Unsolved Problems

 

The Conjecture is not the only unsolved problem emerging in the frame of the Unified View as presented here at the Taylor Center. Below is a list topped by the Conjecture, but including also other unsolved problems associated with the topics of the Taylor Center, all waiting attention from the mathematical community.  

 

1) The Conjecture (slides 6-8) represents a gap in the Unifying View and the foundation of Automatic Differentiation. It claims that ...

 

For any component, say x(t), of an explicit system in x(t), y(t), z(t), ... of m 1st order rational ODEs regular at a given point t


x'(t) = r
1(t, x, y, z, ...)
y'(t) = r
2
(t, x, y, z, ...)
z'(t) = r
3
(t, x, y, z, ...)
. . . . . . . . . . .

 

there exists an n-order explicit rational ODE 

 

x(n) = R(t, x, x', x", ..., x(n-1))


satisfied by x(t) and regular at the same initial point t.

Or...

 

For any component, say x(t), of an explicit system of m 1st order polynomial ODEs


x'(t) = p
1(t, x, y, z, ...)
y'(t) = p
2
(t, x, y, z, ...)
z'(t) = p
3
(t, x, y, z, ...)
. . . . . . . . . . .

obviously regular at any point t of its phase space, for any initial point t there exists an n-order explicit rational ODE

 

x(n) = R(t, x, x', x", ..., x(n-1))


satisfied by x(t) and regular at point t.

 

A not yet established equivalency between two competing definitions (below) of general elementary functions depends on the Conjecture.  

 

First elementary functions were defined by Liouville in the 19th century as a matter of convention: a few well studied functions and their finite compositions.

 

In the 1960s Ramon Moore generalized those conventional elementary functions introducing a definition based on a property rather than a convention.

 

Moore defined so to say vector-elementariness based on a property of a vector-function to satisfy a system of explicit rational 1st order ODEs. Most properties and results in the "Unifying View" rely on namely this definition.

 

A competing definition of stand-alone elementariness is based on a property of a function to satisfy one explicit rational ODE of order n. Are both definitions equivalent? The answer is "Yes" if the Conjecture is true.

 

For the stand-alone elementariness it was discovered that certain holomorphic functions lose their stand-alone elementariness at some isolated points, earlier known as "regular singularities" or "removable singularities". As it was shown, though the singularity at such point is removable, the violation of stand alone elementariness is not, being a new type of special points in some functions (discussed in item 2).
 

 

2) How to establish that (stand alone) elementariness of a function is violated at a certain point? One way of how to do that was found in my article.

 

A function x(t) is stand alone elementary at a point a if it satisfies a polynomial ODEs   P(t, X, X1,…, Xm)=0  regular at a, i.e. P/Xm|t=a¹0  (here Xm  denotes x(m)).

 

Any holomorphic function x(t) is defined by its analytic element, i.e. by the sequence of derivatives x(n)|t=a, and its stand alone elementariness at a point a follows from the property of the sequence of its derivatives x(n)|t=a also. For the type of functions and examples considered in the article the sequences of derivatives are rational numbers rn=x(n)|t=a.

 

The method of proof is based on considering polynomial ODEs   P(t, X, X1,…, Xm)=0  over integer coefficients at a regular point where P/Xm =c ¹ 0. By applying differentiation, we may obtain an infinite sequence of polynomial ODEs  Pk(t, X, X1,…, Xm,…, Xm+k)=0  and then to establish that for all leading derivatives  P/Xm+k =c ¹ 0.  

 

As a first step, the study was made for a particular function

x(t) =

et 1

t

at  t=0  where all derivatives exist and   x(n) = 1/(n+1). The idea of the proof was in that while k grows to infinity, the denominator 1/(n+k) passes through all prime numbers. Therefore, for a big enough prime n+k, the fraction 1/(n+k) in the equation Pk =0 cannot cancel with other fractions leading to a contradiction. That way it was proven that x(t) at t=0  can satisfy no regular polynomial ODE, meaning that its stand alone elementariness is violated at  t=0 – followed merely from the fact that the sequence of derivatives x(n)|t=a are fractions (in lower terms) with growing denominators!

 

This observation allowed uncovering a wide class of functions whose sequence of derivatives is too represented by fractions with infinitely growing denominator (see Table 1 in the article), and here are examples of the respective functions all having t=0 as a point of violation of elementariness:

 

 

et 1

 

sin t

 

ln(t + 1)

 

cos t½

t

t

t

 

And here comes an unsolved part. There are functions – suspects for having points of violation of elementariness, whose sequence of fractions representing the derivatives at the point in question do not fall into the above pattern of fractions with denominator passing throw all prime numbers. For example, the family of Bessel functions u(t) satisfies the known ODE  t2x"+tx'+(t2-p2)x=0  singular at  t=0, and it is not known whether it satisfies a regular ODE at t=0. For every function of the Bessel family Jp the expansion at  t=0  is known and its general term is

Jp(n) =

(-1)kCkp+2k

2p+2k

for   n=p+2k,  k=0, 1, 2,…, and Jp(n) =0 otherwise. However it's impossible to draw a controversy from these fractions following the method of proof above (the denominators are not growing primes).

 

Do the Bessel functions have violation of elementariness at t=0?


Another unsolved part of the problem is whether the examples of functions above having t=0 as a point of violation of their stand alone elementariness violate also their vector elementariness at that point (the answer depending on the equivalency of the two definitions, or on the Conjecture).

 

 

3) More suspects for points of violation of elementariness appear in quite unexpected situations. For example, a transcendental function y(x) which is a non-trivial solution of the equation of commuting powers xy=yx  also satisfies an ODE which is singular at the point x=y=e:   

 

y"x2y2(y-x) – (y')3x4 + (y')2x2y(3x-2y) + y'xy2(3y-2x) –  y4 = 0

 

and it is not known whether y(x) satisfies a regular ODE at x=y=e. It is proven that y(x) is holomorphic at x=y=e, however no general term in closed form is known for its expansion (its derivatives at this point are obtainable only via a complicated recurrent equations) – see the article.

 

Does the y(x)  have violation of elementariness at x=y=e?      

 

There is another unsolved problem associated with the commuting powers: the function y(x) happens to be closely approximated by a hyperbola (in red) 

h(x) = 1 +

(e 1)2

x 1

   

 

and it was proved that h(x) £ y(x) in the neighborhood of the point x=e. The proof for all x>1 is not known.

 

 

4) Scenarios of evaluation of formal Taylor coefficients at a singular point of a polynomial n-order ODE.


If polynomial ODE P(x(m) …, x', x, t) = 0  is singular at a point t = a, the solution x(t) may or may not exist at this point, and may be not unique. At a singular point we speak about a process of formal evaluation of the
Taylor coefficients, and this process may have different outcomes outlined below.


The formal evaluation means applying the rules of formal differentiation to the source ODE not even expecting that the solution exists and has derivatives:
 

 

This process creates an infinite generating system for obtaining a vector of coefficients of the formal derivatives. Both in regular and singular cases the generating system is a triangular system.


In the regular case the generating system is linear, delivering a unique solution-vector for the
Taylor expansion representing the unique holomorphic solution x(t) and the vector of its true derivatives (x, x', x",…, x(m), x(m+1),…).


In the singular case the scenario becomes much more complicated as displayed in the table below. If the process for evaluation of formal derivatives does not stop, the result of the generation is a finite or infinite set of solution vectors S ={(x, x', x",…, x(m), x(m+1) ,…)} comprised of formal derivatives allowing to write down the respective formal
Taylor expansions (convergent or divergent). This set S has a sophisticated tree-like structure showed in the following Table 2.

 

 

At a regular point...

At a singular point...

The unique holomorphic solution x(t) exists, and its Taylor coefficients comprise the unique solution-vector of the generating system.

A holomorphic solution of the ODE does not necessarily exist.

The generating system has a unique solution-vector representing the derivatives x(n) of the solution x(t).

The generating system may have no solution-vectors, or to have more than one, or infinitely many of them.

The generating system is triangular and linear in the leading unknowns with the same nonzero coefficient at every leading unknown.

The generating system is triangular but nonlinear in the leading unknowns.

 

 

 

Some of the equations may degenerate into a numeric relation true or false. If n-th numeric relation is false, the respective branch of the tree ends (marked by stop). If n-th numeric relation is true (i.e. a zero identity), the respective unknown is a free parameter (like c(m+2)).  

If one or several unknowns in the previous equations are free parameters, the subsequent equations become algebraic nonlinear equations with parametric (rather than numeric) coefficients, so that generally it gets impossible to numerically explore and evaluate them.

If the set of the solution-vectors of the generating system is non-empty, some
of the vectors still may comprise a divergent
Taylor series (with zero radius) representing no solution.

Table 1

       

 

 

The values of the formal derivatives obtainable from the equations of the generating system

x

 

x'

x"

x(m)

x(m+1)

 a1(m+1)

a2(m+1)

a3(m+1)

 x(m+2)

 

a1(m+2)

a2(m+2)

c(m+2)

 a3(m+2)

a4(m+2)

x(m+3)

 

stop

a1(m+3)

a2(m+3)

a3(m+3)

 

a4(m+3)

a5(m+3)

a6(m+3)

 

 

a7(m+3)

a8(m+3)

a9(m+3)

a10(m+3)

 

Table 2

 

Here  x, x', … , x(m) are given as the initial values, while x(m+1), x(m+2),… are to be obtained in the process of solution of algebraic equations in one unknown of some degree k ³0. If k=0 the respective equation degenerated into a numeric relation true or false. If false, this branch ended with a controversy. If true, this equation does not yield a specific solution value allowing the respective x(n) be an arbitrary constant (like c(m+2)). Therefore, the set of branches growing from every node of the tree is either…

 

§       empty (end of the branch), or…

§        infinite uncountable of the power continuum, or…

§        finite with the number of branches k representing different roots of the respective algebraic equation.

 

I have illustrative examples for many (though not all) cases of this evaluation scenario.

 

Where (in which textbook or monograph) this setting and scenario is considered?