General Formulas in the Table of
for
DHURATA VALERA1,*, AGRON TATO2
1Department of Mathematics, Faculty of Natural Sciences,
University of Elbasan “Aleksandër Xhuvani”,
Elbasan,
ALBANIA
2Polytechnic University of Tirana,
Tirana,
ALBANIA
Abstract: - The consistent system of pseudo-arithmetical operations and its generator are applied to
that are derived as solutions of some functional equations. Some interesting classes of
are built and the for real functions marks a new development over the years. But,
opens a new perspective highlighting the fundamental properties of these
functions. Based on the fundamental properties of these , we have further outlined our study in
verifying other properties for pseudo-linearity, pseudo-nonlinearity and generalization of the table of
for these transformed functions, with some pseudo-derivative identities as Pseudo-Basic
Properties/Pseudo-General Rules. The table of for the is built as a first attempt
and equipped with the Pseudo-Linearity, the Constant Pseudo-Term, the Pseudo-Product, the Pseudo-Quotient
and the Pseudo-Chain Rule as grouped into eight cases. Further, it will be completed for
different cases, as more modified functions take part in pseudo-nonlinear combinations, Elementary
Transcendental Functions or for of etc., showing once again the
importance of generated Pseudo-Analysis with the broad field of its applications as a generalization of Classical
Analysis.
Key-Words: - Pseudo-arithmetical operations, generator, , , ,
table of , functional equations.
Received: April 15, 2023. Revised: December 2, 2023. Accepted: December 17, 2023. Published: December 31, 2023.
1 Introduction
As a generalization of the Classical Analysis,
Pseudo-analysis, [1], is based on a semiring
([, where this structure is defined on a
real interval , denoting the
corresponding operation ( respectively as
pseudo-addition and pseudo-multiplication, [2], [3],
[4], [5], [6], [7].
The generator of the binary operation , was
extended into the odd function , such that, [7], [8],
[9], [10], [11], [12]:
The pseudo–arithmetical operations (PAO) are
extended to the whole extended real line
, [4], and introduced the operations of
pseudo-subtraction and pseudo-division in [13],
[14], [15], so the system of PAO, generated by a
special generator , is the consistent SPAO
[3], [4].
The role of the consistent SPAO
generated by the generator , is
shown directly by taking the rational functions, [2],
[3], but is a further development
of
The extended forms of [3], [10]
are:
.
The function corresponding to a function
introduced by the [2] (called
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and denoted by in general) are derived
as solutions of some functionals equations using
several results of [16], [17], [18] and four interesting
classes of are built in [2]. The
for real functions introduced in [3],
and investigated in [7], mark a new development in
the field of Pseudo-Analysis. Based on the
fundamental properties of these , [2],
[9], [10], [19], [20], [21], for the first time in this
paper, we have studied and verified other properties
for pseudo-linearity/nonlinearity of
and generalization of the table of
[3] of transformed functions, [2]. The eight
exceptional cases are considered
for some functions’ pseudo-
linear and pseudo-nonlinear combinations with
some conditions. Also, the table of
for these is build and equipped with
pseudo-derivative identities as Pseudo-Basic
Properties/Pseudo-General Rules in the same way as
in the Classical Analysis, [22], [23], for the
derivative function.
2 Problem Formulation
The definition of presented in [3],
opens a new perspective highlighting the basic
properties of all classes of as
transformed functions by that are
treated in [2], [10], [11]. Based on the basic
properties of for these
, [2], we have further outlined our study
in verifying other properties for pseudo-linearity,
pseudo-nonlinearity of for
combinations of some modified functions and
generalization of the table of with
general formulas. Some pseudo-derivative identities
are found in the form of “The Pseudo-Rules” for
eight specific cases treated in this paper and are
presented as which are listed in a
table of .
The Table of Pseudo-Derivative for
functions has been built and equipped
with several formulas as for the
Pseudo-Linearity Rule, the Constant Pseudo-Term
Rule, the Pseudo-Product Rule, the Pseudo-Quotient
Rule, the Pseudo-Chain Rule, etc. This opens the
line for further studies in for
and even more for the corresponding
table of , [17], [18], [19], [22].
2.1 Definition and Some Relations for
In this paper, the real functions are continuous from
to.
Definition 2.1.1. Let be a function on
and the function be a generator of the
consistent system of pseudo-arithmetical operations
[2], [9], [10].
The function given by
for every is said to be
corresponding to the function .
Based on the definition 2.1.1. of , can
consider the pseudo-arithmetical operations
generated by generator , as a modified function of
arithmetic operations by , [3], [10]:
Definition 2.1.2. Let and be two continuous
functions , and let
be extended on (perhaps with some
undefined values). The
from to , for is a
function satisfying [9]:
The composition of functions is not commutative,
but associative.
Definition 2.1.3. If the function is differentiable
on and with the same
monotonicity as the function a generator of the
consistent SPAO
, then we can
define the of f at the point
as
, when the
right part is meaningful [3].
3 Problem Solution
The generalization of the table of
with general formulas for and
finding some Pseudo-General Rules are the main
problems treated in this session. Eight specific cases
are treated for several combinations of
and some pseudo-derivative identities
are found as “The Pseudo-Rules”. These groups of
identities are presented as which are
listed in, Table 2, “ of
for the (Appendix 2).
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We applied the definitions of [3],
[9], [20], [21], , derivative and
[3], for some linear/nonlinear
combinations of [2], and also
considering some conditions for constants, functions
and pseudo-operations that participate in relations
for sum, difference, product, quotient or
composition of etc.
The domain (D) of the sum function ,
difference function or product function
is the intersection of the individual
domains of the two functions in each combination
[23],
[24], [25]. The same request for the domain of the
quotient function, except the values that make the
function in denominator equal to zero, so the
domain of the quotient function is
where
The cases considered in the paper
are described in Leibniz's notation, [23], [24], [25].
3.1 The Constant Pseudo-Factor Rule
( of a Constant Pseudo-
Multiple with a )
The pseudo-derivative definition is applied to a
pseudo-multiple expression of a constant with a
modified function by ( [2], [3],
[9], with respect to x, as below:
This formula is "schematically the same formula" as
for the derivative function of “the multiple of a
constant with a function/the constant factor”, [23],
[24], [25].
The result 3.1. is of
a Constant Pseudo-Multiple with a
.
3.2 The Constant Pseudo-Term Rule
( of a Constant)
As a particular case, the Pseudo-derivative is
applied to a constant (any constant), [2], [3], [9],
and we find:
where .
This formula is "schematically the same formula" as
for the derivative function of “a constant”, [23],
[24], [25].
The result 3.2. is of
“a constant”.
3.3 The Pseudo-Sum Rule
( of the Pseudo-Addition of
two )
The pseudo-derivative, applied to a pseudo-sum
expression of two modified functions [2],
[3], [9], with respect to x, follows these steps as
follows, giving us an important conclusion:
The formula is "schematically the same formula" as
for the derivative function of “the addition of two
functions/the sum function”, [23], [24], [25].
The result 3.3 is for
the Pseudo-Addition of two .
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3.4 The Pseudo-Difference Rule
( of the Pseudo-Subtraction
of two )
The pseudo-derivative, applied to a pseudo-
difference expression of two modified functions
[2], [3], [9], with respect to x, follows
these steps below:
=
The formula is "schematically the same formula" as
for the derivative function of “the subtraction of two
functions/ the difference function”, [23], [24], [25].
The result 3.4. is for
the Pseudo-Subtraction of two .
3.5 The General Pseudo-Linearity Rule/
Property for two
( of the Pseudo-Linear
Combination of one/two and any
constant)
For pseudo-linear combinations of two modified
functions and any constant [2], [3],
[9], the pseudo-derivative is applied so it can be
easily verified and shown in the forms below:
We are following the evidence shown in the four
cases above. We can consider the exceptional cases
of modified functions and constants
in the pseudo-linear combinations, thus
reaching the cases treated in points 3.1. to 3.4. as
particular cases. Remind here the derivat function of
the linear-combination function or
by the derivate table in Classical
Analysis, [23], [24], [25].
The formula is "schematically the same formula/
property" as for the derivative function of “the
linear combinations”, [23], [24], [25].
The result 3.5. is for
pseudo-linear combinations of two modified
functions and for any constant
.
3.6 The Pseudo-Product Rule
( of the Pseudo-Multiplication
of two/ three
3.6.1
of the Pseudo-
Multiplication of two
For the pseudo-multiple expression of two modified
functions concerning x, [2], [3], [9], we
calculate the pseudo-derivative function and find:
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The formula is "schematically the same formula" as
for the derivative function of “the product of two
functions”, [23], [24], [25].
The result 3.6.1. is
of the Pseudo-Multiplication of two
.
3.6.2
of the Pseudo-
Multiplication of three
First, we must note that the domain of the product
function is the intersection of the
individual domain (
) of the three
functions
, [23],
[24], [25]. The pseudo-derivative, applied to a
pseudo-multiplication expression of three modified
functions with respect to x, [2], [3], [9],
follows the steps below, giving us an important
conclusion:
= in Appendix 1.1, see the full proof of case 3.6.2 =
The formula is "schematically the same
formula" as for the derivative function of “the
multiplication of three functions/the product
function”, [23], [24], [25].
The result 3.6.2. is
of the Pseudo-Multiplication of three
.
3.7 The Pseudo-Quotient Rule
( of the Pseudo-Division of
two )
We calculate the pseudo-derivative function for the
pseudo-division expression of two modified
functions with respect to x, [2], [3], [9],
with the conditions for values of function for each
values of , and
get:
= in Appendix 1.2, see the full proof of case 3.7=
Again we have found an interesting conclusion for
3.7. The formula is "schematically the same
formula" as for the derivative function of “the
division of two functions/ the quotient function”,
[23], [24], [25].
The result 3.7 is of
the Pseudo-Division of two .
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3.8 The Pseudo-Chain Rule
( for the composition of two
)
If the function is differentiable on
and also, the function is
differentiable with respect to x, then the composite
function is differentiable, and we
recall, [9], the relationship as a
:
We note with the composition of two
so our function is presented as
or
and in this case, [9].
The Pseudo-Chain Rule is:
The pseudo-derivative definition is applied to a
pseudo-composition expression of two functions or
composition of two modified functions by
( [2], [3], [9], as below:
The formula is is "schematically the same
formula" as for the derivative function of “the
composition of two functions”, [23], [24], [25].
The result 3.8. is of
the Composition of two , or of
the .
4 Results and Discussion
The Pseudo-Derivative function for some pseudo-
linear or pseudo-nonlinear combinations of
, [2], with some
conditions for constants, functions and PAO that
participate in relations for sum, difference, product,
quotient or composition of directed
us to Pseudo-Rules for Pseudo-Linearity, the
Constant Pseudo-Term, the Pseudo-Product, the
Pseudo-Quotient, the Pseudo-Chain cases.
Based on the results we found from the
implementation of the definition
2.1.2, [3], for each case in this study, we
record that all the pseudo-identities formulas are
"schematically the same formula" as in Classical
Analysis, [23], [24] for the derivative function of:
“the multiple of a constant with a function”/
“The Constant Factor or Multiple Rule”;
“a constant”/”The Constant Term Rule”;
“the addition of two functions”/“The Sum
Function” (case for two functions);
“the subtract of two functions”/“The
Difference Function” (case for two functions);
“the linear- combination” (case for one or two
functions)/“Linearity Property”;
“the multiplication of two functions”/“The
Product Function” (case for two and three
functions);
“the division of two functions”/“The Quotient
Function”;
“the composition of two functions”/“The Chain
Rule”.
We applied for a pseudo-linear
combination of two (case 3.5) and
some pseudo-derivative identities as
are founded for The Pseudo-Linearity Rule, after
four cases treated before ( because we
tried to follow the same line with the table of
derivative functions in Classical Analysis, [23],
[24], as well as the sequence in the consistent SPAO
.
But, we emphasize that we can take into
consideration the rules below:
The Constant Pseudo-Factor Rule (case 3.1);
The Constant Pseudo-Term Rule (case 3.2)
The Pseudo-Sum Rule (case 3.3);
The Pseudo-Difference Rule (case 3.4);
as exceptional cases for “The Pseudo-Linearity
Rule”, [24]. All the results for each treated case
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are arranged in a table as
in appendix. We can present this
as a generalization form of the
derivatives table for Classical Analysis, [23], [24].
An interesting problem will be applying the
definition for more than three modified
functions in pseudo-linear/nonlinear combinations
or their , as for Elementary
Transcendental Functions etc. These cases will be
the perspectives of our study.
5 Conclusion
The main problems treated in this paper are the
generalization of the table of for
with general formulas and finding
some Pseudo-General Rules as to
equip the . Eight specific pseudo-
derivatives cases are treated for several
combinations of and some pseudo-
derivative identities are found in the form of “The
Pseudo-Rules”. These pseudo-derivative identities
are arranged in five groups and presented as
listed in a table of for
the as a first attempt in, Table 2,
“ of for the
(Appendix 2). The table of
for the [2], [9], is
equipped explicitly with several pseudo-derivative
identities as Pseudo-Basic Properties/ Pseudo-
General Rules:
The Pseudo-Linearity Rule (The Constant
Pseudo-Factor Rule, The Pseudo-Sum Rule,
The Pseudo-Difference Rule);
The Constant Pseudo-Term Rule;
The Pseudo-Product Rule (case for two and
three functions as pseudo-nonlinearity
formulas);
The Pseudo-Quotient Rule;
The Pseudo-Chain Rule (pseudo-combination
of two pseudo-functions).
In the following, the Table 2, of
for the (Appendix 2),
will be completed with more ,
showing once again the importance of generated
Pseudo-Analysis with the broad field of its
applications, [19], [20], [21], further using
mathematic induction for more modified functions
to take part in pseudo-nonlinear combinations, for
Elementary Transcendental Functions, etc. A
perspective line is open in for
and their [17], [18], [19],
[20], [21], [22], as a generalization form.
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Contribution of individual authors to the
creation of a scientific article (Ghostwriting
policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of funding for research presented in
a scientific article or scientific article itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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Appendix 1: The full proof for two cases 3.6.2 and 3.7 of session 3.
1.1 The full proof of case 3.6.2, for
of the Pseudo-Multiplication of three
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1.2 The full proof of case 3.7, for
of the Pseudo-Division for two
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Appendix 2:
The Table of
for the
Table 2
of
for the
No.
of:
1
a Constant
Pseudo-Multiple
with a
The Constant Pseudo-Factor Rule
2
a Constant
The Constant Pseudo-Term Rule
3
the Pseudo-Addition
of two
The Pseudo-Sum Rule
4
the Pseudo-Subtraction
of two
The Pseudo-Difference Rule
5
the Pseudo-Linear
Combination
of one/two
and
any constant
The General Pseudo-Linearity Rule/Property for two
The General Pseudo-Linearity Rule/ Property for one
6
the
Pseudo-Multiplication
of two/three
The Pseudo-Product Rule for two
The Pseudo-Product Rule for three
7
the Pseudo-Division
of two
Condition:
The Pseudo-Quotient Rule
8
the Composition
of two
Conditions:
,
The Pseudo-Chain Rule
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