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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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Teacher education. Teacher's profession --- Didactics of mathematics --- Mathematics teachers --- Training of

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These new studies of Wittgenstein's 'Tractatus' represent a significant step beyond recent polemical debate. They cover a wide range of themes, and show that close investigation into the composition of the work, and into the various influences on it has much to yield in revealing the complexity and fertility of Wittgenstein's early thought.

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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Teacher education seeks to transform prospective and/or practicing teachers from neophyte possibly uncritical perspectives on teaching and learning to more knowledgeable, adaptable, analytic, insightful, observant, resourceful, reflective and confident professionals ready to address whatever challenges teaching secondary mathematics presents. This transformation occurs optimally through constructive engagement in tasks that foster knowledge for teaching secondary mathematics. Ideally such tasks provide a bridge between theory and practice, and challenge, surprise, disturb, confront, extend, or provoke examination of alternatives, drawn from the context of teaching. We define tasks as the problems or activities that, having been developed, evaluated and refined over time, are posed to teacher education participants. Such participants are expected to engage in these tasks collaboratively, energetically, and intellectually with an open mind and an orientation to future practice. The tasks might be similar to those used by classroom teachers (e.g., the analysis of a graphing problem) or idiosyncratic to teacher education (e.g., critique of videotaped practice). This edited volume includes chapters based around unifying themes of tasks used in secondary mathematics teacher education. These themes reflect goals for mathematics teacher education, and are closely related to various aspects of knowledge required for teaching secondary mathematics. They are not based on the conventional content topics of teacher education (e.g., decimals, grouping practices), but on broad goals such as adaptability, identifying similarities, productive disposition, overcoming barriers, micro simulations, choosing tools, and study of practice. This approach is innovative and appeals both to prominent authors and to our target audiences. This book may inspire researchers who engage in the study of design principles and characteristics of productive tasks for secondary mathematics teacher education.

Teacher education. Teacher's profession --- Didactics of mathematics --- didactiek --- wiskunde --- lerarenopleiding --- lesgeven --- Systèmes d'exploitation (ordinateurs) --- Mathematics—Study and teaching . --- Teaching. --- Mathematics Education. --- Teaching and Teacher Education. --- #ZR ////// ZAIREES RECHT ////// --- 681.3*D45 --- 681.3*D46 --- 681.3*D48 --- 681.3*H24 --- 681.3*H24 Systems: concurrency; distributed systems; query processing; transaction processing (Database management) --- Systems: concurrency; distributed systems; query processing; transaction processing (Database management) --- 681.3*D48 Performance: measurements; modeling and prediction; monitors; operational analysis; queuing theory; simulation; stochastic analysis (Operating systems)--See also {?681.3*C2}; {681.3*D28}; {681.3*I6} --- Performance: measurements; modeling and prediction; monitors; operational analysis; queuing theory; simulation; stochastic analysis (Operating systems)--See also {?681.3*C2}; {681.3*D28}; {681.3*I6} --- 681.3*D46 Security and protection: access controls; authentication; cryptographic controls; information flow controls; security kernels; verification (Operating systems) --- Security and protection: access controls; authentication; cryptographic controls; information flow controls; security kernels; verification (Operating systems) --- 681.3*D45 Reliability: backup procedures; checkpoint/restart; fault-tolerance; verification (Operating systems) --- Reliability: backup procedures; checkpoint/restart; fault-tolerance; verification (Operating systems) --- Didactics --- Instruction --- Pedagogy --- School teaching --- Schoolteaching --- Education --- Instructional systems --- Pedagogical content knowledge --- Training --- Mathematics teachers --- Mathematics --- Training of. --- Study and teaching (Secondary)

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This book explores how professionals can engage and inspire parents to support their young children’s mathematics learning. Bringing together international experts, researchers and scholars, it proposes a framework for engaging with and supporting parents, including those who are less aware of the crucial development of children’s mathematical skills in the early years. Focusing on mathematics learning from birth to 5 years, the book’s underlying assumption is that it is possible to offer guidance to professionals working with families with young children concerning how to engage and support families in the area of mathematics learning, including those families who seem alienated from education services. Specifically, the respective chapters present a framework for understanding children’s early mathematical development and the important role of families in this regard. They describe effective strategies for engaging families in their children’s mathematics learning, including those who are marginalised and experience multiple disadvantages, so that all families can best support their children’s mathematical learning and their development of positive attitudes towards learning. In closing, hurdles and opportunities within the systems surrounding family engagement are addressed.

Mathematics—Study and teaching . --- Child development. --- International education . --- Comparative education. --- Mathematics Education. --- Early Childhood Education. --- International and Comparative Education.