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"The author considers homomorphisms H to K from an affine group scheme H over a field k of characteristic zero to a proreductive group K. Using a general categorical splitting theorem, Andrâe and Kahn proved that for every H there exists such a homomorphism which is universal up to conjugacy. The author gives a purely group-theoretic proof of this result. The classical Jacobson-Morosov theorem is the particular case where H is the additive group over k. As well as universal homomorphisms, the author considers more generally homomorphisms H to K which are minimal, in the sense that H to K factors through no proper proreductive subgroup of K. For fixed H, it is shown that the minimal H to K with K reductive are parametrised by a scheme locally of finite type over k."--Publisher's description.

Group theory --- Linear algebraic groups. --- Group theory. --- Commutative rings. --- Algebraic varieties. --- Geometry, Algebraic. --- Groupes linéaires algébriques --- Groupes, Théorie des --- Anneaux commutatifs --- Varietes algebriques --- Géométrie algébrique --- 51 <082.1> --- Mathematics--Series --- Théorie des groupes --- Groupes linéaires algébriques --- Théorie des groupes --- Géométrie algébrique --- Algebraic varieties --- Commutative rings --- Geometry, Algebraic --- Linear algebraic groups --- Algebraic groups, Linear --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Algebraic geometry --- Geometry --- Rings (Algebra) --- Varieties, Algebraic --- Groupes algébriques linéaires

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Naturalism --- Transcendentalism --- Kant, Immanuel

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Dummett, Michael

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This volume of newly written chapters on the history and interpretation of Wittgenstein's Tractatus represents a significant step beyond the polemical debate between broad interpretive approaches that has recently characterized the field. Some of the contributors might count their approach as "new" or "resolute," while others are more 'traditional', but all are here concerned primarily with understanding in detail the structure of argument that Wittgenstein presents within the Tractatus, rather than with its final self-renunciation, or with the character of the understanding that renunciation might leave behind. The volume makes a strong case that close investigation, both biographical and textual, into the composition of the Tractatus, and into the various influences on it, still has much to yield in revealing the complexity and fertility of Wittgenstein's early thought. Amongst these influences Kant and Kierkegaard are considered alongside Wittgenstein's immediate predecessors in the analytic tradition. The themes explored range across the breadth of Wittgenstein's book, and include his accounts of ethics and aesthetics, as well as issues in metaphysics and the philosophy of mind, and aspects of the logical framework of his account of representation. The contrast of saying and showing, and Wittgenstein's attitude to the inexpressible, is of central importance to many of the contributions. By approaching this concern through the various first-level issues that give rise to it, rather than from entrenched schematic positions, the contributors demonstrate the possibility of a more inclusive, constructive and fruitful mode of engagement with Wittgenstein's text and with each other.

Logic, symbolic and mathematical --- Language and languages --- Philosophy --- Wittgenstein, Ludwig, --- Logic, Symbolic and mathematical --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Logic. --- Argumentation --- Deduction (Logic) --- Deductive logic --- Dialectic (Logic) --- Logic, Deductive --- Intellect --- Psychology --- Science --- Reasoning --- Thought and thinking --- Methodology --- Language and languages - Philosophy --- Wittgenstein, Ludwig, - 1889-1951. - Tractatus logico-philosophicus

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"This book describes research undertaken by leading Australian researcher in Indigenous communities. While the chapters are Australian in their focus, the issues that are discussed are similar to those in other countries where there are indigenous people. In most cases, in Australia and internationally, Indigenous learners are not succeeding in school, thus making the transition into work and adulthood quite tenuous in terms of mainstream measures. The importance of being literate and numerate are critical in success in school and life in general, thus making this collection an important contribution to the international literature. The collection of works describes a wide range of projects where the focus has been on improving the literacy and numeracy outcomes for Indigenous students. The chapters take various approaches to improving these outcomes, and have very different foci. These foci include aspects of literacy, numeracy, curriculum  leadership, ICTs, whole school planning, policy, linguistics and Indigenous perspectives. Most of the chapters report on large scale projects that have used some innovation in their focus. The book draws together these projects so that a more connected sense of the complexities and diversity of approaches can be gleaned."--

Teaching --- Didactics --- Adult education. Lifelong learning --- geletterdheid --- didactiek --- onderwijs --- evidence-based methodiek --- alfabetisering --- analfabetisme --- Aboriginal Australians --- Education and state --- Literacy --- Literacy programs --- Indigenous peoples --- Ethnology --- Literacy campaigns --- National literacy campaigns --- National literacy programs --- Reading programs (Literacy) --- Illiteracy --- Education --- General education --- Education. --- Government policy