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This chapter is about parallel computation, in which many processors collectively perform a lot of work simultaneously in each time step. Fast parallel algorithms for arithmetic are presented. Then the chapter shows close relationships between space, parallel time, and logic formulas. The chapter concludes with a surprising equivalence between parallel time and narrow branching programs.
This chapter is about the prospect of proving that certain problems do not have logarithmic-space algorithms (are not in L). Results proven in this chapter include PSPACE-completeness and NL-completeness of classic problems, and the counterintuitive theorems PSPACE=NPSPACE and NL=coNL.
This chapter introduces the fundamental concepts of computational problems, algorithms, and time and space efficiency. These concepts are formally defined and illustrated with examples. The major time and space complexity classes P, EXP, L, and PSPACE are introduced. The chapter concludes by defining reductions, which are tools for relating complexities of different problems.
This article explores Israeli indie music festivals as sites for understanding contemporary cultural cosmopolitanism through a spatial lens. Drawing on ethnographic research, we conceptualize ‘spatial cosmopolitanism’ using the Möbius strip metaphor, highlighting the continuous folding of local and global spheres and the relational nature of boundaries. These dynamics are further complicated by recent escalations of violence in Israel and Palestine which underscore the tensions inherent in these spaces and challenge claims of perceived borderlessness amid an ongoing territorial conflict. Yet, rather than undermining our theoretical perspective, these conditions reveal how cosmopolitan practices are enacted within politically and territorially constrained environments. As we show, festival organisation is deeply rooted in the national ethos of pioneering, even as it draws on global models and connects organisers to international cultural networks. Revisiting ethnographic material demonstrates that spatial cosmopolitanism entails simultaneously engaging with and critically reflecting on local realities.
I begin by introducing Sutherland’s account of why Kant takes mathematics to require a non-conceptual source of knowledge of bare numerical difference. Given that objective space and time are wholes, however, I wonder why, according to Sutherland, such a source of knowledge can’t be circumvented by taking each spatial or temporal position to have conceptualizable features sufficient for distinguishing them. I then point to an issue with Sutherland’s appeals to “construction” and argue that if Sutherland more sharply distinguished between the syntheses involved in mathematical construction and empirical apprehension, then he could avoid some problems that his reading faces.
The exiles, landowners, bookworms, and kings of early medieval English texts all faced a common problem: how do you know yourself, or anything else, in a relational and therefore constantly changing world? They answered that problem with relational reiteration: sometimes literally walking the edges of property bounds, and sometimes figuratively inviting a reader to walk the edges of a riddle’s solution, or an exile’s lost home. The reiteration of relational landmarks is a temporary solution; each time a relationship is altered or challenged, the bounds of relational identity must be traced anew. And in fact this problem of mutable identity changes only when literary characters or hagiographical narratives achieve Christian salvation, positioning God in these texts as an eternal relationship that cannot be lost, changed, or forgotten. But in the age of their making, most Old English and Anglo-Latin texts and characters relied upon reiteration to stabilize (or sometimes reimagine) their unsaved, ephemeral, relationally-constructed worlds.
Chapter 9 focuses on how students develop foundational understandings of Measurement and Space in the early years (Foundation to Year 2). It explores how young learners engage with concepts such as length, area, time, and mass through hands-on experiences and everyday contexts. You will consider how to structure learning experiences that build conceptual understanding, support spatial reasoning, and introduce key mathematical vocabulary and thinking processes.
This chapter considers sacred space and the battles surrounding its possession, uses and understandings of the sacred more generally, including the wider significance of the landscape itself as a sacred space. Irish Catholic approaches were nuanced, such as the distinctions drawn between newly-built Protestant churches and older buildings that could be ‘re-catholicised’. The numerous rites and practices that this entailed are explored, showing that while notions of the holy and control of space were important to Irish Catholics, there was, nonetheless, great variety in how that was expressed. Spaces such as churches and churchyards are also examined as locations where both sacrality and ethnic tensions overlapped, with disputes surrounding burial showcasing Irish Catholic hostility to Protestants and Protestantism as heresy but also as foreign bodies contaminating Irish soil. The politics of church possession is another theme, which is shown to have explosive potential for intra-confessional conflict, as Irish Catholics debated and fought over the control of churches as a central demand of their cause, leading to conflict among themselves.
Chapter 10 builds on earlier learning and explores how students extend their understanding of Measurement and Space in the middle and upper primary years (Years 3 to 6). It focuses on the development of key concepts such as units of measure, angles, transformations, and geometric properties. You will investigate strategies for using precise mathematical language, addressing common misconceptions, and applying spatial and measurement thinking to solve purposeful, real-world problems.
This chapter includes three parts. The first part provides a brief introduction to rural China during the Song period. This is followed by a focused discussion on the spatial and administrative organization of Changshu County (in Jiangsu) near Lake Tai. Part three presents a case study of another county, Shanyin (Shaoxing, Zhejiang) in the southeast, aiming to illustrate the various aspects of rural life for both elites and ordinary people during the Southern Song.
Chapter 8 looks at a critical moment in Keats’s life in order to trace the way, in his letters, he works through a crucial decision about his future as a writer. Focusing on a series of interlinked and in some ways ‘porous’ letters written during a single week in September 1819, the chapter discusses Keats’s sense that he is, or soon will be, ‘unpoeted’ – that he can no longer be a poet. Alone in the small city of Winchester, Keats writes a series of often overlapping letters that ultimately move him towards a decision concerning whether or not to end his career as a poet. The chapter aligns the specific circumstances of the limited space of the cathedral city in which Keats is temporarily staying with the limited space of the letter-page itself and examines how he resolves a critical life choice in and through correspondence.
In Mumbai, the live slaughter of meat by Muslim butchers is an intergenerational caste practice and a source of livelihood and labour, supplying ritually slaughtered halal meat for Muslims, and fresh quality meat for others in the city. Yet in the neoliberal Hindu Nationalist imagination, the very sights, smells, and sounds of butchering are a sign of Muslim abjection, a source of disgust to be eradicated from the “world-class” Hindu city. Muslim butchers are further subject to an intra-Muslim caste politics of disdain, in which their contact with blood is thought to render them impure (na-pak) and untrustworthy. At butcher stores and meat markets, economic development, urban sanitation, and ritual ideologies recruit the senses into a politics of disgust and disdain. These affects are reflexively embodied in ways that reveal the intimate relation between individualising forms of Islamic piety, Muslim infrastructural marginalisation, and the making of a sanitised neoliberal Hindu city.
Paul Guyer, winner of the 2024 International Kant Prize, is one of the world's leading Kant scholars. This volume collects ten of his essays on Kant's critical approach to metaphysics and epistemology that have been published since his path-breaking Kant and the Claims of Knowledge (Cambridge 1987). These essays resolve long-running debates about the meaning of Kant's doctrine of transcendental idealism while criticizing Kant's fundamental argument for the position; show what is nevertheless enduringly valuable in Kant's transcendental method; and situate Kant's work in theoretical philosophy in his teleological approach to philosophy in general. The essays clarify Kant's inheritance from and disagreements with two of his most important predecessors, John Locke and Gottfried Wilhelm Leibniz, and analyse the difference between Kant's philosophical method and that of his most influential modern interpreter, Peter Strawson.
Our Forum envisions East Asia as part of Islamic Asia, treating it as a space where Muslim communities have forged cross-border networks across time, episodically, and where discourses about Islam have circulated and been appropriated in interconnection with Muslim-majority regions of the continent (that is, ‘Islamic’ Asia). We hold that Islam, as a constellation of religious, political, cultural, and social formations, questions the spatial and conceptual boundaries of East Asia, while East Asia expands the known geographies of Islamic Asia. The articles in this Forum show that Islam was a shared paradigm of meaning-making across inter-Asian geographies, and offered alternative modes and axes of spatial production and political idioms that both Muslims and non-Muslims latched onto across Asia, including its easternmost reaches.
Education policy is always at risk of working at cross-purposes toward education goals. Using a meta-ethnographic methodology and Massey’s geometry of space theory, the present article addresses this in relation to a particular policy realisation problem of teaching for sustainability in schools in depopulated rural areas with identified population challenges. Specific attention has gone to research addressing the enacted curriculum and teachers’ experiences of working with sustainability goals. The results highlight features for goal realisation such as the presence of and attention to rural natural and cultural environmental heritage, having local access and giving curriculum attention to local employment and sustainable vocations and professions, and having community support from the local community and engagement of the school in the community. Working against sustainability were global epistemic rural marginalisation, performative curriculum relations, market competition and competitive exclusions from market participation, tepid community involvement in schools, and socially isolated schools insulated from the local community.
Does the way people talk about time affect how they think about it? Whereas English speakers describe the duration of events most often in terms of spatial length (e.g., a long night), Greek speakers tend to talk about duration in terms of multidimensional spatial size (e.g., mia megali nychta, tr. a big night) or amount (e.g., poli ora, tr. much time). After quantifying these linguistic patterns, we gave non-linguistic tests of duration estimation to English and Greek speakers. English speakers’ estimates were influenced more strongly by irrelevant length information and Greek speakers’ by irrelevant amount information, consistent with verbal metaphors for duration in English and Greek. Next, we tested duration estimation with concurrent verbal interference, to confirm that the observed effects did not depend on participants verbally labeling the stimuli during the task. Finally, we trained English speakers to use Greek-like metaphors for duration, which resulted in Greek-like performance on a non-linguistic duration estimation task. Results show that (a) people who talk about time differently also think about it differently, (b) these effects are not due to participants’ using verbal labels during the task, and (c) language can play a causal role in shaping even basic non-linguistic mental representations of time.
This article explores how pre-service music teachers in Norway reflect on their future professional identities and career trajectories during the final year of a five-year generalist teacher education programme. We analyse two group interviews with eight participants – one conducted during the writing of their master’s theses and one shortly after submission. The study is framed by the concept of spatially situated possible selves, combining Markus and Nurius’ theory of possible selves with Massey’s spatial theory to examine how imagined futures could be shaped by institutional, geographic, and social contexts. Thematic analysis reveals four key areas of reflection: career awareness, the influence of past experiences, the shaping role of music teacher education, and the participants’ hybrid positioning between student and teacher roles. Findings suggest that the master’s thesis serves as a transitional tool for professional development and identity formation. We argue that music teacher education can be understood as a contested and evolving space – a multiplicity of ‘stories-so-far’ – where future selves are imagined, negotiated, and constrained.
Across her fiction and non-fiction, Elizabeth Bowen is consistently intrigued by hotels. From the grand Italian Riviera establishment of her debut novel, The Hotel, to the series of dingy ‘back rooms in hotels … with no view’ occupied by Portia Quayne and her mother in The Death of the Heart, many of Bowen’s characters occupy, however briefly, the transitory, impermanent space of the hotel. Although characters move through hotel space, they are never left unmarked by it. Portia’s teacher observes her ‘hotel habits’, which she cannot shake. This chapter explores Bowen’s preoccupation with the space of the hotel in her writing, and demonstrates her acute sense not only of its unique spatiality, but also of the intricacies of hotel temporality. More specifically, I argue that Bowen is a writer who is profoundly sensitive to the relationship between people and the spaces they occupy, and this sensitivity comes to the fore in the hotel phenomenologies of her characters.