Self-Hyper-Root-Logarithm Function Theory
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
Mathematical Journal
of the "Collegium" All World Academy of Sciences
Munich (Germany)
12 (2012), 14
Keywords: Fundamental, mega-overmathematics, power tower, commutative hyperoperation, exponentiation, tetration, super-root, super-logarithm, iterated logarithm, negative base power theory, number scale transformation, general power-exponential function hyperefficiency theory, self-hyper-root-logarithm function theory, chaos theory, fractal theory.
Introduction
Numbers with very small and very large absolute values [Wikipedia Large_numbers] are extremely important for real world modeling. Moreover, their role exponentially increases because of computer science evolution which requires the so-called scientific number representation, as well as the storage and handling of such numbers to avoid the permanent danger of "computing overflow".
Classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]) widely uses
exponentiation as raising numbers to powers by Michael Stifel [1544],
power functions y = xn with constant exponents n and
exponential functions y = ax with constant bases a ,
as well as their inverse functions, namely
root functions y = x1/n (the nth root of a number x is a number y which, when raised to the power of n , equals x , i.e. yn = x) and
logarithmic functions y = loga x (the logarithm of a number x to a base a is a number y so that raising a base a to the power of y gives x , i.e. ay = x) introduced by John Napier [1614, 1619] and as a notion and notation by Leonhard Euler.
Some power-exponential functions with variable bases and variable exponents such as
y = xx
and also iterated (nested) exponentials (power towers)
a^b^c^... = a^{b^[c^(...)]}
with multiply (repeatedly) raising bases to powers so that power exponents are powers themselves are well-known, see also [Wikipedia Tetration]. Leonhard Euler [1777] introduced the notation
expa(x) = a^x = ax ,
which can be combined with function iteration notation fn(x) giving
expan(x) = a^a^...^a^x
(with a used n times on the right-hand side). He also showed that the infinite power tower
a^a^...
defined as the limit of
a^a^...^a
(with a used n times), converges for e-e ≤ x ≤ e1/e as n goes to infinity, which roughly gives the interval from 0.066 to 1.44. In particular, at a = 21/2 , this limit equals 2. Hans Maurer [1901] already used modern tetration notation
na = a^a^...^a (with a used n times on the right-hand side).
Donald Ervin Knuth [1976] introduced his up-arrow notation
a↑n = a^n = an ,
a↑↑n = a^a^...^a (with a used n times on the right-hand side),
a↑↑n(x) = expan(x) = a^a^...^a^x (with a used n times on the right-hand side)
interpreting super-powers and super-exponential functions via using m arrows in expression a↑m n(x). John Horton Conway [1996] chained arrow notation
a→n→2 = a^a^...^a (with a used n times on the right-hand side)
provides similar generalization via increasing the number 2 and, more powerfully, by extending the chain.
Albert Arnold Bennett [1915] proposed commutative hyperoperations sequence defined by the recursion rule
Fn+1(a , b) = exp(Fn(ln(a), ln(b))
beginning with
F0(a , b) = ln(e^a + e^b) = ln(ea + eb),
addition (I)
F1(a , b) = a + b ,
multiplication (II)
F2(a , b) = ab = eln(a) + ln(b) ,
a commutative form of exponentiation (III)
F3(a , b) = eln(a) ln(b) ,
F4(a , b) = e^{e^[ln(ln(a))ln(ln(b))]}
not to be confused with tetration [Wikipedia Hyperoperation].
Wilhelm Ackermann [1928] defined the function
φ(m , n , p)
resembling the hyperoperation sequence with reproducing such basic operations as addition, multiplication, and exponentiation at p = 0, 1, 2, respectively:
φ(m , n , 0) = m + n ,
φ(m , n , 1) = mn ,
φ(m , n , 2) = m^n = mn ,
φ(m , n , p) = m↑p-1 (n + 1)
for p > 2 with extending these basic operations using Knuth's up-arrow notation.
Reuben Louis Goodstein [1947] introduced the hyperoperations sequence of operations extending succession (the 0th) 1 + b , addition (the 1st) a + b , multiplication (the 2nd) ab , and exponentiation (the 3rd) ab and gave the extended operations beyond exponentiation the Greek names tetration (the 4th)
a↑↑b ,
pentation (the 5th)
a↑↑↑b = a↑3 b ,
hexation (the 6th)
a↑↑↑↑b = a↑4 b ,
etc., where each operation is defined by iterating the previous one.
[Wikipedia Tetration, Super-logarithm, Iterated_logarithm] also represents
super-root functions which can be denoted as y = 1/nx (the nth super-root of a number x is a number y so that tetration ny equals x , i.e. ny = x), e.g. the 2nd-order super-root, square super-root, or super square root ssrt(x) which has no real values by 0 < x < e-1/e , two positive real values by e-1/e < x < 1, and one positive real value by x ≥ 1,
super-logarithm functions y = sloga x (the super-logarithm of a number x to the base a is a number y so that tetration ya equals x , i.e. ya = x), and
iterated logarithm functions y = log*a x (the iterated logarithm of a number x to the base a is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1) so that by positive x ,
log*a x = [sloga x].
Nota bene: Unlike the possibility to represent the usual nth root of a number x as x1/n , it is generally inadmissible to represent the nth super-root of a number x as 1/nx even if
srtn(x) = srtn(x) = 1/nx
holds for n = 2 and x = 4 as an exception.
Examples:
srt2(4) = srt2(4) = 2 because 22 = 22 = 4,
1/24 = 41/2 = 2 = srt2(4) = srt2(4)
whereas
srt2(27) = srt2(27) = 3 because 23 = 33 = 27,
1/227 = 271/2 ≠ 91/2 = 3 = srt2(27) = srt2(27),
and
srt3 4256 = srt3 4256= 4 because 34 = 4^4^4= 4^256= 4256 ,
1/3(4256) = (4256)1/3 = 4256/3 ≠ 4 = srt3 4256 = srt3 4256.
All this is used for numbers with so-called very small and very large absolute values [Wikipedia Large_numbers].
But common approaches have many disadvantages:
1) investigating already available possibilities is much less efficient than concertedly creating new possibilities;
2) positive number bases only are usually considered;
3) bases between 0 and 1 are not efficiently used for representing numbers with so-called very small and very large absolute values;
4) a uniform number scale is not suitable for creating hyperoperation hierarchy;
5) known number scale transformations such as using logarithmic scales cannot provide suitably simultaneously representing numbers both with very small and very large absolute values of the both signs;
6) natural numbers (positive integers) of multiple (combined, composite) power exponents only are usually considered;
7) multilevel placing multiple power exponents brings many typesetting difficulties and misunderstanding, especially by text transformation via software including browsers;
8) already usual exponentiation ab is noncommutative and nonassociative, e.g.
23 = 8 ≠ 9 = 32,
2^3^4 = 2^(3^4) = 281 ≠ 212 = (2^3)^4,
because in
ab = eb ln(a)
the roles of a and b are very different;
9) a commutative form of exponentiation (III)
F3(a , b) = eln(a) ln(b) = aln(b)
by Albert Arnold Bennett [1915] provides noninteger values by natural a , b > 1 and growth much more slower than that of ab by great a , b , which is a very important disadvantage when applying this commutative form of exponentiation to representing great numbers;
10) individual quantities of operands and operation results are not considered at all.
Therefore, in classical mathematics, super-root, super-logarithm, and iterated logarithm functions have very bounded domains of definition and efficiency.
Hence classical mathematics cannot (and does not want to) regard (adequately solve and even consider) very many typical urgent problems in science, engineering, and life. This generally holds when solving valuation, estimation, discrimination, control, and optimization problems, as well as in particular by measuring very inhomogeneous objects and rapidly changeable processes [Encyclopaedia of Physics 1973]. This is also very important for chaos theory (Ilya Prigogine [1993, 1997]) and fractal theory (Benoît Mandelbrot [1975, 1977, 1982]).
Mega-overmathematics by Lev Gelimson [1987-2012] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further developing, extending, and applying mega-overmathematics fundamental sciences systems. Among them are, in particular,
negative base power theory which defines raising a negative number to a power,
general power-exponential function hyperefficiency theory which creates principally new possibilities providing number scale transformation,
and present self-hyper-root-logarithm function theory which creates sign-conserving self-hyper-root-logarithm functions unifying the advantages of super-root, super-logarithm, and iterated logarithm functions and providing number scale transformation with suitably simultaneously representing numbers both with very small and very large absolute values of the both signs due to efficiently using bases also between 0 and 1.
Principal Ideas
A self-hyper-root-logarithm function y = lh(x) has to explicitly or implicitly extend a uniform number scale between -1 and 1 and compress it by (-∞ , -1] and [1, +∞).
For a function y = lh(x) which has the whole real-number axis (-∞ , +∞) both as a domain (of definition) and range, we want to have the following properties:
|y| > |x| by x ∈ (-1, 0) ∪ (0, 1),
|y| < |x| by x ∈ (-∞ , -1) ∪ (1, +∞),
limx→±∞ |y|/|x| = 0,
limx→0 |y|/|x| = +∞ .
It is also natural and desirable that such a function y = f(x) is:
a) sign-conserving, i.e.
sign y = sign x , x ∈ (-∞ , +∞);
b) continuously differentiable if possible;
c) strictly increasing:
x1 < x2 implies f(x1) < f(x2);
d) strictly convex by [0, +∞) and strictly concave by (-∞ , 0].
One possible idea is very natural: to search for such a function y = lh(x) to be an inverse function to a function y = f(x) (in general power-exponential function hyperefficiency theory) which explicitly or implicitly compresses a uniform number scale between -1 and 1 and extends it by (-∞ , -1] and [1, +∞).
For a function y = f(x) which has an inverse function and the whole real-number axis (-∞ , +∞) both as a domain (of definition) and range, we want to have the following properties:
|y| < |x| by x ∈ (-1, 0) ∪ (0, 1),
|y| > |x| by x ∈ (-∞ , -1) ∪ (1, +∞),
limx→±∞ |y|/|x| = +∞ ,
limx→0 |y|/|x| = 0.
It is also natural and desirable that such a function y = f(x) is:
a) sign-conserving, i.e.
sign y = sign x , x ∈ (-∞ , +∞);
b) continuously differentiable if possible;
c) strictly increasing:
x1 < x2 implies f(x1) < f(x2);
d) strictly convex by [0, +∞) and strictly concave by (-∞ , 0].
There are well-known power functions with odd exponents greater than 1
y = x2n+1 , n ∈ N = {1, 2, 3, ...}
which have all the above properties. The same holds for sign-conserving power functions with positive even exponents
y = x"2n , n ∈ N = {1, 2, 3, ...}
where
a"b = |a|b sign a
due to negative base power theory by Lev Gelimson [1987-2012].
But no constant exponent by power functions
y = xn , n ∈ N = {1, 2, 3, ...}
can provide such growth by x→+∞ as by any exponential functions (which could be satisfactory by x > 1 only)
y = ax , a > 1:
limx→+∞ xn/ax = 0.
Therefore, these power functions are sufficient but not very efficient.
The well-known power-exponential function
y = f(x) = xx = 2x = ex ln x
is still more suitable by x > 1 only because the exponent x here grows together with the base x . On the contrary, by 0 < x < 1, we have
xx > x
and even
limx→0+ xx = 1
instead of the required relations
y < x ,
limx→0 |y|/|x| = 0.
Therefore, by 0 < x < 1, the equality of the base x and the exponent x plays a negative role, and any power function
y = xn , n ∈ N = {1, 2, 3, ...}
works here better even if
y = x
(by n = 1) does not provide
y < x ,
limx→0 |y|/|x| = 0
and is insufficient. Hence it seems to be natural to construct piecewise power-exponential functions differently defined on (-∞ , -1], (-1, 1), and [1, +∞), namely with distinct relations between the bases and exponents.
Tetration with Possibly Noninteger Multiplicity
To begin with, use power-exponential function definition
y = f(x) = ax = x^^a = x^2 a = expx[a]+1({a}) = x^x^...^x^{a}
with any positive a and x used [a] + 1 times on the right-hand side
where
a = [a] + {a}
is a positive (possibly noninteger) number,
[a] = floor(a) = entier(a) = max{z ∈ Z | z ≤ a} ≤ a
(Z is the set of all the integers)
is the integer part of a as the greatest integer not exceeding a , and
{a} = a - [a] ∈ [0, 1),
i.e.
0 ≤ {a} < 1,
is the fractional part of a as a sawtooth function.
In particular, by 0 ≤ a ≤ 1, we simply have
y = f(x) = ax = x^^a = expx[a]+1({a}) = x^a = xa ,
which behaves not better than y = x and is hence uninteresting.
To understand the naturalness of this sophisticated definition, consider the following example for
a = 1.5
with
[a] = 1,
{a} = 0.5:
ax = 1.5x = x^x^0.5.
This is natural because
1x = x1 = x = x^x^0,
2x = xx = x^x = x^x^1
and the power exponent
x^0.5 = x0.5
in
1.5x = x^x^0.5
is the geometric mean value of the power exponents x^0 = 1 in
1x = x1 = x = x^x^0
and x^1 = x in
2x = xx = x^x = x^x^1.
To further generalize this result, take a with the same [a] = 1 (which is here inessential because the only two last power exponents are relevant) and any {a} with
0 ≤ {a} < 1:
a = 1 + {a}
where
1 ≤ {a} < 2.
1+{a}x = x^x^{a}.
This is natural because
1x = x1 = x = x^x^0,
2x = xx = x^x = x^x^1
and the power exponent
x^{a}= x{a}
in
1+{a}x = x^x^{a}
is the weighted geometric mean value of the power exponents
x^0 = 1
with its natural weight 1 - {a} in
1x = x1 = x = x^x^0
and
x^1= x
with its natural weight {a} in
2x = xx = x^x = x^x^1.
In fact,
[(x^0)1-{a}(x^1){a}]1/[(1-{a})+{a}] = [11-{a}x{a}]1 = x{a} ,
quod erat demonstrandum.
Tetration Transformation Algorithm
To begin with, consider the well-known tetration notation
nx = x^^n = x^x^...^x
with x used n times (n ∈ N = {1, 2, 3, ...}) on the right-hand side, namely always one (the first) time as a base and (if n > 1) further n - 1 times as exponents.
Namely, use the following algorithm:
1) separate the sign of argument (variable) x from its modulus (absolute value) |x|;
2) directly and explicitly assign sign x to the function value itself;
3) replace the argument (variable) x with its modulus (absolute value) |x|;
4) replace each exponent |x| with the maximum max(|x|, 1/|x|) of |x| and its inverse 1/|x|;
5) consider the product whose first factor
sign x = 0 (x = 0)
to vanish independently of the second factor, or, alternatively, which is sufficient, take its (zero) limit as its value.
General Power-Exponential Function Notation and Sense
Let us introduce:
a notation for powers and exponentials with single-level placing multiple power exponents via separating them with the backslash sign \ , e.g.
a^b^c^d = ab\c\d ;
a space-saving notation for the sign function
a° = sign a ;
a space-saving notation for the function
a? = max(a , 1/a).
Also consider the well-known tetration notation
nx = x^^n = x^x^...^x
(with x used n times on the right-hand side) by n = 2:
2x = x^^2 = x^x = xx .
The above tetration transformation algorithm leads to the function
y = f(x) = x°|x||x|? = (sign x) |x|max(|x|, 1/|x|) = (sign x) |x|(|x|+1/|x|+||x|-1/|x||)/2
where by x° = sign x = 0 (x = 0), the second factor
|x||x|? = |x|max(|x|, 1/|x|) = |x|(|x|+1/|x|+||x|-1/|x||)/2
is not considered at all, or, alternatively, its (zero) limit is taken as its value.
We have piecewise y = f(x) =
xx by x ∈ [1, +∞),
x1/x by x ∈ (0, 1],
0 by x = 0,
-(-x)1/(-x) by x ∈ [-1, 0),
-(-x)(-x) by x ∈ (-∞, -1].
Now, beginning with the tetration notation nx and using the parenthesis " (as well as in negative base power theory by Lev Gelimson [1987-2012]) between the base x and the number n of the base and the exponents each of which equals the base x , naturally introduce the simple notation for this function
y = f(x) = 2"x = x"^^2
and, more generally, for a function
y = f(x) = n"x = x"^^n
with giving it sense further.
Notata bene:
1. In negative base power theory, e.g. in
x"x = x"^x = x°|x|x = (sign x) |x|x ,
the parenthesis " (placed to the right from the base x between the base x and the exponent which equals the base x in this case only) designates the following:
1.1) in the base x only, separate the sign of argument (variable) x from its modulus (absolute value) |x|;
1.2) directly and explicitly assign x° = sign x to the function value itself;
1.3) in the base x only, replace the argument (variable) x with its modulus (absolute value) |x|.
2. In power-exponential function hyperefficiency theory, e.g. in
n"x = x"^^n ,
the parenthesis " (placed to the left or to the right, respectively, from the base x between the base x and the optional operation signs with the number n of the base and the exponents each of which equals the base x) designates the following:
2.1) separate the sign of argument (variable) x from its modulus (absolute value) |x|;
2.2) directly and explicitly assign sign x to the function value itself;
2.3) replace the argument (variable) x with its modulus (absolute value) |x|;
2.4) replace each exponent |x| with the maximum
|x|? = max(|x|, 1/|x|)
of |x| and its inverse 1/|x|;
2.5) consider the product whose first factor
x° = sign x = 0 (x = 0)
to vanish independently of the second factor, or, alternatively, which is sufficient, take its (zero) limit as its value.
Power-Exponential Functions y = x"^^x = (sign x) |x|^max(|x|, 1/|x|)^^(x-1), y = x^^x = x^[x^^(x - 1)], and y = x^[(1/x)^^(x - 1)]
Let us investigate naturally generalizing the above functions
y = f(x) = xx = 2x = x^x ,
y = f(x) = x1/x = x^(1/x),
respectively, via iterating the exponents x and 1/x namely x - 1 and 1/x - 1 times, respectively, and these numbers of times are possibly noninteger.
To begin with, take power-exponential function
y = f(x) = xx = x^^x = expx[x]+1({x}) = x^x^...^x^{x}
with x used [x] + 1 times on the right-hand side
where
x = [x] + {x}
is a positive (possibly noninteger) number,
[x] = floor(x) = entier(x) = max{z ∈ Z | z ≤ x} ≤ x
(Z is the set of all the integers)
is the integer part of x as the greatest integer not exceeding x , and
{x} = x - [x] ∈ [0, 1),
i.e.
0 ≤ {x} < 1,
is the fractional part of x as a sawtooth function.
By x ∈ [1, +∞), power-exponential function
y = f(x) = xx = x^^x = expx[x]+1({x}) = x^x^...^x^{x}
behaves as desired (and much better than y = x2n-1 and y = ax) and has a suitable inverse.
For example, by 1 ≤ x < 2, we obtain
y = f(x) = xx = x^^x = expx[x]+1({x}) = x^x^(x-1) = exp[e(x-1) ln x ln x].
Its first derivative is
y' = df(x)/dx = exp[e(x-1) ln x ln x] [e(x-1) ln x ln x]' = exp[e(x-1) ln x ln x] e(x-1) ln x /x [1 + (x - 1)ln x + x ln2 x]
= x^x^(x-1) xx-2 [1 + (x - 1)ln x + x ln2 x],
f'(1) = 1.
By 0 ≤ x ≤ 1, we simply have
y = f(x) = xx = x^^x = expx[x]+1({x}) = x^x = xx ,
which brings nothing:
takes value 1 at x = 1;
has limit
limx→0+ f(x) = 1;
has the minimum
f(1/e) = (1/e)(1/e).
To provide a suitable extension of this function to x ∈ [0, 1], it is logical to replace
y = f(x) = xx = x^^x = x^(x-1x)
with
y = f(x) = x^[(1/x-1)(1/x)] = x^[(1/x)^^(1/x-1)] = x^exp1/x[1/x]({1/x}).
For example, by 1/2 < x ≤ 1, we obtain
1 ≤ 1/x < 2,
[1/x] = 1,
{1/x} = 1/x - 1,
y = f(x) = x^[(1/x-1)(1/x)] = x^[(1/x)^^(1/x-1)] = x^exp1/x[1/x]({1/x}) = x^(1/x)1/x-1 = x^(1/x)^(1/x - 1) = exp[e(1-1/x) ln x ln x].
Its first derivative is
y' = df(x)/dx = exp[e(1-1/x) ln x ln x] [e(1-1/x) ln x ln x]' = exp[e(1-1/x) ln x ln x] e(1-1/x) ln x /x2 [x + (x - 1)ln x - ln2 x]
= x^x^(1-1/x) /x1+1/x [x + (x - 1)ln x - ln2 x],
f'(1) = 1.
Therefore, function y = f(x) = x"x = x"^^x =
xx = expx[x]+1({x}) by x ∈ [1, +∞),
x^[(1/x-1)(1/x)] = x^exp1/x[1/x]({1/x}) by x ∈ (0, 1],
0 by x = 0,
- (-x)^{[1/(-x)-1][1/(-x)]} = (-x)^exp1/(-x)[1/(-x)]({1/(-x)}) by x ∈ [-1, 0),
- (-x)(-x) = exp(-x)[(-x)]+1({-x}) by x ∈ (-∞, -1]
compresses a uniform number scale between -1 and 1 and extends it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 1:
Fig. 1
Using the sign and maximum functions provides unifying the above piecewise representations as
y = f(x) = x"x = x"^^x = x°|x|^[|x|?^^(x-1)] = (sign x) |x|^[max(|x|, 1/|x|)^^(x-1)] = (sign x) |x|^{[(|x|+1/|x|+||x|-1/|x||)/2]^^(x-1)}
= (sign x) |x|^(x-1)max(|x|, 1/|x|) = (sign x) |x|^(x-1)[(|x|+1/|x|+||x|-1/|x||)/2]
= x°|x|^exp|x|?[x]({x}) = (sign x) |x|^expmax(|x|, 1/|x|)[x]({x}) = (sign x) |x|^exp(|x|+1/|x|+||x|-1/|x||)/2[x]({x})
where by
x° = sign x = 0 (x = 0),
the second factor
|x|^[|x|?^^(x-1)] = |x|^[max(|x|, 1/|x|)^^(x-1)] = |x|^{[(|x|+1/|x|+||x|-1/|x||)/2]^^(x-1)}
= |x|^(x-1)max(|x|, 1/|x|) = |x|^(x-1)[(|x|+1/|x|+||x|-1/|x||)/2]
= x°|x|^exp|x|?[x]({x}) = |x|^expmax(|x|, 1/|x|)[x]({x}) = |x|^exp(|x|+1/|x|+||x|-1/|x||)/2[x]({x})
is not considered at all, or, alternatively, its (zero) limit is taken as its value.
Examples:
10"10 = 10"^^10 = 1010 = 10^^10 = 10^10^10^10^10^10^10^10^10^10,
1/10"(1/10) = (1/10)"^^(1/10) = (1/10)^(10-1)[1/(1/10)] = (1/10)^910 = (1/10)^10^10^10^10^10^10^10^10^10 = 1/1010,
0"0 = 0,
-1/10"(-1/10) = - 1/10"(1/10) = - (1/10)"^^(1/10) = - (1/10)^(10-1)[1/(1/10)] = - (1/10)^910 = - (1/10)^10^10^10^10^10^10^10^10^10 = - 1/1010,
(-10)"(-10) = - 10"10 = - 10"^^10 = - 1010 = - 10^^10 = - 10^10^10^10^10^10^10^10^10^10.
And also generally, it is natural to define
-nx = - nx (n > 0).
Notata bene:
1. The new exponent |x|? = max(|x|, 1/|x|) is
1/|x| = |x|-1 by 0 < |x| ≤ 1,
|x| = |x|1 by 1 ≤ |x| < +∞
with the clear mirror symmetry (-1 and 1) of the exponent in the first additional level about |x| = 1.
2. The new exponent |x|? = max(|x|, 1/|x|) of the base |x| has its minimum 1 by |x| = 1 whereas
limx→0 |x|? = limx→0 max(|x|, 1/|x|) = limx→±∞ |x|? = limx→±∞ max(|x|, 1/|x|) = +∞ ,
which provides much more efficiency than it is possible due to using both power and exponential functions.
3. This function y = f(x) is continuous together with its first derivative.
4. It is reasonable to also try other possibilities via increasing the complexity of power-exponential functions.
Self-Hyper-Root-Logarithm Function y = lh x Inverse to Power-Exponential Function y = x"^^x = (sign x) |x|^max(|x|, 1/|x|)^^(x-1)
Now systematically consider self-hyper-root-logarithm function
y = lh x
to be inverse to power-exponential function
y = f(x) = x"x = x"^^x = (sign x) |x|^max(|x|, 1/|x|)^^(x-1).
Therefore, quanti-hyper-root-logarithm function
y = lh x
extends a uniform number scale between -1 and 1 and compresses it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), see Fig. 2:
Fig. 2
Examples:
lh 1010 = 10 because 10"10 = 10"^^10 = 1010 = 10^^10 = 10^10^10^10^10^10^10^10^10^10,
lh 1/1010 = 1/10 because 1/10"(1/10) = (1/10)"^^(1/10) = (1/10)^(10-1)[1/(1/10)] = (1/10)^910 = (1/10)^10^10^10^10^10^10^10^10^10 = 1/1010,
lh 0 = 0 because 0"0 = 0,
lh(-1/1010) = -1/10 because -1/10"(-1/10) = - 1/10"(1/10) = - (1/10)"^^(1/10) = - (1/10)^(10-1)[1/(1/10)] = - (1/10)^910 = - (1/10)^10^10^10^10^10^10^10^10^10 = - 1/1010,
lh(-1010) = -10 because (-10)"(-10) = - 10"10 = - 10"^^10 = - 1010 = - 10^^10 = - 10^10^10^10^10^10^10^10^10^10.
Basic Results and Conclusions
1. Self-hyper-root-logarithm function theory is advanced on the base of the proposed ideas.
2. Well-known super-root, super-logarithm, and iterated logarithm functions cannot provide suitably simultaneously representing numbers both with very small and with very large absolute values of the both signs and have very bounded domains of definition and efficiency.
3. Self-hyper-root-logarithm functions (inverse to general power-exponential functions which algorithmically transformate tetrations) extend a uniform number scale between -1 and 1, compress it by x ∈ (-∞ , -1] and by x ∈ [1, +∞), and are sign-conserving, continuously differentiable, and strictly increasing.
4. Self-hyper-root-logarithm functions provide suitably simultaneously representing numbers both with very small and with very large absolute values of the both signs.
5. Self-hyper-root-logarithm function theory in mega-overmathematics by Lev Gelimson [1987-2012] is universal and very efficient.
References
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[Gelimson 1996] Lev Gelimson. General Implantation Theory in the New Mathematics. Second International Conference "Modification of Properties of Surface Layers of Non-Semiconducting Materials Using Particle Beams" (MPSL'96). Sumy, Ukraine, June 3-7, 1996. Session 3: Modelling of Processes of Ion, Electron Penetration, Profiles of Elastic-Plastic Waves Under Beam Treatment. Theses of Reports
[Gelimson 1997a] Lev Gelimson. Hyperanalisis: Hypernumbers, Hyperoperations, Hypersets and Hyperquantities. Collegium International Academy of Sciences Publishers, 1997
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[Gelimson 2001f] Lev Gelimson. Objektorientierte Mathematik in der Messtechnik. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001
[Gelimson 2001g] Lev Gelimson. Measurement Theory in Physical Mathematics. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2001. Also published by Vuara along with a number of references to Lev Gelimson's scientific works.
[Gelimson 2003a] Lev Gelimson. Quantianalysis: Uninumbers, Quantioperations, Quantisets, and Multiquantities (now Uniquantities). Abhandlungen der WIGB (Wissenschaftlichen Gesellschaft zu Berlin), 3 (2003), Berlin, 15-21
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[Gelimson 2004b] Lev Gelimson. General Problem Theory. The Second International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 26-27 October 2004
[Gelimson 2004c] Lev Gelimson. Quantisets Algebra. The Second International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 26-27 October 2004
[Gelimson 2005a] Lev Gelimson. Providing helicopter fatigue strength: Flight conditions [Megamathematics]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Vol. II, Dalle Donne, C. (Ed.), Hamburg, 2005, p. 405-416
[Gelimson 2005b] Lev Gelimson. Providing Helicopter Fatigue Strength: Unit Loads [Fundamental Mechanical and Strength Sciences]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 589-600
[Gelimson 2006a] Lev Gelimson. Quantisets and Their Quantirelations. The Third International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 24-25 October 2006
[Gelimson 2006b] Lev Gelimson. Quantiintervals and Semiquantiintervals. The Third International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 24-25 October 2006
[Gelimson 2006c] Lev Gelimson. Multiquantities (now Uniquantities). The Third International Science Conference "Contemporary methods of coding in electronic systems", Sumy, 24-25 October 2006
[Gelimson 2006d] Lev Gelimson. Sets with Any Quantity of Each Element. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2006
[Gelimson 2009a] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2009
[Gelimson 2009b] Lev Gelimson. Overmathematics: Principles, Theories, Methods, and Applications. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2009
[Gelimson 2010] Lev Gelimson. Uniarithmetics, Quantialgebra, and Quantianalysis: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2010
[Gelimson 2011a] Lev Gelimson. Uniarithmetics, Quantianalysis, and Quantialgebra: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations (Essential). Mathematical Journal of the "Collegium" All World Academy of Sciences, Munich (Germany), 11 (2011), 26
[Gelimson 2011b] Lev Gelimson. Science Unimathematical Test Fundamental Metasciences Systems. Monograph. The "Collegium" All World Academy of Sciences, Munich (Germany), 2011
[Gelimson 2011c] Lev Gelimson. Overmathematics Essence. Mathematical Journal of the "Collegium" All World Academy of Sciences, Munich (Germany), 11 (2011), 25
[Gelimson 2012] Lev Gelimson. Fundamental Mega-Overmathematics as Revolutions in Fundamental Mathematics: Uniarithmetics, Quantialgebra, and Quantianalysis: Uninumbers, Quantielements, Quantisets, and Uniquantities with Quantioperations and Quantirelations. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2012
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