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Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers

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  • Sergeyev, Yaroslav D.

Abstract

The paper considers a new type of objects – blinking fractals – that are not covered by traditional theories studying dynamics of self-similarity processes. It is shown that the new approach allows one to give various quantitative characteristics of the newly introduced and traditional fractals using infinite and infinitesimal numbers proposed recently. In this connection, the problem of the mathematical modelling of continuity is discussed in detail. A strong advantage of the introduced computational paradigm consists of its well-marked numerical character and its own instrument – Infinity Computer – able to execute operations with infinite and infinitesimal numbers.

Suggested Citation

  • Sergeyev, Yaroslav D., 2007. "Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 50-75.
    • Handle: RePEc:eee:chsofr:v:33:y:2007:i:1:p:50-75
    DOI: 10.1016/j.chaos.2006.11.001
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    References listed on IDEAS

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    1. Iovane, G., 2006. "Cantorian spacetime and Hilbert space: Part I—Foundations," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 857-878.
    2. Iovane, G., 2006. "Cantorian space–time and Hilbert space: Part II—Relevant consequences," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 1-22.
    3. El Naschie, M.S., 2005. "A guide to the mathematics of E-infinity Cantorian spacetime theory," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 955-964.
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    Cited by:

    1. Tohmé, Fernando & Caterina, Gianluca & Gangle, Rocco, 2020. "Computing Truth Values in the Topos of Infinite Peirce’s α-Existential Graphs," Applied Mathematics and Computation, Elsevier, vol. 385(C).
    2. Lolli, Gabriele, 2015. "Metamathematical investigations on the theory of Grossone," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 3-14.
    3. Sergeyev, Yaroslav D., 2009. "Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3042-3046.
    4. Caldarola, Fabio & Maiolo, Mario, 2021. "A mathematical investigation on the invariance problem of some hydraulic indices," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    5. Kauffman, Louis H., 2015. "Infinite computations and the generic finite," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 25-35.
    6. Amodio, P. & Iavernaro, F. & Mazzia, F. & Mukhametzhanov, M.S. & Sergeyev, Ya.D., 2017. "A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 24-39.
    7. Cococcioni, Marco & Pappalardo, Massimo & Sergeyev, Yaroslav D., 2018. "Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 298-311.
    8. Margenstern, Maurice, 2016. "Infinigons of the hyperbolic plane and grossone," Applied Mathematics and Computation, Elsevier, vol. 278(C), pages 45-53.
    9. Renato De Leone & Giovanni Fasano & Yaroslav D. Sergeyev, 2018. "Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming," Computational Optimization and Applications, Springer, vol. 71(1), pages 73-93, September.
    10. De Leone, Renato, 2018. "Nonlinear programming and Grossone: Quadratic Programing and the role of Constraint Qualifications," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 290-297.
    11. Caldarola, Fabio, 2018. "The Sierpinski curve viewed by numerical computations with infinities and infinitesimals," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 321-328.

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