I have noticed bernstein, and bezier polynomials/splices; although I am not sure if they can be weighted very easily. THere are gleason polynomials, athough that would presumably require some way of re-describing the domain [0,1] in terms of its binary expansions, or otherwise will have to multidimensional

Dear Peter,

I have since resolved this question but yes I do need to correct this text.

Yes, I mean the global max is greater or equal to the global min, for any GIVEN function. It obviously sounds tautologous. I just mean to say it allows for the case where the same function has multiple points in the domain under which it reach its global maximum (preferably there is only one such point; but this is not necessary. Although its preferable.

That statement is also there to allow for the cases where the  functions FI(x) reduce to a constant (a flat line) (f(x)=constant=max=min), ie when the two (global min, global max) are equal.

And when this occurs, all of the functions, do so as well. That is all of thefunctions educe to a constant function forall(x()F^i(x)=c^i,although of course it may not the same constant, between the functions hence the i superscript;

That is, \forall_IF^i(x) =global max=global min. So for example for all (x)F1(x)=0.6  f2(x)=0.3, f3(x)=0.1 This occurs when, and only when the sum of the global minima=1; otherwise it does not, and the sum of the global minima= some value\in (0,1).

And yes, I do mean, that at SOME point at which a function attains its/a global maximum, ALL of the other functions attains their global minimum. At very least, it must be SOME, but preferably all. I have since resolved  it for ALL, however; And likewise for any number of such functions I believe, and for all possible global minima and global maxima values consistent with the below

(1)global max fi=1-\sum_{j\neqi}[global min (fj)].

(2){ \sum_i [global min] ]\leq=1;

(3) the flat line notion when  \sum_i global min=1, or when maximum of any function equals its minima

(4)\forallx\in[domain]\sum_i(fi(x)=1;

(5) and the possibility either way, that when the maxima of a function is set to one, so its minima, and when its minima is set to zero, likewise is its maxima, and conversely; although it also allow for this possibility not obtaining as well,

(6) All functions hit their global maxima, iff all the other distinct function hit their global minima, and conversely, and when any given function hits its global minima, all of the other functions bar one at their global minima, bar one such function, and it will be at its global maxima (as opposed to the other functions being at some midpoint)

(7) for all consistent max, min assignments for all fi, compatible with the above; so it allows for any given function (taken in isolation), to have any sub-interval, [0,1] as its range (taken in isolation) and thus any possible probability value for any given any given x value of the domain (taken in isolation from the other functions.

8. The functions are surjective and uniformly continuous. All values in the range [minfi, maxfi} occur for at least one value in the domain x.

This is given, one can set the sum of the minimums to one, if one wishes to do so, whilst also allowing for the possibility that one need not do so.

Thus, its always possible to find an assignment of  sets of, minfi, s.t that for \forall fi, and  for all (specific')x values, there is always an assignment of [minf1, minf2.......], which will guarantee that \fi(x)  can take any value in [0,1], at that x/domain value, and for that fi function s,t, for all fi, and for all x (taken in isolation, which is why I said specific) , fi(x) can take any value in [0,1]if one wishes to do, without having to change the function

Of course the global maxima and global minima of the distinct functions, f1, and  f2,.. need not be equal; nonetheless,  it should allow for this to be the case, or not to be case, which it appears to.

Nonetheless if the global min are distinct, then so will be the global max; ie in this framework the global-max-global min (in any given case) for any set of such functions, and any specific set of minfi assignments,  will be a constant. These could be: [Gminf1, Gmaxf1] =[0,4,0.6] for f1 example and Gminf2, Gmaxf2=[0,5,0.7] for f2... although they could just as well be f1:[0,1]->[0.3, 0.7], f2:[0,1]->[0.2,0.6] for example

Nor should I rule out that all the local  maxima of any given function,  be equal or not-equal. And likewise between the functions with regards to their global maxima being equal or non-equal; and conversely with relata to the global minima. Of course this is possible, but need not be the case (it should allow, and it does, for either situation). THis should not have to hold, of necessity, either way; and thus like-wise it need not be that all of the maxima for that same function are equal to THAT function's global maxima .

It should allow for either case. Preferably there is only one point at which any given function reach its global maximum, although have not done this much, buts it not necessary.

As currently I have that some functions do indeed reach their global maxima at multiple points; nonetheless it satisfies the constraint, that

(1)  when any function is at its global maxima,all the other functions reach their global minima,  and conversely

(2) When any function, is at its global minima,  its global minima, all bar one of the functions are at their global minima, and the other remaining function is at its global maxima). Although it would be interesting to allow this constraint to be relaxed.To allow for midpoints, so long as they still sum to one.

When one uses the trigonometric functions (some function sin, cos ^2 etc)  one generally get multiple global maxima points, and the functions appear a little non-symmetric, and  as n grows larger. Some have more global max points then other functions, and some are more peaked the others This can be somewhat corrected.

It would be interesting to see if other families of functions such as the exponentials, logarithmics, can even this out or hyperbolics (presumably these would be akin to the trigonometrics). BUt again its not necessary, the main desiderata are currently satisfied

With regards to the function; I have had to use the trigonometric type; They are all of a very similar ilk. There are uncountably many such functions of that class, but they are all too similar for more liking, to count as distinct dispositions of sets of functions. But at least they do the job. The polynomials I mentioned were the bernstein polynomials and the weighted versions theirof; bezair curves. Maybe some of these will work for n=3, and n= some small value, but they are inherently unstable; and cannot be very easily weighted to ensure they retain their invariant properties. The other polynomials that I mentioned, which not so many know which might do the job are much rarer; and they are already multi-dimensional, or (almost as if they were quantum like, or in polar coordinates),.

These are the Gleason- Heywood-MacWilliams Polynomials,(chromatic polynomials I think they are called). These are self-dual and/or duals of each other when they are distinct in some cases (strongly self orthogonal); although they apply to coding. They were developed by the one and only  Gleason (A.M GLeason) of (hilberts 5th, and 'born rule proof 'fame) during ww2 whilst working in berkeley, and later on whilst working on classified issues for the US government with regards to code creating, and code cracking.

I was thinking I could somehow, convert the [0,1] values into their binary/decimal expansion and employ them, although it seems like an impossible feat, and I doubt it would work. And then one has to deal with inner-products, vector calculus and much other added complexity. Apparently these polynomials, are error-correcting in the non-trivial fashion; and can be made additive, or rather the codes can be.

They, that is these polynomials,  do not just cancel out to a constant or to a divisor or an invariant feature, when they have some invariant feature such as being additive or having a common divisor (at least with relata to the codes generated by them or the other way around); that is, or rather, the weight functions used to generate such polynomials for said codes are self-correcting, in this non-trivial fashion, although I may well be wrong, I have not read enough about them

Now the sinosoidal functions, arguably do not do this either, depending on what your view is on the proof equivalent/self evident divide, with relata to sin^2(x)+cos^2(x)=1. There is some distinction between this and berstein polynomails, where its just a case x-x+1=1. One was something had to be proven back in the day, whilst I presume we always knew the latter. Its obviously has to with their transcendental nature, their mth order derivatives do not reduce, or vanish at some point.

As i said I know how to that, already, I had already figured that out before your post in any case. Likewise when they range have to be surjective and range between all values in [0,1] or rather are nonethess surjective and range between all values in [m,]n. in any case I have resolved the problem.

Ok then, I should have said the global max> globalmin, in my case when the \sum(mini) globalmin' is not inherently tautologous, as they may be equal

(2) I never said that original my statement was not a tautology.  I send that it sounds tautologous; that does not imply that it is not tautologous. Technically speaking, whilst the maxims of conversation might seem to indicate by making that kind of statement, the speaker is attempting to indicate that the statement is not tautologous, logic alone does not entail this.

ie, I never said that the statement was not tautologous. And moreover, at least some tautologous self evident statement also sound or even appear tautologous, presumably .

Unless that is, you are committed to the claim that there are no tautologous statements at all that also do no appear or sound tautologous.  Presumably' A or ~A' is true, 'a bachelor is a bachelor',are both  considered a tautologous statements, but they also sound and appear tautologous as well.

Perhaps it is yourself that is out of line with logic, in some strict sense of the word.

(3) thirdly logic alone does not entail that because something is tautologous, one ought not say it. It does not take a stand on that issue. I might be being picky, but if you start talking about logic, then talk about logic.

Now of course I shouldnt have said. that statement, I am just being pedantic I suppose. So i apologize for that