Generalized algebra within a nonextensive statistics

We have applied the concept of fractional derivatives/integrals in several specific electromagnetic problems, and have obtained promising results and ideas that demonstrate that these mathematical operators can be interesting and useful tools in electromagnetic theory. We give a brief review of the general principles, definitions, and several features of fractional derivatives/integrals, and then we review some of our ideas and findings in exploring potential applications of fractional calculus in some electromagnetic problems.

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On the role of fractional calculus in electromagnetic theory

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

The theory of scale relativity is a new approach to the problem of the origin of fundamental scales and of scaling laws in physics, that consists of generalizing Einstein's principle of relativity (up to now applied to motion laws) to scale transformations. Namely, we redefine space-time resolutions as characterizing the state of scale of the reference system and require that the equations of physics keep their form under resolution transformations (i.e. be scale covariant). We recall in the present review paper how the development of the theory is intrinsically linked to the concept of fractal space-time, and how it allows one to recover quantum mechanics as mechanics on such a non-differentiable space-time, in which the Schrodinger equation is demonstrated as a geodesies equation. We recall that the standard quantum behavior is obtained, however, as a manifestation of a “Galilean” version of the theory, while the application of the principle of relativity to linear scale laws leads to the construction of a theory of special scale relativity, in which there appears impassable, minimal and maximal scales, invariant under dilations. The theory is then applied to its preferential domains of applications, namely very small and very large length- and time-scales, i.e. high energy physics, cosmology and chaotic systems.

Scale relativity and fractal space-time: applications to quantum physics, cosmology and chaotic systems.

Redundant manipulators have some advantages when compared with classicalarms because they allow the trajectory optimization, both on the freespace and on the presence of obstacles, and the resolution ofsingularities For this type of arms the proposed kinematic controlalgorithms adopt generalized inverse matrices but, in general, thecorresponding trajectory planning schemes show important limitationsMotivated by these problems this paper studies the chaos revealed by thepseudoinverse-based trajectory planning algorithms, using the theory offractional calculus

Chaotic Phenomena and Fractional-Order Dynamics in the Trajectory Control of Redundant Manipulators

In the frame of a generalized functions analysis, it is shown that the light scattered by fractal objects has fractal properties. The difference between mass diffraction and surface diffraction is presented and illustrated through clear and easy experiments based on the fundamental difference between the concepts of dimension and codimension.

http://iopscience.iop.org/article/10.1088/0963-9659/1/1/005/pdf

Optical wave diffraction on fractal objects

A stochastic action principle for random dynamics is revisited. Numerical diffusion experiments are carried out to show that the diffusion path probability depends exponentially on the Lagrangian action A = ∫ a b L d t . This result is then used to derive the Shannon measure for path uncertainty. It is shown that the maximum entropy principle and the least action principle of classical mechanics can be unified into δ A ¯ = 0 where the average is calculated over all possible paths of the stochastic motion between two configuration points a and b. It is argued that this action principle and the maximum entropy principle are a consequence of the mechanical equilibrium condition extended to the case of stochastic dynamics.

Reformulation of a stochastic action principle for irregular dynamics

We look at chaotic systems evolving in fractal phase space. The entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement. Due to the incompleteness, at any scale, of the information calculation in fractal support, the incomplete normalization ∑ i p i q =1 is applied throughout the paper. It is shown that the information growth is nonadditive and is proportional to the trace-form ∑ i p i −∑ i p i q so that it can be connected to several nonadditive entropies. This information growth can be extremized to give, for nonequilibrium systems, power law distributions of evolving stationary state which may be called “maximum entropic evolution”.

Measuring information growth in fractal phase space

The nonextensive statistics based on the q-entropy S q = - ∑ i = 1 v ( p i - p i q ) 1 - q has been so far applied to systems in which the q value is uniformly distributed. For the systems containing different q’s, the applicability of the theory is still a matter of investigation. The difficulty is that the class of systems to which the theory can be applied is actually limited by the usual nonadditivity rule of entropy which is no more valid when the systems contain non uniform distribution of q values. In this paper, within the framework of the so called incomplete information theory, we propose a more general nonadditivity rule of entropy prescribed by the zeroth law of thermodynamics. This new nonadditivity generalizes in a simple way the usual one and can be proved to lead uniquely to the q-entropy.

Applying incomplete statistics to nonextensive systems with different q indices

The nonextensive statistics based on the $q$-entropy $S_q=-\frac{\sum_{i=1}^v(p_i-p_i^q)}{1-q}$ has been so far applied to systems in which the $q$ value is uniformly distributed. For the systems containing different $q$'s, the applicability of the theory is still a matter of investigation. The difficulty is that the class of systems to which the theory can be applied is actually limited by the usual nonadditivity rule of entropy which is no more valid when the systems contain non uniform distribution of $q$ values. In this paper, within the framework of the so called incomplete information theory, we propose a more general nonadditivity rule of entropy prescribed by the zeroth law of thermodynamics. This new nonadditivity generalizes in a simple way the usual one and can be proved to lead uniquely to the $q$-entropy.

A. Le Méhauté

Papers

Optical wave diffraction on fractal objects

Reformulation of a stochastic action principle for irregular dynamics

Measuring information growth in fractal phase space

Applying incomplete statistics to nonextensive systems with different q indices

Applying incomplete statistics to nonextensive systems with different $q$ indices