Valerii Salov
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Notation for Iteration of Functions, Iteral
2012, arXiv (Cornell University)
https://doi.org/10.48550/ARXIV.1207.0152
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Abstract
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations of the notation include definitions of constant points, periodic points, a filled-in Julia set, the Mandelbrot set, iterations of a logistic map, the double-approximating procedure for solving the Lorenz equations, a description of a financial time series, and reordering nonnegative integers useful for the investigation of the Collatz's (3x+1)/2 convergence problem. The terms iteral and iteral of function are suggested to name the new denomination.
FAQs
AI
What implications does new iteral notation have for iterative function analysis?
The proposed iteral notation facilitates clarity in the expression of iterative sequences and improves ease of application in complex environments, such as high-frequency trading analysis. Specifically, it could enhance the understanding of iterative processes in dynamical systems and chaos theory, where conventional notations prove ambiguous.
How does the proposed iteral notation improve on existing function iteration notations?
The iteral notation explicitly indicates the initial value, the number of iterations performed, and can include nested iterals, enhancing clarity. This distinctiveness allows the notation to accommodate complex algebraic expressions without losing unambiguous structure, unlike earlier notations which often conflated functions and iterations.
When was the concept of iteration in rational functions first introduced?
The study of iteration of rational functions dates back to the mid-19th century, with foundational ideas potentially attributed to mathematicians like Niels Henrik Abel. This historical context frames the evolution of iterative notation leading up to modern applications.
What findings support the relationship between iterations and chaos theory?
Research indicates that iterations are integral to chaos theory, as demonstrated by Edward Lorenz's work with the Lorenz equations, revealing sensitivity to initial conditions. Such findings underscore the chaotic behavior observed in non-linear dynamical systems and highlight the relevance of iterative notations in modeling these systems.
How did various mathematicians influence notation for iteration over time?
Influential figures like Leonard Euler and Carl Friedrich Gauss contributed foundational symbols for summation and factorials, respectively, during the 18th and early 19th centuries. These contributions shaped the evolution of mathematical notation, culminating in current systems which lack consistent representation for functions' iterative processes.
References (35)
- Arnold, V., Underestimated Poincaré, Uspehi Matematicheskih Nauk, Volume 61, N. 1, January -February, 2006, pp. 3 -24 (Russian); Also known as Forgotten and Neglected Theories of Poincaré, Russ. Math. Surv., Vol. 61, 2006, pp. 1 -18.
- Boyer, C., A History of Mathematics, 2nd ed., Revised by Merzbach U., John Wiley, New York, 1991.
- Cajori, F., A History of Mathematical Notations, Vol. II Notations Mainly in Higher Mathematics, The Open Court Publishing Company, Chicago, 1929.
- The Clay Mathematics Institute, Riemann's Manuscript, available at http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/.
- Devaney, R., Chaos, Fractals, and Dynamics, Computer Experiments in Mathematics, Addison-Wesley Publishing Company, New York, 1990.
- Engle, R., The Econometrics of Ultra-High-Frequency Data, Econometrica, Vol. 68, N. 1, 2000, pp. 1 -22.
- Feigenbaum, M., Quantitative Universality for a Class of Nonlinear Trans- formations, Journal of Statistical Physics, Vol. 19, N. 1, 1978, pp. 25 -52.
- Goldman J., The Queen of Mathematics, A Historically Motivated Guide to Number Theory, A K Peters Wellesley, Massachusetts, 1998.
- Knuth, D., The Art of Computer Programming, Vol. 2 Seminumerical al- gorithms, 3rd ed., Addison Wesley, New York, 1998.
- Korn, G., Korn T., Mathematical Handbook for Sceintists and Engineers. Definitions, Theorems, and Formulas for Reference and Review, 2nd ed., McGraw-Hill Book Company, New York, 1968.
- Kuczma, M., Choczewski, B., Ger, R., Iterative Functional Equations, Cambridge University Press, Cambridge, 1990.
- Kumaraswamy, P., A Generalized Probability Density Function for Double- Bounded Random Processes, Journal of Hydrology, Vol. 46, 1980, pp. 79 - 88.
- Li, T., Yorke, J., Period Three Implies Chaos, The American Mathematical Monthly, Vol. 82, No. 10, December, 1975, pp. 985 -992.
- Lagarias, J., The 3x + 1 Problem and Its Generalizations, The American Mathematical Monthly, Vol. 92, N. 1, 1985, pp. 3-23.
- Lorenz, E., Deterministic Nonperiodic Flow, Journal of the Atmospheric Sciences, Vol. 20, 1963, pp. 130 -141.
- Mandelbrot, B., Fractals and Chaos. The Mandelbrot Set and Beyond. Springer, New York, 2004.
- Misiurewicz, M., Remarks on Sharkovsky's Theorem, The American Math- ematical Monthly, Vol. 104, N. 9, 1997, pp. 846 -847.
- Rogers, L.C.G., Williams, D., Diffusions, Markov Processes, and Martin- gales, Volume 2: Itô Calculus, 2nd ed., Cambridge University Press, Cam- bridge, 2000.
- Rumbaugh, J., Jacobson, I., Booch, G., The Unified Modeling Language Reference Manual, Addison-Wesley as imprint of Addison Wesley Longman, Inc., Massachusetts, 1999.
- Salov, V., Market Profile and the Distribution of Price, Futures Magazine, Vol. XL, N. 6, 2011, pp. 34 -36.
- Saltzman, B., Finite amplitude free convection as an initial value problem, Journal of the Atmospheric Sciences, Vol. 19, 1962, pp. 329 -341.
- Sharkovsky, O., Co-existence of the cycles of a continuous mapping of the line into itself, Ukrain. Mat. Zh., Vol. 16, N. 1, 1964, pp. 61 -71 (Russian).
- Sharkovsky, A., Partly ordered system of attracting sets, Dokl. Akad. Nauk of USSR, Mathematics, Vol. 170, N. 6, 1966, pp. 1276 -1278 (Russian).
- Shiryaev, A., Fundamentals of Stochastic Financial Mathematics, Volume I, Facts, Models, FAZIS, Moscow, 1998 (Russian). See also in English: Shiry- aev, A., Essentials of Stochastic Finance: Facts, Models, Theory. Translated by Kruzhilin N. 1st ed., World Scientific Publishing Co, Pte. Ltd., New Jer- sey, 1999.
- Sinai, Y., Chaos Theory Yesterday, Today and Tomorrow, Journal of Stat- istical Physics, Vol. 138, N. 1, 2010, pp. 2 -7.
- Sierpinski, W., What do we known and do not know about the prime numbers, Translated from Polish by Melnikov, I., State Publisher of Phys- Math Literature, Moscow, Leningrad, 1963 (Russian).
- Stroustrup, B., The C++ Programming Language, Special ed., Addison- Wesley, New York, 2000.
- Targoński, Gy., Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981.
- Tung, K., Topics in Mathematical Modeling, Princeton University Press, Princeton, 2007.
- Tversky, A., Kahneman, D., Advances in Prospect Theory: Cumulative Representation of Uncertainty, Journal of Risk and Uncertainty, Vol. 5, 1992, pp. 297 -323.
- Ullian, J., Splinters of Recursive Functions, The Journal of Symbolic Logic, Vol. 25, N. 1, March, 1960, pp. 33 -38.
- Weibull, W., A statistical Distribution Function of Wide Applicability, ASME Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers, Vol. 18, September 1951, pp. 293 -297.
- The History of Mathematics from the Ancient Times to the Beginning of the XIX Century, the Academy of Sciences of the USSR, Editor Yushkevich, A., Vol 1, From the Ancient Times to the Beginning of the New Time, 1970;
- Vol. 2, Mathematics of the XVII Century, 1970; Vol. 3, Mathematics of the XVIII Century, Nauka, Moscow, 1972 (Russian).
- Valerii Salov received his M.S. from the Moscow State University, Department of Chemistry in 1982 and his Ph.D. from the Academy of Sciences of the USSR, Vernadski Institute of Geochemistry and Analytical Chemistry in 1987. He is the author of the articles on analytical, computational, and physical chemistry, the book Modeling Maximum Trading Profits with C++, John Wiley and Sons, Inc., Hoboken, New Jersey, 2007, and papers in Futures Magazine. v7f5a7@comcast.net