UDK 621.74.047:519.68
Original scientific article/Izvirni znanstveni članek
ISSN 1580-2949 MTAEC9, 41(5)213(2007)
A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS DURING CONTINUOUS STEEL CASTING
NOVA TOPOLOGIJA TRAJEKTORIJ MENISKUSA PRI NEPREKINJENEM LITJU JEKLA
Ice B. Risteski
2 Milepost Place # 606, Toronto, Ontario, Canada M4H 1C7 ice@scientist.com
Prejem rokopisa — received: 2007-04-13; sprejem za objavo - accepted for publication: 2007-07-10
The theoretical basis for a new computational method is given for the topology of the meniscus trajectories during continuous steel casting. The method is based on the solution of the meniscus equation in K3. Here the topology is treated only in the sense of the categorization of trajectories in an orientation space. This suggests a new type of efficient self-adaptive scheme suitable for the solution of the shape of the meniscus. In the present work a new approach is used to overcome previously unknown pathological, non-physical predictions in various constitutive models derived using closure approximations. The generalized meniscus equation as well as its stability is solved. Here it is shown, for the first time, that the cyclic change of the shape of the meniscus depends on the coordinates, while up to now the cyclic change of the meniscus was presented only as a function of the time over the expression of the mould velocity.
Key words and phrases: topology of trajectories, shape of the meniscus, meniscus equation, generalized meniscus equation, meniscus stability.
Dana je teoretična podlaga za nov izračun topologije trajektorij meniskusa pri neprekinjenem litju jekla. Podlaga metode je rešitev enačbe meniskusa K3. Topologija je obravnavana le v smislu kategorizacije trajektorij v nekem orientacijskem prostoru. To navaja na novo shemo, ki se sama primerno prilagaja za rešitev oblike meniskusa. V tem delu je uporabljen nov način, da bi se obvladalo preje neznane patološke, nefizikalne napovedi v različnih konstitutivnih modelih, ki so bili razviti z uporabo končnih aproksimacij. Rešeni sta splošna enačba meniskusa in njena stabilnost. Prvič je prikazano, da je ciklična sprememba oblike meniskusa odvisna od koordinat, medtem ko se je dosedaj ciklične spremembe meniskusa prikazalo samo kot funkcijo razmerja čas proti hitrosti kokile.
Ključne besede in stavki: topologija trajektorij, oblika meniskusa, enačba meniskusa, splošna enačba meniskusa, stabilnost meniskusa
1 INTRODUCTION
The study of the changes of the meniscus during continuous steel casting is neither an easy nor a simple task. It is, in fact, very complicated and requires a serious approach and hard work, because the meniscus's appearance depends on many factors of the continuous steel casting process. Generally speaking, this type of multidisciplinary research looks for a sufficient knowledge of steel metallurgy as well as the highest level of mathematics.
Many efforts have been spent to describe the shape of the meniscus during continuous steel casting. For instance, in 1 and 26 the authors considered the shape of the meniscus as a linear function and developed a model based on the Navier-Stokes equation for a hydrodynamic fluid. In 12, by virtue of complex functions, the movement of molten powder between the strand and mould wall is presented as a Newtonian fluid flowing between two parallel plates by neglecting the thermal contact resistance between the solidifying metal and the mould 11. Particular cases of this complex model appear
in the results given in 13,18. In later research works 14,16 the meniscus is shaped with an exponential function by virtue of the meniscus's dimensions 17.
In 24,25 the authors developed a dimensionless model for the meniscus, introducing Reynolds' lubrication theory. A model closely related to the free coating problem 27, is solved numerically and is compared with the published data.
On other hand, in the simulation model 2 a fixed shape of the meniscus is used to calculate the fluid flow and heat transfer. The authors in 5 account for the interdependence of the shape of the flux gape and the fluid flow therein, but still require some parameters to be selected rather arbitrarily, if impossible, to determine the experimental measurements.
Research in 6,7 showed that the movement of molten powder in the flux space may be determined by a pseudo-transient analytical solution of the Navier-Stokes equation. The validity of this solution is verified using an explicit finite-difference discretization method and the MATLAB software package. The simulation and behavior of interfacial mould slag layers in the continuous casting of steel are investigated in 8.
In 28,29,30 the authors modified the model for lubrication on the meniscus, given in 1416, with the difference between steel and flux density and extended it with the heat-transfer phenomena. They do not use the natural logarithm with base e for the description of the expo-
Materiali in tehnologije / Materials and technology 41 (2007) 5, 231-236
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ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
nential shape of the meniscus, and use instead the decade logarithm with base 10, without considering the correlation ln x « 2.303 lg x.
The newest research 410 is directed to cold model experiment on the infiltration of mould flux during the continuous casting of steel, neglecting the mould oscillations and the infiltration phenomena of molten powder derived from an analysis using the Reynolds equation.
Generally speaking, up to now in the literature the meniscus changes during continuous steel casting were approximately treated with one-dimensional mathematical models by virtue of some real function as a fixed shape. The treatment was adopted because the meniscus equation as a function of several variables was not known. In the present work, the introduced meniscus equation as a quasicyclic real function equation, i.e., its solution, can be used for all the possible cases of meniscus changes occurring cyclically during the mould cycle. In this way the mathematical description of the shape of the meniscus is much better, because the shape of the meniscus is closer to its own real shape.
Up to now in relevant references the form of the meniscus was presented as a cyclical change independent of time over the mould speed only, while, in this article, it is shown that the change of the form of the meniscus depends on other coordinates too.
The present work gives a new approach to meniscus vicinity in a sufficiently sophisticated way, which is more complete than previous treatments. With the goal to shed new light on this topic, with this article a new shape of the meniscus is introduced by virtue of the solution of the meniscus equation. With the intention to better understand this approach, emphasis is given to the engineering experience and the theoretical knowledge of mathematical modeling of the continuous steel casting process.
2 PRELIMINARIES
Let A = [a.ij\nxn be a real matrix. Suppose that by ele-
mentary transformations the matrix A is transformed into A = P1DP2, where P1 and P2 are regular matrices and D is
a diagonal matrix with diagonal entries 0 and 1, such that
the number of units is equal to the rank of the matrix A.
The matrix B = P21DP11 satisfies the equality ABA = A. This means that the matrix equation AXA = A has at least one solution for X.
If A satisfies the identity
Ar + k1Ar
• + kr-1A = O
where kr-1 ^ 0 and O is the zero nxn matrix, then the matrix
X = - (Ar-2 + k1Ar-3 + • • • + kr-2I)/kr-1
where I is the unit nxn matrix, is also a solution of the equation AXA = A.
Now we recall the following theorem proven in 20.
Theorem 2.1. If B satisfies the condition ABA = A, then
1° AX = O o X = (I - BA)Q
(X and Q are n x m matrices), 2° XA = O o X = Q(I - AB)
(X and Q are m x n matrices), 3° AXA = A o X = B + Q - BAQAB
(X and Q are n x n matrices), 4° AX = A o X = I + (I - BA)Q, 5° XA = A o X = I + Q(I - AB).
Throughout this paper, ^ is a finite-dimensional real vector space. Vectors from ^ will be denoted by Xi = (x1;,x2;,...,xn;)T, and also we denote with O = (0,0,...,0)T the zero vector in Let ® denote the exterior product in ^ and let k (1 < k < n) be an integer. With respect to the canonical basis in the k-th exterior product space ®k the k-th additive compound matrix A® of A is a linear operator on ®k ^ whose definition on a decomposable element x1 ® • • • ® xk is k
A[k(x1® • • • ®xk) = Y, x! ® • • • ®Ax® • • • ®xk. (2.1)
i=1
For any integer i = 1, 2,..., n!/k!(n - k)!, let (i) =
(i1,___,ik) be the i-th member in the lexicographic ordering
of integer k-tuples such that 1 < i'1 < • • • < ik < n. Then the (i, j)-th entry of the matrix A[k] = [q^ is
q, = ai1,i1 + • • • + aiKik if (i) = (j)
q,= (-1)^	(2.2)
if exactly one entry is of (i) does not occur in (,) and ,m does not occur in (i),
q, = 0
if (i) differs from (j) in two or more entries.
As special cases, we have Aw = A and An = trA 9. Let o(A) = {Ai, 1 < i < n} be the spectrum of A. Then the spectrum of Aik is a(Ak) = {An + • • • + An ,1 < ^ < • • • < ik < n}.
Let H denote a vector norm in The Lozinskii measure p on with respect to |-| is defined by
p(A) = lim (II + pAI -1)/p	(2.3)
p^ 0+
The Lozinskii measures of A = [ajnxn with respect to the three common norms
|xL = sup; |x;|
|x| 1 = Si W
|x|2 = (Si |xj2)1/2
are
where
ß„(A) = sUP; (aü + SKkfi Kl)
P1(A) = supt (akk +	\ak\)
p2(A) = stab [(A + At)/2] stab(A) = max {A, Aea(A)}
(2.4)
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ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
is the stability modulus of the matrix A, and AT denotes the transpose of A 3,p 41.
Definition 2. 2. A stable system is that system in which after the transitive action appearance a constant position is achieved 15,p38.
3 TOPOLOGY OF THE SOLUTIONS OF THE MENISCUS EQUATION
In this section we will give a complete analysis of the meniscus equation represented by a quasicyclic real functional equation for all possible cases. For that purpose we will use techniques for the solution given in
19,21,23
Let us consider now the equation
af(x1,x2,x3) + a2/(x2,x3,x1) + a3/(x3,x1,x2) =
= a/Xx^) + a/fe^^) + a/fe^x) (3.1)
f: ^ where ai, ai (1 < i < 3) are real constants. For equation (3.1) we suppose that |a1 + + |a3 > 0.
If we permute cyclically the variables in the equations (3.1), we obtain
af(x2,x3,x1) + a2/(x3,x1,x2) + a3/(x1 ,x2,x3) =
= a/Xxx) + a/(x3,x3,x1) + a/fox^) (3.2)
af(x3,x1,x2) + a2/(x1,x2,x3) + a3/(x2,x3,x1) =
= a/(x3,x3,x1) + a/fox^) + affex^) (3.3)
The determinant for the system of the equations (3.1), (3.2) and (3.3) is
A, =
Let us note the identity
A = (a1 + a2 + a3)[(a1 - a2)2 + (a2 - a3)2 +
+ (a3 - a1)^/2	(3.4)
First we consider the case
1° Let a1 = a2 = a3 = 0. Now the system (3.1), (3.2) and (3.3) takes the form
a/(x1,x2,x3) + a/(x2,x3,x1) + a/fex^) = 0 a/(x1,x2,x3) + a/x^^x) + a/fex^) = 0 (3.5)
a/x ,x2,x3) + a/(x2,x3 ,x1) + affex^) = 0
If A * 0, then the system (3.5) impliesf(x1,x2,x3) = 0.
Now let A = 0. According to (3.4), this is possible if the real constants a1, a2, a3 satisfy either a1 = a2 = a3 or a1 + a2 + a3 = 0.
First, let us suppose that a1 = a2 = a3 (* 0). Then system (3.5) is equivalent to the equation
/x^x) + f(x1,x3,xl) + /(x^x^) = 0 (3.6) whose general solution according to 22 is
/(x^x,) = Fx^x) - F^x,^) (3.7)
where F: ^ is an arbitrary function.
Now let us suppose that A = 0, although the real constants are not all equal. Then necessarily a1 + a2 + a3
= 0 and we can suppose, without any loss of generality, that a1 * a2. In this case we set a3 = - a1 - a2 and system (3.5) can be written in the form
atfxxx) - /(x^x)] = a/xxxd - /(x^x)] a/x^x) - f^x^)] = afx^x.) - f(x1,x3,xl)] afx^xxj - f(xl,x1,x3)] = a1\f(x1,x3,xl) - f(x3,x1,x2)]
From this we derive easily
(a13 - a23)f(x1,x2,x3) - /fo,^)] = 0 With the assumption that a1 * a2 and they have real values, equation (3.4) reduces to
/x^x) -f(x2,x3,x1) = 0	(3.8)
According to 20, the general solution of the above functional equation is
f(x1,x2,x3) ^ F(x1,x2,x^) + F(x2,x",x1) + F(x^,x1 ,x2) (3.9)
where F: ^ is an arbitrary function.
The above results concerning the cyclic functional equation
a1f(x1,x2,x3) + a2/(x2,x3,x1) + a3/(x3,x1,x2) = 0
can be derived from those in 20, where a1, a2, a3 are real constants.
From now we suppose that |a1 + |a2 + a > 0. We can distinguish the following two cases:
2° Let A * 0, from (3.1), (3.2) and (3.3) we obtain
a1F(x1,x2) + a2F(x2,x3) + a3F(x3,x1) a2 a3 f(x1,x2,x3) = a1F(x2,x3) + a2F(x3,x1) + a3F(x1,x2) a1 a2 a1F(x3,x1) + a2F(x1,x2) + a3F(x2,x3) a3 a1
where F: ^ ^ is defined by F(u,v) = f(u,u,v)/A. If we introduce the notations
	a! a 2 a 3		a1 a 2 a 3		a1 a 2 a 3
A, =	a 3 a 1 a 2	A 2 =	a 2 a 3 a1	A 3 =	a, a2 a3
	a 2 a 3 a 1		a 1 a 2 a 3		a 3 a1 a 2
then we can write
f(x1,x2,x3) = AF(x1,x2) + A2F(x2,x3) + A3F(x3,x1) (3.10)
For (3.10) to be a solution of the functional equation (3.1), the following condition must be satisfied:
a^(A - A2)F(x1,x2) - A3F(x2,xO - A^x^)] + + a^(A - A2)F(x2,x3) - A3F(x3,x2) - AFtexi)] + + a3[(A - A2)F(x3,x1) - A3F(x1,x3) - A1F(x3,x3^ = 0
(3.11)
By a cyclic permutation of the variables x1, x2, x3 in (3.11) we obtain two new equations. The system of these three equations has a nontrivial solution with respect to (A - A2)F(xi,xi+1) - A3F(xi+1,xi) - A^xx), i = 1, 2, 3 (with the convention x4 = x1) if the following condition is satisfied
= 0
(3.12)

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By virtue of an equality of the type of (3.4) this is true if
a1 + a2 + a3 = 0 or a1 = a2 = a3
First, we will consider the case
a1 + a2 + a3 = 0	(3.13)
Since ü1 + Ü2 + a.3 ^ 0 (because of the assumption A ^ 0 and (3.4)), by putting into equation (3.1) X1 = X2 = X3 we derive F(X1,X2) = 0. By using the last equality, the equation (3.11) for X3 = X1 becomes
[ai(A - A2) - a2A3]F(xi,x2) -
-	[a1A3 - a2(A- A2)]f(x2,x1) = 0 (3.14)
If we change the places of X1 and X2, the equation (3.14) is transformed into
-	[a1A3 - a2(A - A2)]F(x1,x2) +
+ [a1(A - A2) - a2A3]F(x2,^) = 0 (3.15)
Let
(a12- a/)[(A - A2)2- A32] * 0 then from (3.14) and (3.15) it follows that F(X1,X2) = 0 and then from (3.10)/(X1,X2,X3) = 0. The condition
(a12- a22)[(A - A2)2- A32] = 0 (3.16) implies (A - A2)2 - A32 = 0. Let us suppose that the last equality is not true. Now, if we set X3 = X2 into (3.11), we obtain
(a12- a32)[(A - A2)2- A32] = 0
The last equality, with (3.16), gives a12 = a22 = a32 which, by virtue of the assumption (3.13), yields a1 = a2 = a3 = 0, and this contradicts the hypothesis a1 + a2 + |a3 > 0.
Thus, we have A - A2 = ± A3. For the case A - A2 = A3 0), equation (3.14) yields
(a1 - a2)AJF(x1,x2) - A3F(x2,x1^ = 0
so that, for a1 ^ a2, we have
f(x1,x2) = g(x1,x2) + g(x2,x1)	(3.17)
where G is an arbitrary function ^ ^ such that
G(X1,X1) = 0
If a1 = a2, then necessarily we must have a1 ^ a3, (because otherwise we would have a 1 = a 2 = a 3 = 0) and by a procedure analogous to the one above we obtain
(3.17).
Let A - A2 = - A3 (^ 0), from (3.14) we obtain
(a1 + a2)AJF(x1,x2) + F(x1,xi)\ = 0 (3.18)
For a1 + a2 ^ 0, the general solution of equation
(3.18)	is given by
f(x1,x2) = g(x2,x1) - g(x2,x1), G: ^ ^ (3.19)
If a1 + a2 = 0, then from (3.13) we deduce a3 = 0, and then a1 + a3 = a1 ^ 0 and we obtain (3.19) by an analogous procedure.
The condition (3.11), for the case A3 = A - A2 = 0, is satisfied for every function F(X1,X2) with the property F(X1,X1) = 0.
If (A - A2)2 ^ A32, then, as was mentioned above, equations (3.14) and (3.15) have a trivial solution as a general solution. According to (3.10) we obtain
/(X1,X2,X3) = 0
Now we suppose that (3.12) is satisfied but (3.13) is not. This means that
a1 = a2 = a3 ^ 0	(3.20)
It immediately follows that
A1 = A2 = A3(* 0)
The quasicyclic equation (3.11) implies
(A - A1)F(X1,X2) - AF(X2,X1) - A1F(X1,X1) =
= APX) - APX2)	(3.21)
where P is an arbitrary function ^ ^ ^
For a1 = a2 = a3 and X1 = X2 = X3 equation (3.1) becomes
(a1 + a2 + a3 - 3a1)F(x1,x1) = 0
Let a1 + a2 + a3 = 3a1, then 3A1 = A and the equality (3.21) takes the form
2F(X1X) - F(X2,X1) = P(X1) - P(X2) + R(X1)
where R(X1) = F(X1,X1).
By a permutation of the variables X1 and X2 it follows that
- F(X1,X2) + F(X2,X1) = P(X2) - P(X1) + R(x) From the last two equalities we obtain
F(X1,X2) = [P(X1) - P(x) + 2R(X1) + r(x2)]/3 By using the last equality, from (3.10) it follows that
/(X1,X2,X3) = QX) + Q(X2) + Q(x) (3.22)
where Q(X1) = A1R(X1).
Let a1 + a2 + a-3 ^ 3a1, then F(x1,x0 = 0 and the formula (2.21) yields
(A - A1FX1X2) - A1F(X2,X1) = APX1) - A1P(X2) (3.23)
From the last equality, with a permutation of the variables we obtain
- AFX1,X2) + (A - A1)F(X2,X1) = Apx) - A1P(X1)
The determinant of the system consisting of the last two equations is A(A - 2A1). If A ^ 2A1, the solution of the last two equations is
F(X1,X2) = A^P(X1) - P(X2)]/A
Then the equality (3.10) gives
/(X1,X2,X3) = 0
Let A = 2A1, then from (3.23) we obtain F(X1,X2) - P(X1) = F(X2,X1) - P(X2) The general solution for the last equation is
F(X1,X2) = P(X1) + G(X1,X2) + G(X2,X1) (3.24)
where P: ^ ^ ^ and G: ^ ^ are arbitrary functions such that
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G(X1,X2) = - P(x1)/2
According to the last relation, the equality (3.24) takes the form
F(x1,x2) = G(x1,x2) + G(x2,x1) - 2G(x1,x1)
and the equality (3.10) becomes
fix^x?) = G(x1,x2) + G(x2,x1) - 2G(x1,x2) + G(x2,x3) + + G(x3,x2) - 2G(x2,x2) + G(x3,x1) + G(x1,x3) - 2G(x3,x3)
(3.25)
where we have replaced G(x1,x2)A1 by G(x1,x2). For
* 0
from (3.10) we obtain
(A - A2)F(x1,x2) - A3F(x2,xO - A^xO = 0 (3.26)
First we suppose that A - A1 - A2 - A3 ^ 0. In this case, with the substitution x2 = xu equation (3.26) reduces to F(x1,x0 = 0. On the basis of the last equality, the equation (3.26) becomes
(A - A2)F(x1,x2) - A3F(x2,xO = 0
From the permutation of the variables x1 and x2, from the above equation it follows that
- A3F(x1x) + (A - A2)F(x2,x1) = 0
The system of the last two equations has a nontrivial solution if and only if the following condition (A - A2)2 = A32 is satisfied.
Let A - A2 = A3 (^ 0), then we obtain
F(x1,x2) = G(x1,x2) + G(x2,x1) where G satisfies G(x1,x0 = 0.
For the case A - A2 = - A3 (^ 0) the general solution
is
F(x1,x2) = G(x1,x2) - G(x2,x0
For A - A2 = A3 = 0, the unique condition that must be satisfied by the function F is F(x1,x1) = 0.
The condition (A - A2)2 ^ A32 gives F(x1,x2) = 0 Next we will pass on to the case A - A1 - A2 - A3 = 0. Now equation (3.26) can be written as
(A1 + A3)F(x1,x2) - A3F(x2,x1) = ARxO
where R(x{) = F(x1,x1). By a permutation of the variables x1 and x2 we obtain
- A3F(x1,x2) + (A1 + A3)F(x2,x1) = A1R(x2) The determinant of this system is (A1 + 2A3)A1. If it is not zero, then
F(x1,x2) = [(A1 + A3)R(x1) + A3R(x2)]/(A1 + 2A3)
From (3.10) it follows that
f(x1,x2,x3) = (A12 + A1A3 + A32)ß(x1) + (A1A2 + A1A3 +
+ A2A3)ß(x2) + A3Aß(x3)
where
Ö(x1) = R(x1)/(A1 + 2A3) Let A1 = 0, A3 * 0. Then
F(x1,x2) = F(x2,x1)
Thus
F(x1,x2) = G(x1,x2) + G(x2,x1)
where G is an arbitrary function K2 ^ K, and f(x1,x2,x3) is given by the formula (3.10).
Now we suppose that A1 = - 2A3 ^ 0. Then
F(x1,x2) + F(x2,x1) = 2R(x1)	(3.27)
By a permutation of the variables x1 and x2 we obtain
F(x2,x1) + F(x1,x2) = 2R(x2)	(3.28)
From (3.27) and (3.28) we get
R(x1) = R(x2) = c
Thus (3.27) takes on the form
[F(x1,x2) - c] + [F(x2,x0 - c] = 0
which implies that
F(x1,x2) = G(x1,x2) - G(x2,x1) + c
where G: K2 ^ K is an arbitrary function and c is an arbitrary real constant. Now from (2.10) we find
f(x1,x2,x3) = - 2A^G(x1,x2) - G(x2,x0] + + (A + A3HG(x2,x3) - Gfe^)] + + A^G(x3,x1) - G(xl,x3)] + c
where c is (another) arbitrary real constant.
In the case A1 = A3 = 0 equation (3.26) is satisfied for every function F: K2 ^ K.
Now we will use the following result. Lemma 3. 1. Let A ^ 0. Then the system
A1 = 0, A2 - A = 0, A3 = 0	(3.29)
implies a = 03, «2 = fl1, «3 = a2.
Proof. The system (3.29) can be written in the form
An(«1 - «3) + A^fe - fl1) + A13(a3 - Ü2) = 0
A21(a - Ü3) + A22(a2 - aù + A23(«3 - a2) = 0 (3.30)
A31(«1 - 03) + A32(a2 - aù + A33(a3 - 02) = 0
where Aij is the cofactor of the element aij (1 < i, j < 3) of the determinant A. The system (3.30) is a homogeneous linear system with respect to a - 03, «2 - fl1, «3 -a2. Its determinant is A2 ^ 0, so that it has only the zero solution.
Thus, the equation
af(x1,x2,x3) + a/fe^,^) + a/fex^) = = afx^x-) + af(x2,x2,x3) + a/fe^x) has the general solution f(x1,x2,x3) = F(x2,x3). 3° Let A = 0. Then from (3.4) it follows that
a1 + a2 + a3 = 0 or a1 = a2 = a3 ^ 0.
First we will consider the case a1 = a2 = a3. From (3.1) and (3.2) we obtain
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ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
(a - a3)F(x1,x2) + («2 - a^Ffe^) +
+ (a3 - a2)F(x3,x1) = 0	(3.31)
with the notation fx1,x1,x2) = F(x1,x2). If a1 = a2 = a3, then the condition (3.31) is satisfied for every function F. For the case a1 = a2 = a3 (^ 0), equation (3.1) takes the form
a f(^1 ,x2,x3) a fx x ,x2) + af(x2,x3,x1 ) - af(x2,x2,x3) + a1fx3,x1,x2) - a1f(x3,x3,x1) = 0 (3.32)
This quasicyclic equation has the general solution
f(x1,x2,x3) = (a1/a1)F(x1,x2) + U(x1 ,x2,x3) - U(x2,x3 ,x1)
(3.33)
with the notation fx1,x1,x2) = F(x1,x2).
By substitution of (3.33) into (3.32) we obtain
F(x1,x2) - (a1/a1)F(x1,x1) - U(x1,x2,x3) + U(x1,x2,x1) + F(x2,x3) - (a^a^Fxx) - Ufe^^) + U(x2,x3,x2) + F(x3,x1) - (a1/a1)F(x3,x3) - U(x3,x3,x1) + U(x3,x1,x3) =0
This quasicyclic equation has the general solution
F(x1,x2) = (a1/a1)F(x1,x1) + U(x1,x1,x2) - U(x1,x2,x1) + + R(x0- Rfe)
where R is an arbitrary function ^ ^
By using the last equality, for a1 = a1, the equality (3.33) becomes
f(x1,x1,x2) = U(x1,x2,x3) - U(x2,x3,x0 + U(x1,x1,x2) -
- U(x1,x2,x1) + 5(x1) - R(x2)	(3.34)
where ^ ^ ^ is such that F(x1,x0 = S(x0 - R(x1).
For a1 ^ a1 it follows from (3.1) that F(x1,x1) = 0. According to the last identity, the equality (3.33) is transformed into
f(x1,x2,x3) = U(x1,x2,x3) - U(x2,x3,x0 + + (a^M U(x1,x1,x2) - U(x1,x2,x1)] + R(x1) - Rfe)
Now we will suppose that the parameters ai (1 < i < 3) are not all equal. Let a1 ^ a3. According to the equality (3.31) for x3 = a (a real constant) we obtain
F(x1,x2) = K(x1) + H(x1)	(3.35)
where we used the notations
K(x1) = [(a2 - a3)/(a1 - a3)]F(a,x0, H(x2) = = [(a1 - a2)/(a1 - a3)] F(x2,a)
If we substitute F(xux2) given by the expression (3.35) into (3.31), and if we set x1 = u, x2 = x3 = b (a real constant) and if, on the other hand, we set x1 = x3 = b, x2 = u, we obtain respectively
(a1 - a3)[K(u) - K(b)] + (a3 - a2)[H(u) - H(b)] = 0
(3.36)
(a2 - a1)[K(u) - K(b)] + (a1 - a3)[H(u) - H(b)] = 0
(3.37)
The determinant of this system is
a. 3 3
= [(a1 - a2)2 + (a - a3)2 + (a3 - a1)2]/2
According to our assumption its value is not 0, then from (3.36) and (3.37) we find K(u) = K(b) and H(u) = H(b), hence
F(x1,x2) = m (a real constant) (3.38) Now the equation (3.1) becomes f(x1,x2,x3) - n + f(x2,x3,x1) - n + f(x3,x1,x2) - n = 0 (3.39) where
n = (a1 + a2 + a3)m/3a1
The general solution of the cyclic functional equation (3.39) is
f(x1,x2,x3) = p(x1,x2,x3) - p(x2,x3,x1) + n (3.40)
From (3.38) and (3.40) we find
m = F(x1,x2) = p(x1,x1,x2) - p(x1,x2,x1) + n
If we put into the last equality x2 = x1, then m = n. This is possible if
a1 + a2 + a3 = 3a1 or n = 0
Moreover,
p(x1,x1,x2) - p(x1,x2,x1) = 0	(3.41)
Now we will use the following result. Lemma 3.2. Let f(x1,x2,x3) be a function of the form
fx^x,) = p(x1,x 2,x3) - p(x2,x3,x1)
such that p(x1,x1,x2) = 0. Then
/(x^^) = U(x1,x2,x3) - U(x2,x1,x3) - U(x2,x3,x1) -
- U(x1,x3,x2)	(3.42)
where U: ^ ^ is an arbitrary function.
Proof. Let p(x1,x2,x3) satisfies equation (3.41). We are looking for p(x1,x2,x3) in the form
p(x1,x2,x3) = k1q(x1,x2,x3) + k2q(x1,x3,x2) + k3q(x2,x1,x3) + + k4q(x2,x3,x1) + k5q(x3,x1,x2) + k6q(x3,x2,x1)
where q: ^ ^ and ki (1 < i < 6) are real constants. By a substitution into (3.41) we find
k5 = k1 - k2 + k3, k6 = k4 - k1 + k2
Thus
f(x1,x2,x3) = k[q(*2,x1,x3) - q(x1,x2,x3^ -l qx^,^) -
- q^^O] + (t- kHq(x3,x2,x1) - q(x3,x1,x2)] where k, are real constants such that k = k3 - k2, = k4 - k1
If we denote
U(x1,x2,x3) = lq(x2,x3,x1) + kq(x3,x1,x2)
we obtain (3.42). Conversely, each function of the form (2.42) satisfies fxuxuxi) = 0 for arbitrary U: ^ Moreover, f(x1,x2,x3) satisfies
/(x^^) + fx^,^) + ffex^) = 0
We note that the representation (3.42) can be obtained just by putting
p(x1,x2,x3) = U(x1,x2,x3) - U(x2,x1,x3)
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Thus the general solution of the equation (3.1) is in this case given by
/(X1,X2,X3) = U(X1,X2,X3) - U(X2,X1,X3) -
- U(x2,x3,x1) + U(x1,x3,x2) + n (3.43)
where U is an arbitrary function S3 ^ S and n is an arbitrary real constant, n = 0 if a1 + a2 + a3 ^ 3a1.
Now we will consider the case that a1, a2, a3 are not all equal. Thus we have
a1 + a2 + a3 = 0	(3.44)
Without any loss of generality, we can assume that a1 ^ a2. Equation (3.1) can be written as
aA /(X1,X2,X3) - /(X3,X1,X2^ - a2^ /(X3,X1 ,X2) - /(X2,X3,X1^ =
= a1F(x1,x2) + a2F(x2,x3) + a3F(x3,x1) (3.45)
where/(X1,X1,X2) = F(X1,X2). Also from (3.2) and (3.3) it follows that
aA /(X2,X3,X1) - /(X1,X2,X3)] - ad /(X1,X2,X3) - /(x1 ,X3,X2)] =
= a1F(x2,x3) + a2F(x3,x1) + a3F(x1,x2) (3.46)
aA /(X3,X1,X2) - /(X2,X3,X1^ - aÄ /(X2,X3,X1) - /(X1,X2,X3)] =
= a1F(x3,X1) + a2F(x1,X2) + a3F(x2,X3) (3.47)
By adding (3.45), (3.46) and (3.47) we obtain
(a1 + a2 + a3^F(x1,x2) + F(x2,x3) + F(x3,x1^ = 0
For a1 + a2 + a3 ^ 0, the following condition must be satisfied
F(X1,X2) + F(X2,X3) + F(X3,X1) = 0
This cyclic functional equation has the general solu-
tion
(3.48)
F(X1,X2) = P(X1) - P(X)
where P is an arbitrary function S ^ S.
From the equation (3.45), (3.46) and (3.47), if we take into account (3.48), we get
/(X1,X2,X3) + [1/(a13 - a23y {a12[a1P(x1) + a2P(x2) + + a3P(x3)] + a22[a3P(xO + a1P(x2) + a2P(x3)] + + a^fePfe) + a3Pfe) + a^fe)]} = = /(X1,X2,X3) + [1/(a13 - a23)]{a^aPX) + a2P(x3) + + a3P(x1y + a22[a3P(x2) + a1P(x3) + a2P(xO] +
+ a^fePfe) + a3P(x3) + a1P(x1)]} The last equation has the general solution
/(X1,X2,X3) + [1/(a13 - a23^ {a12[a1P(x2) + a2P(x3) + + a3P(x1y + a22[a3P(x2) + a1P(x3) + a2P(xO] + + a^fePfe) + a3Pfe) + a^^)]} = = P(X1,X2,X3) + P(X2,X3,X1) + P(X3,X1,X2) (3.49)
where p is an arbitrary function S3 ^ S.
By virtue of (3.48) /(X1,X2,X3) = P(X1) - P(X2), then from (3.49) it follows that
P(X1) - P(x2) + [1/(a13 - a/)]{aA(a1 + a3)P(xO + + a2P(x2^ + a22[(a2 + a3)P(x1) + aPfe)] + a^tfe + a2)P(x1) + a3P(x2^ = = P(X1,X1,X3) + P(X1,X2,X1) + P(X2,X1,X1) (3.50) For X2 = X1 this equality takes the form
[(a1 + a2 + a3)(a12 + a22 + a1a2)/(a13 - a23)]P(x1) = = 3p(x1,x1,x1)
which implies
P(x1) = [3(a1 - a2)/(a1 + a2 + a3^p(x1,x1,x1)
Now from (3.49) we find the general solution in the form
/(X1,X2,X3) = p(x1,x2,x3) + p(x2,X3,X1) + p(x3,X1,X2) -
- [3/(a12 + a23 + a1a2)(a1 + a2 + a3)] {a1^a1p(x2,x2,x2) + + a2p(x3,x3,x3) + a3p(x1,x1,x1y + a2^a1p(x3,x3,x3) + + a2p(x1,x1,x1) + a3p(x2,x2,x2)] + a1a2[a1p(x1,x1,x1) +
+ apX2,X2,X2) + a3p(x3,X3,X3^ (3.51)
where p: S3 ^ S must satisfy the following condition derived from (3.50)
[3(a1 - a2)/(a1 + a2 + a3)] |p(x1,X1,X1) - p(x2,X2,X2)] +
+ [3/(a12 + a23 + a1a2)(a1 + a2 + a3)] {a12[(a1 + + a2)p(x1,X1,X1) + a2p(x2,x2,x2)] + a22[fe + a3)p(x1,X1,X1)+
+ a1p(x2,X2,X2)] + a^tfe + a2)p(x1,X1,X1) + + a^(x2,X2,X2^U = p(x1,X1,X2) + p(x1,X2,X1) + p(x 2,X1 ,X1)
(3.52)
It is easy to see that (3.52) is an equation of the form
p(x1,X1,X2) + p(x1,X2,X1) + p(x2,X1,X1) =
= (3 - y)p(x1,X1,X1) + yp(x2,X2,X2) (3.53)
where the real constant y is given by
Y — - 3(a1 - a2)/(a1 + a2 + a3) + 3(a12a2 + a22a1 + + a1a2a3)/[(a12 + a22 + a1a2)(a1 + a2 + a3)].
Lemma 3. 3. Let f(x1,x2,x3) be a function of the form
/(X1,X2,X3) = p(x1,X2,X3) + p(x2,X3,X1) + p(x3,X1,X2) (3.54) such that f(x1,x2,x3) = 0. Then
/(X1,X2,X3) = U(X1,X2,X3) + U(X2,X3,X1) + U(X3,X1,X2) -
- U(X2,X1,X3) - U(X1,X3,X2) -U(X3,X2,X1) (3.55)
where U: S3 ^ S is an arbitrary function.
Proo/. We are looking for a function of the form
p(X1,X2,X3) = k^q(X1 ,X2,X3) + ^2^(X2,X3,X1) + ^3^(X3,X1 ,X3) + + k4q(x2,x1,x3) + k5q(x1 ,X3,X2) + k6q(x3,x2,x1)
where k (1 < i < 6) are real constants, satisfying
p(x1,X1,X2) + p(x1,X2,X1) + p(x2,X1,X1) = 0 (3.56)
for any function q: S3 ^ S. By a substitution into (3.56) we find
k + k2 + k3 — k^ + k5 + k6
Thus
/(X1,X2,X3) — (k1 + k2 + k3)[ q(X1,X2,X3) + q(X2,X3,X1) + + q(x3,X1,X2^ - (k1 + k2 + k3)[q(x2,X1,X3) + q(x1,X3,X2) + + q(x3,X2,X1)]
If we put
U(X1,X2,X3) — (k1 + k2 + k3) q(x1,x2,x3)
we obtain the representation (3.55).
We note that the representation (3.55) can be obtained from (3.54) by putting
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p(x1,x2,x3) = U(x1,x2,x3) U(x2,x1,x3)
Let us suppose that p(x1,x2,x3) = S(xO ^ 0. The equation
p(x1,x1,x2) + p(x1,x2,x1) + p(x2,x1,x1) = (3 - y)S(x^ + ySfe)
has a no constant solution of the form
p(x1,x1,x2) = S(x0
or, more generally,
p(x1,x2,x3) = m1S(x1) + m2S(x2) + (1 - m1 - m2)S(x3)
only if y = 1. Indeed, we have
(y - 1)[S(x1) - Sfe)] = 0
On the other hand, any S(x0 = a, where a is a real constant, satisfies the last equality. Let us put
p(x1,x2,x3) = Ü^xx) + S(x1)
Then Ü(x1,x2,x3) satisfies an equation of the form (3.56) and we have proved this result.
Corollary 3. 4. Let fx1,x2,x3) be a function of form
(3.54)	such that p(x1,x2,x3) satisfies (3.53). Then
f(x1,x2,x3) = U(x1,x2,x3) + U(x2,x3,x1) + U(x3,x1,x2) -
U(x2,x1 ,x3) U(x1 ,x3,x2) U (x3,x1,x2) +
+ S(x1) + Sfe) + S(x3)
where U: K3 ^ K is an arbitrary function and S is an arbitrary function K ^ K for y =1, S is equal to a constant a eK otherwise.
Thus from (3.51) we find that fix^x) is given by
(3.55)	if y * 1
fx^x,) = S(x0 + S(x2) + S(x3) - [3/(a12 + a22 + + a1a2)(a1 + a2 + a3)] {a12[a3S(x1) + a1S(x2) + + a2S(x3)] + a22[a2S(x1) + a3S(x2) + a1S(x3)\ + + a1a^a1S(x1) + a2S(x2) + a3S(x3)]} + U(x1,x2,x3) + U (x2,x3,x1) + U(x3,x1,x2) U (x2,x1,x3)
U(x1,x3,x2) - U(x3,x2,x1).
Now we pass on the case a1 + a2 + a3 = 0. Then from (3.45), (3.46) and (3.47) we obtain
f(x3,x1,x2) - [1/(a13 - a23^{a12[a1F(x3,x1) - a2F(x2,x3)]+ + a2^a1F(x1,x2) - a2F(x3,xi)] + a^ta^fe^) -- a2F(x!,*2)]} = = fx^X) - [1/(a13 - a2')]{ a2 [aF^x) -- a2F(x3,x1^ + a^aFxx) - a2F(x1x)] +
+ a1a^a1F(x3,x1) - a2F(x2,x3)]} (3.57)
The general solution of equation (3.57) is given as
f(x1,x2,x3) = [1/(a13 - a23)]{a^aF^x) - a2F(x3,x1)] + + a22[alF(x2,x3) - a2F(x1,x2)] + aalaF^xi) -- a2F(x2,x3)]} + q^xx) + q(x2,x3,x1) + qfexx)
(3.58)
where q: K3 ^ K is an arbitrary function. From the equality (3.58) we obtain
F(x1,x2) = [1/(a13 - a23)]{a12[a1F(x1,xO - a2F(x2,xO] -+ a22[a1F(x1,x2) - a2F(x1,xO] + a^Ia^fex) -
- a2F(x1,x2^ + q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1)
(3.59)
For x2 = x1 (3.59) yields
F(x1,x2) = [(a1 - a2)/(a1 - a2)]F(x1,x1) + 3q(x1,x1,x1)
(3.60)
If a1 - a2 = a1 - a2, then q(x1,x1,x1) = 0 and F(x1,x0 = P(x1), where P is an arbitrary function K ^ K. Now we have
[1 + a2(a1 - a1)/(a12 + a22 + a1a2^ F(x1,x2) + + [a1(a1 - a1)/(a12 + a22 + a1a2^ F(x2,x1) = = [a1(a1 + a2) + a22] P(x1)/(a12 + a22 + a1a2) + + q(x1,x1,x2) + q(x1,x2,x0 + q(x2,x1,x1) (3.61) First we consider the particular case, a1 = a1 (then a2 = a2, a3 = a-3 = - (a1 + a2)) Now
F(x1,x2) = P(xJ + q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) Thus, we find that the functional equation
a1f(x1,x2,x3) + a2f(x2,x3,x1) - (a1 + a2)f(x3,x1,x2) = = af(x1,x1,x2) + a2f(x2,x2,x3) - (a1 + a2)f(x3,x3,x1)
has the general solution
f(x1,x2,x3) = P(x0 + q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) +
+ q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2)
where P: K ^ K is an arbitrary function and q: K3 ^ K is an arbitrary function satisfying q(x1,x1,x1) = 0.
Now we consider equation (3.61) in the general case. With a permutation of the variables x1 and x2 we derive the equation
[a1(a1 - a1)/(a12 + a22 + a1a2^F(x1,x2) + + [1 + a2(a1 - a1)/(a12 + a22 + a1a2^ F(x2,x1) = = [a1(a1 + a2) + a22] P(x2)/(a12 + a22 + a1a2) + + qfexx) + q(x2,x1,x2) + q(x1,x2,x2) (3.62) The determinant of the system (3.61), (3.62) is
(a22 + a1a1 + a2a1)(2a12 + a22 - a1a1 + a2a1)/
/(a12 + a22 + a1a2)2	(3.63)
If this expression is not 0, then the solution of this system is
F(x1,x2) = [(a12 + a22 + a2a1)P(x1) - a1(a1 - a^Pfe)]/ /(2a12 + a22 - a1a1 + a2a1) + [(a12 + a22 + a1a2)/ /(a22 + a1a1 + a2a1)(2a12 + a22 - a1a1 + a2a^] x x{(a^ + a22 + a2a1Hq(x1,x1,x2) + q(x1,x2,x1) + + q(x2,x1,x1)] - a1(a1 - a^q^x,^) + + q(x2,x1,x2) + q(x1,x2,x2)]}
Now let us suppose that the expression (3.63) is 0. First let
a22 + a 1(a1 + a2) = 0
If a1 + a2 = 0, then a1 = a2 = a3 = 0, which is contradiction. Thus
a1 = - a22/(a1 + a2), a2 = - a12/(a1 + a2)
Now (3.58) takes the form
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ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
f(x1,x2,x3) = [alF(x3,xl) + a2F(x2,x3)]/(al + a2) +
+ q(xi,x2,x3) + q(x2,x3,xi) + q(x3,xi,x2)
while (3.61) becomes
[aj/(aj + a2)] [F(xi,x2) - FfeXi)] = = q(xi,xi,x2) + q(xi,x2,xi) + qfeXiXi) (3.64) Equation (3.64) implies
q(xi,xi,x2) + q(xi,x2,xi) + qfexiXi) = 0
and
F(xi,x2) - F(x2,xi) = 0
i.e.,
q(xi,x2,x3) + q(x2,x3,xi) + q(x3,xi,x2) =
= U(xi,x2,x3) + U(x2,x3,xi) + U(x3,xi,x2) - U(x2,xi,x3) -
—	U(xi,x3,x2) - U(x3,x2,xi)
where U: ^ ^ is an arbitrary function, and
F(xi,x2) = G(xi,x2) + G(x2,xi)
where G: ^ ^ is an arbitrary function.
Thus the general solution of the functional equation
af(xi,x2,x3) + a2f(x2,x3,xi) - (ai + a2)f(x3,xi,x2) =
= - [a22/(ai + a2)f(xi,xi,x2) - [ai2/(ai + a2)]fx2,x2,x3) -
- [(ai2 + a22)/(ai + ajtf&^ji) is given by the relation
f(xi,x2,x3) = {ai[ G(xi,x3) + G(x3,xi) + aj_ Gfexs) + + G(x-3,x2)]}/(ai + a2) + U(xi,x2,x3) + U(x2,x3,xi) + + U(x3,xi ,x2) U(x2,xi ^3) U(xi ,x3,x2) U (x3,x2,xi ) Next we suppose that
2ai2 + a22 - (ai - a2)ai = 0 Since ai ^ a2, we have ai = (2ai2 + a22)/(ai - a2), a2 = (ai2 + 2aia2)/(ai - a2) Now (3.58) takes the form
fx^x,) = [2aiF(xi,x2) - aiF(x3,xii) - a2F(x2,x3)]/ /(ai - a2) + q(xi,x2,x3) + q(x2,x3,xi) + q(x3,xi,x2)
while (3.6i) becomes
[ai/(ai - a2)] [F(xi,x2) + F(x2,xi) - 2F(xi,xi)] = = q(xi,xi,x2) + q(xi,x2,xi) + qfeXiXi) (3.65) Equation (3.65) implies
q(xi,xi,x2) + q(xi,x2,xi) + qfexiXi) = 0
and
F(xi,x2) + F(x2,xi) - 2F(xi,xi) = 0
i.e.,
q(xi,xi,x2) + q(xi,x2,xi) + q(x2,xi,xi) = U(xi,x2,x3) +
+ U(x2,x3,xi) + U(x3,xi,x2) - U(x2,xi,x3) -
-	U(xi,x3,x2) - U(x3,x2,xi)
where U: ^ ^ is an arbitrary function, and
F(xi,x2) = G(xi,x2) -G(x2,xi) + c
where G: ^ ^ is an arbitrary function and c e^ is an arbitrary constant.
Thus, the general solution of the functional equation
af(xi,x2,x3) + a2f(x2,x3,xi) - (ai + a2)f(x3,xi,x2) =
= [(2ai2 + a22)/(ai - a2)] ffex^) + + [(ai2 + 2aia2)/(ai - a2)] fxx^x) -- [(3ai2 + 2aia2 - a22)/(ai - a2)] fx^x^xi)
is given by the relation
fix^x) = [i/(ai - a2^{2a^G(xi,x2) - GfeXi)] + + ai[G(xi,x3) - Gfexi)] - a^G(x2,x3) - G(x3,x2)]} + c+ + U(xi,x2,x3) + U(x2,x3,xi) + U(x3,xi,x2) - U(x2,xi,x3) -
- U(xi,x3,x2) - U(x3,x2,xi)
Now let [(ai - a2)/(ai - a2)] = Y ^ i. Then from (2.60) we find
F(xi,xi) = [3/(i - y)] q(xi,xi,xi) (3.66)
Now we have
[i + a2(ai- aiy)/(ai2 + a22 + aia2^F(xi,x2) + + [ai(ai - aiy)/(ai2 + a22 + aia2^F(x2,xi) = = ^ai(ai + a2) + a22 y]q(xi,xi,xi)/(ai2 + a22 + aia2) x x (i - y) + q(xi,xi,x2) + q(xiX2,xi) + q(x2,xi,xi) (3.67)
By a permutation of the variables xi and x2 we derive the equation
[ai(ai- aiy)/(ai2 + a22 + aia2^F(xi,x2) + + [i + a2(ai - aiy)/(ai2 + a22 + aia2^F(x2,xi) = = ^ai(ai + a2) + ai y]q(x2,x2,x2)/(ai2 + a22 + aia2) x x (i - y) + q(x2,x2,xi) + q(x2,xi,x2) + qfex^) (3.68)
The determinant of the system (3.67) and (3.68) is
[(ai2 + aia2)(i - y) + a22 + (ai + a2)aj x Xai2(i + y) + aia2(i - y) + a22 - (ai _ a2)aj/
/(ai2 + a22 + aia2)2	(3.69)
If the above expression is not 0, then
F(xi,x2) = 3[ai(ai + a2) + a22 y]/[(ai2 + aia2)(i - y) +
+ a22 + (ai + a2)aj {[ai2 + a22 + aia2(i - y) + a2aj x x q(xi,xi,xi) - ai(ai - aiY)q(x2,x2,x2)}/
/[ai2(i + y) + aia2(i - y) + a22 - (ai - a2)aj +
+ (ai2 + a22 + aia2)^(ai2 + aia2)(i - y) + a22 + + (ai + a2)aj {[ai2 + a22 + aia2(i - y) + a2aj x x [q(xi,xiX2) + q(xi,x2,xi) + q^x^)] -- ai(ai - aiY)[q(x2,x2,xi) + qfex^) + q^x^)]}/ /[ai2(i + y) + aia2(i - y) + a22 - (ai - a2)aj
Now let us suppose that the expression (3.69) is 0. First let
(ai2 + aia2)(i - y) + a22 + (ai + a2)ai = 0 Then
ai = - [a22 + (ai2 + a^Xi - Y)]/(ai + a2), a2 =
= - [ai2 + (a22 + aia2)(i - Y)]/(ai + a2) Now (3.58) takes the form
fxiX2X3) = (i - Y)F(xi,x2) + [aiF(x3,xi) + a2F(x2,x3)]/ /(ai + a2) + q(xi,x2,x3) + q(x2,x3,xi) + qfex^),
while (3.67) becomes
ai[F(xiX2) - F(x2,xi)y(fli + a2) = qfex^) + + q(xi,x2,xi) + q(x2,xi,xi) - 3q(xi,xi,xi)
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ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
The last equation implies
q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2) = U(x1,x2,x3) + + U(x2,x3,x1) + U(x3,x1,x2) - U(x2,x1,x3) - U(x1,x3,x2) -- U(x3,x2,x1) + P(x1) + P(x2) + P(x3)
F(x1,x2) = G(x1,x2) + G(x2,x1) - (a1 + a2)P(x1)/a1
where U: ^ G: ^ ^ and P: ^ ^ ^ are arbitrary functions. The condition (3.66) yields
2G(x1,x1) = [(a1 + a2)/a1 + 3/(1 - y)] P(x1)
Next we suppose that
a12(1 + y) + a1a2(1 - y) + a22 - (a1 - a2)a1 = 0
In this case we have
a1 = [a12 + a12(1 + y) + a1a2(1 - y^/(a1 - a2) a2 = [a12 + a22(1 - y) + a^U + y)]/(a1 - a2)
Now (3.58) takes the form
f(x1,x2,x3) = {[a1(1 + y) + a2(1 - y)]Ffo,^) -
- a1F(x3,x1) - a2F(x2,x-3)}/(a1 - a2) + q(x1,x2,x3) + + q(x2,x3,x0 + q(x3,x1,x2),
while (3.67) becomes
[a1/(a1 - a 2)] [F(x1,x2) + Ffe,^)] = = [3/(a1 - a2)][(1 + y)a1/(1 - y) + ajq(x1,x1,x1) + + q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1)
If
[3/(fl1 - a2)][(1 + Y)a1/(1 - y) + aj * - 1
i.e.,
(2 + Y)a1 + (1 - y)a2 * 0,
as above we find that
q(x1 ,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2) = U(x1,x2,x3) + + U(x2,x3,xl) + U(x3,x1,x2) - U(x2,x1,x3) -- Ufe^x) - U(x3,x2,x1)
F(x1,x2) = G(x1,x2) - G(x2,x1) and
f(x1,x2,x3) = { [a1(1 + y) + a2(1 - y)] [G(x1,x2) - G(x2,x1)] + a^G(x1,x3) - G(x3,x1)] - a^G(x2,x3) - Gfe,^)]}/ /(a1 - a2) + U(x1,x2,x3) + U(x2,x3,x1) + Ufex^) -- U(x2,x1,x3) - U(x1,x3,x2) - U(x3 ,x2,x1)
where G: ^ ^ and U: ^ ^ are arbitrary functions.
If, however
(2 + Y)a1 + (1 - y)a2 = 0
then
q(x1,x2,x3) + q(x2,x3,x0 + q(x3,x1,x2) = = U(x1,x2,x3) + U(x2,x3,xl) + U(x3,x1,x2) - U(x2,x1,x3) -- U(x1,x3,x2) - U(x3,x2xi) + a^Pfe) + Pfe) + P(x3)]
F(x1,x2) = G(x1,x2) - G(x2,x1) + a - a2)P(x1)
where P: ^ ^ ^ and U: ^ ^ are arbitrary functions, and
fx^xs) = {a^- G(x1,x2) + G(x2,x1) + G(x1,x3) -- G(x3,x0] - a^G(x2,x3) -G(x3,x2^}/(a1 - a2) + + (a1 - a2)P(x2) + U(x1,x2,x3) + Ufa^x) +
+ U(x3,x1,x2) - U(x2,x1,x3) - U(x1,x3,x2) -U(x3,x2,xl)
Example 3. 5. Now we will assume as a meniscus the relation (3.6). Let assume further as arbitrary its general solution (3.7) is the function
F(x1,x2,x3) s (x^)23 + (x2/Z2)2/3 + fe/y2'
1
where Zi (1 < i < 3) are real constants, then the shape of the meniscus will be given by the expression
f(x,,x2,x3) = (Zf2/3 - Z3-2/3)x12/3 + (Z2-2/3 - Z1-2/3)x22/3 +
+ (Z3-2/3- Z2-2/3)x32/3 This shows that the shape of the meniscus changes cyclically during the mould cycle.
Remark 3. 6. Also, the above results hold for the vector extension of equation (3.1) of the form
af(X1,X2,X3) + afX2X3X1) + af(X3,X1,X2) =
= af(X1,X1,X2) + a2f(X2,X2,X3) + af^^XO
f: ^ where Xi = (x1i,x2i,x3i)T are real vectors and ai, ai (1 < i < 3) are real constants.
4 GENERALIZED RESULTS
As a natural consequence of the previous considered meniscus equation, we will give the following more general result.
Theorem 4. 1. The generalized meniscus equation
E(f — Y a if(xi'xi+1,.,xi+n-1) =
i =1
n
= Y, a if(xi,xi,xi+1,.,xi+n-2) (xn+i — xi, n > 1)
(4.1)
where ai, ai (1 < i < n) are real constants, has a solution if the right-hand side of (4.1) satisfies
(AC + I)A[ g(x1,x2,...,xn-1), g(x2,x3,...,xn),...,g(xn,x1,...,x„.2)]T = O (4.2)
where A = cycl(a1,a2,...,a„), A = cycl(a1,a2,...,a„), g(x1,x2,.,x„-1) = /(x1,x1,x2,.,x„-1), C is any non-zero nxn cyclic matrix with constant real entries satisfying ACA + A = O, O is the nxn zero matrix and I is the nxn unit matrix.
If the equality (4.2) holds for some C, then the general solution of equation (4.1) is given by the following formula
■^ti)^^,^,. ■ ■,xn>x1),. ■ ■f(xn>x1,. ■ ■,xn-1)'\ = = B [h(x1,x2,.,xn),h(x2,x3,.,xn,x1),.,h(xn,x1,.,xn-1^ -
- A[g(x1,
(4.3)
where the non-zero nxn cyclic matrix B given by
B = cycl(b1,b2,...,bn) satisfies the condition
AB = O
214
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ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
and h is an arbitrary real function Kn ^ K.
Proof. By a cyclic permutation of the variables in (4.1) we get
a1 f(x1,x2,.,xn) + a2f(x2>x3,-• •,xn,x1) + ' ' ' + + anfxn>x1,^,xn-1) = a1g(x1>x2, — ,xn-1) + a2g(x2,x3,.,xn) + + ■ ■ ■ + ang(xn>x1,^,xn-2)
anf(x1>x2,^,xn) + a1f(x2>x3,^,xn>x1) + ' ' ' + + an-1fxn>x1,^,xn-1) = a«g(x1>x2,^,xn-1) + a1g(x2>x3,'•-,xn) + + ■ ■ ■ + an-1g(xn>x1,^,xn-2)
a2 f(x1>x2,'•',xn) + a3^(x2>x3,^,xn>x1) + ' ' ' + + af(xn,x1,...,xn-1) = a2g(x1,x 2,.",xn-1) + a3g(x2>x3,.,xn) + + ■ ■ ■ + a1g(xn,x1,.,xn-2)
i.e., in matrix form
AF = AG	(4.4)
where
F = f(x1>x2,.,xn)^vx2>x3,.,xn>x1),.^xn>x1,.,xn-1^T
and
G = g(x1,
We suppose that equation (4.4) has a solution F and C satisfies ACA + A = O. Then
(AC + I)AG = (AC + I)AF = (ACA + A)F = O
i.e., equation (4.2) must be satisfied. Conversely, let equation (4.2) hold for a cyclic matrix C. Then - CAG is easily seen to be a solution of equation (4.4):
A(- CAG) = - (AC + I)AG + IAG = I AG = AG
Now let us prove that equality (4.3) gives the general solution of equation (4.1).
Let f be a solution of equation (4.1), which we will write in the form
E(f) = L(g)	(4.5)
We denote by fh the general solution of the equation E(f) = 0, and by fp we denote a particular solution of equation (4.5).
Then f = fh + fp is the general solution of equation (4.5). Indeed
E(fh + fp) = E(fh) + E(fp) = E(g)
On the other hand, let f be an arbitrary solution of equation (4.5). Then
E(f - fp) = E(f) - E(fp) = L(g) - L(g) = 0
i.e., f - fp is a solution of the associated homogeneous equation. So there exists a specialization fh* of the expression fh such that
f - fp = fh\ f = fh* + fp Thus fh + fp includes all the solutions of equation (4.5).
The general solution of the homogeneous equation E(f) = 0 given in matrix form is BH, where
H = [h(xbx
2,.,xn),h(x2>x3,.,xn>x1),.,h(xn>x1,.,xn-1)]
and a particular solution of the equation E(f) = L(g) in matrix form is - CAG, then F = BH - CAG includes all the solutions of the nonhomogeneous equation.
On the other hand, every function of the form (4.3) satisfies the functional equation (4.1).
Remark 4. 2. The same results hold for the vector extension of equation (4.1) of the form
Ef) = £ aif(Xi,XM,...,Xi+n_1) =
i =1
= £ a;f(X;,X;,X;+1,...,X;+n.2) Ä+i = X, h >1)
i=1
f: Kh ^ K, where Xi = (x1i,x2i,.,xni)T are real vectors and ai, ai (1 < i < n) are real constants.
5 MENISCUS STABILITY
The meniscus stability was considered for the first time in 1516, but only according to definition 2.2 for the solution of the Navier-Stokes equation, including the pressure gradient. Here, we will give a completely new matrix approach to the solution of the meniscus stability problem.
Now we will derive a necessary and sufficient condition for the stability of the meniscus given by the quasicyclic functional equation (4.1), i.e., its matrix form (4.4) using a simple spectral property of compound matrices.
Let detA * 0, then relation (4.4) takes the form
F = A"1 AG = SG	(5.1)
where S is also a cyclic matrix.
Definition 5. 1. The quasicyclic functional equation (5.1) is stable if stab(S) < 0.
Proposition 5.2. For any cyclic matrix SeK it holds
stab(S) = inf{|a(S), ^ is a Lozinskii measure on Kn}
(5.2)
Proof. The relation (5.2) obviously holds for diagoni-zable matrices in view of
ßT(S) = ß(TST(T is an invertible matrix)	(5.3)
and the first two relations in (2.4). Furthermore, the infimum in (5.2) can be achieved if S is diagonizable. The general case can be shown based on this observation, the fact that S can be approximated by diagoni-zable matrices in K and the continuity of ^( ), which is implied by the property
IMA)- MB)| < |A - B|
Remark 5. 3. From the above proof it follows that
stab(S) = inf{^(TST-1), T is invertible}.
The same relation holds if ^ is replaced by ^1. Corollary 5. 4. Let SeK. Then stab(S) <0 o ß(S) < 0 for some Lozinskii measure ^ on Kn.
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ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
Theorem 5. 5. For stab(S) < 0 it is sufficient and necessary that stab(S[2]) < 0 and (-1)n det(S) > 0.
Proo/. Using the spectral property of S[2], the condition stab(S[2]) < 0 implies that at least one eigenvalue of S can be nonnegative. We may thus suppose that all eigenvalues are real. It is then simple to see that the existence of one and only one non-negative eigenvalue is precluded by the condition (-1)ndet(S) > 0.
Theorem 5.5 and Corollary 5.4 lead to the following result.
Theorem 5. 6. Suppose that (-1)ndet(S) > 0. Then S is stable if and only if p(S[2) < 0 for some Lozinskii measure p on SN, N — n!/2!(n - 2)!.
Theorem 5. 7. If stab(S2l(ß)) < 0 for ß e (a, b), then (a, b) contains no Hopf bifurcation points of S(ß).
Proo/. Let ß ^ S(ß)eS be a function that is continuous for ße (a, b). A point ßö e (a, b) is said to be a Hopf bifurcation point for S(ß) if S(ß) is stable for ß < ß0, and there exists an eigenvalue 1(ß) of S(ß) such that 1(ß) > 0, while the rest of the eigenvalues of S(ß) are nonzero for ß > ß0. From the proof of Theorem 5.5 we see that stab(S[2]) < 0 precludes the existence of a non-negative eigenvalue of S.
Let S and P be nxn real cyclic matrices. A subspace ^ eS is invariant under S if S(Q) e S is said to be stable with respect to an invariant subspace ^ if the restriction of S to S|Q: ^ ^ ^ is stable. Let the matrix P be such that rank P — r (1 < r < n) and
PS — O	(5.4)
Then KerP — {xeS, Px — 0} satisfies S(S) e KerP. In particular, KerP is an (n-r)-dimensional invariant space of S. It is of interest to study the stability of S with respect to KerP when (5.4) holds.
Lemma 5. 8. Let ^ e S be a subspace such that S(S) e ^ and dim^ — k < n. Then 0 is an eigenvalue of S, and there exist n - k null eigenvectors that do not belong to
Proo/. Let 3 be the quotient space S/^. Then S = ^ 8 3 and S(3) — {0} since S(S) e This establishes the lemma.
Theorem 5. 9. Suppose that P and S satisfy (5.4) and rankP — r (1 < r < n). Then for S to be stable with respect to KerP, it is necessary and sufficient that
1°	stab(S[r+2]) < 0
and
2°	lim sup sign [det(£l + S)] — (-1)n r
0 +
Proo/. Let 1; (1 < i < n - r) be eigenvalues of S|KerP. By Lemma 5.8, the eigenvalues of S can be written as
11,12, . . . , 1n-r, 0, . . . , 0, (r zeros)
and thus {1; +1j, 1 < i < j < n - r} e a(Sr+2) by the spectral property of additive compound matrices discussed in Section 2. It follows that stab(Sr+2) < 0
precludes the possibility of more than one non-negative 1; (1 < i < n - r). For £ >0 sufficiently small
sign[det(e/ + S)] — sign(£r11 ■ ■ ■ 1n-r)
The theorem can be proved using the same arguments as in the proof of Theorem 5.5.
Remark 5.10. If r — n in (5.4), then P is of full rank and hence S — O. If r — n -1, then KerP is of dimension 1 and thus the eigenvalues of S are 11 and 0 of multiplicity n - 1. From the above proof we know that Theorem 5.9 still holds in this case, if condition 1° is replaced by tr(S) < 0.
Corollary 5. 11. Suppose that S and P1 satisfy
P1S — ßP1	(5.5)
and rankP1 — r (1 < r < n). Thus S is stable with respect to KerP1 if and only if the following conditions hold:
1°	stab(S[r+2]) < (r + 2)ß
and
2°	(sign ß)r(-1)n-rdet(S) > 0
Proof. Let the matrix P1 be such that rankP1 — r (1 < r < n) and (5.5) holds for some scalar ß ^ 0. Then KerP1 is an invariant subspace of S. Noting that (5.5) is equivalent to P1(S - ßI) — O, one can apply Theorem 5.9 to S - ßI and obtain the proof.
6 DISCUSSION
The previous results may be extended in two different ways.
1° A first possible generalization of equation (3.1) is the following functional equation
a/1(x1,x
2,X3) + a2/2(x2,X3,X1) + a3^3(x3, X1,X2) — a/1(x1,X1,X2) + a/i(x2,x 2,X3) + a3/3(x3' X3,X1)
/;: S3 ^ S, where a;, a; (1 < i < 3) are real constants.
In other words, it means to extend the representation of the meniscus with three unknown functions / (1 < ; < 3) instead of by one unknown function /, as was shown in Section 3. This kind of meniscus representation is really much better, but this problem is very hard and it requires a new method of solvability.
If we continue in this way, it will hold for the generalization of equation (4.1) given by the formula
n
E(f) = ^ aif (xi,xi+l,...,xi+n-1) —
;=1
n
— ^ ai/i (x;,X;,X;+1,. •-^tn^) (xn+i = X;, n > 1)
;=1
/;: Sn ^ S, where a;, a; (1 < ; < n) are real constants. This equation is extremely hard and its solution is unknown up to now.
2° A second generalization of equation (3.1) is the vector equation
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ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
ai/i(Xi»X2»X3) + a2/2(X2»X3»Xi) + a3/3(X3»X1»X2) = = afi(Xi,Xi,X2) + «2/2X2, X2X3) + a/3(X3,X3,Xi)
fi: — where Xi = (xi;,X2iX3i)T are real vectors and ai, ai (i < i < 3) are real constants.
For equation (4.i) its generalized form is given by the equation
E(f) = %aj{ (Xi,XM,...,Xi+n_i) =
i=i
= f af (Xi,Xi,XM,.,Xi+n-2) (Xn+i = Xi, n > i)
i=i
fi: — where Xi = (xii,x2i,...,xni)T are real vectors and ai, ai (i < i < n) are real constants.
Really, in this section the considered equations are more sophisticated than the solved equations in previous sections, but their solutions are extremely difficult and up to now unknown to the author. In any case, their solutions will describe the meniscus form in a way that will be much closer to reality.
7	CONCLUSION
In this work the analyzed meniscus equation shows that it is possible to interpret the meniscus shape with a quasicyclic real functional equation. During continuous steel casting, the shape of the meniscus changes according to the mould cycle. The derived results are appropriate for use in a huge mathematical model for a description of all the appearances on the meniscus considering the technical characteristics of the process. The mathematical results are summarized as follows:
1)	The meniscus equation is completely solved in for all possible cases. The induced topology only categorizes the trajectories in an orientation space.
2)	The extended form of the meniscus equation is derived too in by a compact matrix approach, different from the method used for the solution of the meniscus equation in
3)	The meniscus stability problem is solved by using a simple spectral property of the compound matrices and Lozinskii measure on
In this work the shape of the meniscus during continuous steel casting is considered for the first time as a non-fixed characteristic according to the cyclic operation of the mould. The analysis has shown that it is more appropriate to use this kind of meniscus modeling than some approximate form. It is also shown that the old one-dimensional interpretations of the meniscus may only be used as an approximation.
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