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Given a finite set of formal languages (𝔏1, …, 𝔏n), one may be interested in analyzing the correspondences of short sentences in one language with coherent sheaves of words in another. In this paper, we work towards the goal of doing so by extending the set-theoretic notion of a “transitive inner model” to the more amenable setting of abstract algebra.

Prior studies have identified the impact beliefs have on mathematics instructors’ instructional practice, such as their choice to (or not to) lecture. However, the role of instructional context role in influencing beliefs and instruction has not been thoroughly researched. This paper explores how course context and beliefs could impact mathematics instructors’ propensity to lecture by investigating two very different instructional contexts in undergraduate mathematics in the United States: Calculus and Abstract Algebra. The results of our regression analyses were significant in both data sets and, we did find beliefs in each context that predicted the amount of time spent lecturing. For instance, the more calculus instructors believed in the effectiveness of teacher-centered instructional practices, the more likely they were to lecture. Whereas the more abstract algebra instructors believed in their student’s capacity to learn the less likely they were to lecture. However, while the regression model for the abstract algebra instructors accounted for 37.8% of the variability in the reported amount of time spent lecturing, the model for Calculus instructors only accounted for 2.7% of the variability. Thus our analyses indicate that there are contextual differences, such as course coordination, student demographics, and the job security of the instructors, that may be mitigating the extent to which beliefs impact instructional practice.

We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings.

The goal of this study is to identify the multiple ways in which instructors take advantage of a feature designed into university textbooks that seeks to invite students to get acquainted with the content prior to attending the class in which such content will be discussed. We present an analysis of teacher instrumentation of this feature, which we call questioning devices, using data from 15 instructors who taught calculus, linear algebra, or abstract algebra over one semester. The instructors taught at 14 different universities in the United States. We identified four utilization schemes of the questioning devices in which instructors: completed questioning devices for pre-planning, required students to complete the questioning devices for the purpose of lesson planning, used the questioning devices for the purpose of instruction, and required students complete the questioning devices for the purpose of assessment. These schemes are supported by various operational invariants related to self-perception as competent instructors and implicit theories of teaching and learning. The identified utilization schemes inform textbook developers and author-designers, making them aware of whether these features fulfill their design purposes, and possibly think about changes that might be needed to support instructors in achieving their instructional goals and improve learning. We suggest some further areas of inquiry.

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, and topological spaces. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader framework of fuzzy setting. In this paper, we introduce the notions of int-softm,n-ideals, int-softm,0-ideals, and int-soft0,n-ideals of semigroups by generalizing the concept of int-soft bi-ideals, int-soft right ideals, and int-soft left ideals in semigroups. In addition, some of the properties of int-softm,n-ideal, int-softm,0-ideal, and int-soft0,n-ideal are studied. Also, characterizations of various types of semigroups such asm,n-regular semigroups,m,0-regular semigroups, and0,n-regular semigroups in terms of their int-softm,n-ideals, int-softm,0-ideals, and int-soft0,n-ideals are provided.

The most essential thing in mathematics is proof, it makes mathematics being different with other subjects. One of subject in mathematics that always need prove to understand the concept is abstract algebra. In studying abstract algebra, student need various abstract concepts to include in its concepts. It is hard for student to understand the structures in abstract algebra and prove some of mathematical object that satisfy the structures. Group and its properties is the first structure in abstract algebra that has an abstract concept. It is hard for student to understand some objects, that is proven satisfy a structure and why the proof steps just flow. By giving explanation and reason in every proofing step, we try to increase student proving level and reduce the abstraction level of the concepts. To see how this module reduces the abstraction level in teaching group, this module is applied to university students and evaluated by interviewing and questionnaires to the students. Base on student response and by some perspectives, student proving ability increase and the abstraction level of the concept is diminished in some aspects.

This paper presents a PVS development of relevant results of the theory of rings. The PVS theory includes complete proofs of the three classical isomorphism theorems for rings, and characterizations of principal, prime and maximal ideals. Algebraic concepts and properties are specified and formalized as generally as possible allowing in this manner their application to other algebraic structures. The development provides the required elements to formalize important algebraic theorems. In particular, the paper presents the formalization of the general algebraic-theoretical version of the Chinese remainder theorem (CRT) for the theory of rings, as given in abstract algebra textbooks, proved as a consequence of the first isomorphism theorem. Also, the PVS theory includes a formalization of the number-theoretical version of CRT for the structure of integers, which is the version of CRT found in formalizations. CRT for integers is obtained as a consequence of the general version of CRT for the theory of rings.

We describe the design and implementation of lessons in undergraduate abstract algebra that integrate applications to teaching high school mathematics. Each lesson consists of a pre-activity, class...

Funding: Ajay Kumar is supported by CSIR-UGC JRF, New Delhi, India, through Ref No.: 19/06/2016(i)EU-V/Roll No: 417267. Lavanya Selvaganesh is financially supported by SERB, India, through Grant No.: MTR/2018/000254 under the scheme MATRICS. T. Tamizh Chelvam is supported by CSIR Emeritus Scientist Scheme of Council of Scientific and Industrial Research (No.21 (1123)/20/EMR-II), Government of India. Algebraic graph theory is the study of the interplay between algebraic structures (both abstract as well as linear structures) and graph theory. Many concepts of abstract algebra have facilitated through the construction of graphs which are used as tools in computer science. Conversely, graph theory has also helped to characterize certain algebraic properties of abstract algebraic structures. In this survey, we highlight the rich interplay between the two topics viz groups and power graphs from groups. In the last decade, extensive contribution has been made towards the investigation of power graphs. Our main motive is to provide a complete survey on the connectedness of power graphs and proper power graphs, the Laplacian and adjacency spectrum of power graph, isomorphism, and automorphism of power graphs, characterization of power graphs in terms of groups. Apart from the survey of results, this paper also contains some new material such as the contents of Section 2 (which describes the interesting case of the power graph of the Mathieu group M_{11}) and subsection 6.1 (where conditions are discussed for the reduced power graph to be not connected). We conclude this paper by presenting a set of open problems and conjectures on power graphs. Publisher PDF

The concept of abstract algebra on intuitionistic fuzzy sets were introduced and some basic theorems were proved by authors in 2017. In this study, homomorphism between intuitionistic fuzzy abstract algebras is defined, intuitionistic fuzzy function is examined and then intuitionistic fuzzy congruence relations are defined on intuitionistic fuzzy abstract algebra. First and third isomorphism theorems on intuitionistic abstract algebras are introduced.

We propose the equivalence between one Wolfram model and the qubit permutation semigroup.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Santiago Zarzuela, [en] In this work, we study some of the correspondences between commutative algebra and graph theory. We start with an introduction on edge and cover ideals that lead into introducing the algebra basic tools that lets us state Fröberg’s Theorem on linear resolutions of edge ideals and chordal graphs. The work ends with a proof of Fröberg’s Theorem given by Adam Van Tuyl, based on monomial splittings and some properties of chordal graphs.

As a system of abstract algebra, evolution algebras are commutative and non-associative algebras. There is no deep structure theorem for general non-associative algebras. However, there are deep structure theorem and classification theorem for evolution algebras because it has been introduced concepts of dynamical systems to evolution algebras. Recently, in [25], it has been studied some properties of nilpotent evolution algebra with maximal index (dim E2 = dim E − 1). This paper is devoted to studying nilpotent finite-dimensional evolution algebras E with dim E2 =dim E − 2. We describe Lie algebras related to the evolution of algebras. Moreover, this result allowed us to characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras are found.

Polynomial-time attacks designed to run on quantum computers and capable of breaking RSA and AES are already known. It is imperative to develop quantum-resistant algorithms before quantum computers become available. Computationally hard problems defined on lattices have been proposed as the fundamental security bases for a new type of cryptography. The National Institute of Standards and Technology (NIST) recently hosted the Post-Quantum Cryptography Standardization project, aiming to create a roster of innovative post-quantum cryptosystems. These candidates have been publicly available for testing since early 2017. As they are currently under analysis, new proposals are still desirable. As such, we use the ring learning with errors (RLWE) problem combined with arithmetic functions to propose the O2MD2 cryptosystem, which provides a one-to-many private/public key architecture having a distributed key refresh for a network of users while working on multiple polynomial rings over different prime order fields. Our solution has three different frameworks that reach AES-256 equivalent security, and provides message integrity and message authenticity verifications. We compare our solution’s speed against the speed of the twenty-six different implementations from seven popular candidates in the NIST project, and our cryptosystem performs from 2 to 4 orders of magnitude faster than them. We also propose six different implementations that reach the security levels 1, 3 and 5 proposed in the NIST competition. Finally, we used the NIST Statistical Test Suite to verify the indistinguishability of our produced ciphertexts against randomly generated noise.

This article is based on the study, which tries to unpack strategies of reduction of abstraction in learning abstract algebra from learners’ perspective. Ethnography was used to collect the required information. The study found the strategies of reduction of abstraction in abstract algebra are: making sense and meaning through previous experiences and existing knowledge an analogical creation of mental image, using first person language in course of doing mathematics by students as teachers do in the classroom for logical arguments, focusing on “symbol” or some mathematical entity to manage abstraction for their idiosyncratic understandings of abstract mathematical structure rather than the reflective thinking, using students own idiosyncratic figures to reduce the degrees of complexity of mathematical concepts. This study can lead teachers of abstract algebra to a new awareness of their teaching strategies and their practices.

Connecting and comparing across student strategies has been shown to be productive for students in elementary and secondary classrooms. We have recently been working on a project converting such pr...

The teaching of mathematics does not only require the teacher to have knowledge about the subject, but the teacher also needs mathematical knowledge that is useful for the teaching and explaining thereof, as the teacher’s knowledge effects the students’ knowledge. A teacher should use appropriate mathematical explanation to be understood well by her/his students. In the study reported on here we investigated how prospective mathematics teachers defined the concept of prime number and which strategies they employed to explain the concept. The study was a descriptive survey within qualitative research. Forty-eight participants took part in the study and all completed the abstract algebra courses where they learned about the concept in question. The data collection tool was a form comprising 3 open-ended questions challenging what the concept of prime number was and how this concept could be explained to secondary/high school students. The data were analysed and the results show that the preservice teachers experienced great difficulty in defining the concept of prime number and that they used rules to explain prime numbers. Keywords: explanatory strategies; prime number; prospective mathematics teachers

Mathematics education students’ ability on developing evidence needs to be reviewed by lecturers. 50% of the subjects in the mathematics education departement require students' accuracy in analyzing mathematical statements. This is increasingly important because the Indonesian National Qualifications Framework (KKNI) for the Bachelor level requires graduates to become technicians / analysts. The diversity of levels of students’ ability, gender, and learning styles are assumsed to be the discrepancy abilities to develop evidence. This study aims to describe the evidence developing ability based on the levels of achievement, gender, and student learning styles. The subjects of this study were abstact algebra students in the mathematics education department. Type of this research was descriptive qualitative with data collection techniques using test and non-test techniques. Students were given five questions about abstact algebra that demanded the evidence developing ability. The achivement levels and gender were obtained from students’ academic achievement data. The learning styles were attained from questionnaires. The results of the study is higher the students’ academic achievement, better the evidence developing ability on Abstract Algebra. The gender does not affect the evidence developing ability on Abstact Algebra. Students with visual learning styles have the ability to develop evidence better than students with other learning styles.

The Abstract Algebra is one of the most difficult subjects for students. In this course, students are required to have several textbooks as their reading source. However, the existing textbooks do not guide students in carrying out the process of preparing evidence and tend to speak non-Indonesian languages. The purpose of this research is to design and develop textbooks on abstract algebra courses which contain proofs in full step by step that can improve the ability to organize evidence. This type of research is development research with formative evaluation design consisting of self-evaluation, prototyping (expert reviews, one-to-one, and small groups), and field tests. The validity of the development of abstract algebra textbook is passed through the stages of self-evaluation and expert reviews. The results showed that the prototype of abstract algebra teaching books had a very high level of validity (89.29%).

MLsub extends traditional Hindley-Milner type inference with subtyping while preserving compact principal types, an exciting new development. However, its specification in terms of biunification is difficult to understand, relying on the new concepts of bisubstitution and polar types, and making use of advanced notions from abstract algebra. In this paper, we show that these are in fact not essential to understanding the mechanisms at play in MLsub. We propose an alternative algorithm called Simple-sub, which can be implemented efficiently in under 500 lines of code (including parsing, simplification, and pretty-printing), looks more familiar, and is easier to understand. We present an experimental evaluation of Simple-sub against MLsub on a million randomly-generated well-scoped expressions, showing that the two systems agree. The mutable automaton-based implementation of MLsub is quite far from its algebraic specification, leaving a lot of space for errors; in fact, our evaluation uncovered several bugs in it. We sketch more straightforward soundness and completeness arguments for Simple-sub, based on a syntactic specification of the type system. This paper is meant to be light in formalism, rich in insights, and easy to consume for prospective designers of new type systems and programming languages. In particular, no abstract algebra is inflicted on readers.

The sudden change caused by the global pandemics of COVID-19 leads to the classroom transformation from classical face-to-face meetings into virtual. The development of technology enables classroom variation by providing various applications that can be employed to facilitate learning activities. Nonetheless, not all situations suitable to use advanced technology during distance learning. Some students are living in remote areas with limited internet connection. This study aims to offer an alternative if most of the students were having difficulties with the internet and minimum devices to download heavy applications. The alternative is by using a chat group with an emphasis on the students’ interaction during the lesson. This descriptive study was conducted at a university in Mataram, Indonesia. The subject was 17 students in the Mathematics Education Study Program who follow the course of Abstract Algebra. The data were gathered from students’ observation during the lessons and students’ written work in the middle semester test. The data were analyzed by using descriptive qualitative method. From the analysis, it was found that the students’ activity during distance learning was 83.5%. Furthermore, 88.23% of students achieved the minimum score for the middle semester test (more than 56). The study showed that mathematics teaching and learning could be done with a secure and straightforward access application to gain good results.

This paper aims to reveal the difficulties as experienced by graduate students while learning abstract algebra at master's degree in mathematics education in the first month of their enrollment. For, I adopted a case study of five students, interviewed them with the help of interview guidelines and observed their behaviors in the classroom, and triangulated this information with researcher’s experiences to explore the difficulties in learning abstract algebra. I used inductive method to analyze the information and concluded that graduate students have experienced several difficulties in learning abstract algebra including the difficulties in conceptualizing algebraic facts, constructing examples and non-examples; and proving theorems. Finally, the study suggests that graduate teaching need to focus on conceptual and procedural understanding of students, and emphasizing to construct examples and non-example as much as possible to improve students’ learning.

Los profesores de Topología y algebra moderna manifestaron interés en la necesidad de crear un espacio que permita profundizar el proceso de Generalización Matemática desde la articulación de algunos conceptos de la Teoría de Nudos con el desarrollo de habilidades del Pensamiento Matemático Avanzado (PMA). Objetivo: Ofrecer a los estudiantes un espacio adicional de formación disciplinar que les permita profundizar el proceso de Generalización matemática. Diseño: La metodología utilizada tiene un enfoque cualitativo, como estrategia asumimos la investigación-acción desde la propuesta de Whitehead (1991) desde tres fases. Entorno y participantes: estudiantes del programa Licenciatura en Matemáticas que cursan de tercer a sexto semestre. Recopilación y análisis de datos: enfatizamos en la segunda fase (de intervención), dado que nos permitió articular el esquema holístico de la teoría de nudos con el PMA como se muestra en las tablas 2, 3 y 4 (sección resultados). Resultados: El resultado fue la creación de un sílabo y guía de asignatura para un seminario electivo, que se oferta a los estudiantes de la licenciatura. Conclusión: desde el 2019 se oferta a los estudiantes este seminario electivo que otorga 3 créditos. The professors of Topology and modern algebra expressed interest in the need to create a space that allows deepening the process of Mathematical Generalization from the articulation of some concepts of Theory of Knots with the development of Advanced Mathematical Thinking (PMA) skills. Objective: To offer students an additional space for disciplinary training that allows them to deepen the process of Mathematical Generalization. Design: The methodology used has a qualitative approach, as a strategy we take action research from the Whitehead (1991) proposal from three phases. Setting and participants: students of the Bachelor of Mathematics program who take the third to sixth semester. Data collection and analysis: we emphasized in the second phase (intervention), since it allowed us to articulate the holistic scheme of knot theory with the PMA as shown in Tables 2, 3 and 4 (results section). Results: The result was the creation of a syllabus and subject guide for an elective seminar, which is offered to undergraduate students. Conclusion: since 2019 this elective seminar is offered to students, which awards 3 credits.

Mathematical proof is a logically formed argument based on students' thinking process. A mathematical proof is a formal process which needs the ability of analytical thinking to solve. However, researchers still find students who complete the mathematical proof process through intuitive thinking. Students who have studied mathematical proof in the early semester should not have completed abstract algebraic proof intuitively. Therefore, the aim of this research is to explore students' thinking process in conducting mathematical proof based on Mason's framework. The instrument used to collect data was mathematical proof problems test related to abstract algebra and interviews. There are three out of 25 students who did abstract algebra through intuitive thinking as they only used two stages of the Mason's thinking framework. Then, two out of three students were chosen as the subjects of the study. The selection of research subjects is based on the student's ability to express intuitive thinking verbally process which were conducted while completing the test. It is found that students can form structural-intuitive warrant that they use to complete the mathematical proof of abstract algebra. Structural-intuitive warrant formed by students at the stage of attack and review are in the form of: institutional warrant and evaluative warrant, while at the entry and attack stage are a priori warrant and empirical warrant.

Almost all finite groups encountered by undergraduates can be represented as multiplicative groups of concise block-diagonal binary matrices. Such representations provide simple examples for beginn...

Mathematical proving is an important ability to learn abstract algebra. Many students, however, found difficulties in solving problems involving mathematical proof. This research aims to describe the students' mathematical proving ability and to find out the difference of the ability among students in private universities with three different levels of accreditation – A, B, and C. We used descriptive and comparative methods to reach the goals by involving mathematics education department students from A, B, and C-accredited private universities as its subjects. We used a test and interview to collect the data. The data of the students' mathematical proving ability were then statistically described and then compared among the three subject categories using the Kruskal-Wallis test and U Mann Whitney post hoc test. The results suggest that the students' mathematical proving ability from the A, B, and C-accredited universities respectively were 77.14 (high category), 39.32 (low category), and 36.78 (low category). Furthermore, the comparison results suggest that the significant differences only happen between universities with A and B accreditation level, and between the ones with A and C accreditation. Based on these findings, the mathematical proving ability of the students from B and C-accredited universities still needs to be improved by making the students accustomed to exercising with proof problems, motivating them to learn, and providing them learning materials that are easy to understand.

These notes are from lectures given in 2015 at University College Cork. They aim to explain the most concrete and fundamental aspects of algebra, in particular the algebra of the integers and of polynomial functions of a single variable, grounded by proofs using mathematical induction. It is impossible to learn mathematics by reading a book like you would read a novel; you have to work through exercises and calculate out examples. You should try all of the problems. More importantly, since the purpose of this class is to give you a deeper feeling for elementary mathematics, rather than rushing into advanced mathematics, you should reflect about how the simple ideas in this book reshape your vision of algebra. Consider how you can use your new perspective on elementary mathematics to help you some day guide other students, especially children, with surer footing than the teachers who guided you.

The Following is a Research Thesis by the author of the 8T. The new Infinitytheory aims to present the subject of Infinities as analyzedfrom a sync of set theory and the author forte, Calculus of Variations/Lagrangians. Insert V1.1 - Decays of Infinities - Page 73. The file include three parts, first the Variational approach on Infinities, the second part -classification of Infinitiesand the third andlast part is the Nature of Infinites. Not all theorems are .proven, this will be presented at later versions. The Theory contains four main equations, whichare (1),(1.2) and equation(2) The Thesis is written in a form of self refection, allowing the reader to experience how the ideas varied over time.&nbsp

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES This work is a qualitative case study which aims to research the relevance of the contents of the disciplines that address rings and groups in the pre-service teacher course in Mathematics of the Federal University of Santa Maria. It aims to view the opinions of the undergraduate students. The reason for such research is due to the high rate of failure in these disciplines, even though changes have been made over the years. To attain the objective, a survey of researches was conducted in the Catalog of Theses and Dissertations and in the Brazilian Digital Library of Theses and Dissertations, and analyzed the menus of these disciplines, available on the institution’s menu portal. From the survey and analysis, a questionnaire was elaborated, to be applied to students, and analyzed through the Content Analysis Theory. Thus, it was possible to observe that most of the subjects consider abstract algebra important for the achievement of their degree, and can see relations with the content of Basic Educatio. However, they question the way the discipline approaches the content, because when enrolled in the discipline few had contact with this relationship. O presente trabalho é um estudo de caso, qualitativo, que tem como objetivo pesquisar a respeito da relevância dos conteúdos das disciplinas que abordam anéis e grupos no curso de Licenciatura em Matemática da Universidade Federal de Santa Maria para a formação do professor na visão dos acadêmicos e egressos do curso. A justificativa para tal pesquisa deve-se à elevada taxa de reprovação nessas disciplinas, mesmo tendo sido feitas modificações ao longo dos anos. Para cumprir o objetivo efetuou-se um levantamento de pesquisas no Catálogo de Teses e Dissertações e na Biblioteca Digital Brasileira de Teses e Dissertações. Também foram analisadas as ementas das disciplinas que abordam anéis e grupos do curso de Matemática Licenciatura da Universidade Federal de Santa Maria, disponíveis no portal do ementário da instituição. A partir do levantamento e da análise elaborou-se um questionário para ser aplicado aos acadêmicos e egressos do curso de Matemática Licenciatura dessa universidade. Os dados foram analisados por meio da Análise de Conteúdo de Bardin. Assim, foi possível observar que a maioria dos sujeitos considera a Álgebra Abstrata importante para a formação do Licenciando em Matemática e consegue ver relações com conteúdo da Educação Básica. Porém, questionam a forma que a disciplina aborda o conteúdo, pois quando matriculados na disciplina poucos tiveram contato com esta relação.

[EN] The study of Complex Systems is one of the scientific fields that has had the highest productivity in recent decades and has not ceased to fascinate the community dedicated to studying its properties. In particular, Network Science has proven to be one of the most prolific areas within Complex Systems. In recent years, his methods have been applied to model multiple phenomena in real life, both naturally generated, such as in biology, and due to the actions and interactions of man, such as social networks or communication networks. Recently, it has been seen how the methods of Network Science can be applied in the context of mathematics, as is the case of Number Theory. One of the most studied cases is networks whose elements are numbers and which are related through the divisibility relation. The main objective of this thesis is to extend these studies to other sets of numbers. On the one hand, we study the divisibility in natural numbers when we obtain these from Pascal matrices of increasing size, which allows us to extract non-sequential sets of numbers with non-constant increments between them. On the other hand, we study the case of the divisibility relation of rational numbers. Cantor's diagonal argument provides a way to order all rational numbers, which allows us to check to what extent some of the properties observed for the divisibility of natural numbers are extensible to a more general context. The thesis is divided into 4 Chapters. Chapter 1 contains a general introduction to the thesis and it is structured into 6 sections. In Sections 1.1 and 1.2, we briefly introduce Network Science, show some application examples, and motivate the study of networks of numbers generated from the divisibility property. In Section 1.3, we define the objectives of this PhD thesis and its scope. In Section 1.4, we present the notion of network, its representations, and some measures that can be calculated on them, such as nodes degrees, their distribution, the assortativity and the clustering coefficients. In another hand, in Section 1.5, we review the best-known network models such as Erdo¿s and Re'nyi random networks, Watts and Strogatz small-world networks, Baraba'si and Albert scale-free networks, and hierarchical networks. Finally, at the end of this Chapter 1, we show in Section 1.6 a review of various studies carried out in order to apply Network Science methods to problems and properties that arise in Number Theory, such as divisibility networks or networks generated from Collatz's Conjecture. or Goldbach's Strong Conjecture. In Chapters 2 and 3, we show the results obtained and that have been published to date. Finally, in Chapter 4, we summarize the conclusions obtained and indicate some related problems that we consider of interest to address in the future., [ES] El estudio de los Sistemas Complejos es uno de los campos científicos que ha tenido mayor productividad en las últimas décadas y no ha dejado de fascinar a la comunidad que se dedica al estudio de sus propiedades. En particular, la Ciencia de Redes se ha mostrado como una de las áreas más prolíficas dentro de los Sistemas Complejos. En los últimos años, sus métodos han sido aplicados para modelar múltiples fenómenos de la vida real tanto generados de manera natural, como puede ser en el caso de la biología, como debidos a las acciones e interacciones del hombre, como puede ser el caso de las redes sociales o las redes de comunicaciones. Recientemente, se ha visto cómo los métodos de la Ciencia de Redes pueden ser aplicados en el contexto de las matemáticas, como es el caso de la Teoría de Números. Uno de los casos que más se han estudiado es el de las redes cuyos elementos son números y que se relacionan mediante la relación de la divisibilidad. El objetivo principal de esta tesis es extender estos estudios a otros conjuntos de números. Por una parte, estudiamos la divisibilidad en los números naturales cuando obtenemos estos a partir de subconjuntos de números naturales extraídos de matrices de Pascal de orden creciente, lo que nos permite extraer conjuntos de números de manera no secuencial y con incrementos no constantes entre ellos. Por otra parte, estudiamos el caso de la relación de divisibilidad de los números racionales, dado que a partir del argumento diagonal de Cantor se pueden ordenar, lo que nos permite comprobar hasta qué punto algunas de las propiedades observadas para la divisibilidad de los números naturales son extensibles a un contexto más general. La tesis se divide en 4 capítulos. El capítulo 1 contiene una introducción general a la tesis y está estructurado en 6 secciones. En las secciones 1.1 y 1.2, presentamos brevemente la Ciencia de Redes, mostrando algunos ejemplos de aplicación y motivamos el estudio de redes de números generadas a partir de la propiedad de divisibilidad. En la Section 1.3, definimos los objetivos de esta tesis doctoral y su alcance. En la sección 1.4, presentamos la noción de red, sus formas de representación y algunas medidas que se pueden calcular sobre ellas, como son los grados de los nodos, la distribución de estos grados, la asortatividad y los coeficientes de clustering. Por otro lado, en la Sección 1.5, revisamos los modelos de redes más conocidos como son las redes aleatorias de Erdös y Rényi, las redes de pequeño mundo de Watts y Strogatz, las redes libres de escala de Barabási y Albert y las redes jerárquicas. Mostramos en la Sección 1.6, una revisión de diversos estudios realizados con el fin de aplicar métodos de la Ciencia de Redes a problemas y propiedades que surgen en la Teoría de Números, como son las redes de divisibilidad o redes generadas a partir de la Conjetura de Collatz o la Conjetura Fuerte de Goldbach. En los Capítulos 2 y 3, mostramos los resultados obtenidos y que han sido publicados hasta la fecha y, finalmente, en el Capítulo 4, resumimos las conclusiones obtenidas e indicamos algunos problemas relacionados que consideramos de interés abordar en un futuro., [CAT] L'estudi dels Sistemes Complexos és un dels camps científiques que ha tingut major productivitat en les últimes dècades i no ha deixat de fascinar a la comunitat que es dedica a l'estudi de les seues propietats. En particular, la Ciència de Xarxes s'ha mostrat com una de les àrees més prolífica dins dels Sistemes Complexos. En els últims anys, els seus mètodes han sigut aplicats per a modelar múltiples fenòmens de la vida real tant generats de manera natural, com pot ser en el cas de la biologia, com deguts a les accions i interaccions de l'home, com pot ser el cas de les xarxes socials o les xarxes de comunicacions. Recentment, s'ha vist com els mètodes de la Ciència de Xarxes poden ser aplicats en el context de les matemàtiques, com és el cas de la Teoria de Números. Un dels casos que més s'han estudiat és el de les xarxes els elements de les quals són números i que es relacionen mitjançant la relació de la divisibilitat. L'objectiu principal d'aquesta tesi és estendre aquests estudis a altres conjunts de números. D'una banda, estudiem la divisibilitat en els nombres naturals quan obtenim aquests a partir de matrius de Pascal de grandària creixent, la qual cosa ens permet extraure conjunts de números de manera no sequëncial i amb increments no constants entre ells. D'altra banda, estudiem el cas de la relació de divisibilitat dels nombres racionals, atés que a partir de l'argument diagonal de Cantor es poden ordenar, la qual cosa ens permet comprovar fins a quin punt algunes de les propietats observades per a la divisibilitat dels nombres naturals són extensibles a un context més general. La tesi es troba dividida en 4 Capítols. El capítol 1, conté una introducció general a la tesi i está estructurat en 6 seccions. En les seccions 1.1 i 1.2, presentem breument la Ciència de Xarxes, mostrant alguns exemples d'aplicació i motivem l'estudi de xarxes de números generades a partir de la propietat de divisibilitat. En la Section 1.3, definim els objectius d'aquesta tesi doctoral y el seu abast. En la Secció 1.4, presentem la noció de xarxa, les seves formes de representació i algunes mesures que es poden calcular sobre elles, com són els graus dels nodes, la distribució d'aquests graus, la asortatividad i els coeficients de clustering. En la Sección 1.5, revisem els models de xarxes més coneguts com són les xarxes aleatòries de Erdös i Renyi, les xarxes de xicotet món de Watts i Strogatz, les xarxes lliures d'escala de Barabási i Albert i les xarxes jeràrquiques. Mostrem en la Sección 1.6 una revisió de diversos estudis realitzats amb la finalitat d'aplicar mètodes de la Ciència de Xarxes a problemes i propietats que sorgeixen en la Teoria de Números, com són les xarxes de divisibilitat o xarxes generades a partir de la Conjectura de Collatz o la Conjectura Forta de Goldbach. En els Capítols 2 i 3, vam mostrar els resultats obtinguts i que han sigut publicats fins hui i, finalment, en el Capítol 4, resumim les conclusions obtingudes i indiquem alguns problemes relacionats que considerem d'interés abordar en un futur.

By analyzing the 8T framework the author analyzes the features which can not be predicted and considered as uncertainties. These go beyond the conjugate relation of QM and serve as integral feature of the theory. There is also major uncertainty considering the Question of deriving the masses from a variational principle, identical to the primorial, without measurement.&nbsp

The author uses the setting which allowed the construction of the Primorial coupling series within the idea on N-tuples and ordered sets. Such an idea allows us to expend the Bosons as net variations and present them as product type of tuples.

The author uses the setting which allowed the construction of the Primorial coupling series within the idea on N-tuples and ordered sets. Such an idea allows us to expend the Bosons as net variations and present them as product type of tuples.&nbsp

We discuss the role of a heuristic principle known as the Principle of Permanence of Forms in the development of mathematics, especially in abstract algebra. We try to find some analogies in the development of modern formal logic. Finally, we add a few remarks on the use of the principle in question in mathematical education.

Diagrammatic reasoning, based on visualization and analogy, is the foundation for reasoning in ordinary language and the most esoteric theories of mathematics. Long before they write a formal proof, mathematicians develop their ideas with diagrams, visualize novel patterns, and discover creative analogies. For over two millennia, Euclid’s diagrammatic methods set the standard for mathematical rigor. But the abstract algebra of the 19th century led many mathematicians to claim that all formal reasoning must be algebraic. Yet C. S. Peirce and George Polya recognized that Euclid’s diagrammatic reasoning is a better match to human thought patterns than the algebraic rules and notations. A combination of Peirce’s graph logic, Polya’s heuristics, and Euclid’s diagrams is a better candidate for a natural logic than any algebraic formalism. Psychologically, it supports Peirce’s claim that his existential graphs (EGs) provide "a moving picture of the action of the mind in thought." Logically, EGs have a formal mapping to and from the ISO standard for Common Logic. Computationally, algorithms for Cognitive Memory (CM) and virtual reality (VR) can support cross-modal analogies between language-like and image-like representations.

Formal verification technology has been widely applied in the fields of mathematics and computer science. The formalization of fundamental mathematical theories is particularly essential. Axiomatic set theory is a foundational system of mathematics and has important applications in computer science. Most of the basic concepts and theories in computer science are described and demonstrated in terms of set theory. In this paper, we present a formal system of axiomatic set theory based on the Coq proof assistant. The axiomatic system used in the formal system refers to Morse-Kelley set theory which is a relatively complete and concise axiomatic set theory. In this formal system, we complete the formalization of the basic definitions of sets, functions, ordinal numbers, and cardinal numbers and prove the most commonly used theorems in Coq. Moreover, the non-negative integers are defined, and Peano's postulates are proved as theorems. According to the axiom of choice, we also present formal proofs of the Hausdorff maximal principle and Schröeder-Bernstein theorem. The whole formalization of the system includes eight axioms, one axiom schema, 62 definitions, and 148 corollaries or theorems. The “axiomatic set theory” formal system is free from the more apparent paradoxes, and a complete axiomatic system is constructed through it. It is designed to give a foundation for mathematics quickly and naturally. On the basis of the system, we can prove many famous mathematical theorems and quickly formalize the theories of topology, modern algebra, data structure, database, artificial intelligence, and so on. It will become an essential theoretical basis for mathematics, computer science, philosophy, and other disciplines.

This article presents a brief and new solution to the problem known as the “Fermat’s Last Theorem”. It is achieved without the use of abstract algebra elements or elements from other fields of modern mathematics of the twentieth century. For this reason it can be easily understood by any mathematician or by anyone who knows basic mathematics. The important thing is that the above “theorem” is generalized. Thus, this generalization is essentially a new theorem in the field of number theory.

This paper will discuss level of conceptual understanding of 18 mathematics students in learning elementary group theory during abstract algebra course 2016-2017 academic year at Andalas University. Participants were asked to answer three proof tests in relation to group theory. Students' solutions to the proof test were taken as the key source of data used to: (i) classify students to one of the four levels of conceptual understanding and (ii) analyze students errors in learning elementary group theory. One student for each level was interviewed to provide additional information about common students' errors on the proof task and to aid the process of understanding the underlying cause of these errors. The finding shows that: (1) Students' achievement in proof task is still problematic; (2) Most students have difficulties in verifying the existence of identity and inverse element; (3) Factors that contribute to errors in proof task are: lack of conceptual understanding and that student treated binary operations on a group as a binary operations on real numbers.

Locales, the module system of the theorem prover Isabelle, were designed so that developments in abstract algebra could be represented faithfully and concisely. Whether these goals were met is assessed through a case study. Parts of an algebra textbook, Jacobson’s Basic Algebra, that are challenging structurally were formalised. Key parts of the formalisation are presented in greater detail. An analysis of the work from both qualitative and quantitative perspectives substantiates that the design goals were met. In particular, the size ratio of formal to “pen and paper” text does not increase when going further into the book. The analysis also yields guidance on locales including patterns of use, which are identified and described.

In this paper the linear representation of a graph is defined. A linear representation of a graph is a subgroup of $GL(p,\mathbb{R})$, the group of invertible matrices of order $ p $ and real coefficients. It will be demonstrated that every graph admits a linear representation. In this paper, simple and finite graphs will be used, framed in the graphs theory's area

We establish quasi-isometric rigidity for a class of right-angled Coxeter groups. Let Γ 1 \Gamma _1 , Γ 2 \Gamma _2 be joins of finite thick generalized m m -gons with m ∈ { 3 , 4 , 6 , 8 } m\in \{3,4,6,8\} . We show that the corresponding right-angled Coxeter groups are quasi-isometric if and only if Γ 1 \Gamma _1 , Γ 2 \Gamma _2 are isomorphic. We also give a construction of commensurable right-angled Coxeter groups.

When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially c...

The analysis of the co-orbit inter-satellite-link (ISL) topology plays an important role. ISL routing problem can be achieved by using difference topological method. Therefore, the choice of scientific, rigorous topology methods can have advantages of concepts clear and simple operation. Traditional ISL topology analysis was conducted using permutations and enumeration method. It can be easy to get the number of the ISL connecting types, but not intuitive enough for ISL connections with special configurations. In this paper, those issues are analyzed from a mathematical view, where all the ways of connecting are considered as a set M, while different angles of rotation of the track form a group G. We can see that the rotation caused by G makes a transformation of M, then G has a group action on M. Burnside theorem of modern algebra is used to calculate the number of tracks in this paper. As it can be seen from the theoretical analysis and calculation, the proposed method on the co-orbit ISL topology is clear and intuitive, easy operation, which can be well used in the satellite constellation network design.

Background In the US, there is significant interest from policy boards and funding agencies to change students’ experiences in undergraduate mathematics classes. Even with these reform initiatives, researchers continue to document that lecture remains the dominant mode of instruction in US undergraduate mathematics courses. However, we have reason to believe there is variability in teaching practice, even among instructors who self describe their teaching practice as “lecture.” Thus, our research questions for this study are as follows: what instructional practices are undergraduate mathematics instructors currently employing and what are the factors influencing their use of non-lecture pedagogies? Here, we explore these questions by focusing on instruction in abstract algebra courses, an upper-division mathematics course that is particularly well positioned for instructional reform. Results We report the results of a survey of 219 abstract algebra instructors from US colleges and universities concerning their instructional practices. Organizing our respondents into three groups based on the proportion of class time spent lecturing, we were able to identify 14 instructional practices that were significantly different between at least two of the three groups. Attempting to account for these differences, we analyzed the individual and situational factors reported by the instructors. Results indicate that while significant differences in teaching practices exist, these differences are primarily associated with individual factors, such as personal beliefs. Situational characteristics, such as perceived departmental support and situation of abstract algebra in the broader mathematics curriculum, did not appear to be related to instructional differences. Conclusions Our results suggest that personal bounds in general, and beliefs in particular, are strongly related to the decision to (not) lecture. However, many of the commonly cited reasons used to justify the use of extensive lecture were not significantly different across the three groups of instructors. This lack of differentiation suggests that there may be relevant institutional characteristics that have not yet been explored in the literature, and a transnational comparison might be useful in identifying them.