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Neuro-Amorphic Function
(NAF)
[P-S Standard]
Egger Mielberg
egger.mielberg@gmail.com
07.08.2020
Abstract.
As a good try, we took liberty for formula derivation that would
allow describing many physical phenomena. In Nature, we face
many situations when a single magnitude is a reason for drastic
changes in the whole physical process. A single flexible instrument
for describing processes of any kind would help a lot.
We propose a function for generating mathematical models for a
process behavior.
We introduce special parameters that will help researchers find an
acceptable solution for their tasks. The Dynamics coefficient, as
well as dynamic function, is crucial for graph change. It can be
used for the dynamic corrections of the whole physical process.
1. Introduction
In practice, we see many tries to describe a complex process by usage
of many formulas. But many of those formulas are good for one single
process and not acceptable for another.
With high probability, there is no single solution for many complex
processes and uniform formula as well. But it looks possible to formulate
a mathematical transformer with the possibility to change its properties
depending on changes in the process. Mathematically it can be realized
as a transition from one formula to another. In other words, we use the
core formula that changes their properties with changing dynamic
function and relevant parameters. Unlike the spline function, the neuro-
amorphic function (NAF) includes the unchangeable core part that is
defined on the whole space of real numbers.
2. Problem
Describing the physical process, we need correct information
about:
1. all factors that affect the process.
2. limits of factor space.
3. existing invariants.
4. space entropy
5. etc.
In the case when the process changes their properties
drastically in time the task of mathematical description
becomes difficult. The problem is to find a single relevant
approach for many states of phenomena in question. We will
make a small try to do so below.
3. Solution
(1)
where
The formula (1) was derived from a number of practical
calculations for physical quantities that cannot be described by
the normal distribution law.
In other words, in nature there are many cases when most of
the values of the investigated quantity x do not lie in the vicinity
of .
For example, the physical state of certain bodies changes
dramatically when the limit values are reached. Accordingly, the
distribution law of the studied values of these states changes.
The formula (1) can describe a physical phenomenon that can
have both one limit value and a number of limit values.
Accordingly, the graph of a function can be symmetric
both with respect to one limit value (axis of symmetry) of a
given function and with respect to the numerical series of such
limit values.
Properties of (
):
1. The function is defined and continuous on the entire
number axis X.
(2)
.
2. On the intervals , the function is
strictly monotone (increasing) for .
(3)
3. On the intervals the function is strictly monotone
(decreasing) for .
(4)
4. For two sets and , where
(5)
5. is defined on the entire number
axis X.
6. Provided that the form factor coefficient is zero ( ),
the graph of the function on the interval is
symmetric to the graph of the function on the interval
about the y-axis. In other words, under these
conditions, the function is even.
7. , is not an even or odd function.
8. The dynamic function
is defined and continuous on the
entire positive x-axis which has a singular point or a break
point of the first kind, .
9. of the dynamic function
is
defined on the interval . This coefficient plays an
extremely important role in the dynamic response of
the function to changes in the described
phenomena.
10. is defined on the interval
and defines the limiting maximum value of the function
.
11.
.
Properties of (
):
1. The function is defined and continuous on the entire
x-axis.
2.
.
3. The dynamic function
is defined and continuous on the
entire positive x-axis.
4.
.
5. .
6. At a certain value of the dynamics coefficient and a shift
of the graph of the function relative to the x-axis, the
function approximates with a certain accuracy the
probability density function of the distribution of ideal gas
molecules over velocities.
7. At a certain value of the dynamics coefficient the
function approximates with a certain accuracy the
brick heating speed function (refractory, ceramic brick).
8. At a certain value of the coefficients the
function approximates with a certain accuracy the
product life cycle function in the market.
Convolution function
The convolution operation shows the degree of influence of two
functions on each other, forming the third function.
It is important not to confuse the convolution of two different
functions with an autocorrelation function. In the case of
convolution, the range of one function may differ from the range
of another one.
The integrand convolution function is a function defined on a
fixed set M which is formed by multiplying two functions f and g
defined on a set K and L, respectively.
Neuro-amorphic function can be represented as two
continuous functions:
, (6)
(7)
However the convolution for the functions
(8)
will be a complex expression to calculate.
The convolution function
(9)
will tend to the Gaussian function with shift x, provided
.
The Gaussian function and form an integrand
convolution function
, (10)
where ,
that degenerates into a straight line at
.
The product of the two functions and
, where
, (11)
gives a function tending to .
Transformation formulas
(12)
(13)
4. Conclusion
The mathematical expectation cannot always qualitatively
describe the behavior of the values of a quantity in a particular
sample. As in many practical cases, the normal distribution can
only be used to generalize the average results of the
considered quantity. A good examples, the quantum physics or
human neural net.
We offer a function that has a fairly flexible approximation
mechanism as well as the ability to build dynamic systems of
varying complexity.
We hope that our decent work will help other researchers in
their life endeavors.
References
[1] Levy P, A Special Problem Of Brownian Motion, And A General
Theory Of Gaussian Random Functions,
https://digitalassets.lib.berkeley.edu/math/ucb/text/math_s3_v2_article-
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M. Rodríguez-Dagnino, A practical procedure to estimate the shape
parameter in the generalized Gaussian distribution,
https://www.cimat.mx/reportes/enlinea/I-01-18_eng.pdf
[3] Zhanyu Ma, Non-Gaussian Statistical Models and Their Applications,
https://www.diva-portal.org/smash/get/diva2:455048/FULLTEXT01.pdf
[4] Alberts T, Khoshnevisan D, Calculus on Gauss Space: An
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http://www.math.utah.edu/~davar/math7880/F18/GaussianAnalysis.pdf
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ualities.pdf
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http://www.math.iisc.ernet.in/~manju/GAF_book.pdf
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