Here at this fundamental section we commute ourselves to describing the some attributes of new field in mathematics - Hierarchical Mathematics. This field itself will grow along the subjects of Heterogeneous, Scaled, and Hierarchical media, spaces and problems in almost any physical and other sciences (biology, social sciences, chemistry, etc.). As long as in any of those would be raised the serious need to construct the model, mathematical model not of semantics nature, for the scaled phenomena - this is the starting point to look for the right tool. The mathematical right tool.
We have to assert that our Hierarchical Mathematics system constructed on the basis of accepted knowledge foundation with regard of the each separately taken subspace. The knowledge pertained to the separate physical system of scales has been taken as it had been developed and existing in the conventional one-scale homogeneous physics. In this conceptual way we do not see so far a threat that might be formed by application of the Gödel's theorem - that any well developed axiomatic system - and we have for each of taken homogeneous physics scale the rich logical system, exist some statements that can not be found as being correct or incorrect.
Our mathematical system's correctness of some statements and results (can be said of any) relias on few cornerstones.
One of them is that we constantly verify our findings with the data system existing in each scale coordinate system that have been developed through the centuries in physics. Nevertheless, that is not the final "god's" given truth, we might agree on that.
Another system of knowledge coordinates we constantly checking our concepts with, is the transition of data from one scale (Upper, say) to the Lower and even to the second from the Upper scale - the Top-Down scaling.
Reflecting on what reasonings were expressed by B.Davies (2005) with regard of usefulness or success in finding a solution for the two-scale optimization problem that in an interesting way is tight to the Kepler's problem. The search is about optimization of the multibody, with various intraphase and interphase and scaled interface action fields, we might say that these kinds of problems are from the scaled HSP-VAT optimization field. And as soon as someone find enough courage and strength to tackle upon this task - he might succeed, but only within the field of scaled Hierarchical Optimization.
What is or might be uploaded onto this section, naturally, not of the comprehensive set of methods and procedures allowing someone to realize some of the techniques straightly and immediately. There should be a substantial road to study while updating to the new physics and mathematics. This is the guide rather for navigating in the large ocean of new concepts of multiscaled, hierarchical physics and mathematics.
The substantial difference in our texts from the few known today mostly of physical content monographs, including written by mathematicians, on the Upper scale mathematics is that, among other features, we are applying the theory and procedures to both (or more) scales physics and mathematics. This I am mentioning in few places of this website already. I will be referring to those publications in the appropriate areas.
It looks like nowadays - beginning of the 21st century - only lazy postdocs don't write on "multiscaling". Saying this we noticed that some years ago - there is happen the splash of almost pure mathematical studies regarding the "multiscaling."
That's good on the one hand, and bad on the other, because anybody with a degree who started to do this - writes his/her own "Theory". I have written on this in a few places on this website and in papers. Meanwhile, mathematicians are supposed to be more educated on the subject. It appears that is not. They don't know or right now ignore the definition what is the scale? We guess that don't know what it is. What is the physical, biological model pertinent to the scale? What is averaging? Local-nonlocal variables? Etc.
Reading those texts can be said that they continue to dwell in an Ivory Tower Mathematical (ITM) when pretending that there are nobody have been working in the field of "multiscaling" but some mathematicians.
They do not "know" that the field of Multiscaling, Heterogeneous Multiscaling, Local-Nonlocal phenomena description happen to be born in the 70-80s without pure mathematicians.
When in 2003 I have been sending to few of them - mathematicians, the message with the note that the first time scaled problems were solved strictly and straightly in 90s-2003 they did not confirm openly on the event.
Few places in this website speak on the misconception of What is the correct, good scaling procedures:
What is really the Heterogeneous Multiscale Theory and transport description-
and What are the differences with other "multiscaling" procedures including ones that sprang-up mostly after 2000 -
Well - here is the good piece of analysis of "multiscale" schemes designed throughout the physics and biology (almost without straight mathematicians) to mimic the multiscaling in biology -
1) Some mathematical workers are giving a misleading coin to the suggested mathematical procedure as, for example, - "Heterogeneous multiscale" method (). Well, all these methods are applied to materials, matter, most of them can be characterized as heterogeneous. Meanwhile, that mathematical procedure is not about heterogeneous materials?!
2) They do not differentiate the homogenization theory and the HSP-VAT and affiliated methods, theories and mathematical procedures!? That is true, and I've already written on this - they just copy one from another the affirmative statements without digging in. For what reason to dig other works as soon as works of those people to analyze are from another inappropriate to accept camp?
3) They do not refer and analyze the same type of strive for averaging and scaling that being suffered by their older colleagues mathematicians in the previous century, with an uphold in the 60-70s.
There is a great number of studies on averaging from that time. New wave of "multiscale" people doesn't bother to study and refer to those works. They think they are developing something new.
4) They do suggest new mathematical algorithms, often based on successes of other disciplines, and even do not recognize that what they are doing is the repetition after technical advancements got in some other "multiscale" physics, materials science and physical chemistry.
5) They do cross-referring to each other publications making it looking as the only house in town is the Ivory Tower Mathematical (ITM). At last, they do review of each other works and everything is thought as a good work in progress. That progress has been going on for 70 or so years and hasn't achieved proper solutions, but with the HSP-VAT only, by the way.
We would like to state here what is said many times in public at presentations and lectures and in this site in a lot of spots, that there is - No way to solve or analyze correctly the Heterogeneous Matter Physical problems if Homogeneous Tools Applied, because that says the scaling physics and mathematics. Mathematicians should comprehend that among of the first.