Algebraic Language Research Articles

Didactic Explanation and School Mathematical Discourse: The Case of Variation

Context: An important area of research in educational mathematics is social communicative practices in classroom organization. Objective: To locate and analyse the forms of introduction and development of variation in teaching situations. Design: Using a qualitative - interpretive approach, specifically an ethnographic study, for the analysis of socially shared practices among teachers, when they use explanation in the classroom and its correlation in the extended classroom. Environment and participants: The research participants were three professors (one physicist and two mathematicians) who taught the subject Mathematics I. On average, thei r groups consisted of 37 students. Data collection and analysis: Information was collected through audio - recorded and transcribed classroom observations. A detailed sequential study was carried out on teaching situations to describe the work done in each i ntervention that precedes or proceeds to yet another situation and thus construct the categories of analysis. Results: Due to the interactive nature, the construction of explanations is seen as an object of analysis and this implies that the minimum units are sequences of interactions, since the construction of discursive resources and meanings for variation was addressed. Conclusions: During the classes we recorded different types of explanation, models in which the notion of variation is modelled, the tea ching representations when explaining the contents through numerical, algebraic, and natural language representation.

TurboBC: A Memory Efficient and Scalable GPU Based Betweenness Centrality Algorithm in the Language of Linear Algebra.

Betweenness centrality (BC) is a shortest path centrality metric used to measure the influence of individual vertices or edges on huge graphs that are used for modeling and analysis of human brain, omics data, or social networks. The application of the BC algorithm to modern graphs must deal with the size of the graphs, as well with highly irregular data-access patterns. These challenges are particularly important when the BC algorithm is implemented on Graphics Processing Units (GPU), due to the limited global memory of these processors, as well as the decrease in performance due to the load unbalance resulting from processing irregular data structures. In this paper, we present the first GPU based linear-algebraic formulation and implementation of BC, called TurboBC, a set of memory efficient BC algorithms that exhibits good performance and high scalability on unweighted, undirected or directed sparse graphs of arbitrary structure. Our experiments demonstrate that our TurboBC algorithms obtain more than 18 GTEPs and an average speedup of 31.9x over the sequential version of the BC algorithm, and are on average 1.7x and 2.2x faster than the state-of-the-art algorithms implemented on the high performance, GPU-based, gunrock, and CPU-based, ligra libraries, respectively. These experiments also show that by minimizing their memory footprint, the TurboBC algorithms are able to compute the BC of relatively big graphs, for which the gunrock algorithms ran out of memory.

Multi-Agent Based Optimal Operation of Hybrid Energy Sources Coupled with Demand Response Programs

Electricity is an indispensable commodity on which both urban and rural regions heavily rely. Rural areas where the main grid cannot reach make use of distributed energy resources (DER), especially renewable energy sources (RES), in an islanded microgrid. Therefore, it is necessary to make sure there is a sufficient power supply to balance the demand and supply curve and meet people’s demands. The work done in this paper aims to minimize the daily operating cost of the hybrid microgrid while incorporating a demand response strategy built on an incentive-based demand response (IBDR) model. Three case studies were constructed and analyzed to derive the best, most reduced daily operational cost. This was achieved using the CPLEX solver embedded in algebraic modeling language in the Advanced Interactive Multidimensional Modeling Systems (AIMMS) software with multi-agent system (MAS); the MAS was used to make sure that the developed intelligent-based agents work independently to achieve an optimal microgrid system. The sensitivity analysis employed established that case study 2 gave the most reduced daily operation cost (USD 119), which represents an 8% reduction in the daily operational cost from case study 1 and a 9% reduction from case study 3. Then, we achieved 17% and 25% reductions, as compared to specific other approaches.

The complexity of homomorphism factorization

We investigate the computational complexity of the problem of deciding if an algebra homomorphism can be factored through an intermediate algebra. Specifically, we fix an algebraic language, \(\mathcal L\), and take as input an algebra homomorphism \(f:X\rightarrow Z\) between two finite \(\mathcal L\)-algebras X and Z, along with an intermediate finite \(\mathcal L\)-algebra Y. The decision problem asks whether there are homomorphisms \(g:X\rightarrow Y\) and \(h:Y\rightarrow Z\) such that \(f=hg\). We show that this problem is NP-complete for most languages.

TurboBFS: GPU Based Breadth-First Search (BFS) Algorithms in the Language of Linear Algebra.

Graphs that are used for modeling of human brain, omics data, or social networks are huge, and manual inspection of these graph is impossible. A popular, and fundamental, method used for making sense of these large graphs is the well-known Breadth-First Search (BFS) algorithm. However, BFS suffers from large computational cost especially for big graphs of interest. More recently, the use of Graphics processing units (GPU) has been promising, but challenging because of limited global memory of GPU's, and irregular structures of real-world graphs. In this paper, we present a GPU based linear-algebraic formulation and implementation of BFS, called TurboBFS, that exhibits excellent scalability on unweighted, undirected or directed sparse graphs of arbitrary structure. We demonstrate that our algorithms obtain up to 40 GTEPs, and are on average 15.7x, 5.8x, and 1.8x faster than the other state-of-the-art algorithms implemented on the SuiteSparse:GraphBLAS, GraphBLAST, and gunrock libraries respectively. The codes to implement the algorithms proposed in this paper are available at https://github.com/pcdslab.

The IDAES process modeling framework and model library—Flexibility for process simulation and optimization

AbstractEnergy systems and manufacturing processes of the 21st century are becoming increasingly dynamic and interconnected, which require new capabilities to effectively model and optimize their design and operations. Such next generation computational tools must leverage state‐of‐the‐art techniques in optimization and be able to rapidly incorporate new advances. To address these requirements, we have developed the Institute for the Design of Advanced Energy Systems (IDAES) Integrated Platform, which builds on the strengths of both process simulators (model libraries) and algebraic modeling languages (advanced solvers). This paper specifically presents the IDAES Core Modeling Framework (IDAES‐CMF), along with a case study demonstrating the application of the framework to solve process optimization problems. Capabilities provided by this framework include a flexible, modifiable, open‐source platform for optimization of process flowsheets utilizing state‐of‐the‐art solvers and solution techniques, fully open and extensible libraries of dynamic unit operations models and thermophysical property models, and integrated support for superstructure‐based conceptual design and optimization under uncertainty.

FROM BOOLE’S LOGIC TO BOOLEAN APPLICATIONS IN COMPUTER SCIENCE

The developments of an algebraic logical language of thoughts by G. Boole are considered using historical and theoretical perspectives. The technical implementations of Boolean logic in combinational circuits and in modern cryptography show strong influences of a 19th century logic on the latest technologies of computing.

The six books of Diophantus’ Arithmetic increased and reduced to specious: the lost manuscript of Jacques Ozanam (1640–1718)

The introduction of a new analytical method, due fundamentally to Francois Viete and Rene Descartes and the later dissemination of their works, resulted in a profound change in the way of thinking and doing mathematics. This change, known as process of algebrization, occurred during the seventeenth and early eighteenth centuries and led to a great transformation in mathematics. Among many other consequences, this process gave rise to the treatment of the results in the classic treatises with the new analytical method, which allowed new visions of such treatises and the obtaining of new results. Among those treatises is the Arithmetic of Diophantus of Alexandria (approx. 200–284) which was written, using the new algebraic language, by the French mathematician Jacques Ozanam (1640–1718), who in addition to profusely increasing the original problems of Diophantus, solved them in a general way, thus obtaining many geometric consequences. The work is handwritten, it has never been published, it has been lost for almost 300 years, and the known references show its importance. We will show that Ozanam’s manuscript was quoted as an important work on several occasions by others mathematicians of the time, among whom G. W. Leibniz stands out. Once the manuscript has been located, our aim in this article is to show and analyze this work of Ozanam, its content, its notation and its structure and how, through the new algebraic method, he not only solved and expanded the questions proposed by Diophantus, but also introduced a connection between the algebraic solutions and what he called geometric determinations by obtaining loci from the solutions.

Examen colegiado de cálculo diferencial

Esta investigación analiza el examen colegiado de cálculo diferencial en los ciclos escolares 2018-2 y 2019-1 a un total de 3,751 estudiantes de primer semestre de las carreras de ingeniería de una universidad pública mexicana. Se estudia la actividad cognitiva requerida para la resolución de cada reactivo y los registros de representación presentes, además del carácter estructural o procesal del reactivo. Con los resultados se observa que el examen colegiado es una prueba confiable y con poder de discriminación satisfactorio, con carga mayoritaria de reactivos procedimentales y con representación algebraica que identifican también el esquema de enseñanza. Se determinan los reactivos con mayor dificultad para los estudiantes, los cuales tienen presente la actividad cognitiva de conversión, son procedimentales, con lenguaje natural presente y abundante lenguaje algebraico.

Determination of the optimal distillation sequence of a ternary mixture incorporating heat integration by means of Microsoft Excel Solver

AbstractNowadays, the computer skills acquisition by Chemical Engineering (ChE) students is based on practical uses of specialized software packages in different applications. Process optimization courses use high‐level algebraic programming languages, such as GAMS (General Algebraic Modeling System) or AMPL (A Mathematical Programming Language), which are usually reserved for advanced programs in ChE degrees or ChE M.Eng. Commercial software for process optimization has several benefits, such as the possibility of facing complex numerical problems in a more efficient and straightforward way. By contrast, they typically present expensive license fees, and students usually require intense training to get the proper user‐level skills. The use of simpler and economic software tools may help overcoming these limitations. In this context, Microsoft (MS) Excel Solver tool is a widespread software with the potential to be used worldwide by any ChE student, and with the power to solve complex optimization problems at the academic level. In this paper, we introduce the application of MS Excel Solver for the optimization of a distillation sequence for the separation of a ternary mixture, including the heat integration in the definition of the problem. This is an MINLP problem, which usually requires specialized software performing stochastic methods, such as Simulated Annealing, to achieve a solution. In this regard, the approach to solve an optimization problem using the MS Excel Solver tool shown in this study was successfully employed by a student of ChE M.Eng. at Rey Juan Carlos University and the Autónoma University of Madrid in the development of her master thesis.

A comparative analysis of generation and transmission expansion planning models for power loss minimization

In modern power systems, renewable energy resources are becoming more relevant due to their eco-friendly nature and sustainable electrification. Therefore, it is important to consider the effects of the renewable energy resources (RERs) on the generation expansion planning and transmission expansion planning (GTEP) simultaneously. The proper assessment of power losses is essential in order to adequately evaluate the consequence of renewable energy resources in power systems. This paper presents multi-objective optimization problems that are solved using DC, modified DC and AC modelling methods in order to accurately and efficiently compute electrical losses and its impacts on the expansion planning procedure. This paper develops a Mixed Integer Non-linear Programming (MINLP) problem and was solved using CONOPT and CPLEX solvers embedded in Algebraic Modelling Language. The proposed methods are evaluated and validated on three case studies: IEEE Garver’s 6 bus system, IEEE 24 bus test system and a real-world Nigeria Power System. A comparative analysis of the three modelling approaches and a sensitivity analysis of the simulation results indicate that AC solution method provide an accurate computation of power losses and efficient optimal planning strategy as compared to the modified DC method.

Quasi-Nelson algebras and fragments

AbstractThe variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

Wisdom begins in wonder: Implementing identity-based information literacy instruction across the content areas

In this article, we describe a collaboratively developed information literacy curriculum that was embedded into three core 8th grade subjects: Algebra, American History, and English Language Arts. The curriculum was rooted in the concept of generalist literacy and supported students as they developed their curiosity, skepticism, commitment to accuracy, and persistence around online information. Each quarter, we included three new information literacy lessons and time for students to work on an independent passion project, the Wonder Project. We discuss the benefits of engaging in this work across multiple content areas and conclude with recommendations for future work.

Geometrically closed positive varieties of languages

A recently introduced operation of geometrical closure on formal languages is investigated from the viewpoint of algebraic language theory. Positive varieties V containing exclusively languages with regular geometrical closure are fully characterised by inclusion of V in W, a known positive variety arising in the study of the commutative closure. It is proved that the geometrical closure of a language from the intersection of W with the variety of all star-free languages SF always falls into RLT, which is introduced as a subvariety of R, the variety of languages recognised by R-trivial monoids. All classes between RLT and W∩SF are thus geometrically closed: for instance, the level 3/2 of the Straubing-Thérien hierarchy, the DA-recognisable languages, or the variety R. It is also shown that W∩SF is the largest geometrically closed positive variety of star-free languages, while there is no largest geometrically closed positive variety of regular languages.

Seymour's second‐neighborhood conjecture from a different perspective

AbstractSeymour's Second‐Neighborhood Conjecture states that every directed graph whose underlying graph is simple has at least one vertex such that the number of vertices of out‐distance two from is at least as large as the number of vertices of out‐distance one from it. We present alternative statements of the conjecture in the language of linear algebra.

On the Commutative Equivalence of Algebraic Formal Series and Languages

The problem of the commutative equivalence of context-free and regular languages is studied. Conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated.

Experimental Analysis of Algebraic Modelling Languages for Mathematical Optimization

In this work, we perform an extensive theoretical and experimental analysis of the characteristics of five of the most prominent algebraic modelling languages (AMPL, AIMMS, GAMS, JuMP, and Pyomo) and modelling systems supporting them. In our theoretical comparison, we evaluate how the reviewed modern algebraic modelling languages match the current requirements. In the experimental analysis, we use a purpose-built test model library to perform extensive benchmarks. We provide insights on which algebraic modelling languages performed the best and the features that we deem essential in the current mathematical optimization landscape. Finally, we highlight possible future research directions for this work.

Mathematical foundation of system design regulatory controls based on ARP4754A

The article provides an interpretation and formal mathematical foundation in the language of modern algebra for V-process of designing complex systems recommended by the ARP4754A manual, in the context of a new paradigm in mathematical systems theory developed on the basis of the General principle of isomorphism. This provides a scientific basis for the widely used design approach

Higher Kac–Moody algebras and symmetries of holomorphic field theories

We introduce a higher dimensional generalization of the affine Kac-Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of currents associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras recently proposed by Faonte-Hennion-Kapranov. A central goal of this paper is to witness higher Kac-Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field realization of an affine Kac-Moody algebra and also develop the theory of q-characters for this class of algebras in terms of factorization homology. Finally, we exhibit the large N behavior of higher Kac-Moody algebras and their relationship to symmetries of non-commutative field theories.

Ensino da álgebra em uma turma do oitavo ano de uma escola pública de São Mateus - ES

O presente trabalho, visa investigar possíveis contribuições da utilização de dois jogos educativos em aulas de matemática, para ensinar as operações de adição, subtração e multiplicação de polinômios para uma turma do 8º ano do Ensino Fundamental, que tiveram um papel investigativo, mostrando que os alunos possuem dificuldades de interpretação e portanto, dificuldades na passagem da linguagem natural para a algébrica e geométrica, bem como em conceitos importantes como os de perímetro e área. Por este motivo, se torna importante metodologias de ensino que visem enfatizar a interpretação de problemas e a contextualização desse conteúdo a realidade dos alunos, visto que alguns deles não conseguem visualizar a utilidade desses conceitos no dia a dia.