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Students’ questions play an important role in meaningful learning and scientific inquiry. They are a potential resource for both teaching and learning science. Despite the capacity of students’ questions for enhancing learning, much of this potential still remains untapped. The purpose of this chapter, therefore, is to examine and review the existing research on students’ questions and to explore ways of advancing future work into this area. The chapter begins by highlighting the importance and role of students’ questions and the ways in which they have been categorized to argue that there are limitations to each of these. It then seeks to show, drawing on sets of classroom videos, that a schema based on the epistemic function of the question for constructing knowledge would suggest that there are really three categories of question – ontic questions, causal questions, and epistemic questions. The chapter then explores which programs of research offer promise for helping teachers to scaffold students at producing epistemic and better questions.
We prove that the category of (rigidified) Breuil–Kisin–Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil–Kisin–Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a ‘
-adic Serre group’.
We begin by proving that if K is a triangulated category and S ⊆ K is a denominator set of cohomological origin, then the localized category KS is triangulated and the localization functor Q : K → KS is triangulated. In the case of the triangulated category K(A,M) and the set of quasi-isomorphisms S(A,M) in it, we get the derived category D(A,M) := K(A,M)S(A, M) and the triangulated localization functor Q : K(A,M) → D(A,M). We look at the full subcategories of K(A, M) corresponding to boundedness conditions and the corresponding derived categories. We prove that the obvious functor M → D(M) is fully faithful. The section ends with a study of the triangulated structure of the opposite derived category D(A,M)op .
Moduli spaces of stable objects in the derived category of a
surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O’Grady’s filtration on the
-group of the
surface. This solves a conjecture of O’Grady and improves on previous results of Huybrechts, O’Grady, and Voisin. Second, we propose a candidate for the Beauville–Voisin filtration on the
-group of the moduli space of stable objects. We discuss its connection with Voisin’s recent proposal via constant cycle subvarieties, and prove a conjecture of hers on the existence of special algebraically coisotropic subvarieties for the moduli space.
This paper relates Kant'saccount of pure apperception to the agential approach to self-knowledge. It argues that his famous claim ‘The I think must be able to accompany all of my representations’ (B131) does not concern the possibility of self-ascribing beliefs. Kant does advance this claim in the service of identifying an a priori warrant we have as psychological persons, that is, subjects of acts of thinking that are imputable to us. But this warrant is not one to self-knowledge that we have as critical reasoners. It is, rather, an a priori warrant we have, as thinkers, to prescribe to given representations their conformity to principles of thinking inherent in our capacity of understanding itself.
As a quasi-judicial body, the WGAD operates without a formal set of rules but is instead guided in implementing its mandate by its Methods of Work, which explain the overall process by which to submit a case for consideration.1 That said, however, the WGAD considers individual cases brought to it in closed sessions of its members and staff. And the day-to-day operations of the WGAD are run by a small secretariat of staff of the Office of the UN High Commissioner for Human Rights (OHCHR) in its Special Procedures Branch, led by its Secretary. This chapter will illuminate and elucidate how an individual case can most effectively be brought to the WGAD, supplementing the procedures described in the Methods of Work with the author’s practical experience gained from having taken more than forty-five cases to the WGAD and having interviewed many current and former WGAD members and staff.
In keeping with the nature of the Cambridge Critical Concepts series, the introduction establishes decadence as a concept. We show how the concept emerges from a combination of etymology and history, and how decadence cuts across and calls into question traditional literary categories, such as genre and periodization. We articulate the relevance of decadence to recent literary interests, such as gender politics and queer theory. Finally, we explain the rationale for the organization of the volume as an effort to ‘scale up’ and reset the parameters of decadence as a concept; preview the individual contributions to the collection; and clarify the structure of the volume: the origins of the concept of decadence, its development through nineteenth-century fields, and its application to various twentieth-century disciplines and literary modalities. The introduction concludes with commentary on the contemporary resonance of decadence today.
While Yucatecan elites consistently characterized the Caste War as a racial conflict and labelled the rebels as Indians, the insurgents were in fact a fairly mixed population. Ample evidence from contemporary observers shows that many non-Indians were found in the rebel ranks. The rebels employed terms of self-identification that reflect their mixed social and ethnic composition and religious affiliation, generally referring to themselves as cristiano’b (Christians), otsilo’b (poor), masewalo’b (commoner) or kruso’b (the crosses) and not as Indians or Maya. It comes as no surprise that legally most rank and file rebels were Indians, as revealed by their Maya surnames. Legal Indians were overrepresented among the rural lower classes, the insurgents’ main social base. In addition, the preponderance of Indians simply mirrors Yucatán’s demographic structure, since the bulk of the rebels came from areas where this group outnumbered vecinos by three or four to one.
This introductory chapter by the editor discusses the goals of the book, introduces the questions central to it as well as develops a methodological framework. It relates to existing scholarship on the impact of human rights law upon other branches of international law and on the fragmentation of international law. It outlines the methodology that was applied in working towards this volume by presenting a range of distinctions that provide conceptual tools for detecting and assessing the different ways how international non-human-rights courts may refer to human rights. These include categories of human rights, sources of human rights norms, and three contexts for their application, namely due process rights applied in the proceedings of the court in question, substantive human rights norms as applicable law or basis for subject-matter jurisdiction, and interpretive reliance on human rights through systemic integration. The chapter also relates to the legitimacy of international courts by showing that how international courts relate to human rights norms matters for factors of legitimacy.
In the concluding chapter, the editor engages in a comparative and theory-building exercise across the jurisdictions covered in the book. There are important differences between international non-human-rights courts as to the legal basis for their application of human rights norms. While due process rights of the parties appearing before it, and systemic integration, are available for all courts, there are marked differences in issues such as standing by individuals, the status of human rights norms as applicable substantive law or basis for jurisdiction, and the patterns concerning which categories of human rights have made their way into other international courts. There are also clear examples of ‘other’ courts widening the scope of justiciable human rights, for instance through applying economic, social and cultural rights, or the right to property, or collective rights of peoples beyond the practice of actual human rights courts. In their application of human rights norms, 'other' international courts have at least so far tended to do so reflecting more the trend of humanisation, rather than constitutionalisation, of international law.
We study the two model-theoretic concepts of weak saturation and weak amalgamation property in the context of accessible categories. We relate these two concepts providing sufficient conditions for existence and uniqueness of weakly saturated objects of an accessible category
. We discuss the implications of this fact in classical model theory.
We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin–Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in
Using a recent computation of the rational minus part of
by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Déglise and a version of the Röndigs and Østvær theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor–Witt correspondences in the sense of Calmès and Fasel.
The paper shows how the Scott–Koymans theorem for the untyped λ-calculus can be extended to the differential λ-calculus. The main result is that every model of the untyped differential λ-calculus may be viewed as a differential reflexive object in a Cartesian-closed differential category. This extension of the Scott–Koymans theorem depends critically on unraveling the somewhat subtle issue of which idempotents can be split so that differential structure lifts to the idempotent splitting.
The paper uses (total) Turing categories with “canonical codes” as the basic categorical semantics for the λ-calculus. It develops the main result in a modular fashion by showing how to add left-additive structure to a Turing category, and then – on top of that – differential structure. For both levels of structure, it is necessary to identify how “canonical codes” must behave with respect to the added structure and, furthermore, how “universal objects” must behave. The latter is closely tied to the question – which is the crux of the paper – of which idempotents can be split while preserving the differential structure of the setting.
This paper is the full version of a conference paper and includes the proofs which were omitted from that version due to page-length restrictions.
Several fixed-point models share the equational properties of iteration theories, or iteration categories, which are cartesian categories equipped with a fixed point or dagger operation subject to certain axioms. After discussing some of the basic models, we provide equational bases for iteration categories and offer an analysis of the axioms. Although iteration categories have no finite base for their identities, there exist finitely based implicational theories that capture their equational theory. We exhibit several such systems. Then we enrich iteration categories with an additive structure and exhibit interesting cases where the interaction between the iteration category structure and the additive structure can be captured by a finite number of identities. This includes the iteration category of monotonic or continuous functions over complete lattices equipped with the least fixed-point operation and the binary supremum operation as addition, the categories of simulation, bisimulation, or language equivalence classes of processes, context-free languages, and others. Finally, we exhibit a finite equational system involving residuals, which is sound and complete for monotonic or continuous functions over complete lattices in the sense that it proves all of their identities involving the operations and constants of cartesian categories, the least fixed-point operation and binary supremum, but not involving residuals.
Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index
in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.
Categorical compositional distributional semantics is a model of natural language; it combines the statistical vector space models of words with the compositional models of grammar. We formalise in this model the generalised quantifier theory of natural language, due to Barwise and Cooper. The underlying setting is a compact closed category with bialgebras. We start from a generative grammar formalisation and develop an abstract categorical compositional semantics for it, and then instantiate the abstract setting to sets and relations and to finite-dimensional vector spaces and linear maps. We prove the equivalence of the relational instantiation to the truth theoretic semantics of generalised quantifiers. The vector space instantiation formalises the statistical usages of words and enables us to, for the first time, reason about quantified phrases and sentences compositionally in distributional semantics.