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Lie algebras --- Nilpotent Lie groups

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"We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen's parameter. We illustrate our results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces"--

Heisenberg uncertainty principle --- Heisenberg, Principe d'incertitude d' --- Hausdorff measures --- Hausdorff, Mesures de --- Nilpotent Lie groups. --- Conformal mapping. --- Markov processes. --- Hausdorff measures. --- Thermodynamics --- Mathematical models.

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Stochastic processes --- Brownian motion processes --- Brownse beweging [Proces van de ] --- Limiettheorema's (Waarschijnlijkheidstheorie) --- Limit theorems (Probability theory) --- Mouvement brownien [Processus du ] --- Probability measures --- Théorèmes limites (Théorie des probabilités) --- Nilpotent Lie groups. --- Probability measures. --- Brownian motion processes. --- Wiener processes --- Brownian movements --- Fluctuations (Physics) --- Markov processes --- Probabilities --- Measures, Normalized --- Measures, Probability --- Normalized measures --- Distribution (Probability theory) --- Lie groups, Nilpotent --- Lie groups --- Nilpotent groups --- Nilpotent Lie groups

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Topology --- Fixed point theory. --- Mappings (Mathematics) --- 515.126.4 --- 515.142.22 --- 512.813 --- Fixed point theory --- Homotopy theory --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Transformations (Mathematics) --- Deformations, Continuous --- Fixed point theorems (Topology) --- Nonlinear operators --- Coincidence theory (Mathematics) --- Fixed points and coincidences --- Study of topological spaces and continuous mappings by homological methods. Homological dimension theory. Homological theory of fixed points and coincidence points --- Special classes of Lie groups. Compact, semisimple, solvable, nilpotent Lie groups --- Homotopy theory. --- Mappings (Mathematics). --- 512.813 Special classes of Lie groups. Compact, semisimple, solvable, nilpotent Lie groups --- 515.142.22 Study of topological spaces and continuous mappings by homological methods. Homological dimension theory. Homological theory of fixed points and coincidence points --- 515.126.4 Fixed points and coincidences

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This self-contained text provides an introduction to modern harmonic analysis in the context in which it is actually applied, in particular, through complex function theory and partial differential equations. It takes the novice mathematical reader from the rudiments of harmonic analysis (Fourier series) to the Fourier transform, pseudodifferential operators, and finally to Heisenberg analysis. Within the textbook, the new ideas on the Heisenberg group are applied to the study of estimates for both the Szegö and Poisson–Szegö integrals on the unit ball in complex space. Thus the main theme of the book is also tied into complex analysis of several variables. With a rigorous but well-paced exposition, this text provides all the necessary background in singular and fractional integrals, as well as Hardy spaces and the function theory of several complex variables, needed to understand Heisenberg analysis. Explorations in Harmonic Analysis is ideal for graduate students in mathematics, physics, and engineering. Prerequisites include a fundamental background in real and complex analysis and some exposure to functional analysis.

Electronic books. -- local. --- Harmonic analysis. --- Nilpotent Lie groups. --- Harmonic analysis --- Operations Research --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Lie groups, Nilpotent --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Mathematics. --- Group theory. --- Approximation theory. --- Fourier analysis. --- Functions of complex variables. --- Mathematical models. --- Abstract Harmonic Analysis. --- Fourier Analysis. --- Mathematical Modeling and Industrial Mathematics. --- Approximations and Expansions. --- Several Complex Variables and Analytic Spaces. --- Group Theory and Generalizations. --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Lie groups --- Nilpotent groups --- Differential equations, partial. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Partial differential equations --- Math --- Science --- Analysis, Fourier --- Complex variables --- Elliptic functions --- Functions of real variables --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Models, Mathematical --- Simulation methods