If A and B are really existing objects, then the real distance from A to B, which is r(A,B) , can be measured using such sample criteria as:
It is clear that, in general, r(A,B) is not equal to r(B,A). Distances with similar properties are called anisotropic. In the real world we usually deal with anisotropic distances instead of the real ones, and the awareness of existence of such distances in our everyday practice could significantly change our lives, making us more economically motivated. In particular, it could affect the world energy consumption. Therefore, our proposal can be subtitled as Project for Global Energy Saving.
- time spent on reaching point B if walking;
- amount of gasoline needed for an actual car model if driving;
- total cost of the gasoline, etc.
Problems related to efficient energy use become now a priority because of the ongoing accidents at the nuclear power plants. A nuclear tragedy, when occurred in a geographically small country, can put an end to the history of this country. Cessation of operations at the nuclear power stations is a matter of time.
How could we spend less energy? First of all, by revision of our energy consumption appetites. Second, by ability to calculate instantly in any geographic location with the help of high-speed super-computers. Such possibility puts mathematics in the leading position in the development of new energy saving algorithms.
For instance, one possible approach to implementation of this energy saving project is to create additional departments within existing GPS and GLONASS systems.
As an alternative approach, we suggest the following sequence of actions:
The number of possible applications to the proposed theory is very big. For example, there will be an opportunity to make regular public transportation schedules interactive, where the ticket price could vary depending on quantity of passengers in every particular moment. This practical task will also reveal a large cluster of subsequent problems, related to coordination of routes and schedules.
- acquisition of super-computers;
- construction of systematically updated database of human settlements in the region and the
roads connecting them (taking into account anisotropy of the bonds);
- building of an algorithm calculating the optimal “distance” from point A to point B;
- building of the Internet-on-demand platform for the real world distances and high speed
responses, how it is done for example at
maps.google.com to show isotropic distances.
Our main goal is to develop a mathematical theory that provides the above mentioned algorithm. In particular, within the next 2-3 years, we are planning to construct basis of mathematical analysis in anisotropic spaces. First steps in this direction were taken in my book Weight Sobolev's Classes, Anisotropic Metrics, and Degenerate Quasiconformal Maps (in Russian). There is program for a longer period of time (for now, 173 pp. in Russian).
The simplest examples of such spaces are the Minkowski space-time and the Finsler space. Now let's move to the task of development of mathematical instruments needed (in isotropic case) for the problems of building grids on surfaces and surface triangulation. For more expanded approach we need systematic contacts with elaborationers of concrete navigational methods. At the same time, collective work on the project can be started immediatelly (beginning with the simplest systems - metro, railway, etc.)
Already existing (and very advanced!) research findings in the fields of informational geodesy and global navigation satellite systems will be certainly in demand in the development of general approaches to navigation in anisotropic spaces.
At the same time, let's mention that even most ordinary differential equations with partial derivatives can describe non-isotropic processes in the fields of Euclidean space if their coefficients depend on their derivatives. Thus, methods for studying non-isotropic processes are far from being exhausted now and require the use the full range of existing mathematical theories. This is especially important because even reformulation of basic physical postulates in newly-created mathematical language will take some time before the methods of anisotropic analysis could be applied.
Here (in Russian) we present a model program of the studies we plan for the coming years. At the end of each chapter, we formulate a group of most important problems.
Vladimir M. Miklyukov
Doct. of Sci., Prof., Honored Scientist of Russian Federation
Independent Scientific Laboratory Uchimsya, LLC
PO Box 1272, Yonkers, NY, 10702, USA
June of 2011.
О некоторых математических проблемах, возникающих при описании микро- и нанопотоков.
|In September of 2011, Superslow Processes laboratory launches a new weekly scientific webinars program. Information about our previous workshops is available in the Announcements section. For participation in our webinars, please check the updates on our website, Facebook, or VKontakte.|
Распространено мнение, что математика - изобретение халдейских магов, практиковавших на территории Южной Мессопотамии в первой половине первого тысячелетия до Р. Х. В определенной степени с этим трудно не согласиться. Действительно, и сегодня математика обладает многими характерными чертами, свойственными чародейству и колдовству, с чем безусловно согласится каждый, кто хотя бы раз соприкасался с ней в процессе работы. Вместе с тем, переходя к исследованиям микро- и наномира и расширяя сферу применений математики, невольно задаешься вопросом - а как выглядела бы сегодня математика, если бы халдеи владели микроскопом? Читать дальше.