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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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Using classroom mathematics tasks to support student learning is the topic of this timely volume. Employing research-based data, the authors focus on teacher practice as well as teacher and student learning and knowledge creation to demonstrate the use of mathematics tasks which promote effective student understanding. Unique in the field, the book provides a thorough, comprehensive guide to the nature of tasks for researchers, teacher educators, curriculum designers, administrators and teachers. Chapters on the effective implementation of mathematics tasks in the classroom, distinct pedagogical concepts related to teaching with mathematics tasks, and sample lessons that clearly demonstrate successful uses for mathematics tasks in the classroom are included.  The book is designed to provide a mix of cutting-edge data on task use with concrete examples of successful tasks and implementation tactics.  All of the lesson plans and illustrative examples provided have been extensively evaluated and tested in actual learning situations and feature specific suggestions for combating student difficulties and promoting solution pathways. This is a book that is essential for anyone hoping to understand both the importance of mathematics tasks for enhancing student learning and ways in which mathematics tasks can be applied in the classroom to achieve learning goals and objectives.     .

Mathematics -- Study and teaching (Secondary). --- Mathematics -- Study and teaching. --- Mathematics. --- Mathematics --- Education --- Physical Sciences & Mathematics --- Social Sciences --- Education, Special Topics --- Mathematics Teaching & Research --- Study and teaching --- Mathematics teachers. --- Study and teaching. --- Education. --- Teaching. --- Mathematics Education. --- Teaching and Teacher Education. --- Teachers --- Math --- Science --- Mathematics—Study and teaching . --- Didactics --- Instruction --- Pedagogy --- School teaching --- Schoolteaching --- Instructional systems --- Pedagogical content knowledge --- Training