6 research outputs found

    Collected Papers (on Neutrosophic Theory and Its Applications in Algebra), Volume IX

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    This ninth volume of Collected Papers includes 87 papers comprising 982 pages on Neutrosophic Theory and its applications in Algebra, written between 2014-2022 by the author alone or in collaboration with the following 81 co-authors (alphabetically ordered) from 19 countries: E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al-Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al-Tahan, Mehmat Ali Ozturk, Minghao Hu, S. Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel-Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K. Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al-Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok-Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang

    Investigation of students' reasoning with examples and non-examples of function in abstract algebra

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    The focus of this dissertation is on students' thinking regarding the function concept in the context of abstract algebra, focusing on the properties of well- and everywhere-definedness. This dissertation follows a three-paper format, where each paper has a different yet related focus.In the first paper, I analyze (non-)examples found in textbooks and those used during instruction related to function in abstract algebra to gain insight into how students are expected to reason about functions in this context. I investigate experts' thinking about functions by conducting a textbook analysis and semi-structured clinical interviews with mathematicians. I then elaborate the essential properties of well- and everywhere-definedness into four categories of non-examples that students are expected to successfully reason about in an introductory abstract algebra course.In the second paper, I explore students' reasoning with examples and non-examples of function related to the four categories by conducting task-based clinical interviews. I provide characteristics of a coordinated way of understanding functions in abstract algebra and illustrate how such a coordinated way of understanding functions enables students to reason productively with function tasks and the properties of well- and everywhere-definedness. This paper addresses what is entailed in reasoning productively about well-definedness and everywhere-definedness.In the last paper, I present a functions activity focused on recovering an example from a non-example of a function by prompting students to modify the domain, the codomain, and/or the rule. I argue that such an activity can help instructors have a clearer image of how they might support their students in developing a coordinated view of function and is a tool that abstract algebra instructors can use to help students attend to well-definedness and everywhere-definedness

    Guide to Discrete Mathematics

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    Topological materials discovery by large-order symmetry indicators.

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    Crystalline symmetries play an important role in the classification of band structures, and their richness leads to various topological crystalline phases. On the basis of our recently developed method for the efficient discovery of topological materials using symmetry indicators, we explore topological materials in five space groups ( S G s), which are diagnosed by large-order symmetry indicators (ℤ8 and ℤ12) and support the coexistence of several kinds of gapless boundary states in a single compound. We predict many candidate materials; some representatives include Pt3Ge ( S G 140 ), graphite ( S G 194 ), XPt3 ( S G 221 , X = Sn, Pb), Au4Ti ( S G 87 ), and Ti2Sn ( S G 194 ). As by-products, we also find that AgXF3 ( S G 140 , X = Rb, Cs) and AgAsX ( S G 194 , X = Sr, Ba) are good Dirac semimetals with clean Fermi surfaces. The proposed materials provide a good platform for studying the novel properties emerging from the interplay between different types of boundary states

    The Design and Validation of a Group Theory Concept Inventory

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    Within undergraduate mathematics education, there are few validated instruments designed for large-scale usage. The Group Concept Inventory (GCI) was created as an instrument to evaluate student conceptions related to introductory group theory topics. The inventory was created in three phases: domain analysis, question creation, and field-testing. The domain analysis phase included using an expert consensus protocol to arrive at the topics to be assessed, analyzing curriculum, and reviewing literature. From this analysis, items were created, evaluated, and field-tested. First, 383 students answered open-ended versions of the question set. The questions were converted to multiple-choice format from these responses and disseminated to an additional 476 students over two rounds. Through follow-up interviews intended for validation, and test analysis processes, the questions were refined to best target conceptions and strengthen validity measures. The GCI consists of seventeen questions, each targeting a different concept in introductory group theory. The results from this study are broken into three papers. The first paper reports on the methodology for creating the GCI with the goal of providing a model for building valid concept inventories. The second paper provides replication results and critiques of previous studies by leveraging three GCI questions (on cyclic groups, subgroups, and isomorphism) that have been adapted from prior studies. The final paper introduces the GCI for use by instructors and mathematics departments with emphasis on how it can be leveraged to investigate their students\u27 understanding of group theory concepts. Through careful creation and extensive field-testing, the GCI has been shown to be a meaningful instrument with powerful ability to explore student understanding around group theory concepts at the large-scale

    Integer Sparse Distributed Memory and Modular Composite Representation

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    Challenging AI applications, such as cognitive architectures, natural language understanding, and visual object recognition share some basic operations including pattern recognition, sequence learning, clustering, and association of related data. Both the representations used and the structure of a system significantly influence which tasks and problems are most readily supported. A memory model and a representation that facilitate these basic tasks would greatly improve the performance of these challenging AI applications.Sparse Distributed Memory (SDM), based on large binary vectors, has several desirable properties: auto-associativity, content addressability, distributed storage, robustness over noisy inputs that would facilitate the implementation of challenging AI applications. Here I introduce two variations on the original SDM, the Extended SDM and the Integer SDM, that significantly improve these desirable properties, as well as a new form of reduced description representation named MCR.Extended SDM, which uses word vectors of larger size than address vectors, enhances its hetero-associativity, improving the storage of sequences of vectors, as well as of other data structures. A novel sequence learning mechanism is introduced, and several experiments demonstrate the capacity and sequence learning capability of this memory.Integer SDM uses modular integer vectors rather than binary vectors, improving the representation capabilities of the memory and its noise robustness. Several experiments show its capacity and noise robustness. Theoretical analyses of its capacity and fidelity are also presented.A reduced description represents a whole hierarchy using a single high-dimensional vector, which can recover individual items and directly be used for complex calculations and procedures, such as making analogies. Furthermore, the hierarchy can be reconstructed from the single vector. Modular Composite Representation (MCR), a new reduced description model for the representation used in challenging AI applications, provides an attractive tradeoff between expressiveness and simplicity of operations. A theoretical analysis of its noise robustness, several experiments, and comparisons with similar models are presented.My implementations of these memories include an object oriented version using a RAM cache, a version for distributed and multi-threading execution, and a GPU version for fast vector processing
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