Algebraic Language Research Articles

The work of a prospective high school teacher in pre-professional training in highlighting mathematical knowledge

This article aims to identify the knowledge that enables a prospective secondary school teacher to mobilize number sense components during a pre-professional practicum. Using a qualitative case study approach, four classroom episodes focused on fractions, percentages, and algebraic language are analyzed. The study involves one prospective teacher and 23 students from a multi-grade school in central-southern Chile. The analysis integrates teacher’s specialized knowledge and mathematical working spaces to explore how these frameworks inform the teacher’s knowledge. The findings reveal that the teacher employs personal theories, real-life examples, and graphical aids to enhance students’ understanding, though there is a reliance on rules and algorithms. While specialized knowledge aids in teaching, there is less emphasis on fostering students’ conceptual understanding and judgement. The study recommends improving teacher training programs to better equip prospective teachers in grasping number sense components, highlighting the need for more focused training on its development.

Adding an implication to logics of perfect paradefinite algebras

Abstract Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion () order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix. We studied these logics extensively in Gomes et al. ((2022). Electronic Proceedings in Theoretical Computer Science357 56–76.) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) Set-Set and order-preserving and $\top$ -assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil’s “symmetric modal logic.” For the order-preserving Set-Set logic, in particular, we obtain a Set-Set axiomatization that is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.

Coalgebraic methods for Ramsey degrees of unary algebras

Abstract In this paper, we prove the existence of small and big Ramsey degrees of classes of finite unary algebras in an arbitrary (not necessarily finite) algebraic language Ω. Our results generalize some Ramsey-type results of M. Sokić concerning finite unary algebras over finite languages. To do so, we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat unary algebras as Eilenberg-Moore coalgebras for a functor with comultiplication, and using pre-adjunctions transport the Ramsey properties, we are interested in from the category of finite or countably infinite chains of order type ω. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.

Ontological description method for building service-oriented distributed learning systems

The subject of the research is the analysis and justification of the use of software engineering processes based on ontologies of data access and representation of access to knowledge bases and knowledge reuse in the language of algebra of finite predicates, which is a class of descriptive logics. This approach allows you to use the advantages of ontologies and logic programming in the refactoring of distributed distance learning systems. The purpose of the work is to create a formalism for describing the interaction of services and algorithms for building web service interfaces for the implementation of an effective SOA system using the paradigm of access to data based on ontologies. To achieve the goal, the following tasks were solved: an analysis of the main types of interaction structures of SOA services, an overview of implementations of systems working with a family of descriptive logic languages, namely, algebra of finite predicates, and a description of the structure of the ontology necessary for the operation of service-oriented systems, which has the markup of the Semantic Web standard using finite predicate algebra equations. The task is an urgent task of ensuring the organization of interaction of Internet resources in the development of distributed virtual learning systems, and such interaction is enabled by the service-oriented architecture of software development. The article examines modern technologies of the Semantic Web and their role in ensuring the creation of applied software systems of distributed learning resources built using distributed interactive services. The considered methods and materials include methods of algebra of finite predicates, theory of algorithms, object-oriented design, theory of unification. The results. The semantics of the SOA system is obtained, which makes it possible to describe SOA systems in the languages of descriptive logic. The effectiveness of the mathematical formalism of the algebra of finite predicates for the tasks of logical analysis of applied ontologies, the use of knowledge reuse methods, and the description of service-oriented systems is shown. The necessity of using algorithms for automatic construction of web interfaces is shown. The conclusions of the work highlight the advantages of the researched solution, namely, algorithms for the automated construction of web service interfaces for SOA architecture, which make it possible to automatically build web service interfaces. Algorithms differ in a more general model and speed of work and require an assessment of complexity.

Coupling of photovoltaic thermal with hybrid forward osmosis-membrane distillation: Energy and water production dynamic analysis

This study proposes a novel solar photovoltaic thermal (PVT)-driven forward osmosis (FO) and membrane distillation (MD) system for desalinating brackish water, addressing water scarcity issues by combining the advantages of FO's low operating cost with MD's ability to treat hypersaline water. A physics-based dynamic model was developed for the system, and the model equations were discretized and solved using an open-source algebraic modeling language. Model validation involved verifying the consistency of outputs between successive unit operations and conducting sensitivity analysis by varying one operating condition at a time. The analysis considered varying weather conditions while maintaining constant operating conditions. Increasing the flow rate of brackish water (cooling fluid) from 19.79 to 59.38 kg/h improved thermal efficiency by 28.7 %; however, FO and MD water fluxes decreased by 1.12 LMH and 7.77 kg/m2/h, respectively. Peak thermal efficiency (81.2 %) was achieved at the highest flow rate. Incorporating heat transfer analysis into FO models goes beyond the common focus on mass transfer, allowing for a more comprehensive understanding of energy dynamics within the system. Both FO and MD water fluxes improved with irradiance and ambient temperature but decreased with cooling fluid flow rate and wind velocity. Increasing feed concentration to 10,000 mg/l reduced water flux by 0.66 LMH, and for each 0.5 M increase in draw solution concentration, FO water flux increased by >10 LMH. Compared to MD water flux, FO water flux is more sensitive to changes in concentration than to changes in weather conditions.

Elements of Timed Pattern Matching

The rise of machine learning and cloud technologies has led to a remarkable influx of data within modern cyber-physical systems. However, extracting meaningful information from this data has become a significant challenge due to its volume and complexity. Timed pattern matching has emerged as a powerful specification-based runtime verification and temporal data analysis technique to address this challenge. In this paper, we provide a comprehensive tutorial on timed pattern matching that ranges from the underlying algebra and pattern specification languages to performance analyses and practical case studies. Analogous to textual pattern matching, timed pattern matching is the task of finding all time periods within temporal behaviors of cyber-physical systems that match a predefined pattern. Originally we introduced and solved several variants of the problem using the name of match sets, which has evolved into the concept of timed relations over the past decade. Here we first formalize and present the algebra of timed relations as a standalone mathematical tool to solve the pattern matching problem of timed pattern specifications. In particular, we show how to use the algebra of timed relations to solve the pattern matching problem for timed regular expressions and metric compass logic in a unified manner. We experimentally demonstrate that our timed pattern matching approach performs and scales well in practice. We further provide in-depth insights into the similarities and fundamental differences between monitoring and matching problems as well as regular expressions and temporal logic formulas. Finally, we illustrate the practical application of timed pattern matching through two case studies, which show how to extract structured information from temporal datasets obtained via simulations or real-world observations. These results and examples show that timed pattern matching is a rigorous and efficient technique in developing and analyzing cyber-physical systems.

Modal algebra of multirelations

Abstract We formalize the modal operators from the concurrent dynamic logics of Peleg, Nerode and Wijesekera in a multirelational algebraic language based on relation algebras and power allegories, using relational approximation operators on multirelations developed in a companion article. We relate Nerode and Wijesekera’s box operator with a relational approximation operator for multirelations and two related operators that approximate multirelations by different kinds of deterministic multirelations. We provide an algebraic soundness proof of Goldblatt’s axioms for concurrent dynamic logic and one for a multirelational Hoare logic based on Nerode and Wijesekera’s box as applications.

Simulation limitations of affine cellular automata

Cellular automata are a famous model of computation, yet it is still a challenging task to assess the computational capacity of a given automaton; especially when it comes to showing negative results. In this paper, we focus on studying this problem via the notion of CA intrinsic simulation. We say that automaton A is simulated by B if each space-time diagram of A can be, after suitable transformations, reproduced by B.We study affine automata – i.e., automata whose local rules are affine mappings of vector spaces. This broad class contains the well-studied cases of linear automata. The main result of this paper shows that (almost) every automaton affine over a finite field Fp can only simulate affine automata over Fp. We discuss how this general result implies, and widely surpasses, limitations of linear and additive automata previously proved in the literature.We provide a formalization of the simulation notions into algebraic language and discuss how this opens a new path to showing negative results about the computational power of cellular automata using deeper algebraic theorems.

Fluid Maxwell’s equations in the language of geometric algebra

Fluid Maxwell’s equations in the language of geometric algebra

Modified bosonic integrable hierarchy

In this paper, we construct a new integrable hierarchy by using charged free boson and Friedan–Martinec–Shenker (FMS) bosonization, called modified bosonic integrable hierarchy, which is an anologue of the classical (l−l′)–th modified KP hierarchy. Firstly, the representation of glˆ∞ is given by using charged free bosons. Then the modified bosonic integrable hierarchy is proposed. Next after discussing FMS bosonizations in the language of lattice vertex algebra, bilinear equations of modified bosonic integrable hierarchy are constructed by FMS bosonization, which are rewritten into Hirota bilinear equations by using Hirota derivatives. At last, we consider solutions of this modified bosonic integrable hierarchy.

$\alpha$-induction for bi-unitary connections

The tensor functor called \alpha -induction produces a new unitary fusion category from a Frobenius algebra object, or a Q -system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor N\subset M of finite index and finite depth, which gives a braided fusion category of endomorphisms of N . It is also understood in terms of Ocneanu’s graphical calculus. We study this \alpha -induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain 4 -tensors appearing in recent studies of 2 -dimensional topological order. We show that the resulting \alpha -induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a local Q -system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.

Coordinate-free Lie-group-based modeling and simulation of a submersible vehicle

<abstract><p>Submersible vehicles may be regarded as complex systems because of their complex interaction with the surrounding fluid. This paper presents a mathematical model of a submersible vehicle formulated in a coordinate-free manner through the language of Lie groups and Lie algebras. The d'Alembert virtual-work principle was applied in conjunction with the minimal-action principle for a rigid body in order to incorporate into the mathematical model external influences such as fluid-current-induced deflection and control inputs. Such a method from mathematical physics can also take into consideration how a vehicle interacts with the fluid it is immersed in under the form of added (or virtual) mass. The resulting equations of motion were given over the Lie group of three-dimensional rotations as (non-pure) Euler-Poincaré relations. A numerical simulation technique based on Lie-group integrators was also briefly recalled and deployed to simulate the behavior of such mathematical model of an existing, academic-design-type submersible vehicle.</p></abstract>

Learning the Parabola Mediated by Problem-Solving, Semiotic Representations, and Applets in GeoGebra

In this article we evaluate the contribution of the problem-solving approach, semiotic representations and the use of applets designed in GeoGebra when first-semester science and engineering students study the mathematical concept and application of the parabola. At the end of the classroom intervention, a significant improvement was noted in the students' performance, especially in their cognitive thinking activities; that is, in the use of algebraic language, in the appropriation of geometric concepts, and in the problem-solving performance.

Preservice teachers’ errors in the creation and extension of sequences using color tiles: An exploratory study

AbstractRecognizing, building, describing, extending, and analyzing patterns are key components of algebraic reasoning. Oftentimes pattern‐based instruction is used to support the understanding of functions and variables. In this study, elementary preservice teachers (n = 23) were asked to explore, extend, explain, create equations for, and analyze linear, geometric, and quadratic integer patterns using color tiles. Then they created, extended, described, and analyzed their own patterns while working in groups (n = 8). The research question was: What algebraic errors do PSTs demonstrate in the process of constructing, explaining, and analyzing their own linear, geometric, and quadratic patterns using color tiles? The analysis, which focused on errors and misunderstandings preservice teachers demonstrated in their final unit, revealed three specific areas of difficulties: figural errors, explanation challenges, and improper use of algebraic language. Participants failed to connect their algebraic images in a coherent manner, there was confusion in articulating their algebraic thought processes, and algebraic terminology was improperly used. Implications for teaching and further research are described.

Language Outlay for the Sokoto Cement Production System Using Deterministic Finite Automata Scheme

This paper constructs the compact, detailed and extended models, and focuses on the algebraic theoretic strings and language of each transition of the finite automata scheme. It was discovered that from the initial stage to the final stage of cement production processes, each transition or production process can have a particular language. In addition, a language scheme is developed for each of the sub-states (sub-model) that leads to a theoretic study of language scheme and semantics of the model. It can be deduced that when represented as binary codes, the established schemes in the sub-states can be studied as a Boolean algebraic scheme.

Optimizing a Multi Period Deterministic Inventory Routing Problem in Agriculture Industries

This paper develops a solving model for multi-period deterministic inventory routing problem (MP-DIRP) in supply chain focus on agriculture industries. The issue is how to improve the perishable product delivery process from a distribution centre (DC) to a collection of customers over a planning horizon. This paper proposed a model to achieve optimization in terms of costs respectively. The transportation problem is solved by using a simulation technique and developed using an algebraic modelling language which is known as a mathematical programming language (AMPL) to find an optimal solution within the period. The route for a distribution centre (DC) is set up by a supplier depending on the number of products assigned to each customer, which contributes to determining the best inventory routing problem (IRP) for each period. As a result, by applying the transportation model developed in this study, it is expected that the industries can achieve optimization by minimizing their total inventory and transportation costs.

Problemas de concentración común de soluciones y le campo conceptual multiplicativo: un análisis en los libros de texto de química

The present investigation aimed to analyze and classify problem situations of Common Concentration of Solutions present in Chemistry textbooks, according to the theory of Conceptual Fields. For the analyses, the six Chemistry works approved by the PNLD 2018 were considered and the volume of the 2nd year of each work was selected, as they include the content of Common Concentration of Solutions. The research analyses, carried out in the light of the Conceptual Fields theory, showed that situations of Common Concentration are associated with the Multiplicative Conceptual Field and are classified as Simple Proportion. It was also identified that situations may vary in subclasses: Partition, One-to-Many Multiplication, Quota and Fourth Proportional. Therefore, the results showed that the problem situations of Common Concentration of Solutions are presented in the textbooks analyzed through contexts close to the reality of the students and that mathematical notions such as multiplication, division, operations with decimal numbers, fractions, proportion, volume and conversion of measurement units, as well as different mathematical representations, such as algebraic, numerical, tabular, graphic and natural language are essential for solving Common Concentration problem situations in Chemistry.

Knowledge representation of mathematical optimization problems and constructs for modeling

We introduce two new knowledge bases that we have developed to categorize mixed-integer linear programming (MILP) problems and standardize the element definitions of MILP models. MILP is a commonly used mathematical programming technique for modeling and solving real-life scheduling, routing, planning, resource allocation, and timetabling optimization problems providing optimized business solutions for industry sectors such as manufacturing, agriculture, defense, healthcare, medicine, energy, finance, and transportation among others. Despite the numerous real-life combinatorial optimization problems (COPs) found and solved, and many yet to be discovered and formulated, the number of types of constraints (the building blocks of a MILP) is relatively small. In the search for a uniform categorization of MILP problems and a machine-readable knowledge representation structure for MILP models, we have developed an optimization modeling tree (OMT) and a MILP model ontology. The two knowledge structures can serve as a standardized, uniform representation in understanding MILP problems and developing MILP models for combinatorial business optimization problems. While there are several algebraic modeling languages (AMLs) for developing and solving MILP models, the semantic correctness of such models cannot be guaranteed using the syntactic grammar of such AMLs. The MILP model ontology will act as the main resource for semantic validation of MILP models through ontology instantiations and axiom assertions.

Use of Gamification to Approach the Algebraic Language and Equations in 1st Grade of Secondary Education

Active methodologies, as innovative agents, are becoming more and more frequent in education, but they are not a majority. However, it is a good resource to adapt the educational system to the current society based on knowledge and information.
 This article shows a useful and effective tool to connect with the needs and interests of students through an active methodology. The proposal, a gamification (teaching-learning methodology based on the mechanics of games), approaches the students of 1st grade of secondary education to the algebraic language and equations, combining cooperative learning activities.
 Working through gamification means a higher academic performance, an increase in motivation and the development of a positive attitude allowing, in addition, to attend to diversity in the classroom and the use of ICT promoting collaborative work.
 Gamification, based on the use of ICT, facilitates the acquisition of knowledge in a more attractive way, favoring their performance.

GBOML: a structure-exploiting optimization modelling language in Python

Mixed-Integer Linear Programs (MILPs) have many practical applications. Most modelling tools for MILPs fall in two broad categories. Indeed, tools such as algebraic modelling languages allow practitioners to compactly encode models using syntax close to mathematical notation but usually lack support for special structures, while other tools instead provide predefined components that can be easily assembled but modifying or adding new components is difficult. In this work, we present the inner workings of the Graph-Based Optimization Modelling Language (GBOML), an open-source modelling tool implemented in Python combining the strengths of both worlds. GBOML natively supports special structures that can be encoded by a hierarchical hypergraph, offers syntax close to mathematical notation and facilitates the modular construction and reuse of time-indexed models. We detail design choices enabling these features and show that they simplify problem encoding, lead to faster instance generation times and sometimes faster solve times. We benchmark the times taken by GBOML, JuMP, Plasmo, Pyomo and AMPL to generate instances of a structured MILP. We find that GBOML outperforms Plasmo and Pyomo, is tied with JuMP but is slower than AMPL. With parallel model generation, GBOML outperforms JuMP and closes the gap with AMPL. GBOML has the smallest memory footprint.