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We define the notion of quantized function algebras at q = 0 or crystallization of the q deformations of the type A n compact Lie groups at the C ∗ -algebra level. The C ∗ -algebra A n (0) is defined as a universal C ∗ -algebra given by a finite set of generators and relations. We obtain these relations by looking at the irreducible representations of the quantized function algebras for q > 0 and taking limit as q → 0 + after rescaling the generating elements appropriately. We then prove that in the n = 2 case the irreducible representations A 2 (0) are precisely the q → 0 + limits of the irreducible representations of the C ∗ -algebras A 2 (q) . [ABSTRACT FROM AUTHOR]

In this paper, we examine how researchers and teachers in a multi-year professional development program shifted their conceptualizations of equity. Following (Grapin et al (2023) Sci Educ 107:999–1032), we ground our analysis in two conceptualizations of equity that exist across fields: equity-as-access (learners should have access to disciplinary knowledge, practices, and career paths) and equity-as-transformation (learners should transform what it means to learn and participate in disciplines). In this study, we describe a professional development (PD) design initially intended to support equitable science teaching and learning by focusing on representations. This initial framing did not distinguish between conceptions of equity-as-access versus equity-as-transformation. As a result, the PD did not provide facilitators or teachers with resources for ideological sensemaking towards equity-as-transformation. Catalyzed by teachers' request for PD focused on multilingual learners (MLs), we noticed aspects of our design that offered only images of equity-as-access. In response, we designed activities for teachers that offered space and resources for considering equity-as-transformation. As a case study (Yin (2014) Case study research: design and methods, SAGE) using interaction analysis (Jordan and Henderson (1995) J Learn Sci 4:39–103) of PD videos, we describe how we PD activities and facilitation strategies to integrate transformative conceptualizations of equity. These findings have implications for both research and practice. In terms of research, they demonstrate the importance of using multiple lenses to consider equity in science classrooms. In terms of practice, they underscore the importance of providing teachers with opportunities to explicitly connect new perspectives of equity with day-to-day experiences of classroom teaching. [ABSTRACT FROM AUTHOR]

The teacher's use of representations is a crucial aspect of instructional quality in mathematics education, given their pivotal role in facilitating mathematics learning. However, in our international research community, perspectives on what constitutes high-quality use of representations may vary. This cross-cultural study aims to explore whether the perspectives from Western literature, emphasizing the importance of explicit connections between symbolic and graphic representations, can be extended legitimately to the East Asian context. Using a situated approach, the study elicited norms of high-quality representation use from researchers in Germany and Taiwan. A total of 31 mathematics education professors from both countries evaluated the use of representations in three secondary mathematics classroom situations presented as text vignettes. The vignettes, designed by the German research team, each depicted a situation where from their perspective, a norm of high-quality representation use, specifically the explicit connection between symbolic and graphic representations, was violated. Qualitative analysis of the researchers' responses revealed that in each situation, at least half of the German researchers expected explicit connections between representations. Conversely, the majority of Taiwanese researchers only expected such connections in one situation, particularly when the graphic representation served as an independent learning objective rather than solely aiding conceptual understanding. These findings indicate easily unnoticed culture-specific differences regarding how a common aspect of instructional quality is expected to unfold in teaching. [ABSTRACT FROM AUTHOR]

The growing diffusion of perspectivism within the debate on information system ontologies [ISOs] does not correspond to a thorough analysis of what perspectivism specifically consists of. This paper aims to fill this void. First, I show what supporting perspectivism in information system ontologies [PISO] means in terms of (minimal) claims and implications; then I argue that the definitions of ISO implicitly assume PISO's (minimal) claims or, in other words, that ISOs presuppose and maintain PISO. Section 2 presents the main definitions of ISO. Section 3 specifies what claims are common to all perspectivists in ISO. Sects. 4–7 analyze the implications of those claims. Section 8 explores the chance of multiple perspectivisms within ISOs' domain. Finally, Sects. 9–10 assume that, if PISO's (minimal) claims and (their) implications can be inferred from ISO's definitions, then ISOs are perspectivist, or PISO's minimal claims are assumptions underlying ISOs. [ABSTRACT FROM AUTHOR]

We conducted a descriptive exploratory study in which we analyzed 313 sixth to eighth grade students' answers to a word problem, accompanied by diagrams, involving generalization in an algebraic functional context. In this research, we jointly addressed two objectives: (a) to determine the strategies deployed by students to generalize and (b) to identify the types of representation used to express their generalizations. We integrated how regularities are produced, evidenced in structures and represented by students. One of the most prominent findings was that functional strategy was used by almost all the students who generalized. They expressed the generalization using verbal, symbolical, or multiple representations. Ways of expressing regularities that are not restricted to algebraic symbolism are also shown. Although the potential to identify functional relationships was observed in sixth graders, seventh and eighth school students were able to represent more varied and structurally complex relationships. However, no relevant differences in generalization strategies were found between students of different ages with and without previous algebraic training. [ABSTRACT FROM AUTHOR]

For a locally compact group G and 1 < p < ∞ , let B p (G) denote the p-analog of the Fourier–Stieltjes algebra B (G) (or B 2 (G)) . Let r : B p (G) → B p (H) be the restriction map given by r (u) = u | H for any closed subgroup H of G. In this article, we prove that the restriction map r is a surjective isometry for any open subgroup H of G. Further, we show that the range of the map r is dense in B p (H) when H is either a compact normal subgroup of G or compact subgroup of an [SIN] H -group. [ABSTRACT FROM AUTHOR]

Given a representation φ : B n → G n of the braid group B n , n ≥ 2 into a group G n , we are considering the problem of whether it is possible to extend this representation to a representation Φ : S M n → A n , where S M n is the singular braid monoid and A n is an associative algebra, in which the group of units contains G n . We also investigate the possibility of extending the representation Φ : S M n → A n to a representation Φ ~ : S B n → A n of the singular braid group S B n . On the other hand, given two linear representations φ 1 , φ 2 : H → G L m (k) of a group H into a general linear group over a field k , we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of S B n which is an extension of the Lawrence–Krammer–Bigelow representation (LKBR) and compute the defect of this extension with respect to the exterior product of two extensions of the Burau representation. Finally, we discuss how to derive an invariant of classical links from the Lawrence–Krammer–Bigelow representation. [ABSTRACT FROM AUTHOR]

We use the crystallised C ∗ -algebra C (S U q (2)) at q = 0 to obtain a unitary that gives an approximate equivalence involving the GNS representation on the L 2 space of the Haar state of the quantum SU(2) group and the direct integral of all the infinite dimensional irreducible representations of the C ∗ -algebra C (S U q (2)) for nonzero values of the parameter q. This approximate equivalence gives a KK class via the Cuntz picture in terms of quasihomomorphisms as well as a Fredholm representation of the dual quantum group S U q (2) ^ with coefficients in a C ∗ -algebra in the sense of Mishchenko. [ABSTRACT FROM AUTHOR]

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This text presents some psycho-sociological comments on the social representation (or SR) of the object "hymen" and its practical counterpart of defloration. The main argument is based on the observation of the recurrence of fetishistic attitudes towards bodily objects and ritual behaviors associated with the initiation into sexuality and the idea of physical and moral purity. The SR of virginity is observed in a heterogeneous corpus: history books, short stories, novels, anthropological archives, secondary data on the conception of virginity among young people, essays, online press, ethnographic reports, etc. This social representation may also include other terms such as "chastity", "purity", "first time", "sexual initiation", "abstinence" or "continence." First, the genesis of the objectified and anatomical meaning of virginity and defloration is analyzed as a scientifically and institutionally constructed imaginary process. Entering the world of sexuality, however, involves bodily techniques and behaviors that are less and less regulated today. The representational, semantic (connotations, effects of cultural matrices) and historical evolution of the purity associated with virginity are also examined. Various themes will be addressed in turn: the disjunctive structure (instrumentalization/idealization) of virginity; the proximity between puritanical and perverse attitudes towards these phenomena; certain mythographies and rituals (magic, novels) and the practice of bodily rituals, remnants of earlier practices. The interpretation of the social representation of the virginity and its anthropological logic are synthesized at the end of the article. Finally, the philosophical and psycho-sociological meanings of the purity fetish are presented. The couple virginity/defloration and more generally any form of bodily organization that conveys eroticism and sexuality, are seen as structures of protraction, quite very close to the notion of apparatus (or device). [ABSTRACT FROM AUTHOR]

Recent research syntheses have identified several potentially high-leverage teaching strategies for improving low-attaining secondary school students' learning of mathematics. These strategies include the structured use of representations and manipulatives and an emphasis on derived facts and estimation. This paper reports on 70 semi-structured interviews conducted with low-attaining students in Years 9–10 (ages 13–15) in England. The interviews addressed the students' perceptions of learning mathematics and the teaching strategies that they experienced and believed were most helpful. Many students reported rarely using number lines, not spontaneously estimating answers and being unfamiliar with derived facts. During the interviews, with minimal direction, students often showed that they were well able to make use of these strategies; however, they did not report making spontaneous use of them independently. We conclude that many of the most well-evidenced and recommended strategies to support low-attaining students in mathematics appear to be unfamiliar and unvalued, and we discuss how this might be addressed. [ABSTRACT FROM AUTHOR]

In today's rapidly changing world, technology has profoundly impacted our daily lives, making it increasingly difficult to predict what skills will be necessary for the future. To address this challenge, 21st-century skills have been proposed as a framework for shaping future education, in which one of the skills is technological literacy. How the latter is understood, positioned, and approached is influenced by discourses produced within and among various societal practices, such as the educational system. This article provides a glimpse of the discourse on technological literacy by exploring how "technological literacy" is portrayed within the visual cultures of four Swedish upper secondary VET programs. Our results indicate that the portrayals of becoming technologically literate for VET students vary among the programmes. However, they all emphasise practical and hands-on activities, where students can be seen engaging with various tools, equipment, and objects relevant to each programme's field of study. Teamwork and peer-to-peer learning are also portrayed as central to these programmes, simultaneously as leadership roles and the distribution of responsibilities among students are commonly featured. While the programmes' visual portrayals reflect diversity in ethnicity and gender, traditional gender roles are still (re)produced in the pictorials. [ABSTRACT FROM AUTHOR]

The good regulator theorem and the internal model principle are sometimes cited as mathematical proofs that an agent needs an internal model of the world in order to have an optimal policy. However, these principles rely on a definition of “internal model” that is far too permissive, applying even to cases of systems that do not use an internal model. As a result, these principles do not provide evidence (let alone a proof) that internal models are necessary. The paper also diagnoses what is missing in the GRT and IMP definitions of internal model, which is that models need to make predictions that represent variables in the target system (and these representations need to be usable by an agent so as to guide behavior). [ABSTRACT FROM AUTHOR]

Different why-questions emerge under different contexts and require different information in order to be addressed. Hence a relevance relation can hardly be invariant across contexts. However, what is indeed common under any possible context is that all explananda require scientific information in order to be explained. So no scientific information is in principle explanatorily irrelevant, it only becomes so under certain contexts. In view of this, scientific thought experiments can offer explanations, should we analyze their representational strategies. Their representations involve empirical as well as hypothetical statements. I call this the "representational mingling" which bears scientific information that can explain events. Buchanan's thought experiment from constitutional economics is examined to show how mingled representations explain. [ABSTRACT FROM AUTHOR]

The exploration of children's drawings as mathematical representations is a current focus in early years mathematics education research. This paper presents a qualitative analysis of 72 kindergarten to Grade 3 (5 to 8 years old) children's drawings produced during problem-solving tasks centred on multiplicative strategies. Existing frameworks for the developmental sequence of mathematical drawings and the progression of children's strategies for multiplicative situations were an interpretive lens through which to analyse the drawings. Children used pictographic and iconic drawing types to represent the "story" in the problem and the multiplicative strategies employed to solve the tasks. Exploration of the children's drawings suggested that as children's drawings become more structural, schematic in nature, it may be easier for children to show their understanding of the structural elements of multiplicative relationships. Results revealed that structural elements of multiplicative relationships were more easily seen in iconic representations; however, both pictographic and iconic drawings were useful to observe counting, additive, and multiplicative strategies when mathematical elements of the problem were visible. Additional representations attached to the drawings (e.g. numerical) were needed to confirm children's strategies when their drawings lacked structure. These findings have implications for how young children's drawings are interpreted by classroom teachers. The interpretation of these drawings suggested that some children may not yet realise how their drawings in mathematics need to shift from illustrations of the problem's story context to representing mathematical ideas and processes — which requires intentional teaching of the purpose of drawings for mathematical contexts. [ABSTRACT FROM AUTHOR]

In previous works, Buskes and the author have made use of representations of Archimedean Riesz spaces in terms of real-valued continuous functions defined on dense open subsets of a topological space in studying tensor products. These representations may be obtained from the Ogasawara–Maeda representation by means of restriction to the set on which representing functions are real-valued, rather than infinite. In this note, we show how to obtain such a representation as a simple consequence of the Krein–Kakutani representation of an order unit space. We conclude by studying the representation of Riesz homomorphisms in this setting. [ABSTRACT FROM AUTHOR]

A renewed interest in combinatorial and arithmetic properties as well as applications to differential equations, identities, formulas, and probability theory has been sparked by the study of degenerate versions of several specific numbers and polynomials. The article aims to explore a 3D unified degenerate class of generalized Fubini polynomials by utilizing 2D generalized degenerate polynomials. The potential of applications are provided by deriving certain computational formulas and identities,recurrence relations and derivative expressions for the 3D degenerated Gould–Hopper–Fubini, 3D degenerate Hermite-Fubini and 3D degenerate 2-iterated Fubini polynomials, which are extracted out of the 3D degenerate generalized Fubini polynomials. Finally, the behaviour of zeros of two concrete degenerate polynomials with some specific set of parameters is shown by drawing graphs using Mathematica [ABSTRACT FROM AUTHOR]

The inf-convolution of risk measures is directly related to risk sharing and general equilibrium, and it has attracted considerable attention in mathematical finance and insurance problems. However, the theory is restricted to finite sets of risk measures. This study extends the inf-convolution of risk measures in its convex-combination form to a countable (not necessarily finite) set of alternatives. The intuitive meaning of this approach is to represent a generalization of the current finite convex weights to the countable case. Subsequently, we extensively generalize known properties and results to this framework. Specifically, we investigate the preservation of properties, dual representations, optimal allocations, and self-convolution. [ABSTRACT FROM AUTHOR]

To explain why, in scientific problem solving, a diagram can be "worth ten thousand words," Jill Larkin and Herbert Simon (1987) relied on a computer model: two representations can be "informationally" equivalent but differ "computationally," just as the same data can be encoded in a computer in multiple ways, more or less suited to different kinds of processing. The roots of this proposal lay in cognitive psychology, more precisely in the "imagery debate" of the 1970s on whether there are image-like mental representations. Simon (1972, 1978) hoped to solve this debate by thoroughly reducing the differences between forms of mental representations (e.g., between images and sentences) to differences in computational efficiency; to carry out this reduction, he borrowed from computer science the concepts of data type and of data structure. I argue that, in the end, his account amounted to nothing more than characterizing representations by the fast operations on them. This analysis then allows me to assess what Simon's approach actually achieves when transported from psychology to the study of scientific representations, as in Larkin and Simon (1987): it allows comparing, not representations in and of themselves, but rather the computational roles they play in particular problem-solving processes—that is, representations together with a particular way of using them. [ABSTRACT FROM AUTHOR]

It was recently found that not only tool-specialized New Caledonian crows, but also Goffin cockatoos can manufacture physical objects in accordance with a mental template. That is, they can emulate features of existing objects when they manufacture new items. Both species spontaneously ripped pieces of card into large strips if they had previously learned that a large template was rewarded, and small strips when they previously learned that a small template was rewarded. Among New Caledonian crows, this cognitive ability was suggested as a potential mechanism underlying the transmission of natural tool designs. Here, we tested for the same ability in another non-specialised tool user–Hooded crows (Corvus cornix). Crows were exposed to pre-made template objects, varying first in colour and then in size, and were rewarded only if they chose pre-made objects that matched the template. In subsequent tests, birds were given the opportunity to manufacture versions of these objects. All three crows ripped paper pieces from the same colour material as the rewarded template, and, crucially, also manufactured objects that were more similar in size to previously rewarded, than unrewarded, templates, despite the birds being rewarded at random in both tests. Therefore, we found the ability to manufacture physical objects relative to a mental template in yet another bird species not specialized in using or making foraging tools in the wild, but with a high level of brain and cognitive development. [ABSTRACT FROM AUTHOR]

Boolean functions are mathematical objects used in diverse domains and have been actively researched for several decades already. One domain where Boolean functions play an important role is cryptography. There, the plethora of settings one should consider and cryptographic properties that need to be fulfilled makes the search for new Boolean functions still a very active domain. There are several options to construct appropriate Boolean functions: algebraic constructions, random search, and metaheuristics. In this work, we concentrate on metaheuristic approaches and examine the related works appearing in the last 25 years. To the best of our knowledge, this is the first survey work on this topic. Additionally, we provide a new taxonomy of related works and discuss the results obtained. Finally, we finish this survey with potential future research directions. [ABSTRACT FROM AUTHOR]

Suppose that A is a finite, nonempty subset of a cyclic group of either infinite or prime order. We show that if the difference set A - A is "not too large", then there is a nonzero group element with at least as many as (2 + o (1)) | A | 2 / | A - A | representations as a difference of two elements of A; that is, the second largest number of representations is, essentially, twice the average. Here the coefficient 2 is best possible. We also prove continuous and multidimensional versions of this result, and obtain similar results for sufficiently dense subsets of an arbitrary abelian group. [ABSTRACT FROM AUTHOR]

Representations are key to mathematical activities and meaning-making processes as they are part of modeling, connecting, communicating, and understanding mathematical ideas and concepts. The current study sought to examine a group of novice algebra learners' interactions with different representations from an onto-semiotic approach. A case study method was employed to understand how different algebraic practices (abstracting, generalizing, justifying, and operating on symbols) and functional thinking types (recursive, covariational, and correspondence) were facilitated through working with multiple representations. Three 6th graders participated in the study by completing 12 algebra tasks and taking part in two interviews. The onto-semiotic approach guided the data analysis process that involved the identification of mathematical objects that emerged in the participating students' mathematical practices. Then, the configuration of objects and semiotic functions established by the students in the functional situations was examined to understand the role of representations in the students' development of algebraic thinking and practices. Findings showed that abstraction is an essential process for generalization. Thinking about far figures facilitated abstraction and generalization through helping students construct non-ostensive concrete/pictorial representations. Verbal representations interacted with all representations and preceded symbolic representations. Working with near figures promoted recursive and covariational thinking while examining the far figures usually resulted in correspondence thinking. Implications for the school curriculum are discussed in the paper. [ABSTRACT FROM AUTHOR]

We give an elementary proof of the following result. Let C be an abelian and irreducible subgroup of the symplectic group Sp(2m, p). Then C is cyclic and embeds in the (multiplicative) subgroup of order p m + 1 of the field of order p 2 m . The proof yields, in fact, a similar result for nonsingular bilinear forms more generally. [ABSTRACT FROM AUTHOR]

In this paper, we consider polyhedral realizations for crystal bases B (λ) of irreducible integrable highest weight modules of a quantized enveloping algebra U q (g) , where g is a classical affine Lie algebra of type A n - 1 (1) , C n - 1 (1) , A 2 n - 2 (2) or D n (2) . We will give explicit forms of polyhedral realizations in terms of extended Young diagrams or Young walls that appear in the representation theory of quantized enveloping algebras of classical affine type. As an application, a combinatorial description of ε k ∗ functions on B (∞) will be given. [ABSTRACT FROM AUTHOR]

Mental model construction is supposed to be a useful cognitive devise for learning. Beyond human capacity of constructing mental models, scientists construct complex explanations about phenomena, named scientific or theoretical models. In this work we revisit three vissions: the first one concern about the polisemic term "model". Our proposal is to discriminate between "mental models" and "explicit models", being the former those "imaginistic" ideas constructed in scientists'—o teachers—minds, and the latter those teaching devices expressed in different languages that tend to communicate any "scientific model". From this point of view, the class is considered a place where teachers' mental models should be learned by novice students by decoding their teaching devices which are expressed in different languages. Other proposal of this work claims to distinguish the term "representation" with respect to its artistical or instrumental origin, highlighting that they are types of teaching devices and that artistical representations are always analogies. Finally, data about the construction of freshmen's wrong mental models related to the use of the analogy between the chemical combustion and the global process of cellular respiration from glucose is presented to reinforce previous epistemological reflections. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to give some construction results of BiHom-post-Lie algebras which are a generalization of both post-Lie-algebras and Hom-post-Lie algebras. They are the algebraic structures behind the weighted O -operator of BiHom-Lie algebras. They can be also regarded as the splitting into three parts of the structure of a BiHom-Lie-algebra. Moreover we develop the representation theory of BiHom-post-Lie algebras on a vector space V. We show that there is naturally an induced representation of its sub-adjacent Lie algebra. We give also all 2-dimensional BiHom-post-Lie algebras. We exhibit in this work some important examples of post-Lie algebras and Hom-post-Lie algebras. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate whether (and how) humans are unique in using tools and artifacts. Non-human animals exhibit some impressive instances of tool and artifact-use. Chimpanzees use sticks to get termites out of a mound, beavers build dams, birds make nests, spiders create webs, bowerbirds make bowers to impress potential mates, etc. There is no doubt that some animals modify and use objects in clever and sophisticated ways. But how does this relate to the way in which humans make and use objects to achieve their goals? To answer this question, this paper first presents a taxonomy of artifacts, identifying four overlapping categories, namely embodied, perceptual, cognitive, and affective artifacts. It then discusses definitions of animal tool-use, arguing that we need a more liberal approach, one that goes beyond the use of tools that are embedded in occurrent perception-action cycles. This paper ends by analysing how instances of animal tool and artifact-use can be classified according to the four identified categories, concluding that some animals use embodied, perceptual, cognitive, and affective artifacts. In this sense, humans are thus not unique in the kinds of tools and artifacts we use. What is unique, however, is our unprecedented flexibility and openness to deeply incorporate a large variety of complex tools and artifacts into our embodied, perceptual, cognitive, and affective systems. [ABSTRACT FROM AUTHOR]

Let p be a prime. We define S(p) the smallest number k such that every positive integer is a sum of at most k squares of integers that are not divisible by p. In this article, we prove that S (2) = 10 , S (3) = 6 , S (5) = 5 , and S (p) = 4 for any prime p greater than 5. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by 5, except the unique positive integer 79. [ABSTRACT FROM AUTHOR]

Search-based software engineering (SBSE) is an emerging research sub-area in the field of software engineering. The concept of SBSE is based on the idea of formulation of software engineering problem as a search-based optimization problem and effective exploitation of metaheuristic search optimizers to solve it. The complex nature of software engineering problems and complex computational behaviour of the metaheuristic search algorithms makes the SBSE approaches challenging to understand and analyze. A variety of multimedia technologies are generally used to make the problem formulation and their computational method more understandable and analyzable. Even after wide application of multimedia in science and engineering, the SBSE got little attention in this direction. To explore and exploit the potential of the multimedia in the SBSE, this work first conducted a research study on the current trends of multimedia in SBSE, then based on this study, the various challenges and opportunities are presented. More specifically, our work mainly focusses on current multimedia trends in various forms of SBSE approaches (e.g., single, multi, and many-objective SBSE). Apart from that, we also explore the various opportunities and challenges in SBSE from the perspective of visualization of software artefacts, software quality metrics, problem formulation, search trajectory, Pareto optimal set and front. [ABSTRACT FROM AUTHOR]

This paper offers a first-in-kind systemic visual content analysis of food systems images.A Google Images search for "food systems" was conducted and the top one hundred were captured. After images were filtered, a total of 82 figures were coded based on their text, pictures, geometric symbols (circles, arrows, etc.), and proportionality. The analysis documented phenomena that were foreground, background, and unrepresented. After reporting on what had been counted through the analysis, discussion turns to addressing the negative data collected—i.e., concepts/phenomena the images failed to represent. The article concludes reflecting on what the analysis means in terms of the food systems thinking and policy deliberations encouraged by these visual representations. [ABSTRACT FROM AUTHOR]

The present article is devoted to measuring a person's cognitive potential on the obtained experimental data to reveal its potentialities, as well as to monitor their dynamics, for example, for the diagnosis of recovery after an illness. This objective is divided in this study into two problems. Namely, estimation of the cognitive potential requires two algorithms to be developed: first, for assessment of the cognitive-complexity level of tasks and, second, for the system of cognitive-potential levels for an individual. The methods rely on a set of experimental techniques, including tailor-made proprietary ones, as well as on mathematical methods for data processing and calculation of introduced specific parameters for formalization of the cognitive potential. On the basis of these methods, methods (and specific formulas) are proposed for calculation of the cognitive potential of an individual using experimental data and tasks of different levels of complexity. As a part of this study, a methodology for determination of the cognitive potential is designed based on the theory of information patterns/representations. For objectification of cognitive skills (including so-called "soft skills"), a special-purpose web toolkit was developed. The obtained values allow one to study how social, genetic, and pathogenetic factors influence the dynamics of cognitive features. A new theoretical and technological platform for digital mapping and optimization of cognitive functions is proposed. [ABSTRACT FROM AUTHOR]

The paper presents a biosemiotic approach to the study of the built environment, its representations and practices in social media. First, it outlines the main developments that make semiotics hold a significant position in the study of urban space and the built environment. It then goes on to overcome the limitations of the binary opposition paradigm: in particular, nature/culture is reconsidered as a category in which the two terms are in a relation of mutual participation rather than being exclusive to each other. Following this, the paper explores three participatory categories that can be useful for the study of the built environment and its social media representations and practices: (a) life/semiosis, (b) natural environment/built environment, (c) text/practice. Finally, it identifies five main topics representing the interplay between the natural and built environment: the interaction of nature and architecture, urban parks, urban agriculture, digital environmentalism and ecotourism. [ABSTRACT FROM AUTHOR]

This study describes how 24 third graders (8–9 years old) relate and represent the relationships between variables when working with a functional thinking problem. This aspect contributes to providing insights about how elementary school students attend properties and relationships between covarying quantities rather than isolated computations. From a functional approach to early algebra, we describe written students' answers when working with a problem that involves a function, which includes questions for specific values and to generalize. Design research guidelines, specifically those set out for Classroom Teaching Experiment were followed. This study addresses the fourth and last Classroom Teaching Experiment session, which involved a function of the type y = ax + b and students had not previously worked it. Students primarily evidenced correspondence relationship, using natural language and numerical representation to express this functional relationship. Our findings let us to state that (a) although students were not used to working with these types of problems, eleven of them go beyond arithmetic computations, finding relationships that relate the variables; and (b) three students generalized using natural language as a useful vehicle, while there are other students who perceived the same regularity for different specific values but they are unable to represent generalization clearly. [ABSTRACT FROM AUTHOR]

Engaging with the significant increase in (post-)apocalyptic stories and representations in science fiction, this contribution aims at showing why a critical reconsideration of this type of cultural expression is essential to deal with this apparent pessimism and how its contents may be productive for future-oriented social and political thought. The reflexive resources present in these fictions are often overlooked for critical uses. To bring them to light, using a primarily literary and cinematic corpus, the analysis will first identify the predominant visions that are presented and characterize the symbolic contents they assemble. By reflecting on the forms of sense-making that are conveyed, it will then distinguish the functions they can perform. Finally, by considering that maintaining the powers to act is what is important in these (post-)apocalyptic situations, the argument will lead to an analysis of how these powers to act are deployed and the reactions and actions that may be thus inspired for an era of ecological crisis. [ABSTRACT FROM AUTHOR]

We introduce a notion of n-Lie–Rinehart algebras as a generalization of Lie–Rinehart algebras to n-ary case. This notion is also an algebraic analogue of n-Lie algebroids. We develop representation theory and describe a cohomology complex of n-Lie–Rinehart algebras. Furthermore, we investigate extension theory of n-Lie–Rinehart algebras by means of 2-cocycles. Finally, we introduce crossed modules of n-Lie–Rinehart algebras to gain a better understanding of their third cohomology groups. [ABSTRACT FROM AUTHOR]

Teachers' knowledge of proportional reasoning is important, particularly in the middle grades in the USA. This exploratory study investigated 32 teachers' use of knowledge resources in two mathematically similar tasks (one a paper and pencil task, the other a dynamic task) around proportional reasoning. The two tasks invoked different knowledge resources by the same teachers. Results suggest questions to the field around how we access or invoke teacher knowledge and the need to more purposefully explore the potential benefits of using a dynamic task to invoke knowledge resources. [ABSTRACT FROM AUTHOR]

This article discusses the characteristics of word problems that are associated with students' use of functional strategies and their ability to represent the generalization of functions. In the context of a broader research project designed to explore and foster functional thinking among elementary school students, twenty-five grade 4 (9- to 10-year-old) students were asked to identify functional relationships in five problems involving specific or indeterminate quantities. Their responses to a number of questions involving the generalization of the relationships in the problems were analyzed and associated to the characteristics of the problems. The type of representation of generalization used (verbal, generic, or symbolic) was also identified. Our findings indicate that grade 4 students showed potential for functional thinking prior to receiving instruction on variables and their notation. Such thinking was most effectively prompted when they worked with word problems that explicitly involved an additive function. When students generalized functional relationships, they represented them verbally or with generic examples. None of the students used symbolic representation. The originality of this study lies in the description of the specific characteristics of word problems that are associated with functional thinking; this information will prove useful to both teachers and curriculum designers. Identifying these characteristics could help build and propose tasks that encourage students to use more than one and more sophisticated strategies. [ABSTRACT FROM AUTHOR]

Primary school topics involving abstract concepts are challenging to teach. Electric circuits can be simply constructed but complex to explain. New approaches in teaching students for understanding are needed to advance practice in primary science. This paper combines the strengths of multimodality research perspectives with variation theory to provide insights into a teaching sequence designed for year 6 students. Application of a representation construction approach (RCA) provides opportunities for multimodal meaning-making of electric circuits. The case study adopted a design-based research method to investigate teaching electric circuits. Data collection involved video capture of classroom practice, teacher and student interviews, student journal entries and assessment artefacts, field notes, and pre-test and post-test results. Design principles included identifying key ideas, devising a lesson sequence emphasising energy transfer and transformation, hands-on exploration using multimodal representations in response to learning challenges, and student journaling. Students' metarepresentational competence was also developed through evaluation, negotiation, and creation of representations and models of electric circuits. Representational challenges followed by strategic teacher-led discussion facilitated students' developing understanding through focusing attention on critical features. Deep learning was evidenced by journal records, formative and summative assessment artefacts, and post-test responses. RCA principles were instrumental in the successful design of an effective teaching sequence through focus on critical aspects of energy. We advocate a RCA for the design of a multimodal learning sequence. Variation theory was a useful analytical framework to understand the enactment of the design sequence. The study contributes to the challenge of rethinking traditional teaching practices in primary science. [ABSTRACT FROM AUTHOR]

The construction of dynamic multimedia products requires the selection and integration of a range of semiotic resources. As an assessment task for preservice teachers, this construction process is complex but has significant potential for learning. To investigate how weaving together multiple representations in such tasks enables learners to develop conceptual understanding, the paper presents an indicative case study of a 2nd-year preservice primary (K-6) teacher who created a digital explanation on the topic of 'transparency' for stage 3 children (ages 11–12). We focus on data gathered during the 3-h construction process including artefacts such as images, online searches, websites accessed and paper records used for planning; the digital explanation as product; audio and video capture of the construction process and pre- and post-construction interviews. Using multimodal analysis, we examine these data to understand how meanings are negotiated as the maker moves iteratively among multiple representations and through semiotic choices within these representations to explain the science concept. The analyses illustrate the complexity of the construction process while providing insight into the creator's decision-making and to her developing semiotic and conceptual understandings. These findings allow us to build on the concept of cumulative semiotic progression (Hoban & Nielsen, Research in Science Education, 35, 1101-1119, 2013) by explicating the role of iterative reasoning in the production of pedagogic multimedia. [ABSTRACT FROM AUTHOR]

In this paper, novel modular representations and diagrams for service robots are introduced based on the work of ISO TC299-Working Group 6, aiming to provide a standard guideline for the users to present the modular service robots with a unified form, taking hardware connectivity and software aspect into account. The suggested modularity representation mainly consists of a line diagram, a circle diagram, a task-based diagram and a block diagram. At first, their concrete forms and introduction are illustrated in detail. Afterwards, some user case examples are provided, using the suggested diagrams, to present the real commercial modular designed service robots. [ABSTRACT FROM AUTHOR]

“The physics of representation” (Poldrack, 2020) aims to (1) define the word “representation” as used in the neurosciences, (2) argue that such representations as described in neuroscience are related to and usefully illuminated by the representations generated by modern neural networks, and (3) establish that these entities are “representations in good standing”. We suggest that Poldrack succeeds in (1), exposes some tensions between the broad use of the term in neuroscience and the narrower class of entities that he identifies in the end, and between the meaning of “representation” in neuroscience and in psychology in (2), and fails in (3). This results in some hard choices: give up on the broad scope of the term in neuroscience (and thereby potentially opening a gap between psychology and neuroscience) or continue to embrace the broad, psychologically inflected sense of the term, and deny the entities generated by neural nets (and the brain) are representations in the relevant sense. [ABSTRACT FROM AUTHOR]

Are there representations in the brain? It depends on what you mean by representations, and it depends on what you want them to do for you—both in terms of the causal role they play in the system, and in terms of their explanatory value. But ideally, we would like an account of representation that allows us to (a) assign a representational role and content to the appropriate mechanistic precursors of behavior that in fact play that role (if any do) and (b) conversely, search for the mechanistic realizers of representational roles that are posited by our models of behavior and cognition. Such an account would be methodologically valuable in neuroscience. Lately, people have started asking similar questions about deep neural networks. What representations do they learn and use, and what is the relation between those representations and the sometimes impressive capacities these networks exhibit? More philosophically puzzling, perhaps, is the question of why the internal activities or dispositions labeled as such in neuroscience and AI research should count as representations at all. I think we can give a unified answer to both sets of questions, in the form of a kind of representational pragmatism. What makes something a representation just is that we can identify and re-identify it as such, and moreover, that we can manipulate it effectively to do whatever it is that we think representations ought to do for us in that particular context. For the first condition, the idea is that so long as we have some probe that allows us to pick out the relevant causally effective candidate, we should consider that to be a candidate representation-relative-to-that-probe. The second condition can be understood as a kind of anti-gerrymandering constraint: one ought to be able to intervene on the candidate, and see effects of that intervention consistent with the functional role the representation is supposed to play. Naturally the details need to be spelled out for different contexts—and I will articulate them for a few illustrative cases—which effectively allows for a kind of pluralism about representation depending on the setting and the kinds of questions being asked. With the extra components filled in, representational pragmatism can make sense of existing practices in neuroscience and AI, as well as their relationship to naturalistic theories of representation in philosophy. [ABSTRACT FROM AUTHOR]

This paper gives the concept of the reduced pro-Banach algebra crossed product associated with inversely pro-Banach algebra dynamical system, and shows that the reduced crossed product is an inverse limit of an inverse system of Banach algebra crossed products. Also, the authors show that if the locally compact group is amenable, then the crossed product and the reduced crossed product are isometrically isomorphic. [ABSTRACT FROM AUTHOR]

The purpose of this study was to examine relationships between students' semiotic representational transformations and students' mathematics performance outcomes in digital math games. The study employed a convergent parallel mixed methods design. Students in Grades 4, 5, and 6 (ages 9–12) played nine digital math games. Researchers administered pretests and posttests and gathered video data. Researchers coded the video data to identify the representations used and students' representational transformations to understand how these related to mathematics performance outcomes. The video data revealed that the most frequent type of semiotic representation used by students was images, followed by symbols. The most common type of representational transformation students made was a conversion (i.e., transforming across two different representational registers), which is more complex than a treatment (i.e., transforming within the same representational register). A difference in proportions test revealed significant differences in test scores when students made representational transformations using language, images, symbols, and gestures. The results showed that some of the digital math games prompted more representational transformations than others; However, all of the games prompted the conversion type of representational transformation. Students who progressed between the pretest and posttest were much more likely to make representational transformations. Students who did not recognize and use representations or make transformations were much more likely to regress between the pretest and posttest. These results show the importance of the recognition and use of semiotic representations and representational transformations to mathematics outcomes when students play digital math games. [ABSTRACT FROM AUTHOR]

We present a study that explores Kindergarten and first-grade students' understandings and representations of arithmetic properties. The 16 students participated in a classroom teaching experiment designed to explore children's algebraic understandings, including their understandings and symbolic representations of three arithmetic properties: additive identity, additive inverse, and commutativity. We characterized students' understandings in terms of Skemp's framework of instrumental (rules without reason) and relational (knowing what to do and why) understandings. Following Vergnaud, we analyzed the types of additive relationships (transformation, comparison, or combination) and representations used by students. Our findings show that students' understandings developed in sophistication over time. We observed the least sophisticated understandings for the commutative property, particularly among Kindergarten students who exhibited instrumental understandings even after instruction. [ABSTRACT FROM AUTHOR]

We report a longitudinal study carried out along 3 years in an early childhood education (ECE) classroom in which we examined children's (aged 3–6) engagement with science representations. The research questions are as follows: (1) How do children's science representations develop from ECE1 to ECE3? (2) What are the features and affordances of the teacher's scaffolding of the production of science representations and how is it facilitated from the first to the third year of ECE? The participants were 21 children and their teacher. The group was involved in long-term science projects that lasted for 5 months each. Sessions (N = 30) were recorded and children's drawings (N = 487) gathered. Data were analyzed using discourse and content analyses, coupled with an analysis of the intensity of scaffolding. The results indicate that children's representations of science phenomena became more complex along several dimensions. We have identified teacher's scaffolding strategies which supported children's increasing autonomy in producing representations. Implications are drawn for teaching science at the ECE level as well as for further research. [ABSTRACT FROM AUTHOR]

Correctly understood, teleosemantics is the claim that "representation" is a function term. Things are called "representations" if they have a certain kind of function or telos and perform it in a certain kind of way. This claim is supported with a discussion and proposals about the function of a representation and of how representations perform that function. These proposals have been retrieved by putting together current descriptions from the literature on neural representations with earlier explorations of the features common to most things we are inclined to call representations (... maps, graphs, human language, signals between animals, stop signs ... etc.) as these were assessed in Millikan (Language, thought and other biological categories. MIT Press, Cambridge, 1984 and following). Of interest is the degree to which these independent sources converge. I conclude that there is no need to employ any new or technical sense of the term "representation" for it to play an important role in neuroscience. [ABSTRACT FROM AUTHOR]

We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction to prove that the fusion reduct of any finite member is a distributive semilattice, and to show that this variety is not locally finite. [ABSTRACT FROM AUTHOR]