Unsolved Problems
1) The Conjecture as a gap in the Unifying View and the foundation of Automatic Differentiation
2) How to establish that
(scalar) elementariness of a function is violated at a certain point
3) A question about the functions having no points of
violation of scalar elementariness
4) Regularization of a polynomial ODE and the Conjecture
5) More suspects for points of violation of elementariness
6) "AD" for
singularities: Scenarios of
evaluation of formal derivatives at a singular point of a polynomial m-order ODE
Here and further on we deal with holomorphic functions and their
derivatives in
the complex plane.
The Conjecture (a brief presentation here (slides 6-8), in details here) is not the only unsolved problem emerging in the frame of the Unifying View, but it is the central open issue affecting the shape of the Unifying View theory. Below are a few problems topped by the Conjecture proper and including other unsolved problems associated with the Conjecture, all awaiting attention of the mathematical community.
1) The
Conjecture as a gap in the
Unifying View and the foundation of
Automatic
Differentiation
In either of the two equivalent forms below, the
Conjecture 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
|
For any component, say x(t), of an explicit system in x(t), y(t), z(t), ... of m 1st order polynomial ODEs(1)
regular at all points of its phase space, for any initial point a |
|
there exists an explicit n-order rational ODE satisfied by x(t)
x(n) = R(t, x, x',
x", ..., x(n-1))
(3)
|
|
So far neither the proof nor
counterexample for this Conjecture is available. However if the Conjecture is weakened by
dropping the requirement
of
regularity in the target ODE (3), there is a proof based on
combinatorial considerations (though impractical for obtaining the
target
ODE) – see the slide 13 or Appendix 1 here) .
___________________
(1) 1st
order rational ODEs regular
at a given point are transformable into a wider polynomial system of ODEs.
First elementary functions were defined by Liouville in the 19th century as a matter of convention: a few well studied vastly used functions and their finite compositions.
In the 1960s Ramon Moore suggested a different approach, replacing the convention about the elementary functions with a definition based on their fundamental property.
Moore defined (vector-) elementariness based on a property of an elementary vector-function to satisfy a system of explicit rational 1st order ODEs.
A competing definition
of stand-alone or
scalar elementariness would be
based on a
property of an elementary function to satisfy one explicit rational ODE of
order n, or one implicit polynomial ODE of
order n.
The vector
elementariness
by Moore easily follows from the scalar elementariness (because
conversion
of an n-order
ODE into a system of 1st order ODEs is trivial). As to the vice versa
statement, the scalar
elementariness by Moore follows from vector elementariness due to the
Theorem proved by me in Appendix 1 here. Therefore the scalar and vector elementariness by Moore
are equivalent.
The Liouville-elementary
functions are also
Moore-elementary. Say cos t is vector-elementary
together with sin t because they satisfy the
system
y' = x .
However each of them is also scalar-elementary satisfying the ODE
x" = x (or satisfying an implicit first
order non-linear ODE (x')2
+ x2 = 1).
In general case however it may be a challenge to obtain an n-order ODE demonstrating the
scalar
elementariness from the system of ODEs demonstrating
the vector-elementariness.
It was understood indeed that
there exist infinitely many ODEs for demonstrating elementariness of a
particular (vector-) function: some ODEs possibly singular at certain
points of their phase space.
It's worth noting that...
-
In their entire domain of existence, and ...
-
With understanding that the rational ODEs may have points of singularity in their phase space unusable as initial points, though sometimes it's possible to replace singular ODEs with the regular ones at the point of interest.
Then in 2008 it was discovered that there exist isolated points in some holomorphic scalar-elementary functions whose Moore-elementariness at such points may be demonstrated only with singular ODEs. For example such is the function
|
x(t) = |
et – 1 |
|
t |
considered in
details below. Such
kind of special points were earlier known as "regular singularities" or
"removable
singularities".
As it was shown, though the singularity in functions at such point is
removable and the functions are holomorphic (with the proper defined
value at such point), it's impossible to rid of singularity in the ODEs
demonstrating their Moore-like scalar-elementariness at such points.
This
new type of
specialty happens to be unremovable (discussed in item 2).
It was namely the discovery
of these special points which gave the reason to refine the Moore
definition of
elementariness as a property of a function at
a neighborhood of a point
(rather than a property in the entire domain
of existence).
Definition 1: a vector-function (x, y, z, ...) is called
vector-elementary at and near
a point t=a if it satisfies a
system of rational ODEs (1) regular at the point t=a, or a
polynomial
system (2).
Definition 2: a function x(t)
is called scalar-elementary at and
near a point t=a if it satisfies a rational ODE (3)
regular
at the point t=a, or if it
satisfies
an implicit polynomial
ODE (4) regular
at a
meaning ∂P/∂Xm|t=a ≠ 0 (where Xm
denotes
x(m)).
The vector
elementariness
easily follows from the scalar elementariness (because conversion
of an n-order
ODE into a system of 1st order ODEs is trivial). As to the vice versa
statement, the scalar
elementariness would follow from the vector elementariness if the
Conjecture is true. Therefore equivalence between these two definitions
depends on the Conjecture.
Most properties
and results in the "Unifying View" are proved only for vector-elementariness, and it is
not known whether they are true for scalar
elementariness. In particular, the fundamental property of closeness of the class of
vector-elementary
functions states that a composition of such vector-functions, or the
inverse vector-function preserves
vector-elementariness. (For the scalar elementariness this is not
known).
2) How to establish
that
(scalar) elementariness of a
function is violated at a certain point?
Our method of proof that scalar elementariness of a function is
violated at a certain point utilizes the specific of the sequence of
derivatives at this point, for example the specific of x(n)|t=0 = 1/(n+1)
(see the article ).
Any holomorphic function x(t) is defined entirely by its analytic element, i.e. by the sequence of derivatives x(n)|t=a at a point. It appears that the scalar elementariness at a point a also depends on the property of the sequence of its derivatives x(n)|t=a . For the type of functions and examples considered in the article, the sequences of derivatives are rational numbers x(n)|t=a =rn .
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 ≠ 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 ≠ 0.
As a first step, the study was made for a particular function
|
x(t)|t=0 = |
et – 1 |
|
t |
where all derivatives exist and x(n)|t=0= 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 scalar 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 enabled uncovering a wide class of functions whose sequence of derivatives is also 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 scalar elementariness:
|
|
et – 1 |
|
sin t |
|
ln(t + 1) |
|
cos t½ |
|
t |
t |
t |
It's notable that
(sin t)/t at t=0 represents the "remarkable
limit" (studied in the foundation of the calculus). It's new meaning is
that it is a point of violation of the scalar elementariness of the
respective entire function. The function (ln
(1+t))/t is
also closely related to the "remarkable
limit" (1+t)1/t (see section 4).
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 does not fall into the above pattern of fractions with
denominator passing throw all prime numbers. For example, the family of
Bessel
functions J(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 zero 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 scalar 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 scalar elementariness violate also their vector elementariness at that point (the answer depending on the equivalency of the two definitions, or on the Conjecture).
3)
A question about the functions having no points of violation
of scalar elementariness
At point t=0 all the functions above can satisfy no regular rational ODE – and consequently no explicit polynomial ODE; but which kind of ODEs can they satisfy at points other than t=0?
Indeed, at points other than t=0 they do satisfy rational ODEs (or implicit polynomial ODEs). As to explicit polynomial ODEs, here is the
at either point of its phase space.
The equivalent form of this Theorem is...
However the vice versa statement is an open Proposition.
If a functions x(t) is scalar elementary in the entire domain of its existence, it means that the denominator g(t, x, x', x",...) in R=f/g of the rational ODE (3) must never disappear on x(t) in its entire domain. This seemingly suggests that there must be no denominator at all – as the image of complex functions usually is the entire complex space including 0 (unless the function is a constant), so that the Proposition seems true. In reality however...
- Either there is no denominator at all because the rational ODE (3) actually happened to be polynomial – as the Proposition claims;
- Or g(t,
x, x', x",...) never reaches zero on x(t) because zero may happen to be
the excluded (lacunary) value of g
(in accordance with the Picard theorem) – see the Example below. It's
because of exactly this possibility the Proposition remains an open
statement.
and its denominator never disappears on x(t). Though in this trivial example we do know an alternative simple linear ODE satisfied by x(t) (in accordance with what the Proposition states), for an arbitrary x(t) we do not know whether the Proposition is true.
4) Regularization of a polynomial ODE and the Conjecture
First consider these examples.
Example 1. The (entire) function x(t) = tet satisfies the ODE
singular at t=0. However x(t) satisfies also the ODE
x" – 2x' + x = 0
regular at t=0.
Example 2. The (entire) function x(t) = (et – 1)/t satisfies the ODE
singular at t=0. However it's proven that there can not exist a polynomial ODE regular at t=0 satisfied by this function. Nor can it satisfy a polynomial ODE with a constant nonzero factor at the leading derivative – because unlike in the Example 1, here function x(t) does have a point of violation of its scalar elementariness.
singular at t=1 (with any initial value for x, in particular with x(1) = e – 1 corresponding to x(t) in Example 2). At that, this x(t) satisfies also the ODE (7) regular at t=1 (though singular at t=0). If we set a goal to rid of singularity at t=1 in the Example 3, ODE (7) is the answer. (The ODE in Example 3 was obtained in a process of an intentional "planting" of a singularity into the ODE (7) at t=1 described here).
P(t, x, x', ..., x(m)) = 0 (8)
and P=0 happened to be singular at t=0. Does a method exist which either offers another poly ODE
Q(t, x, x', ..., x(n)) = 0
satisfied by x(t) but regular at t=0, or it says that x(t) cannot satisfy any poly ODE (8) regular at t=0.
Here is how this question relates to the Conjecture.
If the Conjecture is true and proved, then the answer to the question is this:
5)
More
suspects for
points of violation of elementariness
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 the point 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 scalar elementariness at x=y=e?
Being singular, at this point this ODE
has more than one solution being satisfied also by the trivial solution –
the identity function y=x .
If it were proven that any
ODE satisfied by the nontrivial solution y(x) is satisfied also by the
trivial one (thus having more than one solution), that would prove that
x=y=e is
really the point of violation of the scalar elementariness of the
nontrivial y(x). However this
is not proved. Note, that the ODE y'=1
satisfied by the trivial solution is not satisfied by the nontrivial
one.
Amazingly, a similar open question exists concerning the equation of the associating powers x^(y^z) = (x^y)^z. It's solution y = z1/(z – 1), y(1) = e is proved to be holomorphic at the point z = 1. However it's not known if this is a point of violation of scalar elementariness.
If we denote z–1=t
then
y = z1/(z – 1) =(1 + t)1/t
= u(t), so that u(t)
represents the remarkable
limit: lim u(t) = e when t→0. The
proof that
the function ln u(t) = (ln (1+t))/t loses
scalar elementariness at t=0 is based on the special property
and
simplicity of its derivatives (–1)(n+1)n!/(n+1),
however
the derivatives of u(t) at t=0 are not so simple at all,
the
known proof does not work for them, while the fact that ln u(t)
is scalar non-elementary does not tell
anything whether u(t) is scalar
non-elementary.
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 a small
neighborhood of the point x=e. (The proof for all x>1
was recently obtained by Lajos
Lóczi – August 2020).
6) "Automatic Differentiation" for
singular epxressions
or
scenarios
of evaluation
of formal derivatives at a singular point of a
polynomial m-order
ODE.
As we saw above, a function f(t) holomorphic at some point t=t0 however may be defined with an expression having a removable singularity at this point so that f(t0) must be specially defined as the lim f(t) when t→t0 . Moreover, at certain points (where elementariness is lost) such functions can be defined only via singular rational expressions or ODEs.
Thus, we have holomorphic functions at a point, having all their
derivatives, but the conventional AD is inapplicable for computing
those derivatives. Therefore we developed a process (as a
generalization of AD) capable to compute n-order derivatives of singular
expressions or singular ODEs. Understandably, the AD generalized for
singularities, is more
complicate and less suitable for automatization. That is why the
expression "Automatic Differentiation"
is taken in quotes in the title. Unlike at a regular point, evaluation
of ODEs at the singular points admits a variety of scenarios, which I
first outlined in 2009, here in section
8.2.
All cases when an elementary function in question x(t) is defined via elementary systems of ODEs
or expressions containing singularities may be reduced to one implicit
polynomial ODE P(x(m) …, x', x, t) = 0
singular at some point t = a, (i.e. ∂P/∂Xm
= 0 at t = a). The solution x(t) may or may
not exist
at this point, or may be not unique. At a singular point we speak
about a
process of formal evaluation of the derivatives, and this process may have
different
outcomes outlined below.
The formal evaluation means
applying the rules of differentiation to the
source ODE
not even expecting that the solution exists and has complex derivatives.
This process
creates an
infinite generating system for obtaining a vector of coefficients of
the formal
derivatives x(m+1),
x(m+2),…. 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 formal derivatives 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 (having zero or non-zero
convergence radius). 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 have more than one, or infinitely many of them. |
|
The generating system is triangular whose each equation is linear in one unknown with the same nonzero coefficient at every unknown. |
The
generating system is triangular but not necessarily linear as each of
its equations in one unknown is polynomial with numeric or parametric
coefficients. |
|
|
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 nonlinear polynomial 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 or all of the vectors still may comprise a divergent Taylor series (with zero radius) representing no solution. |
Table 1
This table below illustrates an example of evaluation of a polynomial
ODE at a singular point.
|
|
The values of the formal derivatives obtainable from the equations of the generating system |
||||||||||||||
|
x |
x,
x', x", .... , x(m) are given.
The equation in one unknown x(m+1) happened to be of degree 3 having 3 roots a1(m+1), a2(m+1), a3(m+1) . |
||||||||||||||
|
x' |
|||||||||||||||
|
x" |
|||||||||||||||
|
… |
|||||||||||||||
|
x(m) |
|||||||||||||||
|
x(m+1) |
a1(m+1)
The
equation in one unknown x(m+2)
happened
to be of degree 2 having 2 roots a1(m+2),
a2(m+2)
|
a2(m+1)
The
equation in one unknown x(m+2)
degenerated
into an identity so that x(m+2)
may be an
arbitrary complex number c(m+2) |
a3(m+1) |
||||||||||||
|
x(m+2) |
a1(m+2) |
a2(m+2) The equation in one unknown x(m+3) happened to be of degree 3
|
c(m+2) The
equation in one unknown x(m+3)
|
a3(m+2) |
a4(m+2) |
||||||||||
|
x(m+3) |
stop |
a1(m+3) |
a2(m+3) |
a3(m+3) |
|
a4(m+3)(c(m+2)) |
a5(m+3)(c(m+2)) |
a6(m+3)(c(m+2)) |
|
|
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 ™ 0. 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…
- an infinite uncountable set of the power continuum, or…
- a finite set with
the number of branches k representing k 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) is this setting and scenario considered?