Are there sets of three, or rather for any N>3 sets of N sur-jective uniformly continuous functions, for all N>3, where n denotes the number of function in the sets, such that each function in the set has the same domain [0,1] and the same range [0,1],such that these functions, in a given set, sum to one on all points in the domain [0,1]. ie \forallv\in[0,1] [\sum_{i=1}^{i=1n)fi(v)} =1 in [0,1] .Moro-ever, Are there arbitrarily many such functions for all n>=3.
By non trivial, I mean, -non linear, (and presumably not quadratic) functions which just so happen to be that that their sum give value 1, in [0,1] or perhaps for any domain, and not by way of the algebraic sum cancelling out to a constant 1 as in the case of x, 1-x (ie error correcting) .Presumably due to the nature of the derivatives
That is, so that the functions, would sum to one, regardless of the domain [0,1] or least for if the domain [0,1] is held fixed, and the functions are weighted; without it being a mere artefact that the algebraic sum to cancels out to give a constant,1,for any x; that is error-correcting functions. That is three (or n>3) functions all of which have a maximum range value of one, miniimum range value of zero, and which sum to one for all points in the domain [0,1]; and which are surjective and uniformly continuous, ie for every value in the range ri\in [0,1], these functions have some (at least one) value, ci in the domain [0,1] such that f(ci)=ri such that takes on that .
Where these maximum values and min values for all fi coincide (the same element of the domain for which fi corresponds to 1, is such that other n-1/2 functions correspond to zero etc); where obviously these three/n max points such that fi(c)=1,(one, and only one for each function, fi,i1=3 such functions. So there are three distinct domain points in [0,1] corresponding to a n/3-tuple of function values,f1,f2,f3(c)= ,,f1,f2,f3(c1)=f1,f2,f3(c2)=,; corresponding the three distinct elements/points in the domain [0,1] , c\neqc1\neqc2, c,c1,c2,\in [0,1] . Where the first function is at its maximum value here 1, and the other two are their minimum, zero at c, the second is at its maximum at c1 (1), and first and third are a their mimimum (zero here) at c1 etc
And are continuous (no gaps, and uniformly continuous, ie no spikes).
As one could not make use of these such functions, if one wanted them to be weighted otherwise; if said functions have sums which either (A) as just cancel out to be constant, or two (are such that it their sums are do not cancel out to a constant, but just so happen to line up because the domain is [0,1]
Likwise, the functions a similar form; so that one does not want, one having two maximums whilst the other two for example have one maxima, and two minima. Perhaps Berstein polynomials could be so weighted, but I do not know; the linear forms cancel out but their weighted bezauir forms seem to be a little unstable from what I have read.
I passed through
A set of uniformly continuous surjective functions f1,...,fn from [0,1] to [0,1] is enclosed.
In particular, for n=3:
f1(x) = cos2(2pi.x).cos2(4pi.x),
f2(x) = cos2(2pi.x).sin2(4pi.x),
f3(x) = sin2(2pi.x).
The construction shows that many other convenient sets of functions exist.
I passed through
A set of uniformly continuous surjective functions f1,...,fn from [0,1] to [0,1] is enclosed.
In particular, for n=3:
f1(x) = cos2(2pi.x).cos2(4pi.x),
f2(x) = cos2(2pi.x).sin2(4pi.x),
f3(x) = sin2(2pi.x).
The construction shows that many other convenient sets of functions exist.
f2(x) = cos2(2pi.x).sin2(4pi.x), when i checked the second equation however, it appears not to be surjective with regard to the range[0,1]; It appears to hit a maximum value of ~0.5; although i noticed how to rectify this by changing f3=sin^2(4pix) and f2 to cos2(4pi.x).sin^2(2pi.x), one just has to make sure they all reach 1, and o, and that each function is at 1 when the others are at zero. They did sum to one before, but they the middle term appeared to be squashed in. Nonetheless this has been helpful. . I will have to check how sensitive they are to scaling now, which is what I am doing now. where once sets a min value , yfor all three where the sum of the mins values f1 min+f2min+f3min =1 or [fi min,fimax], where these still sum to one for all combinations or min values that sum to one
Congrats William, you adjusted it excellently!
You are right, I wrote unwillingly the arguments in opposite direction.
Enclosed is the graph of f2 for the case n=4.
The sets of such functions are uncountably infinite. Describing them is another matter.
Dear William Balthes,
The answer to your question is YES. Please take a look at the topic "Partition of Unit" in an (advanced) Real Analysis book.
Dear David Thanks for this; I have since resolved this question. The partition of unity reference was helpful nonetheless- although I am wondering if there is a name for a surjective partition of unity; example the bernstein polynomials sum to one regardless of the range,and are also a partition of unity on [0,1], which is the added requirement that the range of the functions are [0,1]; although these functions are not surjective. Ie the third quadratic polynomial; has a global maxima of 0.5, it never reaches any value larger then that. So am wondering if there is a such a name of 'surjective partitions of unity'
Dear William, I don't think the name "surjective partition of unity" exists. On the other hand, I think that all the functions f_i in the partition can have range [0,1] if the covers U_i of [0,1] (to which the f_i's are subordinated) are chosen in a suitable way. And you can also have the domain of each f_i to be [0,1] by extending f_i as zero outside U_i.
Dear David.
I am using transcendtal functions at the moment and they appear to do the job without having to changing the functions when i decide the to weight the functions differently which is the most important issue. So I have been able to resolve this. It would be interesting to see, how well other non-transcendental functions of this sort fare.
Thanks for this; I just realized that in some sense, the name sur-jective parittion of unity does not in itself make a great deal of sense; as it may imply that each number in the domain in each fi is mapped to the entire unit interval. But yes from what I gather, the a partition of unity would imply given how it is generally defined that the fi must have at least a range of [0,1] for all elements in the domain R, and thus including [0,1] domain whether and that they functions sum to one for all elements of R not just [0,1].
Of course what I meant, and i presume what you meant in answering my question is whether there are such functions which will have some D=domain sub-interval of the real numbers (even if ends up nearly the entirely unit line) over which fi, are surjective relative to their range c= [0,1], ie every element of that interval c, has some such element in D, such f(x)=c. And that moreover, it would be presumably then not too difficult to make that domain interval [0,1].
From what I gather, it is already implicit in the definition of a partition of unity
that (1) the range of the fi has to be [0,1] and that the fi (
2) they sum to one for elements of the Real number line domain;
I presume that this is why the berstein polynomials do not 'count' as partition of unity, despite the fact that they (the set of k+1 polynomials which are kth set of bernstein polynomials) sum to one, for all for elements of the real domain even for those outside [0,1].
This being because not all elements of their range are in [0,1], except insofar that they their domain is restricted to [0,1] They are only partition of unity on that domain or interval [0,1]. I presume that when you speak about covers etc, and what not that these are discussed in some suitable textbooks (i was wondering if there is one that you could recommend), And or whether this pertains to bezair curves.
They look to be a little a unstable when both weighted (given teh requirement that must also form a partition of unity), and their underlying functional form in my case, 'surjective partition of unity' on [0,1] onto [0,1]. I presume this may due to the fact that they are standard sorts of polynomials of the same sort, although I presume some may have to be quartics and or others quadratics etc, at least in my case, as there are other desirata they must meet.
Ie the functional form of the base function that is then weighted, cannot change when I alter the weights; ie if I want one f1 to be onto [0,4,0.6] this is just fed into the same function which is some function of the base function, where that base function (or all of the base function in the set)remain the same.
It does not change in my case (which is the most important thing I need to preserve), which is why the sinosoidal and or other transcendental functions seem to work very well for this, and I have been able to resolve this question to my satisfaction.
And it will allow be use any sub-interval of [0,1] such that \sum_mini
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