Class Of Modules Research Articles

Associated varieties of minimal highest weight modules

Let g \mathfrak {g} be a complex simple Lie algebra. A simple g \mathfrak {g} -module is called minimal if the associated variety of its annihilator ideal coincides with the closure of the minimal nilpotent coadjoint orbit. The main result of this paper is a classification of minimal highest weight modules for g \mathfrak {g} . This classification extends the work of Joseph [Ann. Sci. École Norm. Sup. (4) 31 (1998), 17–45], which focused on categorizing minimal highest weight modules annihilated by completely prime ideals. Furthermore, we have determined the associated varieties of these modules. In other words, we have identified all possible weak quantizations of minimal orbital varieties.

The Relaxed Tutorial Project: distance learning and anxiety in Classical Studies

Abstract ‘Relaxed’ events are now common across the public Arts and Heritage sectors. Although designed with the needs of autistic children and their families in mind, they suit people with a range of access needs. Our project at The Open University asked: what would a ‘relaxed tutorial’ look like, and who might benefit from it? Across two years (2021–2023), tutors in the Department of Classical Studies at The Open University trialled a set of autism-friendly accessibility adjustments to live online tutorials for distance learners on a Roman History module. We found that relaxed tutorials were welcomed by students with anxiety disorders, caring responsibilities, chronic conditions and low confidence, as well as by autistic students. Since the project was concluded in 2023, relaxed tutorials have been rolled out across further Classical Studies modules at The Open University. Their principles and structure offer a new way of looking at accessibility adjustments which could be adapted to other teaching contexts and levels.

Classification of multiplication modules over multiplication rings with finitely many minimal primes

Abstract A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterization of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive modules. It is proven that for a multiplication ring with finitely many minimal primes every faithful, Noetherian, distributive module is a faithful multiplication module, and vice versa.

Development and optimization of dual-objective PVT modules for a smart residential building using inclined fins

According to the sustainable development strategy, air-based photovoltaic-thermal (PVT) modules integrated with smart buildings are an effective and environmentally safe option for producing electricity and heating purposes to overcome the problem of scarcity and environmental pollution resulting from fossil fuels. Despite the importance of PVT modules, this technology has not achieved the required diffusion rates compared to other solar energy technologies due to its low overall efficiency. Therefore, the present study aims to develop the design of a heat transfer system integrated with PVT modules. To achieve this goal, the design of a heat transfer system was developed by adding 45° inclined fins integrated with the cooling channel to generate swirling movement near the back surface of PVT modules and increase the heat transfer area. To illustrate the best location for installing 45° inclined fins that achieve the highest performance improvement of PVT modules, we suggested two positions for installing 45° inclined fins and compared them with the dual-objective classical PVT modules (DO-CPVT) without fins. The first case is dual-objective PVT modules with top-inclined fins (DO-PVT&TIF). The second case is the dual-objective PVT modules with bottom-inclined fins (DO-PVT&BIF). The results presented that, dual-objective PVT modules with top-inclined fins (DO-PVT&TIF) represent a good design that achieves the highest performance. The improvement in thermal efficiency, electricity generation, overall energy efficiency, total exergy efficiency, and sustainability index for using DO-PVT&TIF modules reached 231 %, 11.5 %, 133.8 %, 19.3 %, and 4.3 %, respectively compared to classical DO-CPVT.

Morphogenesis analysis of icosahedral and prismatic tensegrity modules

Morphogenesis is a critical aspect of tensegrity system design. This paper explores icosahedral and prismatic tensegrity modules' pattern formation and foundational theories. First, a material consumption functional and self-strain energy functional are introduced from an engineering perspective, along with a variational principle involving independent dual-variable in the morphogenesis process of tensegrity structures. Next, a Symbol Parsing Method is proposed and successfully applied to the optimal morphology analysis of the icosahedral tensegrity module. Due to memory limitations in personal computers, the Singular Value Decomposition of the symbolic equilibrium matrix for the prismatic tensegrity module is constrained. As an alternative, a discrete numerical solution, the semi-symbolic parsing method, is obtained by directly utilizing a numerical equilibrium matrix. This method was recently employed to analyze and optimize a practical tensegrity footbridge designed by the authors. Lastly, the paper delves into the stability of self-equilibrium states in icosahedral and classical prismatic tensegrity modules by employing a second-order variational analysis of the self-strain energy functional. The findings presented are expected to contribute to future research on the morphogenesis of tensegrity structures.

Big pure projective modules over commutative noetherian rings: Comparison with the completion

Abstract A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add ⁡ ( M ) {\operatorname{Add}(M)} , which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add ⁡ ( M ) {\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V * ⁢ ( M ) {V^{*}(M)} , of isomorphism classes of countably generated modules in Add ⁡ ( M ) {\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V * ⁢ ( M ) {V^{*}(M)} is a submonoid of V * ⁢ ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} , this allows us to make computations with examples and to prove some realization results.

SDGSA: a lightweight shallow dual-group symmetric attention network for micro-expression recognition

Recognizing micro-expressions (MEs) as subtle and transient forms of human emotional expressions is critical for accurately judging human feelings. However, recognizing MEs is challenging due to their transient and low-intensity characteristics. This study develops a lightweight shallow dual-group symmetric attention network (SDGSA) to address the limitations of existing methods in capturing the subtle features of MEs. This network takes the optical flow features as inputs, extracting ME features through a shallow network and performing finer feature segmentation in the channel dimension through a dual-group strategy. The goal is to focus on different types of facial information without disrupting facial symmetry. Moreover, this study implements a spatial symmetry attention module, focusing on extracting facial symmetry features to emphasize further the symmetric information of the left and right sides of the face. Additionally, we introduce the channel blending technique to optimize the information fusion between different channel features. Extensive experiments on SMIC, CASME II, SAMM, and 3DB-combined mainstream ME datasets demonstrate that the proposed SDGSA method outperforms the metrics of current state-of-the-art methods. As shown by ablation experimental results, the proposed dual-group symmetric attention module outperforms classical attention modules, such as the convolutional block attention module, squeeze-and-excitation, efficient channel attention, spatial group-wise enhancement, and multi-head self-attention. Importantly, SDGSA maintained excellent performance while having only 0.278 million parameters. The code and model are publicly available at https://github.com/YZY980123/SDGSA.

A classification of polyharmonic Maaß forms via quiver representations

We give a classification of the Harish-Chandra modules generated by the pullback to SL2(R) of polyharmonic Maaß forms for congruence subgroups of SL2(Z) with exponential growth allowed at the cusps. This extends results of Bringmann–Kudla in the harmonic case. While in the harmonic setting there are nine cases, our classification comprises ten; A new case arises in weights k>1. To obtain the classification we introduce quiver representations into the topic and show that those associated with polyharmonic Maaß forms are cyclic, indecomposable representations of the two-cyclic or the Gelfand quiver. A classification of these transfers to a classification of polyharmonic weak Maaß forms. To realize all possible cases of Harish-Chandra modules we develop a theory of weight shifts for Taylor coefficients of vector-valued spectral families. We provide a comprehensive computer implementation of this theory, which allows us to provide explicit examples.

Hybrid Quantum Neural Network Approaches to Protein–Ligand Binding Affinity Prediction

Drug repositioning is a less expensive and time-consuming method than the traditional method of drug discovery. It is a strategy for identifying new uses for approved or investigational drugs that are outside the scope of the original medical indication. A key strategy in repositioning approved or investigational drugs is determining the binding affinity of these drugs to target proteins. The large increase in available experimental data has helped deep learning methods to demonstrate superior performance compared to conventional prediction and other traditional computational methods in precise binding affinity prediction. However, these methods are complex and time-consuming, presenting a significant barrier to their development and practical application. In this context, quantum computing (QC) and quantum machine learning (QML) theoretically offer promising solutions to effectively address these challenges. In this work, we introduce a hybrid quantum–classical framework to predict binding affinity. Our approach involves, initially, the implementation of an efficient classical model using convolutional neural networks (CNNs) for feature extraction and three fully connected layers for prediction. Subsequently, retaining the classical module for feature extraction, we implement various quantum and classical modules for binding affinity prediction, which accept the concatenated features as input. Quantum predicted modules are implemented with Variational Quantum Regressions (VQRs), while classical predicted modules are implemented with various fully connected layers. Our findings clearly show that hybrid quantum–classical models accelerate the training process in terms of epochs and achieve faster stabilization. Also, these models demonstrate quantum superiority in terms of complexity, accuracy, and generalization, thereby indicating a promising direction for QML.

Dualidad de Matlis generalizado, sobre un anillo conmutativo Noetheriano

Let R be a Noetherian commutative ring and let E be the minimal injective cogenerator of the category of R-modules. Belshoff, Enochs and García Rozas are three people who introduced in [6] the so-called I-reflexive Matlis modules where I is an ideal. In this context we present in the second section a brief introduction of the reflective modules and some of their properties, and in the third and last section we give the classification of reflective modules with respect to a minimal cogenerator E. A node M is said to be reflexive with respect to E if the map of M in Hom(HomR(M, E), E) is an isomorphism and thus we establish the classification of the R-modules M, which are reflexive with respect to E if and only if M has a finitely generated submodule S such that the quotients: M/S and R/anul(M) is complete Artinian and semilocal respectively.

Minimal resolutions of Iwasawa modules

In this paper, we study the module-theoretic structure of classical Iwasawa modules. More precisely, for a finite abelian p-extension K/k of totally real fields and the cyclotomic Zp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_p$$\\end{document}-extension K∞/K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{\\infty }/K$$\\end{document}, we consider XK∞,S=Gal(MK∞,S/K∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}={{\\,\ extrm{Gal}\\,}}(M_{K_{\\infty },S}/K_{\\infty })$$\\end{document} where S is a finite set of places of k containing all ramifying places in K∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{\\infty }$$\\end{document} and archimedean places, and MK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{K_{\\infty },S}$$\\end{document} is the maximal abelian pro-p-extension of K∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{\\infty }$$\\end{document} unramified outside S. We give lower and upper bounds of the minimal numbers of generators and of relations of XK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}$$\\end{document} as a Zp[[Gal(K∞/k)]]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_p[[{{\\,\ extrm{Gal}\\,}}(K_{\\infty }/k)]]$$\\end{document}-module, using the p-rank of Gal(K/k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Gal}\\,}}(K/k)$$\\end{document}. This result explains the complexity of XK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}$$\\end{document} as a Zp[[Gal(K∞/k)]]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_p[[{{\\,\ extrm{Gal}\\,}}(K_{\\infty }/k)]]$$\\end{document}-module when the p-rank of Gal(K/k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Gal}\\,}}(K/k)$$\\end{document} is large. Moreover, we prove an analogous theorem in the setting that K/k is non-abelian. We also study the Iwasawa adjoint of XK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}$$\\end{document}, and the minus part of the unramified Iwasawa module for a CM-extension. In order to prove these theorems, we systematically study the minimal resolutions of XK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}$$\\end{document}.

Simple Harish-Chandra modules over the superconformal current algebra

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra gˆ, which is the semi-direct sum of the N=1 superconformal algebra with the affine Lie superalgebra g˙⊗A⊕CC1, where g˙ is a finite-dimensional simple Lie algebra, and A is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the N=1 Heisenberg-Virasoro algebra.

Reflexive Modules on Normal Gorenstein Stein Surfaces, Their Deformations and Moduli

In this paper we generalize Artin-Verdier, Esnault and Wunram construction of McKay correspondence to arbitrary Gorenstein surface singularities. The key idea is the definition and a systematic use of a degeneracy module, which is an enhancement of the first Chern class construction via a degeneracy locus. We study also deformation and moduli questions. Among our main result we quote: a full classification of special reflexive MCM modules on normal Gorenstein surface singularities in terms of divisorial valuations centered at the singularity, a first Chern class determination at an adequate resolution of singularities, construction of moduli spaces of special reflexive modules, a complete classification of Gorenstein normal surface singularities in representation types, and a study on the deformation theory of MCM modules and its interaction with their pullbacks at resolutions. For the proof of these theorems it is crucial to establish several isomorphisms between different deformation functors, that we expect that will be useful in further work as well.

On the endomorphism algebra of Specht modules in even characteristic

Over fields of characteristic 2, Specht modules may decompose and there is no upper bound for the dimension of their endomorphism algebra. A classification of the (in)decomposable Specht modules and a closed formula for the dimension of their endomorphism algebra remain two important open problems in the area. In this paper, we introduce a novel description of the endomorphism algebra of the Specht modules and provide infinite families of Specht modules with one-dimensional endomorphism algebra.

Quantum Heisenberg enveloping algebras at roots of unity

In this article, the two-parameter quantum Heisenberg enveloping algebra, which serves as a model for certain quantum generalized Heisenberg algebras, has been studied at roots of unity. In this context, the quantum Heisenberg enveloping algebra becomes a polynomial identity algebra, and the dimension of simple modules is bounded by its PI degree. The PI degree, center, and complete classification of simple modules up to isomorphism are explicitly presented. We work over a field of arbitrary characteristic, although our results concerning the representations require that it is algebraically closed.

Examples and tutorials on using Google Colab and Gradio to create online interactive student‐learning modules

AbstractThis work provides online learning modules and instructions on how educators can leverage these technologies to help students learn in a personalized online environment. In particular, we focus on Google Colab, and the features provided by the Gradio Python library to provide interactivity within these modules. The contributions of this work include: (1) Development of a teaching framework using Gradio/Colab that offers automated grading and feedback for both educators and students; (2) Design of a versatile proposal, accommodating beginners with a straightforward interface while addressing the needs of advanced learners; (3) Creation of a comprehensive set of examples tailored for teaching digital logic subjects, with adaptability for application in various computer science areas. (4) A classification of these example learning modules in terms of their learning level for the students; (5) A novel client‐server approach based on Colab/Gradio, allowing teachers to manage the main notebook efficiently while providing a lightweight and reliable interface for students. The goal of this work is to further expose educators to the remarkable capabilities that cloud computing brings to online supplemental education, noting that large language models such as ChatGPT complement this work, in that chatbots will be able to guide students in these dynamic simulations.

PERFECTING GROUP SCHEMES

Abstract We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups and obtains a bijection with the set of classifying spaces of compact connected Lie groups topologically localised away from the characteristic. We also study the representations of perfectly reductive groups. We establish a highest weight classification of simple modules, the decomposition into blocks, and relate extension groups to those of the underlying abstract group.

Classification of simple smooth modules over the Heisenberg–Virasoro algebra

In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra ${\mathfrak {D}}$ , and simple smooth modules over the twisted Heisenberg–Virasoro algebra $\bar {\mathfrak {D}}$ with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak {D}}$ . A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg–Virasoro vertex algebras. We also present a few examples of simple smooth ${\mathfrak {D}}$ -modules and $\bar {\mathfrak {D}}$ -modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak {D}$ and $\bar {\mathfrak {D}}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.

The 3-cyclic quantum Weyl algebras, their prime spectra and a classification of simple modules ($q$ is not a root of unity)

The 3-cyclic quantum Weyl algebra A=A(\alpha, \beta, \gamma;q^{2}) , where \alpha, \beta, \gamma, q^{2} \in \mathbb{K} (a ground field), is a quadratic Noetherian domain of Gelfand–Kirillov dimension 3 that is generated by three subalgebras each of them is either a quantum plane or the quantum Weyl algebra. For the algebras A , their prime, completely prime, primitive and maximal spectra are described together with containments of prime ideals (the Zariski–Jacobson topology on the spectrum) and simple A -modules are classified when q^{2} is not a root of unity. For each prime ideal, an explicit set of ideal generators is given. The centre Z(A) of A is \mathbb{K}[\Omega] where \Omega is a cubic element. A semisimplicity criterion for the category of finite dimensional A -modules is given. Criteria are presented for all ideals of the algebra A to commute and for each ideal of A to be a unique product of primes (up to order).

Failure Mode Classification of IGBT Modules Under Power Cycling Tests Based on Data-Driven Machine Learning Framework

Failure Mode Classification of IGBT Modules Under Power Cycling Tests Based on Data-Driven Machine Learning Framework