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Given an action by a finite quantum group $\mathbb {G}$ on a von Neumann algebra M, we prove that a number of familiar $W^*$ properties are equivalent for M and the fixed-point algebra $M^{\mathbb {G}}$ (i.e., hold or not simultaneously for the two algebras); these include being hyperfinite, atomic, diffuse, and of type I, $II$, or $III$. Moreover, in all cases, the canonical central projections of M and $M^{\mathbb {G}}$ cutting out the summand with the respective property coincide. The result generalizes its classical-$\mathbb {G}$ analog due to Jones–Takesaki.
This chapter focuses on games with unawareness, where the players may be unaware of some of the choices that others can make. The player’s view specifies the choices in the game that he is aware of. The chapter starts by explaining how a game with unawareness can be viewed as a collection of one-person decision problems. Subsequently, it is shown how belief hierarchies about choices and views can be visualized by means of a beliefs diagram, and mathematically encoded by means of an epistemic model with types. This is used to provide a formal definition of common belief in rationality. It is shown that the choices which are possible under common belief in rationality can be characterized by iterated strict dominance for unawareness. The chapter finally turns to the scenario of fixed beliefs on views, where the players hold some pre-specified beliefs about the opponents’ views.
This chapter focuses on standard games. It starts by explaining how a standard game can be viewed as a collection of one-person decision problems. Subsequently, it is shown how belief hierarchies about choices can be visualized by means of a beliefs diagram, and mathematically encoded by means of an epistemic model with types. This is used to provide a formal definition of common belief in rationality – the central line of reasoning which states that a player believes that others choose rationally, believes that others believe that others choose rationally, and so on. It is shown that the choices which are possible under common belief in rationality can be characterized by the iterated elimination of strictly dominated choices.
This chapter focuses on games with incomplete information, where the players may be uncertain about the utility functions of the other players. It starts by explaining how a game with incomplete information can be viewed as a collection of one-person decision problems. Subsequently, it is shown how belief hierarchies about choices and utility functions can be visualized by means of a beliefs diagram, and mathematically encoded by means of an epistemic model with types. This is used to provide a formal definition of common belief in rationality. It is shown that the choices which are possible under common belief in rationality can be characterized by the generalized iterated strict dominance procedure. The chapter finally turns to the scenario of fixed beliefs on utilities, where the players hold some pre-specified beliefs about the opponents’ utility functions.
Stefanie Markovits’ chapter thinks about counting and accountability, and how they inform literary representations of the military man, one of the most visible of war’s outcomes in mid-Victorian Britain. Markovits reflects on this period as one which saw ‘the rise of statistics as a discipline of social science and a method of statecraft in Britain’, and with it the growing need for accountability in public affairs. The figure of the soldier is both hero and statistic, individual and number, in a period where fiction, philosophy, and popular commentary, were preoccupied with how individuals realised their fully individualised potential. The soldier’s cultural and political potency is enabled because his being is aligned with the numbers that account for him. In the work of Tennyson, Harriet Martineau, and Dickens we see how ‘the mid-century soldier becomes such a potent figure precisely because his “type” aligns so closely with numbers’.
This chapter examines how essayistic personae enabled writers and readers to understand personhood as a means of making a unity out of multiplicity. It draws on Thomas Hobbes’s theory of the person to track how essayistic personae both depicted corporate personhood and themselves served as corporate persons, allowing many writers, real or imagined, to write as one. It also uses Locke’s theory of personhood to show how essayistic personae present conscious persons as contingent unities imposed upon multitudinous thoughts and experiences. Essayistic personae not only extended personhood to non-human beings, such as corporations and animals, they also drew attention to the limited nature of personhood for many human beings, including married women and enslaved people.
This chapter provides details of the different kinds of vaccines available (live, attenuated, killed, inactivated, recombinant, subunit, mRNA, DNA and vector) for virus infections (e.g. SARS-CoV-2, HAV, HBV, influenzaviruses, JEE, measlesvirus, mumpsvirus, rubellavirus, HPV, rabies, polio, rotavirus, smallpox, mpox, yellow fever virus) in humans. It details the routes of administration and usage (e.g. pre-exposure or post-exposure, childhood, occupational health, travel and for at-risk groups).
This chapter looks at the period 1700 to 1820, one of profound change in Ireland as technological advances coupled with social and educational developments deeply influenced the intellectual and literary landscape. In the six and a half centuries since the invention of printing many new technologies affected the creation, distribution, consumption, and enjoyment of printed texts. Innovations and developments in printing, typefounding, papermaking, and marketing contributed to the advance of literary culture. The rise in education from the eighteenth century created an audience for literature in its many forms. Imaginative writing developed and attracted new audiences as literacy expanded among different cohorts. The newspaper provided the most comprehensive medium for the dissemination of information. Literacy was not necessarily a requirement as evidence shows that one newspaper could be shared among readers and read aloud to groups of listeners. Print advertising, an eighteenth-century innovation, increased the market for literary works.
Technology in Irish Literature and Culture shows how such significant technologies—typewriters, gramophones, print, radio, television, computers—have influenced Irish literary practices and cultural production, while also examining how technology has been embraced as a theme in Irish writing. Once a largely rural and agrarian society, contemporary Ireland has embraced the communicative, performative and consumption habits of a culture utterly reliant on the digital. This text plumbs the origins of the present moment, examining the longer history of literature's interactions with the technological and exploring how the transformative capacity of modern technology has been mediated throughout a diverse national canon. Comprising essays from some of the major figures of Irish literary and cultural studies, this volume offers a wide-ranging, comprehensive account of how Irish literature and culture have interacted with technology.
This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory (also known as higher-order intuitionistic logic), an important form of type theory based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined.
This chapter introduces students to the study of morphology. We look in a preliminary way at the difficulty inherent in defining what we mean by a word and introduce the term morpheme. We introduce the basic concepts of simple versus complex words. Students learn the distinction between word tokens, word types, and lexemes. We end with a brief introduction to the difference between inflection and derivation.
In this chapter, Alfano connects the Nietzschean constructs of instinct, drive, and type. He argues that a drive is a disposition to engage in a characteristic pattern of actions and evaluations. Drives are distinguished from preferences and desires in that preferences and desires aim at outcomes whereas drives aim at activity. As such, drive-motivated behaviors can lead to consequences that the agent does not prefer. Instincts, according to Alfano, are innate drives, but not all drives are instinctual. A person's type, in the Nietzschean framework, is the full constellation of her instincts and other drives. Different people embody different types, which turns out in the next chapter to be crucial to understanding Nietzsche's person-type-relative unity of virtue thesis.
In this chapter, Alfano connects the Nietzschean constructs of instinct, drive, and type. He argues that a drive is a disposition to engage in a characteristic pattern of actions and evaluations. Drives are distinguished from preferences and desires in that preferences and desires aim at outcomes whereas drives aim at activity. As such, drive-motivated behaviors can lead to consequences that the agent does not prefer. Instincts, according to Alfano, are innate drives, but not all drives are instinctual. A person's type, in the Nietzschean framework, is the full constellation of her instincts and other drives. Different people embody different types, which turns out in the next chapter to be crucial to understanding Nietzsche's person-type-relative unity of virtue thesis.
A Nikishin–Maurey characterization is given for bounded subsets of weak-type Lebesgue spaces. New factorizations for linear and multilinear operators are shown to follow.
We study some questions on numerical semigroups of type 2. On the one hand, we investigate the relation between the genus and the Frobenius number. On the other hand, for two fixed positive integers g1, g2, we give necessary and sufficient conditions in order to have a numerical semigroup S such that {g1, g2} is the set of its pseudo-Frobenius numbers and, moreover, we explicitly build families of such numerical semigroups.
The importance of social networks for young people who have experienced abuse and neglect remains an underdeveloped area of research and practice. The aim of the study was to investigate the relationship between abuse experienced by children and adolescents and subsequent outcomes on their social support networks. The study sample consisted of 85 clients (aged 8–15) of a service specifically for children reported to child protection due to child abuse and neglect. Abuse was measured using the Harm Consequences Assessment (HCA), which recorded the level of abuse experienced in five domains: Abandonment/No Appropriate Carer, Developmental and Medical Harm, Emotional and Psychological Harm, Physical Harm and Injury, and Sexual Harm. This also ranked abuse experienced in terms of severity: concerning, serious or extreme. Social network was measured using the Social Network Map. Analyses revealed a very high level of abuse for most young people across multiple domains. Social support was most evident in the “other family” category, and a relatively high level of perceived support was reported. There were few significant associations between levels of abuse and social support networks. However, one significant effect evident was for those young people who had not experienced developmental abuse who reported a significantly better network quality in work/school area of life than those who had experienced concerning or serious developmental abuse. This study contributes to an important body of emerging evidence on social support networks for children who have experienced maltreatment.
We use the best constants in the Khintchine inequality to generalise a theorem of Kato [‘Similarity for sequences of projections’, Bull. Amer. Math. Soc.73(6) (1967), 904–905] on similarity for sequences of projections in Hilbert spaces to the case of unconditional Schauder decompositions in $\ell _{p}$ spaces. We also sharpen a stability theorem of Vizitei [‘On the stability of bases of subspaces in a Banach space’, in: Studies on Algebra and Mathematical Analysis, Moldova Academy of Sciences (Kartja Moldovenjaska, Chişinău, 1965), 32–44; (in Russian)] in the case of unconditional Schauder decompositions in any Banach space.
The geochemistry and petrology of the type section of the Caradoc Series in the Onny valley indicate that it was deposited on a marginal basin continental shelf similar to the western side of the present Sea of Japan. The lower beds form a transgressive–regressive sequence in which the rocks become less mature upwards. All the coarser sediments above the basal quartzites and conglomerates are greywackes in which the apparent muddy and ferrous matrix is due to the breakdown of unstable minerals and particles. Higher values of Na2O and Na/K ratios are found in the coarser shallow-water sandstones of the Horderley Sandstone Formation and decrease markedly in the succeeding beds, accompanied by an increase in K2O. Higher values of carbonate-corrected (and hence related) other major and minor elements like SiO2, CaO, P2O5, MnO and most trace elements correlate with the transgressive systems and maximum flooding surfaces of the three sequences recognized where they are related to condensation at those horizons. Chemical Indices of Alteration (CIA) suggest that the Horderley Sandstone Formation underwent greater predepositional physical weathering than lower and higher beds, which is compatible with the petrography, and were deposited during a cool phase within overall warm Sandbian–Katian times. Trace element ratios suggest an oxic to suboxic depositional environment.