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Transmutation Theory and Applications

Differential operators. --- Transmutation operators. --- Operators, Transmutation --- Operator theory --- Operators, Differential --- Differential equations

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Chapter 1 presents theorems on differentiable functions often used in differential topology, such as the implicit function theorem, Sard's theorem and Whitney's approximation theorem.The next chapter is an introduction to real and complex manifolds. It contains an exposition of the theorem of Frobenius, the lemmata of Poincaré and Grothendieck with applications of Grothendieck's lemma to complex analysis, the imbedding theorem of Whitney and Thom's transversality theorem. Chapter 3 includes characterizations of linear differentiable operators, due to Peetre and Hormander. The ineq

Differentiable manifolds. --- Complex manifolds. --- Differential operators. --- Operators, Differential --- Differential equations --- Operator theory --- Analytic spaces --- Manifolds (Mathematics) --- Differential manifolds

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This book examines the conditions for the semi-boundedness of partial differential operators, which are interpreted in different ways. For example, today we know a great deal about L2-semibounded differential and pseudodifferential operators, although their complete characterization in analytic terms still poses difficulties, even for fairly simple operators. In contrast, until recently almost nothing was known about analytic characterizations of semi-boundedness for differential operators in other Hilbert function spaces and in Banach function spaces. This book works to address that gap. As such, various types of semi-boundedness are considered and a number of relevant conditions which are either necessary and sufficient or best possible in a certain sense are presented. The majority of the results reported on are the authors’ own contributions.

Differential operators. --- Operators, Differential --- Differential equations --- Operator theory --- Operator theory. --- Differential equations, partial. --- Operator Theory. --- Partial Differential Equations. --- Partial differential equations --- Functional analysis --- Partial differential equations.

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Ordered algebraic structures --- Differential operators. --- Opérateurs différentiels --- Rings (Algebra) --- Anneaux (algèbre) --- Invariants. --- Invariants --- Differential operators --- Algebraic rings --- Ring theory --- Algebraic fields --- Operators, Differential --- Differential equations --- Operator theory --- Opérateurs différentiels.

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With applications in quantum field theory, general relativity and elementary particle physics, this three-volume work studies the invariance of differential operators under Lie algebras, quantum groups and superalgebras. This second volume covers quantum groups in their two main manifestations: quantum algebras and matrix quantum groups. The exposition covers both the general aspects of these and a great variety of concrete explicitly presented examples. The invariant q-difference operators are introduced mainly using representations of quantum algebras on their dual matrix quantum groups as carrier spaces. This is the first book that covers the title matter applied to quantum groups. ContentsQuantum Groups and Quantum AlgebrasHighest-Weight Modules over Quantum AlgebrasPositive-Energy Representations of Noncompact Quantum AlgebrasDuality for Quantum GroupsInvariant q-Difference OperatorsInvariant q-Difference Operators Related to GLq(n)q-Maxwell Equations Hierarchies

Quantum groups. --- Differential invariants. --- Differential operators. --- Operators, Differential --- Differential equations --- Operator theory --- Invariants, Differential --- Continuous groups --- Enveloping algebras, Quantized --- Function algebras, Quantized --- Groups, Quantum --- Quantized enveloping algebras --- Quantized function algebras --- Quantum algebras --- Group theory --- Mathematical physics --- Quantum field theory