Operator Algebras Research Articles

Abstract visual reasoning based on algebraic methods

Extracting high-order abstract patterns from complex high-dimensional data forms the foundation of human cognitive abilities. visual reasoning involves identifying abstract patterns embedded within composite images, considered a core competency of machine intelligence. Traditional neuro-symbolic methods often infer unknown objects through data fitting, without fully exploring the abstract patterns within composite images and the sequential sensitivity of visual sequences. This paper constructs a relation model with object-centric inductive biases, learning end-to-end multi-granular rule embeddings at different levels. Through a gating fusion module, the model incrementally integrates explicit representations of objects and abstract relationships. The model incorporates a relational bottleneck method from information theory, separating the input perceptual information from the embeddings of abstract representations, thereby restricting and differentiating feature processing to encourage relational comparisons and induce the extraction of abstract patterns. Furthermore, this paper bridges algebraic operations and machine reasoning through the relational bottleneck method, extracting common patterns of multi-visual objects by identifying invariant sequences within the relational bottleneck matrix. Experimental results on the I-RAVEN dataset demonstrate a total accuracy of 96.8%, surpassing state-of-the-art baseline methods and exceeding human performance at 84.4%.

Generalized Hecke operators on weakly holomorphic modular forms of non-positive weight

Generalized Hecke operators, originating from the replication formula in Monstrous Moonshine, were extended in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717] to apply to harmonic Maass functions on modular curves of higher genera, building on works in [M. Koike, On replication formula and Hecke operators, Nagoya University, preprint]. Their action was further applied to weakly holomorphic modular functions, deriving numerous arithmetic properties of Fourier coefficients. In this paper, we extend these operators to weakly holomorphic modular forms of arbitrary even non-positive weights. In the process, we show [Proposition 3.1 of L. Beneish and M. H. Mertens, On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras, Math. Z. 297(1–2) (2021) 59–80], which is used to obtain dimension formulas for certain vertex operator algebras, can be derived from our results. Additionally, we identify the conditions under which the action of the generalized Hecke operator preserves holomorphicity. Moreover, we show that this action can be expressed as a linear combination of same-level or lower-level forms, refining the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717]. Finally, we establish more general congruence relations on Fourier coefficients, with the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717] emerging as a special case.

Software Implementation of the Epps-Pulley Criterion in Matlab Modeling Environment

Purpose. Modeling systems and programming platforms provide broad opportunities for the use of statistical tools in research activities. Since the normal distribution is one of the most common distribution laws, the criterion for checking the sample for normality is in high demand among statistical assessment tools, among which the Epps-Pulley test has the status as one of the most powerful tests to check the deviation of the distribution from the normal one. There are a number of implementations of this test in the R and Python languages. However, this test is not implemented in one of the most popular Matlab modeling software. Thus, the purpose of this study is to develop a software implementation of the Epps-Pulley criterion in the Matlab environment and verify the correctness of the performed calculations.Materials and methods. We implemented the calculation of EppsPulley statistics by two methods – classical, using cycles, and matrix-vector, using linear algebra operations. The classical method requires calculating the intermediate values necessary to obtain the criterion statistics using two independent cycles, the second cycle being a double one, in which one cycle is nested within the other. The matrix-vector method requires fewer lines of code by performing calculations using linear algebra operations on matrices and vectors. We obtained critical statistical values for the sample size ranging from 8 to 1000 elements with two-dimensional linear interpolation of tabular values. We used an approximation by a beta function of the third kind for a sample of over 1000 elements.Results. An assessment of the computational efficiency of the methods showed that the cyclic approach is about three times higher than the matrix-vector approach in terms of consumed time, which is presumably associated with the processing of insignificant elements in triangular matrices when performing component-by-component operations. The correctness of the software implementation of the Epps-Pulley criterion was tested on several examples, which confirmed the compliance of the calculated values of the criterion statistics, as well as the critical values of statistics, with known data. We carried out a criterion statistical evaluation based on the empirical values of the error of the first kind. We obtained the error values correspondence to the specified significance levels. We performed comparative estimates of the Epps-Pulley test with the Anders-Darling and Shapiro-Wilk tests in terms of the criterion empirical power. Evaluation results are tabulated. The software implementation of the Epps-Pulley test is published on the MATLAB Central Internet resource and is available for free use.

On Superization of Nonlinear Integrable Dynamical Systems

We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. We checked that the so-called "naive" approach, based on the superization of the related phase space variables via extending the corresponding Poisson brackets upon the related functional supermanifold, fails to retain the dynamical system super-integrability. Moreover, we demonstrated that there exists a wide class of classical Lax-type integrable nonlinear dynamical systems on axes in relation to which a superization scheme consists in a reasonable superization of the related Lax-type representation by means of passing from the basic algebra of pseudo-differential operators on the axis to the corresponding superalgebra of super-pseudodifferential operators on the superaxis.

Accelerating machine learning queries with linear algebra query processing

The rapid growth of large-scale machine learning (ML) models has led numerous commercial companies to utilize ML models for generating predictive results to help business decision-making. As two primary components in traditional predictive pipelines, data processing, and model predictions often operate in separate execution environments, leading to redundant engineering and computations. Additionally, the diverging mathematical foundations of data processing and machine learning hinder cross-optimizations by combining these two components, thereby overlooking potential opportunities to expedite predictive pipelines. In this paper, we propose an operator fusion method based on GPU-accelerated linear algebraic evaluation of relational queries. Our method leverages linear algebra computation properties to merge operators in machine learning predictions and data processing, significantly accelerating predictive pipelines by up to 317x. We perform a complexity analysis to deliver quantitative insights into the advantages of operator fusion, considering various data and model dimensions. Furthermore, we extensively evaluate linear algebra query processing and operator fusion utilizing the widely-used Star Schema and TPC-DI benchmarks. Through comprehensive evaluations, we demonstrate the effectiveness and potential of our approach in improving the efficiency of data processing and machine learning workloads on modern hardware.

Additivity of multiplicative Jordan higher semi-derivations on rings and standard operator algebras

Additivity of multiplicative Jordan higher semi-derivations on rings and standard operator algebras

Modules induced from large subalgebras of the Lie algebra of differential operators of degree at most one

Modules induced from large subalgebras of the Lie algebra of differential operators of degree at most one

A novel Q-neutrosophic soft under interval matrix setting and its applications

Decision-making theory serves as an effective framework to guide decision-makers in solving problems. One notable application of this theory is in the medical field, where it aids doctors in analyzing patient data to determine whether a patient is infected. To enhance this theory with more adaptable mathematical methods, we propose an expanded approach based on previously introduced matrixes of Q-neutrosophic soft under an Interval-valued setting (IV-Q-NSM). This represents a new finding of existing mathematical tools to address the two-dimensional uncertainty prevalent in various life domains. This work explores several algebraic properties and matrix operations associated with IV-Q-NSM. Subsequently, we introduce a new methodology for decision-making (DM) in medical diagnosis selection problems. This approach aims to provide a more flexible and comprehensive framework for evaluating complex medical data and improving diagnostic accuracy.

The Properties of Two-Fold Algebra Based on the n-standard Fuzzy Number Theoretical System

In this paper, we study the two-fold algebra based on the n-standard fuzzy number theoretical system as a special type of two-fold fuzzy algebras, where we study the elementary properties of the algebraic operations defined over this system. Also, we prove many results that describe the relations between two-fold substructures and sub-algebras defined by fuzzy number theoretical systems. On the other hand, we provide many different examples to explain our results.

High‐efficiency multilevel inverter topology with minimal switching devices for enhanced power quality and reduced losses

AbstractThe advent of multilevel inverters (MLIs) has brought significant advancements in their applications across industrial, residential, and renewable energy sectors, as they produce high‐quality output voltage that closely approximates a sinusoid in small voltage steps or levels, resulting in lower total harmonic distortion (THD) and reduced electromagnetic interference (EMI). However, the MLI topologies require more switching unidirectional/bidirectional semiconductor devices with high standing voltage, gate drivers, and complex control strategies while attaining higher voltage levels. From this perspective of reducing component count and gate drivers, the objective is to develop a new MLI topology that overcomes the drawbacks above. In this article, a novel MLI topology is introduced in symmetric and asymmetric configurations aiming to attain fewer power electronic devices for synthesizing more steps in the load voltage in contrast with conventional topologies. The idea behind the approach is coining the series connected voltage source, which imbibes bidirectional current flow with an additional voltage source for performing algebraic operation under asymmetrical modes of operation. The proposed topology uses minimal on‐state switching devices leading to a diminution of power loss and voltage drop. The suggested topology is optimized for a fewer number of power devices, an input DC supply, and auxiliary gate drivers to achieve a maximum voltage level in the load terminals. The suggested topology has been verified in SIMULINK and the laboratory prototype is constructed in line with the simulated response to demonstrate its performance suitable for real‐time applications.

Steenrod operator of the dihedral homology of A_∞-algebras

In this study, we examine the interaction between Steenrod operations and dihedral homology within the framework of A-infinity algebras, aiming to understand how these operations, initial in algebraic topology, contribute to homological invariants and influence dihedral homology structures. We begin by exploring the initial properties of A-infinity algebras as generalizations of associative algebras, focusing on their volume to support Steenrod operations. From there, we delve into the effects of these operations within dihedral homology, uncovering their role in revealing deeper algebraic structures and improving our understanding of homology theories. We also analyze the connections between Steenrod operations and projective varieties over finite fields, emphasizing their actions in derived categories and their significance in the context of α-characteristic fields. By defining Steenrod operators within dihedral homology, we explain the complex relationships between these algebraic structures and operations derived from homology theory. Through specific examples and theoretical models, we demonstrate how these interactions advance our understanding of homological invariants and provide valuable tools and perspectives for the broader fields of algebraic topology and homological algebra.

On holonomy groupoid of vertex operator algebra bundles on foliations

For a foliation [Formula: see text] defined on a smooth complex manifold [Formula: see text] we introduce the category of vertex operator algebra [Formula: see text] bundles with sections provided by vectors of elements of the space of algebraically extended [Formula: see text]-module [Formula: see text]-valued differentials. An intrinsic coordinate-independent formulation for such bundles is given. Finally, we identify the cohomology of the spaces of sections for a vertex operator algebra [Formula: see text] bundle with vertex operator algebra cohomology of the holonomy groupoid [Formula: see text].

Computing With Residue Numbers in High-Dimensional Representation.

We introduce residue hyperdimensional computing, a computing framework that unifies residue number systems with an algebra defined over random, high-dimensional vectors. We show how residue numbers can be represented as high-dimensional vectors in a manner that allows algebraic operations to be performed with component-wise, parallelizable operations on the vector elements. The resulting framework, when combined with an efficient method for factorizing high-dimensional vectors, can represent and operate on numerical values over a large dynamic range using resources that scale only logarithmically with the range, a vast improvement over previous methods. It also exhibits impressive robustness to noise. We demonstrate the potential for this framework to solve computationally difficult problems in visual perception and combinatorial optimization, showing improvement over baseline methods. More broadly, the framework provides a possible account for the computational operations of grid cells in the brain, and it suggests new machine learning architectures for representing and manipulating numerical data.

Amortized Analysis via Coalgebra

Amortized analysis is a cost analysis technique for data structures in which cost is studied in aggregate: rather than considering the maximum cost of a single operation, one bounds the total cost encountered throughout a session. Traditionally, amortized analysis has been phrased inductively, quantifying over finite sequences of operations. Connecting to prior work on coalgebraic semantics for data structures, we develop the alternative perspective that amortized analysis is naturally viewed coalgebraically in a category of cost algebras, where a morphism of coalgebras serves as a first-class generalization of potential function suitable for integrating cost and behavior. Using this simple definition, we consider amortization of other sample effects, non-commutative printing and randomization. To support imprecise amortized upper bounds, we adapt our discussion to the bicategorical setting, where a potential function is a colax morphism of coalgebras. We support algebraic and coalgebraic operations simultaneously by using coalgebras for an endoprofunctor instead of an endofunctor, combining potential using a monoidal structure on the underlying category. Finally, we compose amortization arguments in the indexed category of coalgebras to implement one amortized data structure in terms of others.

Examining tenth-grade students’ errors in applying Polya’s problem-solving approach to Pythagorean theorem

This study uses Polya’s problem-solving strategy to explore student errors and causal factors in Pythagorean theorem problem-solving. The sample was drawn from tenth-grade students at Al Hosn Secondary School in Abu Dhabi, United Arab Emirates in the academic year 2023/2024. Data were collected using two methods: written examinations, as outlined in Polya’s strategy, and interviews with students who committed errors. The research test instrument consists of 3 trigonometric problem-solving. From 30 students of Al Hosn Secondary School, there were 48.9% of data errors, 50% of concept errors, 57.8% of strategy errors, 47.8% of calculation errors, and 14.4% of careless errors. The errors made by the students originated from their inability to comprehend the geometric interpretation of the Pythagorean theorem, difficulties in applying algebraic operations, and challenges in modelling the data provided in the problems.

Boson-Fermion Algebraic Mapping in Second Quantization.

We present an algebraic method to derive the structure at the basis of the mapping of bosonic algebras of creation and annihilation operators into fermionic algebras, and vice versa, introducing a suitable identification between bosonic and fermionic generators. The algebraic structure thus obtained corresponds to a deformed Grassmann-type algebra, involving anticommuting Grassmann-type variables. The role played by the latter in implementing gauge invariance in second quantization within our procedure is then discussed. This discussion includes the application of the mapping to the case of the bosonic and fermionic harmonic oscillator Hamiltonians.

The Case for Adopting the Sequential Jacobi’s Diagonalization Algorithm in Neutrino Oscillation Physics

Neutrino flavor oscillations and conversion in an interacting background (MSW effects) may reveal the charge-parity violation in the next generation of neutrino experiments. The usual approach for studying these effects is to numerically integrate the Schrödinger equation, recovering the neutrino mixing matrix and its parameters from the solution. This work suggests using the classical Jacobi’s diagonalization in combination with a reordering procedure to produce a new algorithm, the Sequential Jacobi Diagonalization. This strategy separates linear algebra operations from numerical integration, allowing physicists to study how the oscillation parameters are affected by adiabatic MSW effects in a more efficient way. The mixing matrices at every point of a given parameter space can be stored for speeding up other calculations such as model fitting and Monte Carlo productions. This approach has two major computation advantages, namely, being trivially parallelizable, making it a suitable choice for concurrent computation, and allowing for quasi-model-independent solutions which simplify Beyond Standard Model searches.

Rings of almost everywhere defined functions

The following representation theorem is proven: A partially ordered commutative ring R is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space X if and only if R is archimedean and localizable. Here we assume that the positive cone of R is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on X is one that is defined on a dense open subset of X. A partially ordered commutative ring R is archimedean if the underlying additive partially ordered abelian group is archimedean, and R is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the σ-bounded case, lattice-ordered commutative rings (f-rings), partially ordered fields, and commutative operator algebras.

Free field realizations for rank-one SCFTs

In this paper, we construct the associated vertex operator algebras for all N = 2 superconformal field theories of rank one. We give a uniform presentation through free-field realizations, which turns out to be a particularly suitable framework for this task. The elementary building blocks of the construction are dictated by the low energy degrees of freedom on the Higgs branch, which are well understood for rank-one theories. We further analyze the interplay between Higgs and Coulomb data on the moduli space of vacua, which tightly constrain the overall structure of the free field realizations. Our results suggest a plausible bottom-up classification scheme for low-rank SCFTs incorporating vertex algebra techniques.

Universal Quantitative Algebra for Fuzzy Relations and Generalised Metric Spaces

We present a generalisation of the theory of quantitative algebras of Mardare, Panangaden and Plotkin where (i) the carriers of quantitative algebras are not restricted to be metric spaces and can be arbitrary fuzzy relations or generalised metric spaces, and (ii) the interpretations of the algebraic operations are not required to be nonexpansive. Our main results include: a novel sound and complete proof system, the proof that free quantitative algebras always exist, the proof of strict monadicity of the induced Free-Forgetful adjunction, the result that all monads (on fuzzy relations) that lift finitary monads (on sets) admit a quantitative equational presentation.