Projective Modules Research Articles

Case Study: Implementation and evaluation of a team-based authentic research project module for large cohorts.

Acquiring skills needed to plan and conduct research and communicate research outcomes are key learning aims in biochemical and biomedical disciplines. Final-year projects/dissertations are high-impact educational activities that commonly feature in undergraduate curricula. When cohort sizes exceed infrastructure and staff capacity, traditional models of supervised projects may not be feasible. This case study aims to share one model of practice with colleagues similarly engaged in design and delivery of final-year projects and research. Here, we outline the implementation and evaluation of a team-based, final-year research module on a transnational joint programme. Investigative Skills module was piloted in 2016-2017 and continues to run annually for >100 students. The research component is conducted over a timetabled, two-week block. In student questionnaires, the majority of respondents agreed that the projects were authentic, interesting and appropriate. The favourite aspect for most of the respondents was performing experimental work/doing research. Over 80% agreed that working in teams was conducive to accomplishing their goals, and their ideal team size is three to five students per team. The majority agreed that there was sufficient experimental work to do, but that more than two weeks practical time would be beneficial. The feedback has given insight into the whole of the student research experience of Investigative Skills, which is a sustainable model for authentic dissertation research for large cohorts.

Relatively flat modules on ring extensions

A ring extension is a ring homomorphism preserving identities. In this article, we establish the relationship among relatively flat modules, relatively projective modules and relatively injective modules on ring extensions. In particular, we prove that relatively projective modules are relatively flat, and that finitely presented and relatively flat modules are relatively projective. Moreover, we give a series of equivalent conditions of relatively flat modules on ring extensions from different perspectives. As applications, we prove that relatively flat modules are closed under extensions with respect to relative exact sequences, and obtain a necessary and sufficient condition for relative factor modules of relatively flat modules to be relatively flat.

Endotrivial modules for finite groups via homotopy theory

Classifying endotrivialkGkG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite groupGG, has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thévenaz, and others, it has been reduced to understanding the subgroup consisting of modular representations that split as the trivial modulekkdirect sum a projective module when restricted to a Sylowpp-subgroup. In this paper we identify this subgroup as the first cohomology group of the orbit category on non-trivialpp-subgroups with values in the unitsk×k^\times, viewed as a constant coefficient system. We then use homotopical techniques to give a number of formulas for this group in terms of the abelianization of normalizers and centralizers inGG, in particular verifying the Carlson–Thévenaz conjecture—this reduces the calculation of this group to algorithmic calculations in local group theory rather than representation theory. We also provide strong restrictions on when such representations of dimension greater than one can occur, in terms of thepp-subgroup complex andpp-fusion systems. We immediately recover and extend a large number of computational results in the literature, and further illustrate the computational potential by calculating the group in other sample new cases, e.g., for the Monster at all primes.

Categorification of Integer Sequences via Brauer Configuration Algebras and the Four Subspace Problem

The four subspace problem is a known matrix problem, which is equivalent to determining all the indecomposable representations of a poset consisting of four incomparable points. In this paper, we use solutions of this problem and invariants associated with indecomposable projective modules with some suitable Brauer configuration algebras to categorify the integer sequence encoded in the OEIS as A100705 and some related integer sequences.

The Decomposition of a Finitely Generated Module over Some Special Ring

This research aims to give the decompositions of a finitely generated module over some special ring, such as the principal ideal domain and Dedekind domain. One of the main problems with module theory is to analyze the objects of the module. This research was using a literature study on finitely generated modules topics from scientific journals, especially those related to the module theory. And by selective cases we find a pattern to build a conjecture or a hypothesis, by deductive proof, we prove the conjecture and state it as a theorem. The main result in this study is the decomposition of the finitely generated module is a direct sum of the torsion submodule and torsion-free submodule. Since the torsion-free module is always a free module over a principal ideal domain, then the torsion-free submodule is a free module. Last, we generalize the ring, from a principal ideal domain, to a Dedekind domain. We found then the torsion-free submodule became a projective module. Then the decomposition of the finitely generated module is a direct sum of the torsion submodule and the projective submodule. These results should help the researchers to analyze the objects on module theory.

On representation-finite gendo-symmetric algebras with only one non-injective projective module

Motivated by the relation between Schur algebra and the group algebra of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang and Steffen Koenig [15,16] addressed some behaviour of the endomorphism algebra of a generator over a symmetric algebra, which they called gendo-symmetric algebra. Continuing this line of works, we classify in this article the representation-finite gendo-symmetric algebras that have precisely one isomorphism class of indecomposable non-injective projective module. We also determine their almost ν-stable derived equivalence classes in the sense of Wei Hu and Changchang Xi [21]. It turns out that a representative can be chosen as the quotient of a representation-finite symmetric algebra by the socle of a certain indecomposable projective module.

A Serre–Swan theorem for coisotropic algebras

Coisotropic algebras are used to formalize coisotropic reduction in Poisson geometry as well as in deformation quantization and find applications in various other fields as well. In this paper we prove a Serre-Swan Theorem relating the regular projective modules over the coisotropic algebra built out of a manifold $M$, a submanifold $C$ and an integrable smooth distribution $D \subseteq TC$ with vector bundles over this geometric situation and show an equivalence of categories for the case of a simple distribution.

RD-projective module whose subprojectivity domain is minimal

A p-indigent module is one that is subprojective only to projective modules. An RD-projective module is subprojective to any torsionfree (and flat) module. An RD-projective module $T$ is called rdp-indigent if it is subprojective only to torsionfree modules. In this work, we consider the structure of SRDP rings whose (simple) RD-projective right $R$-modules are rdp-indigent or torsionfree. Moreover, new characterizations of P-coherent rings and torsionfree rings are presented by subprojectivity domains.

On finitely generated G-flat modules over domains

Let [Formula: see text] be a domain. It is proved that if [Formula: see text], then the class of finitely generated [Formula: see text]-flat modules and the class of finitely generated [Formula: see text]-projective modules coincide. It is also proved that an integrally closed domain [Formula: see text] is a Prüfer domain if and only if [Formula: see text], if and only if [Formula: see text].

UNIVERSITY-SCHOOL PARTNERSHIP IN ACHIEVING THE VISION OF INDONESIA 2045 THROUGH RESEARCH TRAINING ON HIGH SCHOOL STUDENTS

Universities participate in realizing Indonesia's vision 2045 “The Indonesians who are excellent, civilized, and mastering science technology, to bring Indonesia as the center of science technology development in the Asia and the world”. It would need intense effort and synergy, considering the science literacy of Indonesian students was ranked the 10th lowest out of 79 countries in 2018. The innovation capacity is ranked 74 out of 141 countries. Research is an essential activity, encouraging the development of science technology. Research competency should be honed by high school students as part of scientific inquiry. Therefore, this community service was carried out in collaboration with FORISMA FK UNAIR, aimed to increase students’ understanding of the urgency, stages, and procedures of research as well as increase interest in research activities. It was also intended to develop their skills to compile and present research proposals. The partner was Raudlatul Jannah High School Sidoarjo. It was implemented through workshops, coaching, and presentations followed by the teachers and 11th-grade students. The whole activity successfully improved students' knowledge. Increased knowledge score was obtained in 83% of participants. The pre-test evaluation showed 88% of participants had poor knowledge, whilst post-test evaluations were predominantly good. This community service increased students’ interest in research and changed their perceptions of research’s importance. This improved students' capability to identify research problems, search and compile scientific references as a basis for building hypotheses and conceptual frameworks, and determine research methods. The four groups of students were able to produce research proposals of sufficient quality and were able to present and discuss them in the forum. This community service was perceived to help the school in the implementation of curriculum, especially in science literacy. The school was committed to deepening students' knowledge and skills in the field of scientific research by providing research project modules. This community service is part of the contribution of higher education to the achievement of the vision of Indonesia 2045. This can be a model of scientific inquiry learning in secondary education, thus it is necessary to be implemented in other high schools.

3D visualization model construction based on generative adversarial networks.

The development of computer vision technology is rapid, which supports the automatic quality control of precision components efficiently and reliably. This paper focuses on the application of computer vision technology in manufacturing quality control. A new deep learning algorithm is presented, Multi-angle projective Generative Adversarial Networks (MapGANs), to automatically generate 3D visualization models of products and components. The generated 3D visualization models can intuitively and accurately display the product parameters and indicators. Based on these indicators, our model can accurately determine whether the product meets the standard. The working principle of the MapGANs algorithm is to automatically infer the basic three-dimensional shape distribution through the product’s projection module, while using multiple angles and multiple views to improve the fineness and accuracy of the three-dimensional visualization model. The experimental results prove that MapGANs can effectively reconstruct two-dimensional images into three-dimensional visualization models, and meanwhile accurately predict whether the quality of the product meets the standard.

Weak diamond, weak projectivity, and transfinite extensions of simple artinian rings

We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. We prove that if R is small, then the Weak Diamond implies that projectivity of an arbitrary module can be tested at the layer epimorphisms of R.

Height measurement of microbumps using a white-light triangulation method.

As the electronic interconnection between chips, microbumps are crucial components in advanced packaging for the demand of better performance and higher packaging density. The height and coplanarity of microbumps are critical to ensure the reliability of connections. The poor uniformity in bump height will lead to disconnect or insufficient contact, which will directly result in the failure of the chip's function and a lower yield rate. In this paper, we proposed a height measurement method of microbumps using white-light triangulation combined with geometrical characteristics of bumps. A linear light projection module composed of a lens group and LED light source was set up as well as a high-quality imaging module consisting of a CCD camera and microscope objective group. The projection and imaging model of microbumps illuminated by the light plane at different positions during the scanning process was analyzed. The microbump height is computed from a simple formula based on the geometry of the specimen and the system configuration. The measurement results are compared with that obtained from a commercial optical profiler, and the measurement uncertainty is analyzed in detail.

Auslander's formula and correspondence for exact categories

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category modadm(E) of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category E. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of E are reflected in modadm(E), for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe modadm(E) as a subcategory of mod(E) when E is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use modadm(E) to give a bijection between exact structures on an idempotent complete additive category C and certain resolving subcategories of mod(C).

On rings with finite Gorenstein weak global dimension

Let [Formula: see text] be a ring with finite Gorenstein weak global dimension. We characterize Gorenstein projective, injective and flat modules over [Formula: see text]. As an application, it is proved that [Formula: see text] is (strongly) CM-free if and only if [Formula: see text] has finite weak global dimension.

Differential projective modules over differential rings, II

Differential modules over a commutative differential ring R which are finitely generated projective as ring modules, with differential homomorphisms, form an additive category, so their isomorphism classes form a monoid. We study the quotient monoid by the submonoid of isomorphism classes of free modules with component wise derivation. This quotient monoid has the reduced K 0 of R (ignoring the derivation) as an image and contains the reduced K 0 of the constants of R as its subgroup of units. This monoid provides a description of the isomorphism classes of differential projective R modules up to an equivalence.

Bosonic Ghostbusting: The Bosonic Ghost Vertex Algebra Admits a Logarithmic Module Category with Rigid Fusion

The rank 1 bosonic ghost vertex algebra, also known as the beta gamma ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor C_2-cofinite. We identify a module category, denoted category mathscr {F}, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category mathscr {F} has sufficiently many projective and injective modules, give a classification of all indecomposable modules, show that fusion is rigid and compute all fusion products. The fusion product formulae turn out to perfectly match a previously proposed Verlinde formula, which was computed using a conjectured generalisation of the usual rational Verlinde formula, called the standard module formalism. The bosonic ghosts therefore exhibit essentially all of the rich structure of rational theories despite satisfying none of the standard rationality assumptions such as C_2-cofiniteness, the vertex algebra being isomorphic to its restricted dual or having a one-dimensional conformal weight 0 space. In particular, to the best of the authors’ knowledge this is the first example of a proof of rigidity for a logarithmic non-C_2-cofinite vertex algebra.

Training for the Industry: Examining the Use of Real-life Consultancy Projects in Development Studies Programmes

Development degrees have begun experimenting with real-life consultancy projects carried out by students for external clients in the development sector. Students, concerned about their ‘employability’, flock to these programmes due to their promises of hands-on, professionalizing experience. Surprisingly, these developments have not yet triggered a major reflection on what this teaching device means for the way our students are educated. I address this gap by critically examining their functioning, the different actors and their motivations and incentives. I argue that the introduction of development organizations into the teaching relationship challenges three core academic principles: scientificity, fairness and ethics. Drawing on my experience of managing a consultancy project module, I discuss the extent to which departments can address them.

Explicit calculations in an infinitesimal singular block of SLn

AbstractLet $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$. For each $n \geq 1$, there exists a singular block in the category of $G_1$-modules, which contains precisely $n+1$ irreducible modules. We are interested in the ‘lift’ of this block to the category of $G_1T$-modules. Imposing only mild assumptions on $p$, we will perform a number of calculations in this setting, including a complete determination of the Loewy series for the baby Verma modules and all possible extensions between the irreducible modules. In the case where $p$ is extremely large, we will also explicitly compute the Loewy series for the indecomposable projective modules.

Analysis of Deep Entrepreneurial Model in College Students Based on Mixed Space and Time Big Data Stream

Due to the homogenization of massive entrepreneurial undertaking by college students, they face a series of dilemmas such as the low utilization rate, poor user stickiness, failure to support various activities of the teachers and students, inability to maintain the sustainable development and so on. The solution is to establish a deep entrepreneurship mode of college students based on the mixed space and time big data stream. In this paper, a new idea of building an information platform based on the mixed space and time big data stream is explored. A new method for the construction of platform with the data as the core and the project as the orientation is put forward, which is composed of the data source analysis, data processing analysis and model establishment, innovation and entrepreneurship project module. Hence, a system platform is established with the “data” as the center and the functions expanded continuously by using the data mining technology, which will play an active role in the sustainable development of projects, the self-cognition of teachers and students, the discovery of enterprise talents, the decision making of college education and other aspects.