Everywhere differentiable functions without monotonicity intervals and transcendental numbers. (English. Russian original) Zbl 1415.26002
Dokl. Math. 97, No. 3, 219-222 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 480, No. 2, 137-140 (2018).
Summary: The class of everywhere differentiable functions without monotonicity intervals is considered in terms of number theory. A number-theoretic representation of the set of points of the unit interval is constructed using the classification of transcendental numbers proposed by K. Mahler, and a theorem on sufficient conditions for differentiable functions to belong to this class is stated. Results concerning the behavior of derivatives of functions from this class are presented. A mixed problem for the heat equation modeling heat transfer in a distributed system is considered. It is shown that the control function for this system can be everywhere differentiable but having no monotonicity intervals.
MSC:
| 26A30 | Singular functions, Cantor functions, functions with other special properties |
| 26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 49N60 | Regularity of solutions in optimal control |
References:
| [1] | I. P. Natanson, Theory of Functions of a Real Variable (Ungar, New York, 1955; Nauka, Moscow, 1974). · Zbl 0064.29102 |
| [2] | Garg, K. M., No article title, Rev. Math. Pures Appl., 7, 663-671, (1962) · Zbl 0138.03801 |
| [3] | A. M. Bruckner, Differentiation of Real Function (Am. Math. Soc., New York, 1994). · Zbl 0796.26004 · doi:10.1090/crmm/005 |
| [4] | Karasinska, A.; Wagner-Bojakowska, E., No article title, Math. Acta Hungary, 120, 235-248, (2008) · Zbl 1174.26003 · doi:10.1007/s10474-008-7093-y |
| [5] | Sprindzhuk, V. G., No article title, Russ. Math. Surv., 35, 1-80, (1980) · Zbl 0463.10020 · doi:10.1070/RM1980v035n04ABEH001861 |
| [6] | Erdos, P., No article title, Michigan Math. J., 9, 59-60, (1962) · Zbl 0114.26306 · doi:10.1307/mmj/1028998621 |
| [7] | J. C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces (Springer-Verlag, New York, 1971). · Zbl 0217.09201 · doi:10.1007/978-1-4615-9964-7 |
| [8] | W. J. Leveque, J. London Math. Soc. s1-28 (2), 220-229 (1953). · Zbl 0053.36203 |
| [9] | Agadzhanov, A. N.; Butkovskii, A. G., No article title, Dokl. Math., 82, 732-735, (2010) · Zbl 1210.35275 · doi:10.1134/S1064562410050145 |
| [10] | Agadzhanov, A. N., No article title, Dokl. Math., 87, 181-184, (2013) · Zbl 1275.26010 · doi:10.1134/S1064562413020178 |
| [11] | Agadzhanov, A. N., No article title, Dokl. Math., 95, 109-112, (2017) · Zbl 1380.28008 · doi:10.1134/S1064562417020016 |
| [12] | Sakawa, Y., No article title, IEEE Trans. Autom. Control, 9, 420-426, (1964) · doi:10.1109/TAC.1964.1105753 |
| [13] | Katznelson, Y.; Stromberg, K., No article title, Am. Math. Month., 81, 349-354, (1974) · Zbl 0279.26007 · doi:10.1080/00029890.1974.11993558 |
| [14] | Blazek, J.; Borák, E.; Malý, J., No article title, Casopis Pest. Mat., 103, 53-61, (1978) · Zbl 0376.26006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
