A rain drop is a sphere. It may not look like it, what with it's pointy head and rounded bottom, but it is. Fractional calculus can prove it.
Fractional calculus is concerned not only with how quickly and to what extent change develops but also in what order it advances. This is math, with memory. It can be used to recall the past, perhaps to see if the rain drop was once a sphere. It can also predict the future, maybe to determine if the rain drop will become a sphere.
A team of mathematicians from the University of Averio, in Portugal, combined the two operators - the past and future of the change process - into one theory. They published their work in IEEE/CAA Journal of Automatica Sinica (JAS).
"Many real world phenomena are better described by... derivatives," said Dr.Ricardo Alemida, an assistant professor at the University of Averio in Portugal and an author on the paper, referencing the mathematical term for the measuring the sensitivity of a variable to changing circumstances. "In fact, fractional derivatives have unique characteristics that may model certain dynamics more efficiently."
The field began with questions concerning such things as whether a rain drop contains the volume required to make a sphere. The answers resulted in the ability to measure a variable's past reaction to change, as well as predict future reactions, laying the groundwork for a number of technologies and applications.
Reading an article online, receiving a text, taking a picture, listening to the radio - none of these things would be possible without accounting for how each variable changed or will change.
Compare a letter to an email. It doesn't matter if a letter is torn or delivered to the wrong address. The letter inherently remains what it always was: pen on paper. An email is more complicated. The sender's program must be able to not only understand what the sender constructed, but it must also predict the environment to which it's sent. Despite programming advancements such as text only options or mobile reactivity, emails containing a broken image or emails that are difficult to read on a mobile device are common.
Dr.Almeida and his colleagues will continue to expand their work and test the stability of their combined theory by incorporating more and more variables. Work in this field may lead to future improvements for how we exchange information, among other applications.
"[We plan to] generalize the previous results to higher-order derivatives," said Dr.Almeida.
Fulltext of the paper is available:
IEEE/CAA Journal of Automatica Sinica (JAS) is a joint publication of the Institute of Electrical and Electronics Engineers, Inc (IEEE) and the Chinese Association of Automation. JAS publishes papers on original theoretical and experimental research and development in all areas of automation. The coverage of JAS includes but is not limited to: Automatic control/Artificial intelligence and intelligent control/Systems theory and engineering/Pattern recognition and intelligent systems/Automation engineering and applications/Information processing and information systems/Network based automation/Robotics/Computer-aided technologies for automation systems/Sensing and measurement/Navigation, guidance, and control.
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