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ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
UDC 519.2
PROBLEMS OF BUILDING AND APPLICATION OF HYPERCOMPLEX NUMBERS_
DOI: 10.31618/ESU.2413-9335.2021.4.83.1261 Ibrayev1 A. T., Alkhan1 Y.A., Toktar1 A. Y.
1Al-Farabi Kazakh National University, Almaty, Kazakhstan
ПРОБЛЕМЫ ПОСТРОЕНИЯ И ПРИМЕНЕНИЙ ГИПЕРКОМПЛЕКСНЫХ ЧИСЕЛ
А. Т. Ибраев1, Е.А. Алхан1, А.Е. Токтар1
'Казахский национальный университет имени Аль-Фараби,
Казахстан, г.Алматы
ABSTRACT
The article is dedicated to the problems of using multidimensional numbers for mathematical and computer modeling of complex physical processes and the design of knowledge-intensive devices, including digital image processing. The emphasis is on the issues of building the methods for processing three-dimensional signals. It is proposed to use three-dimensional variables presented in the form of hypercomplex numbers to formulate the three-dimensional Fourier transformation forms, which allows to analyze and process three-dimensional signals.
АННОТАЦИЯ
В работе рассматриваются проблемы использования многомерных чисел для математического и компьютерного моделирования сложных физических процессов и проектирования наукоемких технических приборов, в том числе, при цифровой обработке изображений. Основное внимание уделяется вопросам построения методики обработки трехмерных сигналов. Предлагается использовать трехмерные переменные, представленные в виде гиперкомплексных чисел, для формулировки трехмерной формы преобразования Фурье, которые дают возможность анализа и обработки трехмерных сигналов.
Key words: hypercomplex number, quaternion, transformation, multidimensional signal, harmonic function.
Ключевые слова: гиперкомплексное число, кватернион, преобразование, многомерный сигнал, гармоническая функция.
At the moment, vector algebra, algebras of quaternions, octonions and other hypercomplex numbers are actively used in various fields of science and technology for mathematical and computer modeling of complex physical processes and the design of knowledge-intensive devices and systems [1-5]. For example, when designing modern control systems for mobile objects, communication and information transmission systems, much attention is paid to the problems of digital processing of multidimensional signals and images [6,7]. When solving specific problems of transforming multidimensional signals, different authors use branches of mathematics they think most suitable. The effective methods of processing signals with complex structures can also include the method of using hypercomplex numbers [8]. It should be noted that in recent years, a number of scientific papers have also proposed new modifications of hypercomplex numbers, which expands the theoretical possibilities of research.
This article deals with the basic properties of three-dimensional numbers and their use, along with quaternions, in the modeling and digital processing of three-dimensional signals and images.
As we know, Fourier, Laplace and Z-transformations are used most often in processing one-dimensional signals. If the spatial dimension of the signals increases, the mentioned transformations are considered depending on the corresponding number of
variables. For example, when studying two-dimensional signals, the extended Fourier transformation of two variables is used and considered. The use of hypercomplex numbers provides a good range of tools for the study and processing of complex multidimensional signals and images.
We should remind that the most developed systems of hypercomplex numbers are obtained by first doubling the complex numbers, which are usually written as
Z = X + iy ,
(1)
and then the obtained four-dimensional and other further doubled hypercomplex numbers. We should also note that the author of this paper showed the possibility of building the algebra of three-dimensional numbers [9].
The paper [9] also notes that the most well-known extension of complex numbers that form
noncommutative division algebra is the quaternions, For multidimensional hypercomplex numbers, the
which are as follows unit direction numbers may meet the following
conditions
X — Xq + ix1 + jx2 + kx3 ,
i0 — Ï-0 — 1, Ïoi-n — in
(2)
where Xo, Xi, X2, X3 are real numbers, i, j and k are imaginary units that meet the following rules
i2 = j2 = k2 = —l,ij = —ji = k,
jk = —kj = i,ki = —ik = j.
In this work it was also noted that in general, a hypercomplex number q can be as follows
Я S71=0 înXn,
(6)
iji k — Ojk
(3)
where Xn is a real number, in is a unit direction number.
For a complex number, if it is considered as a special case of a hypercomplex number q, the following conditions must be met in the formula (3) for the unit direction numbers
(7)
The values ajk determine the properties of hypercomplex numbers. The formulas (4) and (6) mean that i0 is a unit number of the axis of real numbers, and loXo = Xo is a real component of a hypercomplex number.
For the hypercomplex numbers could form a division algebra, we take the unit direction numbers in as mutually orthogonal and establishing an ordered relationship of direction of any of Xn with X0, i.e. with points of the axis of real numbers, as follows
—
dxp dx„
ïq — i-o — 1, i-oH — h
(8)
From (8) it follows that in are unit vectors.
(4)
Therefore, taking into account the properties of a vector product and using (8) without transformations, we get
= —1 ,
ijik = — i kij = —1 ,
i2 = —1 n .
where a is an arbitrary real number.
(5)
(9)
Any hypercomplex number q = x0 + Y,n=i inxncan be geometrically represented as a point A in the space with the coordinates x^, x-, x^,..., xN. In this case, the coordinate x0 characterizes the scalar component of a hypercomplex number, and the set of other coordinates describes its vector component. A hypercomplex number can also be represented as a vector with a starting point at the origin and an ending point at A. When representing a hypercomplex number in this way, the scalar component x0 and the total vector component can be considered, respectively, as a scalar and vector projections of a hypercomplex number.
The main principle of the theory of limits for hypercomplex numbers is based and proceeds from the fact that there is a single point belonging to all spaces of a given sequence. The concept of a limit point for hypercomplex numbers is similar to the concept of a limit point for complex numbers.
Assume that P = f(q) is a single-valued function defined in the area G of the space of a hypercomplex variable q. The derivative of the function f(q) at the point q (let us denote
f'(q) = —) is defined as follows
dq
(13)
From (12) and (13) it follows that the Cauchy-Riemann conditions for hypercomplex numbers are as follows
ËA = ËA + ËA +
dxn
dx1
dx2
. dfn _ spN afn dxn = Ln=1dxn'
(14)
ah dx0
dfo d x/
dfk dx0
drL, at n=1,2,
dxk
fXq) = limèL = lim fto+w-™
Aq^oAq Aq^O Aq
dh
dxk
(15)
dfk dxi '
(10)
Equation (10) can be represented as
,. AP
lim — =
Aq^O Aq
,. APS+APV ru \
lim —2-- = f(q),
Aqs^0 Aqs+Aqv Aqv^0
(16)
The equation (16) follows from (15). The equation (14) takes into account the expression (16).
Having differentiated (14) with respect to x0 taking into account (15), we get
(11)
d2fo I yN d2fo _ r> dx2 + Ln=1 dxl ■
where Aqs = Ax 0, APs=Afo, AP = AY,Nn=iinfn.
Since Aq--0 both for a separate Aq; separate fulfillment of the condition Aqv-
Aqv = A Yin=i i-n^n,
0 and for a
>0,
lim =
APç+AP,
Axo
Z = lim Af0+Zn=iinAfn = f,(q) Ax0^0 Axo
(17)
As we see, the equation (17) is a Laplace equation. It means that the function f0 is a harmonic function. From the equations (14) - (16) we also get
(12)
l™Ao:^:±Afn = f'(q)
Aqv^0 Zn^i-nAX;
a2fn
^N a2fn dx22 ' Zn=1 dxn 0
(21)
(18)
It follows from the last equation that the functions/- are also harmonic.
Let us summarize the first and main results of the foregoing.
Hypercomplex numbers of the form (3) meeting the conditions (6) and (9) form, in general, noncommutative algebra under multiplication with division over the field of real numbers.
The functions of the considered hypercomplex variables satisfy the Laplace equation. This means that the functions of hypercomplex variables are harmonic.
These results allow us to conclude that the development of the theory of functions of hypercomplex variables will significantly expand the tooling of applied mathematical research.
Since the problems we have to solve in practice most often are the problems in three-dimensional space, we will talk more specifically about the study of functions of three-dimensional hypercomplex numbers.
Having limited the expression (3) to the first three expansion terms, let us write a three-dimensional hypercomplex number as follows
In the expression (21), R is a module of a three-dimensional hypercommplex number.
We should note that these three-dimensional numbers do not form the normalized algebra similar to the quaternion algebra, but can be used to solve a number of applied scientific and technical problems.
We denote the angle between the radius vector and the x-axis as в, the angle between the projection of the radius vector on the >>z-plane, and the >>-axis as ф, then (formula)
x = R cos в, у = R sin в cos ф, z = R sin в sin ф.
A three-dimensional hypercomplex number in the coordinate system R, в, ф is as follows
q = R [cos в + (i cos ф + j sin ф) sin в]
q = x + iy + jz
(19)
where x — x0,y — x1,z — x2, i — i^j — i2. In this case, the condition (9) for three-dimensional hypercomplex numbers in these notations will be as follows
(22)
In (22) R is a module, в and ф are the arguments of a three-dimensional hypercomplex number.
In addition to the presentation in the form (19), a three-dimensional hypercomplex number can be as follows
q = R [cos вх + i cos ву + j cos dz].
r = j2 = —1, ij = —j i = —1.
(23)
(20)
The value of a radius vector of a three-dimensional variable is defined by the expression
R = iqi = jqq = ^x2+y2 + z2.
where 0x is an angle between the radius vector and the x-axis, 6y is an angle between the radius vector and the y-axis, 62 is an angle between the radius vector and the z-axis.
Using the presentation of a three-dimensional number in the form (19), we can also build a three-dimensional version of the Laplace transformation.
As we know, the Laplace integral is as follows
F(p) = J f(z) eXp(-pz) dz.
(24)
Here, the integration is performed over a certain given circuit L in the plane of the complex variable z,
which associates the function f(z) defined on L with the analytical function F(p) from the complex variable p = x + iy.
When using a three-dimensional variable instead of a complex variable in (24), we can get
F(q) = J f(z) exp(-qz) dz = J f(z) exp[-(s + ix + jy)z] dz. (25)
In this equation, z and q are three-dimensional variables of the form
q = + X + y.
(26)
With the value s = 0 and the condition ij = 0 from (25), we can obtain an analog of the two-dimensional Fourier transformation, which has been sufficiently studied and is widely used in practice. A more detailed analysis is not provided here due to the size limits of the article.
To sum up, it should be noted that the use of hypercomplex numbers allows to apply them for mathematical and computer modeling of complex physical processes and the design of knowledgeintensive devices, including the efficient conduction of studies in the field of processing multidimensional signals, the dimension of which coincides with the dimension of hypercomplex numbers.
References
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УДК 519.852.6
5. Ibrayev A.T. Theory of Cathode Lens with Multipole Components of Electrostatic Field and the Space Charge.- Microscopy and Microanalysis, 2015, V. 21, N6, P. 270-275. https://doi.org/10.1017/S1431927615013495.
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Information about authors:
Ibraev Alpamys Tuyakovich - Doctor of Physical and Mathematical Sciences, Professor of the Al-Farabi Kazakh National University.
Alkhan Yerassyl Armanuly - Master's student of Al-Farabi Kazakh National University.
Toktar Adil-Yer Yerboluly - Master's student of Al-Farabi Kazakh National University.
Сведения об авторах:
Ибраев Алпамыс Туякович - доктор физико-математических наук, профессор Казахского национального университета имени Аль-Фараби
Алхан Е.А. - магистрант Казахского национального университета имени Аль-Фараби
Токтар А.Е. - магистрант Казахского национального университета имени Аль-Фараби
НАХОЖДЕНИЕ ГАРАНТИРОВАННОГО СУБОПТИМАЛЬНОГО РЕШЕНИЯ ПО _ОГРАНИЧЕНИЯМ В ЦЕЛОЧИСЛЕННОЙ ЗАДАЧЕ О РАНЦЕ._
РР1: 10.31618/ЕБи.2413-9335.2021.4.83.1263 Мамедов Назим Нариман оглы
Доктор философии по математике, докторант Института Систем Управления НАН Азербайджана.
АННОТАЦИЯ
В работе введены понятия гарантированного решения и гарантированного субоптимального решения для целочисленной задачи о ранце. На основе одной экономической интерпретации разработан метод нахождения гарантированного субоптимального решения. С применением этого метода решена одна конкретная задача.
