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Quaternions --- Àlgebra universal --- Cinemàtica --- Corbes --- Superfícies (Matemàtica) --- Anàlisi vectorial --- Nombres complexos --- Functions, Quaternion. --- Quaternion functions --- Functions of complex variables

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This book surveys the foundations of the theory of slice regular functions over the quaternions, introduced in 2006, and gives an overview of its generalizations and applications. As in the case of other interesting quaternionic function theories, the original motivations were the richness of the theory of holomorphic functions of one complex variable and the fact that quaternions form the only associative real division algebra with a finite dimension n>2. (Slice) regular functions quickly showed particularly appealing features and developed into a full-fledged theory, while finding applications to outstanding problems from other areas of mathematics. For instance, this class of functions includes polynomials and power series. The nature of the zero sets of regular functions is particularly interesting and strictly linked to an articulate algebraic structure, which allows several types of series expansion and the study of singularities. Integral representation formulas enrich the theory and are fundamental to the construction of a noncommutative functional calculus. Regular functions have a particularly nice differential topology and are useful tools for the construction and classification of quaternionic orthogonal complex structures, where they compensate for the scarcity of conformal maps in dimension four. This second, expanded edition additionally covers a new branch of the theory: the study of regular functions whose domains are not axially symmetric. The volume is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general. From the reviews of the 1st edition: "[The authors] document their own very recent theory of quaternionic regular functions, a development that parallels familiar complex function theory spectacularly well. This user-friendly primary source confirms that quaternionic calculus is not a dead end, and clearly answers a popular question regarding the analogy of complex function theory (complex analysis) with quarternionic variables, making it an excellent basis for a capstone course. Summing Up: Highly recommended. Upper-division undergraduates through professionals." (D. V. Feldman, Choice, Vol. 51 (1), September, 2013)".

Algebraic geometry --- Functional analysis --- Mathematical analysis --- Mathematics --- complexe veranderlijken --- reeksen (wiskunde) --- functies (wiskunde) --- wiskunde

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The theory of holomorphic dynamical systems is a subject of increasing interest in mathematics, both for its challenging problems and for its connections with other branches of pure and applied mathematics. This volume collects the Lectures held at the 2008 CIME session on "Holomorphic Dynamical Systems" held in Cetraro, Italy. This CIME Course focused on a number of important topics in the study of discrete and continuous dynamical systems, including both local and global aspects, providing a fascinating introduction to many key problems in current research. The contributions provide an ample description of the phenomena occurring in central themes of holomorphic dynamics such as automorphisms and meromorphic self-maps of projective spaces, of entire maps on complex spaces and holomorphic foliations in surfaces and higher dimensional manifolds, elaborating on the different techniques used and familiarizing readers with the latest findings on current research topics.

Holomorphic functions --- Differentiable dynamical systems --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematical Theory --- Functions, Holomorphic --- Mathematics. --- Dynamics. --- Ergodic theory. --- Functions of complex variables. --- Potential theory (Mathematics). --- Dynamical Systems and Ergodic Theory. --- Functions of a Complex Variable. --- Several Complex Variables and Analytic Spaces. --- Potential Theory. --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Complex variables --- Elliptic functions --- Functions of real variables --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Physics --- Statics --- Math --- Science --- Functions of several complex variables --- Differentiable dynamical systems. --- Differential equations, partial. --- Partial differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Cetraro <2008>

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The work aims at bringing together international leading specialists in the field of Quaternionic and Clifford Analysis, as well as young researchers interested in the subject, with the idea of presenting and discussing recent results, analyzing new trends and techniques in the area and, in general, of promoting scientific collaboration. Particular attention is paid to the presentation of different notions of regularity for functions of hypercomplex variables, and to the study of the main features of the theories that they originate.

Algebraic functions. --- Functions of complex variables. --- Mathematical analysis. --- Functions, Quaternion --- Functions of complex variables --- Clifford algebras --- Quaternions --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematical Theory --- Quaternions. --- Algebraic fields. --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Mathematics. --- Analysis (Mathematics). --- Mathematics, general. --- Analysis. --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra) --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Geometry, Algebraic --- Math --- Science --- 517.1 Mathematical analysis --- Mathematical analysis

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The theory of holomorphic dynamical systems is a subject of increasing interest in mathematics, both for its challenging problems and for its connections with other branches of pure and applied mathematics. This volume collects the Lectures held at the 2008 CIME session on "Holomorphic Dynamical Systems" held in Cetraro, Italy. This CIME Course focused on a number of important topics in the study of discrete and continuous dynamical systems, including both local and global aspects, providing a fascinating introduction to many key problems in current research. The contributions provide an ample description of the phenomena occurring in central themes of holomorphic dynamics such as automorphisms and meromorphic self-maps of projective spaces, of entire maps on complex spaces and holomorphic foliations in surfaces and higher dimensional manifolds, elaborating on the different techniques used and familiarizing readers with the latest findings on current research topics.