metric on the complex projective s…

MaxWave: Rapid Maximum Likelihood Wavelet Reconstruction of Non-Gaussian features in Gravitational Wave Data
Sudhi Mathur, Neil J. Cornish
https://arxiv.org/abs/2508.13377 https…

MaxWave: Rapid Maximum Likelihood Wavelet Reconstruction of Non-Gaussian features in Gravitational Wave Data
Advancements in the sensitivity of gravitational-wave detectors have increased the detection rate of transient astrophysical signals. We improve the existing BayesWave initialization algorithm and present a rapid, low-latency approximate maximum likelihood solution for reconstructing non-Gaussian features. We include three enhancements: (1) using a modified wavelet basis to eliminate redundant inner product calculations; (2) shifting from traditional time-frequency-quality factor (TFQ) wavelet …

Introducing Image-Space Preconditioning in the Variational Formulation of MRI Reconstructions
Bastien Milani, Jean-Baptist Ledoux, Berk Can Acikgoz, Xavier Richard
https://arxiv.org/abs/2507.05312

Introducing Image-Space Preconditioning in the Variational Formulation of MRI Reconstructions
The aim of the present article is to enrich the comprehension of iterative magnetic resonance imaging (MRI) reconstructions, including compressed sensing (CS) and iterative deep learning (DL) reconstructions, by describing them in the general framework of finite-dimensional inner-product spaces. In particular, we show that image-space preconditioning (ISP) and data-space preconditioning (DSP) can be formulated as non-conventional inner-products. The main gain of our reformulation is an embeddin…

Multi-Partitioned Meshfree Quantum Finite Particle Method: A Hybrid Quantum Framework for Fluid Flow
Yudong Li, Wenkui Shi, Yan Li, Chunfa Wang, Ling Tao, Zhuojia Fu, Moubin Liu, Zhiqiang Feng
https://arxiv.org/abs/2509.11276

Multi-Partitioned Meshfree Quantum Finite Particle Method: A Hybrid Quantum Framework for Fluid Flow
This study established a quantum-classical hybrid framework that integrates quantum computing paradigm with meshfree finite particle method. By harnessing quantum superposition and entanglement, it hybridized the critical computational kernels (termed as quantum finite particle method). A resource-efficient quantum computational strategy on multi-partitioned zones was proposed, which leverages a fixed small-scale quantum circuit as a fundamental processing unit to handle inner product for arbit…

The Kaczmarz Algorithm in Hilbert $C^{*}$-modules
Daniel Alpay, Chad Berner, Eric S. Weber
https://arxiv.org/abs/2508.13861 https://arxiv.org/pdf/2508.1386…

The Kaczmarz Algorithm in Hilbert $C^{*}$-modules
The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert $C^*$-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform f…

Certain Upper Bounds on the $A$-Numerical Radius of Operators in Semi-Hilbertian Spaces and Their Applications
M. H. M. Rashid
https://arxiv.org/abs/2507.05398

Certain Upper Bounds on the $A$-Numerical Radius of Operators in Semi-Hilbertian Spaces and Their Applications
Consider a complex Hilbert space $\left(\mathcal{H}, \langle \cdot, \cdot \rangle\right)$ equipped with a positive bounded linear operator $A$ on $\mathcal{H}$. This induces a semi-norm $\|\cdot\|_A$ through the semi-inner product $\langle x, y \rangle_A = \langle Ax, y \rangle$ for $x, y \in \mathcal{H}$. In this semi-Hilbertian space setting, we investigate the $A$-numerical radius $w_A(T)$ and $A$-operator semi-norm $\|T\|_A$ of bounded operators $T$. This paper presents …