Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.

Development of meta-representations: Procedural metacognition and the relationship to Theory of Mind

In several studies it was shown that metacognitive ability is crucial for children and their success in school. Much less is known about the emergence of that ability and its relationship to other meta-representations like Theory of Mind competencies. In the past years, a growing literature has suggested that metacognition and Theory of Mind could theoretically be assumed to belong to the same developmental concept. Since then only a few studies showed empirically evidence that metacognition and Theory of Mind are related. But these studies focused on declarative metacognitive knowledge rather than on procedural metacognitive monitoring like in the present study: N = 159 children were first tested shortly before making the transition to school (aged between 5 1/2 and 7 1/2 years) and one year later at the end of their first grade. Analyses suggest that there is in fact a significant relation between early metacognitive monitoring skills (procedural metacognition) and later Theory of Mind competencies. Notably, language seems to play a crucial role in this relationship. Thus our results bring new insights in the research field of the development of meta-representation and support the view that metacognition and Theory of Mind are indeed interrelated, but the precise mechanisms yet remain unclear.

Explicit description of generic representations for quivers of type An or Dn

We describe explicitly a generic representation for Dynkin quivers of type An or Dn for any dimension vector.

A Mathematical Model for Simplifying Representations of Objects in a Geographic Information System

The study of operations on representations of objects is well documented in the realm of spatial engineering. However, the mathematical structure and formal proof of these operational phenomena are not thoroughly explored. Other works have often focused on query-based models that seek to order classes and instances of objects in the form of semantic hierarchies or graphs. In some models, nodes of graphs represent objects and are connected by edges that represent different types of coarsening operators. This work, however, studies how the coarsening operator "simplification" can manipulate partitions of finite sets, independent from objects and their attributes. Partitions that are "simplified first have a collection of elements filtered (removed), and then the remaining partition is amalgamated (some sub-collections are unified). Simplification has many interesting mathematical properties. A finite composition of simplifications can also be accomplished with some single simplification. Also, if one partition is a simplification of the other, the simplified partition is defined to be less than the other partition according to the simp relation. This relation is shown to be a partial-order relation based on simplification. Collections of partitions can not only be proven to have a partial- order structure, but also have a lattice structure and are complete. In regard to a geographic information system (GIs), partitions related to subsets of attribute domains for objects are called views. Objects belong to different views based whether or not their attribute values lie in the underlying view domain. Given a particular view, objects with their attribute n-tuple codings contained in the view are part of the actualization set on views, and objects are labeled according to the particular subset of the view in which their coding lies. Though the scope of the work does not mainly focus on queries related directly to geographic objects, it provides verification for the existence of particular views in a system with this underlying structure. Given a finite attribute domain, one can say with mathematical certainty that different views of objects are partially ordered by simplification, and every collection of views has a greatest lower bound and least upper bound, which provides the validity for exploring queries in this regard.

Emerging Identities in Colonial Tunisia: "Alliancist" and Zionist Representations in Tunis prior to World War I

By 1900 the Jewish community of Tunisia witnessed the emergence of new competing identities: “assimilationist” of the Alliance Israelite Universelle, termed “Alliancist,” and Zionist. Strikingly, two members of the same family in Tunis, Raymond Valensi, President of the AIU Regional Committee, and Alfred Valensi, President of the Zionist Federation, led the struggle for their separate causes. In his discussion of identity in the modern world, Homi Bhabha asks, "How do strategies of representation or empowerment come to be formulated in the competing claims of communities…where, despite shared histories of …discrimination, the exchange of values, meanings and priorities…may be profoundly antagonistic…?" It is in this context that the claims of the Alliance and Zionism will be examined prior to World War I, based on the Archives of the AIU and on such secondary sources as the indispensable work of Paul Sebag. The tensions between the Alliancists and Zionists continued until the outbreak of World War II, as the French-speaking Jews of Tunisia sought to define their individual and collective identities.

UNDERSTANDING NURSE CREATED COGNITIVE ARTIFACTS: PERSONALLY-CREATED-COGNITIVE-ARTIFACTS AS EXTERNAL REPRESENTATIONS OF DISTRIBUTED COGNITION

Manuscript 1: “Conceptual Analysis: Externalizing Nursing Knowledge” We use concept analysis to establish that the report tool nurses prepare, carry, reference, amend, and use as a temporary data repository are examples of cognitive artifacts. This tool, integrally woven throughout the work and practice of nurses, is important to cognition and clinical decision-making. Establishing the tool as a cognitive artifact will support new dimensions of study. Such studies can characterize how this report tool supports cognition, internal representation of knowledge and skills, and external representation of knowledge of the nurse. Manuscript 2: “Research Methods: Exploring Cognitive Work” The purpose of this paper is to describe a complex, cross-sectional, multi-method approach to study of personal cognitive artifacts in the clinical environment. The complex data arrays present in these cognitive artifacts warrant the use of multiple methods of data collection. Use of a less robust research design may result in an incomplete understanding of the meaning, value, content, and relationships between personal cognitive artifacts in the clinical environment and the cognitive work of the user. Manuscript 3: “Making the Cognitive Work of Registered Nurses Visible” Purpose: Knowledge representations and structures are created and used by registered nurses to guide patient care. Understanding is limited regarding how these knowledge representations, or cognitive artifacts, contribute to working memory, prioritization, organization, cognition, and decision-making. The purpose of this study was to identify and characterize the role a specific cognitive artifact knowledge representation and structure as it contributed to the cognitive work of the registered nurse. Methods: Data collection was completed, using qualitative research methods, by shadowing and interviewing 25 registered nurses. Data analysis employed triangulation and iterative analytic processes. Results: Nurse cognitive artifacts support recall, data evaluation, decision-making, organization, and prioritization. These cognitive artifacts demonstrated spatial, longitudinal, chronologic, visual, and personal cues to support the cognitive work of nurses. Conclusions: Nurse cognitive artifacts are an important adjunct to the cognitive work of nurses, and directly support patient care. Nurses need to be able to configure their cognitive artifact in ways that are meaningful and support their internal knowledge representations.

Learning -dependent plasticity of movement representations in the primate primary motor cortex

Primary motor cortex (M1) is involved in the production of voluntary movement and contains a complete functional representation, or map, of the skeletal musculature. This functional map can be altered by pathological experiences, such as peripheral nerve injury or stroke, by pharmacological manipulation, and by behavioral experience. The process by which experience-dependent alterations of cortical function occur is termed plasticity. In this thesis, plasticity of M1 functional organization as a consequence of behavioral experience was examined in adult primates (squirrel monkeys). Maps of movement representations were derived under anesthesia using intracortical microstimulation, whereby a microelectrode was inserted into the cortex to electrically stimulate corticospinal neurons at low current levels and evoke movements of the forelimb, principally of the hand. Movement representations were examined before and at several times after training on behavioral tasks that emphasized use of the fingers. Two behavioral tasks were utilized that dissociated the repetition of motor activity from the acquisition of motor skills. One task was easy to perform, and as such promoted repetitive motor activity without learning. The other task was more difficult, requiring the acquisition of motor skills for successful performance. Kinematic analysis indicated that monkeys used a consistent set of forelimb movements during pellet extractions. Functional mapping revealed that repetitive motor activity during the easier task did not produce plastic changes in movement representations. Instead, map plasticity, in the form of selective expansions of task-related movement representations, was only produced following skill acquisition on the difficult task. Additional studies revealed that, in general, map plasticity persisted without further training for up to three months, in parallel with the retention of task-related motor skills. Also, extensive additional training on the small well task produced further improvements in performance, and further changes in movement maps. In sum, these experiments support the following three conclusions regarding the role of M1 in motor learning. First, behaviorally-driven plasticity is learning-dependent, not activity-dependent. Second, plastic changes in M1 functional representations represent a neural correlate of acquired motor skills. Third, the persistence of map plasticity suggests that M1 is part of the neural substrate for the memory of motor skills. ^