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● Ф изик а–мате мат ик а
ылымдар ы
9. Skakov M.K, Kurbanbekov Sh.R. Vliyanie electrolitno-plazmennoy cementasii na fazogo-structurnoe
sostoyanie i mahanicheskie svoystva poverhnosti stali 12Х18N10Т // Vestnik КаzNTU №4 (91), - Almaty, 2012, s.197201.
Скаков М.К., Батырбеков Э.Г., Курбанбеков Ш.Р.
Тот баспайтын 12Х18Н10Т болатының беттік модификацияланған қабатының субқұрылымы мен
фазалық құрамын ПЭМ әдісімен зерттеу
Түйіндеме. Бұл жұмыста электролиттік плазмалық өңдеуден кейінгі 12Х18Н10Т аустениттік болаттың
беттік модификацияланған қабатының құрылымы мен фазалық құрамы зерттелген. Электролиттік-плазмалық
цементациядан кейін модификацияланған беттік қабаттағы карбидтік фаза болат құрамындағы орналасуынан
тәуелсіз өзінің құрамында Fe3C химиялық қоспаға ие екендігі анықталды. γ- және α- темірдің қатты
ерітіндісінде көміртегі атомының жоғары концентрациясы болатындығы және дислокациялардың жоғары
тығыздығы бар екендігі анықталды. Сондай-ақ, карбидтік фаза бөлшектерінің және қалдық аустениттік
қабаттардың жоқтығы анықталды. Модификацияланған электролиттік-плазмалық нитроцементация кезінде
12Х18Н10Т болаттың беттік қабатындағы көміртегі негізінен цементитте, карбонитридте M23(C, N)6 және αфазада орналасқан. Ал, азоттың ε- және α- фазада, сондай-ақ карбонитридте M23(C, N) орналасатындығы
анықталды. Болатты электролиттік плазмалық азоттау нәтижесінде үлгілердің жұқа беттік қабатында негізгі
фаза ε- Fe2-3N және онымен белгілі бір ориентациялық заңдылықпен орналасқан CrN фазасы түзілетіндігі және
құрамында карбонитридтік бөлшектер орналасқан инелік мартенсит түзілетіндігі анықталды.
Негізгі сөздер: құрылым,фазалық құрам, модификацияланған беттік қабат.
Скаков М.К., Батырбеков Э.Г., Курбанбеков Ш.Р.
ПЭМ-исследования субструктуры и фазового состава поверхностных модифицированных слоев
нержавеющей стали 12Х18Н10Т
Резюме. В настоящей работе изучены структура и фазовый состав поверхностного модифицированного
слоя аустенитной стали 12Х18Н10Т после электролитно-плазменной обработки. Определено, что после электролитно-плазменной цементации карбидная фаза в модифицированной поверхности, независимо от расположения в структуре стали, имеет химический состав Fe3 C. Выявлена высокая концентрация атомов углерода в
твердом растворе на основе γ- и α-железа, большая плотность дислокаций, присутствие частиц карбидной фазы
и прослоек остаточного аустенита. Выявлено, что в модифицированном при электролитно-плазменной нитроцементации поверхностном слое стали 12Х18Н10Т углерод, в основном, сосредоточен в цементите и карбонитриде М23(С, N)6, а также в α-фазе, азот сосредоточен в ε- и α-фазах и в карбонитриде М23(С, N)6. Обнаружено,
что в результате электролитно-плазменного азотирования стали наблюдается формирование на поверхности
образцов тонкого слоя, основой которого является фаза ε-Fe2-3N, и находящаяся с ней в закономерном ориентационном соотношении CrN-фаза, а также игольчатый мартенсит, внутри которого расположены частицы карбонитридов.
Ключевые слова: структура, фазовый состав, поверхностный модифицированный слой.
Skakov M.K., Batyrbekov E.G., Kurbanbekov Sh.R.
TEM- Research Substructure and Phase Composition of the Surface Modified Layers of Stainless Steel
12Cr18Ni10Ti
Summary. In the present work we have studied the structure and phase composition of the surface the modified
layer austenitic steel 12Cr18Ni10Ti after electrolyte-plasma processing. It is defined that after electrolytic-plasma cementation carbide phase in modified surface, regardless of the location in the structure of steel, has a chemical composition Fe3C. The revealed high concentrations of carbon atoms in the solid solution based γ- and α -iron, much dislocation
density, presence of carbide phase particles and layers of residual austenite. It is revealed, that in a modified at electrolytic-plasma carbonitriding surface layer of steel 12Cr18Ni10Ti carbon, mainly concentrated in the cementite and carbonitride M23(C, N)6, and in α-phase, nitrogen concentrated in ε- and α-phases and carbonitride M23(C, N)6. Found that
as a result of electrolytic-plasma nitriding steel, has been forming at the sample surface a thin layer, the main of which
is the phase ε-Fe2-3N and finds her in a natural orientation ratio phase CrN, and the needle martensite, inside of which
there are particles carbonitrides.
Key words: structure, phase composition, surface modified layer.
УДК 512.28
Bimurat Sagindykov
(Candidate (PhD) of Physical and Mathematical sciences
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401
● Ф и зико –мате мат ичес кие н аук и
Kazakh National Technical University named after K. I. Satpayev,
Almaty, Kazakhstan)
THE INTERNAL STRUCTURE OF A COMPLEX NUMBER
Annotation. The goal of this paper is to introduce generalized complex numbers in the form
. They
are obtained from consideration the square of an imaginary part of a complex number other than
. Representing an imaginary unit by an expression of the form
(where
and
are real numbers) let us divide
generalized complex numbers into elliptic, hyperbolic and parabolic complex numbers depending upon
and
meanings. It means that generalized complex numbers are characterized by the property of the imaginary unit.
Based on the theory of generalized complex numbers we solve some differential equation systems, and obtain
generalized Euler formula, similarity of Cauchy-Riemann conditions. Moreover, generalized complex numbers can be
used in the theory of elasticity, mechanics of fluid, theory of differential equations and other fields. But the most important result is that the solution of differential equation systems can give rise to generalized complex numbers which
are relative to the internal structure of a complex number.
Keywords: generalized complex number, double numbers, dual numbers, Cauchy-Riemann conditions, differential equations
The dear God has made the whole numbers, all the rest is man’s work.
Leopold Kronecker
1 Introduction
Mathematics matters related to solving algebraic equations led to the complex numbers emergence.
Henceforth the complex numbers are widely used not only in mathematics but in physics, mechanics, and
other areas [1], for example, in the theory of differential equations [2]. At the same time the usage of socalled dual and double numbers, i.e. the algebra of generalized complex numbers is very rare. This is because
they contain zero divisors.
In this work all these numbers arise from general theory. We attempt to expand and find new numbers.
To do so, we consider a system
of generalized complex numbers. The system
categorize into elliptic,
hyperbolic and parabolic systems of numbers. In particular, we show that complex, double and dual numbers
are subsystems of the above listed systems of numbers, respectively. In the following on the
system we
construct functions, that are generalized complex variables.
2 Key concepts
Let
denote the system of numbers of the form
(1)
where is a symbol (an object) which commuted with real numbers under multiplication.
Then to make the system (1) to form a group
should be presented in a form
, that
is, composition law is established for the system elements, according to which any pair of elements is associated with an element of the same system, and for all elements of the system
all axioms of the group must
be fulfilled [1].
Under the above assumptions we can call the system
a system of generalized complex numbers.
Let consider special cases to make a term be in accord with the name.
If
a generalized complex number corresponds to a complex number of
form.
If
, then we go to a dual number.
If
, then we get a double number.
Changing control parameters
we obtain different theories.
3 Field of generalized complex numbers
A generalized complex number of
form equipped with the addition
(2)
and the multiplication
(3)
Thus defined addition and multiplication operators are commutative and associative.
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The unit element of the generalized complex number can be defined simply by 0; that is for any
we have
. A generalized complex number
will be opposite to , as
. A generalized complex number
can be denoted simply by 1; that is for any
the multiplication
equals
to
.
The subtraction of generalized complex numbers and is a number , which satisfies next correlation
. It follows that
.
Thus generalized complex numbers represent ordered pairs of the real numbers
, which have the
addition operation according to the rule (2), and multiplied by the rule (3).
It is easy to see that the product of a generalized complex number
by a real number identified as a special case of generalized complex numbers multiplication:
(4)
If we introduce the conjugate of the generalized complex number in the form:
(5)
then the multiplication of element
and its conjugate
, i.e. the absolute
value (the modulus) will be as follows
(6)
The division of generalized complex numbers
by
is a number , which satisfies
next correlation
. It is not difficult to find from this condition
(7)
In what follows that for any generalized complex number
number
which equals to
exist its reciprocal of a
(8)
and the product of
by is 1.
It is easy to check that addition and multiplication of generalized complex numbers are related with
distributive law.
Generalized complex numbers which have the second part is equal zero in operations behave like real
numbers. Hence numbers of
form create a sub field in the field of generalized complex numbers
which can be identified with real numbers field.
4
system classification
It is clear from (6) that depending on the control parameters
the absolute value (the modulus) of
the generalized complex number can be positive, negative or zero, i.e. it may not correspond to a generally
accepted definition of an element modulus. Therefore it is necessary to understand the entity of the generalized complex number. For this purpose we consider modulus square of the generalized complex number as a
quadratic form of two variables
(9)
where
,
,
.
Matrix of coefficients of the quadratic form (9) will be as follows:
(10)
From general theory we know next theorem.
Theorem 1: Sum of coefficients of coordinates square for quadratic form is in next form:
(Spur of matrix)
The determinant of the coefficients of the leading terms is invariant:
(11)
(12)
It means that it does not change in passing from one Cartesian coordinate system to another one, i.e.
when coordinate axes are rotated and transferred in parallel way.
Changing control parameters
in (9) we get different theories. In order to understand intricacies
of these theories we bring the quadratic form to a canonical form, for that we compose a characteristic equa-
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● Ф и зико –мате мат ичес кие н аук и
tion in the form:
(13)
Deployment of the determinant (13) give us:
(14)
According to Viet theorem
,
(15)
(16)
By reason of matrix symmetries characteristic roots of the equation (14) are only real. Then the
meaning of
at any point
will be expressed by next equation
,
(17)
where
new coordinates of
point:
(18)
Let
then
the same correlation for old coordinates:
since
, we have
(19)
This implies that to make quadratic form (9) correspond to generally accepted modulus of the generalized complex number as non-negative real number characteristics roots must be positive. Such form is called
positive defined form and the system of numbers
called an elliptic system of numbers.
Assume that
. From the correlation (19) we conclude that in this case (except
the origin of coordinates)
. Such form is called negatively defined.
Now we consider a quadratic parabolic form. Assume that
; then
. Hence in this case the quadratic parabolic form (9) overall is positive or equals zero, and the system of numbers
is called parabolic system of numbers.
By a same way we find out when
the quadratic form is overall negative or equals zero.
Finally, we consider a quadratic hyperbolic form. Let
. By virtue of the fact that
and
represent borders of
varying of the unit circle the quadratic hyperbolic form is alternating in sign,
and the
system is called hyperbolic system of numbers. By this means that the set of generalized complex numbers divides into systems of numbers [1].
4.1 Elliptic system of numbers
Let be
. Then
, and characteristic roots of the quadratic form (9)
are positive.
In particular, the set of complex numbers is a subset of the elliptic system of numbers because of
.
Among the infinitive variety of number systems a special place is occupied by those that have basic
properties of real numbers: commutativity and associativity of multiplication, division possibility, i.e. unique
solution of
equations, and the possibility of introducing norm (modulus) so that the
norm of the numbers product is equal to the product of multipliers. According to Frobenius and Hurwitz theories only a system of complex numbers have all these properties.
One of the most important property of complex numbers expresses as:
(20)
multiplication modulus equals to multipliers modulus. If we denote and
in forms
,
equation (20) can be rewritten in the form
(21)
Naturally arise a question: whether is such property true for the elliptic number of systems? By direct
check we make sure that not only for the elliptic number of systems but also for generalized complex numbers next equation is true
,
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where
,
,
ылымдар ы
and
,
.
Now we can generalize Frobenius theorem: the generalized complex number gives the only possible
extension of real numbers with preservation of the algebraic properties.
4.2 Parabolic system of numbers
,
(22)
4.3 Hyperbolic system of numbers
(23)
4.4 Generalized Euler formula
By analogy with complex numbers for the generalized complex number we can define the concept of
the argument and introduce the trigonometric form, consider sequences, series and functions, differentiation,
integration, and other operations.
For complex numbers we have well-known Euler formula
,
(24)
For generalized complex numbers Euler formula will be as follows:
(25)
All main elementary functions can be obtained from this formula.
5 Differentiable functions. Cauchy-Riemann conditions
Assume that a function
is continuous at the point
. If there
exists a limit
in which
, then this limit is called derivative of function at the point
and
denoted by
, and the function
is called differentiable at point .
Theorem 2. Function
is differentiable at the point
1.
and
functions are differentiable at the point
2.
Cauchy-Riemann conditions are satisfied at the point
if:
,
5.1 Equivalence of Cauchy-Riemann conditions and
Given
. It is easy to express
its conjugate
:
,
Formally because of this we can consider the function
find by the rule of indirect differentiation:
(26)
condition
and variables in terms of
.
and
(27)
as a function of two variables
.
and . Let
(28)
The Theorem 2 is proved.
5.2 Cauchy-Riemann conditions similarity
Particular importance in the theory of complex variables function have analytic functions for which
the limit of the increment of a function to the increment of the argument does not depend on the ratio of real
and imaginary parts to the increment of the argument. Note that in literature apart from “analytic function”
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● Ф и зико –мате мат ичес кие н аук и
term we use other equivalent terms:
Necessary and sufficient conditions of function
1.
function differentiability in domain;
2. Cauchy-Riemann conditions:
,
,
Then from (26) we have:
regularity in domain :
,
.
,
(29)
, and
.
Sufficient conditions for
function regularity in domain:
1. Morera theorem;
2. Weierstrass’ first theorem.
Let consider a similar requirement in the case of a dual variable. We will follow the classic definition
of a function as a law of transformation a definition range into a value domain. If the definition range and the
value domain are dual numbers domain function can be represented by components:
,
where
, and are two real functions of two arguments.
For dual variable function
differentiability from the form (26) we get the similarity of CauchyRiemann conditions:
,
.
(30)
.
(31)
It arises from the second correlation that the function one variable
from the first one for :
function:
, and
where
is a function of only variable. Then common expression of dual variable function
that satisfies the derivative independence of the argument increment the way long will have form
below:
.
Using Cauchy-Riemann conditions for dual variable function we can get
function derivative:
.
This means that differentiation with respect to the dual variable is reduced to the differentiation of
real variable .
In the domain of double numbers the similarity of Cauchy-Riemann conditions has next form:
.
,
.
(32)
If we differentiate the first equation with respect to , and the second one with respect to , taking into
account derivatives persistence
we get:
Analogical equation can be developed for
.
(33)
.
(34)
:
Last two equations are called wave equations. The solution of these equations is a twice differentiable
function of double variable
, where
.
6 Examples
Newtons second law states that the force of an object is equal to its mass times its acceleration and expressed mathematically as shown below
(35)
One can rewrite equation (35) into scalar form
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(36)
where
.
So, in general, equation (36) shows a variable-coefficient differential equation of second order, which
can be converted into the system of first order equations:
(37)
In the mechanics we know this system as a dynamical system in the plane, and an analytical solution is
not always found. Then, we discuss this case, and first of all we consider homogenous linear system, which
the right hand side is complied with Cauchy-Riemann conditions, i.e.
,
(38)
where and are analytical functions of class
proach.
Example 1. Let and be in next form
. Now we analyze its solution ap-
(39)
To solve the system we multiply second equation by the common denominator i and add to the first
equation term wise. Hence
.
Let represent this equation in next form
,
(40)
where
and
.
Last equation is a first order differential equation with separate variables. Its general solution is
(41)
Summary. If the right hand side satisfies Cauchy-Riemann conditions then equation (41) is its general
solution.
Example 2.
(42)
The right hand side of the system (42) is not in compliance with Cauchy-Riemann conditions since
,
It means
,
(43)
condition is not satisfied. The question is how to find system solution? According
to the above method we rewrite equations system (37) in scalar form, after that we find system solution from
scalar equation solution.
Now let do the opposite: from the equation (40) move to a system equation like (37).
So
now
let
transform
equation
(40)
into
next
form:
. After separating the real
and imaginary parts, we obtain the following system of equations:
(44)
Now by comparing the above system with a standard form system
(45)
We estimate the connection between their coefficients
аз ТУ хабаршысы №4 2014
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Thus
.
If so we find the system (45) solution from formula (41)
(46)
By separating the real and imaginary parts of last equation we get systems general solution:
(47)
Here
Let go back to the Example 2. Where
,
,
,
,
.
,
,
,
.
Example 3. Find a general solution of the system
,
,
,
,
,
,
,
,
,
.
,
.
REFERENCES
1. Lavrentiev M. A., Shabat B.V. Problems of hydeodinmics and their mathematical models. Moscow // Nauka. – 1973.
2. Matveev N. M. Problems and exercises on ordinary differential equations // Minsk, Vysheishaya Shkola. – 1977.
Сағындықов Б.Ж.
Комплекс санның ішкі құрылымы
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ылымдар ы
Әдетте нақты сандар жиынын кеңейте отырып арифметиканың белгілі заңдылықтары орындалатынын
және өріс құратын комплекс сандарды аламыз. Сондықтан бұл мақалада комплекс сандардың тек ішкі
құрылымының өзгерісін және қайсыбір қолдануларын қарастырамыз.
Негізгі сөздер: жалпы комплекс сандар, қосарланған сандар, қос сандар, Коши-Риман шарттары,
дифференциалдық теңдеулер
Сагиндыков Б.Ж.
Внутренняя структура комплексного числа
Единственным обобщением действительных чисел с сохранением известных законов арифметики и образующее поле являются комплексные числа. Поэтому мы в данной статье займемся обобщением только внутреннего строения комплексных чисел. И рассмотрим некоторые применения.
Ключевые слова: обобщенное комплексное число, двойные числа, дуальные числа, условие КошиРимана, дифференциальные уравнения
УДК 533
С.А. Филько1, И.Н. Филько2, А.Е. Абденбаева2
( Жетысуский государственный университет им. И.Жансугурова, Талдыкорган, Республика Казахстан)
2
Международный университет информационных технологий, Алматы, Республика Казахстан)
1
ПЕРЕНОС ЭНЕРГИИ ПО ОДНОМЕРНОЙ ЦЕПОЧКЕ
ЛИНЕЙНО ВЗАИМОДЕЙСТВУЮЩИХ ТЕЛ
Аннотация. Изучены особенности переноса энергии от массивного тела к более легкому по одномерной
цепочке линейно взаимодействующих тел. Задача передачи механической энергии с помощью цепочки абсолютно упругих промежуточных тел обобщена для случая их бесконечно большого количества. Найдены условия передачи максимальной энергии от массивного тела к более легкому по цепочке промежуточных тел для
неупругих соударений. Доказано, что при каждом наборе параметров задачи (отношения масс и коэффициента
передачи энергии) имеется конечное число промежуточных тел, при котором малое тело получает наибольшую
возможную механическую энергию.
Ключевые слова: перенос энергии, абсолютно упругий удар, неупругий удар, одномерная цепочка тел.
Введение. Для изучения процессов передачи энергии в биологических и органических молекулах, анизотропных кристаллах, наноразмерных трубках и т.п. большое внимание уделяется процессу
распространения энергии вдоль одномерных структур [1-4]. Одномерные системы также применяются при решении фундаментальных вопросов теплопроводности. В связи с развитием и внедрением
композиционных и наноструктурированных материалов актуальной становится проблема передачи
энергии путем упругого удара на границах между мезоскопическими образующими элементами [5].
Если сталкиваются тела массами М и т (М > т), доля переданной энергии будет тем меньше, чем
больше отношение масс M / m .
Одна из возможностей увеличения доли передаваемой энергии заключается в размещении
между сталкивающимися телами нескольких «промежуточных» тел с массами т ≤ т1 ≤ т2 ≤… ≤ тп ≤
М. Для случая абсолютно упругого центрального удара значения промежуточных масс т, т1, т2,…,
тп, при которых от движущегося тела массы М будет передано покоящемуся телу массы т наибольшее количество энергии, должны образовывать геометрическую прогрессию – это известная «задача
Гюйгенса» [6].
В работе изучаются возможности передачи механической энергии между телами разных масс
по одномерной цепочке промежуточных тел: 1) при бесконечном количестве абсолютно упругих
промежуточных тел; 2) для случая неупругих промежуточных тел.
1. Задача Гюйгенса. Если тело массы М, движущееся со скоростью υ, ударяется о неподвижное тело массы т, то по законам сохранения импульса и энергии для случая абсолютно упругого центрального удара приобретаемая последним скорость u равна:
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