Archimedes Maps and Optimal location of monitoring points by Bernard

Where to put the monitoring points in a system is a very common problem (environment, industry, health, and so on). We show that Archimedes maps, that is, measure preserving transformations, provide the right tool for this problem. Conversely, we insist that random exploration, often preferred because of simplicity and low cost, should definitely be avoided. Many of our contracts, both in industrial or environmental situations, deal with the following type of question : one wants to define "monitoring points" in a system (simple or complex) and one wants to know where to put them. Or, if the system has been already defined (for historical reasons), people want to know what is its degree of validity. Examples are :  Where to put 10 stations for monitoring water quality on a river ? This is a one-dimensional problem.  For France, where to put 300 stations for air quality surveillance ? This is a two-dimensional problem.  For the Paris area, where should the firemen put their basis stations (these are the places where the vehicles stay, before they are called for fires or assistance to victims) ? This is also a two-dimensional problem.