tailieunhanh - Dual approach to averaged values of functions

Averaged values play major roles in the study of dynamic processes. The use of those values allows transforming varying processes to some constant characteristics that are much easier to be investigated. In order to extend the use of averaged values one may apply the dual approach which suggests the consideration of two different aspects of a problem in question. This short communication proposes new global and local averaged values functions based on the dual conception. | Vietnam Journal of Mechanics, VAST, Vol. 34, No. 3 (2012), pp. 211 – 214 Short Communication DUAL APPROACH TO AVERAGED VALUES OF FUNCTIONS Nguyen Dong Anh Institute of Mechanics, VAST Abstract. Averaged values play major roles in the study of dynamic processes. The use of those values allows transforming varying processes to some constant characteristics that are much easier to be investigated. In order to extend the use of averaged values one may apply the dual approach which suggests the consideration of two different aspects of a problem in question. This short communication proposes new global and local averaged values functions based on the dual conception. Key words: Dual, averaged, global - local. 1. INTRODUCTION Averaged values play major roles in the study of dynamic processes. The use of those values allows transforming varying processes to some constant characteristics that are much easier to be investigated. From the mathematical point of view the averaging approach brings the complicate differential problems more simple algebraic ones. In order to extend the use of averaged values one may apply the dual approach recently proposed and developed in [1-2]. One of significant advantages of the dual conception is its consideration of two different aspects of a problem in question allows the investigation to be more appropriate. The main idea of this short communication is to propose new global and local averaged values functions based on the dual conception. 2. DUAL CONCEPTION TO AVERAGED VALUES Let first reconsider briefly the basic idea of averaged values. In mathematics an average is a measure of the "middle" or "typical" value of a data set. It is thus a measure of central tendency. The most common statistic is the arithmetic mean. The concept of average of a data set can be extended to functions. The global average value (GAV) of an integrable deterministic function x(t) on a domain D : (0, d) is a constant value .