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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

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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

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https://www.diva-portal.org/smash/get/diva2:455048/FULLTEXT01.pdf

[4] Alberts T, Khoshnevisan D, Calculus on Gauss Space: An

Introduction to Gaussian Analysis,

http://www.math.utah.edu/~davar/math7880/F18/GaussianAnalysis.pdf

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ualities.pdf

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Functions and Determinantal Point Processes,

http://www.math.iisc.ernet.in/~manju/GAF_book.pdf

[7] Mielberg E, Sense Theory, https://vixra.org/abs/1905.0105