Acta Polytechnica Vol. 51 No. 4/2011 Ito-Sadahiro numbers vs. Parry numbers Z. Masáková, E. Pelantová Abstract We consider a positional numeration system with a negative base, as introduced by Ito and Sadahiro. In particular, we focus on the algebraic properties of negative bases −β for which the corresponding dynamical system is sofic, which happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of − β β +1 is eventually periodic. We call such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in the same context for the Rényi positive base numeration system. Keywords: numeration systems, negative base, Pisot number, Parry number. 1 Introduction The expansion of a real number in the positional number system with base β > 1, as defined by Rényi [12] is closely related to the transformation T : [0, 1) �→ [0, 1), given by the prescription T (x) := βx − #βx$. Every x ∈ [0, 1) is a sum of the infinite series x = ∞∑ i=1 xi βi , where xi = #βT i−1(x)$ (1) for i = 1, 2, 3, . . . Directly from the definition of the transformation T we can derive that the ‘digits’ xi take values in the set {0, 1, 2, . . . , %β& − 1} for i = 1, 2, 3, . . .. The expres- sion of x in the form (1) is called the β-expansion of x. The number x is thus represented by the infinite word dβ (x) = x1x2x3 . . . ∈ AN over the alphabet A = {0, 1, 2, . . . , %β& − 1}. From the definition of the transformation T we can derive another important property, namely that the ordering on real numbers is carried over to the ordering of β-expansions. In particular, we have for x, y ∈ [0, 1) that x ≤ y ⇐⇒ dβ (x) ) dβ (y) , where ) is the lexicographical order on AN, (order- ing on the alphabet A is usual, 0 < 1 < 2 < . . . < %β& − 1). In [11], Parry has provided a criterion which de- cides whether an infinite word in AN is or not a β- expansion of some real number x. The criterion is formulated using the so-called infinite expansion of 1, denoted by d∗β (1), defined as a limit in the space AN equipped with the product topology, by d∗β (1) := lim ε→0+ dβ (1 − ε) . According to Parry, the string x1x2x3 . . . ∈ AN rep- resents the β-expansion of a number x ∈ [0, 1) if and only if xixi+1xi+2 . . . ≺ d∗β (1) (2) for every i = 1, 2, 3, . . . Condition (2) ensures that the set Dβ = {dβ(x) | x ∈ [0, 1)} is shift invariant, and so the closure of Dβ in AN, denoted by Sβ, is a subshift of the full shift AN. The notion of β-expansion can naturally be ex- tended to all non-negative real numbers: The expres- sion of a positive real number y in the form y = ykβ k + yk−1β k−1 + yk−2β k−2 + . . . , (3) where k ∈ Z and ykyk−1yk−2 . . . ∈ Dβ , is called the β-expansion of y. Real numbers y having in the β-expansion of |y| vanishing digits yi for all i < 0 are usually called β- integers, and the set of β-integers is denoted by Zβ . The notion of β-integers was first considered in [3] as an aperiodic structure modeling non-crystallographic materials with long range order, called quasicrystals. Numbers y with finitely many non-zero digits in the β-expansion of |y| form the set denoted by Fin(β). The choice of the base β > 1 strongly influences the properties of β-expansions. It turns out that an important role among bases is played by such num- bers β for which d∗β (1) is eventually periodic. Parry himself called these bases beta-numbers; now these numbers are commonly called Parry numbers. We can demonstrate the exceptional properties of Parry numbers on two facts: • The subshift Sβ is sofic if and only if β is a Parry number [6]. 59 Acta Polytechnica Vol. 51 No. 4/2011 • Distances between consecutive β-integers take finitely many values if and only if β is a Parry number [15]. Recently, Ito and Sadahiro [5] suggested a study of positional numeration systems with a negative base −β, where β > 1. The representation of real numbers in such a system is defined using the trans- formation T : [lβ, rβ ) �→ [lβ , rβ ), where lβ = − β β + 1 , rβ = 1 + lβ = 1 1 + β , T (x) := −βx − #−βx − lβ$ . (4) Every real x ∈ Iβ := [lβ, rβ ) can be written as x = ∞∑ i=1 xi (−β)i , (5) where xi = #−βT i−1(x) − lβ$ for i = 1, 2, 3, . . . The above expression is called the (−β)- expansion of x. It can also be written as the infi- nite word d−β (x) = x1x2x3 . . . We can easily show from (4) that the digits xi, i ≥ 1, take values in the set A = {0, 1, 2, . . . , #β$}. In this case, the order- ing on the set of infinite words over the alphabet A which would correspond to the ordering of real num- bers is the so-called alternate ordering: We say that x1x2x3 . . . ≺alt y1y2y3 . . . if for the minimal index j such that xj �= yj it holds that xj (−1)j < yj (−1)j . In this notation, we can write for arbitrary x, y ∈ Iβ that x ≤ y ⇐⇒ d−β (x) )alt d−β (y) . In their paper, Ito and Sadahiro have provided a criterion to decide whether an infinite word AN be- longs to the set of (−β)-expansions, i.e. to the set D−β = {d−β (x) | x ∈ Iβ}. This time, the criterion is given in terms of two infinite words, namely d−β (lβ) and d ∗ −β (rβ ) := lim ε→0+ d−β (rβ − ε) . These two infinite words have a close relation: If d−β (lβ ) is purely periodic with odd period length, i.e. d−β (lβ ) = (d1d2 . . . d2k+1) ω, then we have d∗−β(rβ ) =( 0d1d2 . . . (d2k+1 − 1) )ω . (As usual, the notation wω stands for infinite repetition of the string w.) In all other cases we have d∗−β (rβ ) = 0d−β(lβ ). Ito and Sadahiro have shown that an infinite word x1x2x3 . . . represents a (−β)-expansion of some x ∈ [lβ , rβ ) if and only if for every i ≥ 1 it holds that d−β (lβ ) )alt xixi+1xi+2 . . . ≺alt d∗−β (rβ ) . (6) The above condition ensures that the set D−β of in- finite words representing (−β)-expansions is shift in- variant. In [5] it is shown that the closure of D−β defines a sofic system if and only if d−β (lβ ) is even- tually periodic. By analogy with the definition of Parry numbers, we suggest that numbers β > 1 such that d−β (lβ ) is eventually periodic be called Ito-Sadahiro numbers. The relation of the set of Ito-Sadahiro numbers and the set of Parry numbers is not obvious. Bassino [2] has shown that quadratic numbers, as well as cubic numbers which are not totally real, are Parry if and only if they are Pisot. For the same class of num- bers, we prove in [10] that β is Ito-Sadahiro if and only if it is Pisot. This means that notions of Parry numbers and Ito-Sadahiro numbers on the mentioned type of irrationals do not differ. This would support the hypothesis stated in the first version of this pa- per, namely that the set of Parry numbers and the set of Ito-Sadahiro numbers coincide. However, dur- ing the refereeing process Liao and Steiner [9] found an example of a Parry number which is not an Ito- Sadahiro number, and vice-versa. The main results of this paper are formulated as Theorems 4 and 7. Theorem 4 gives a bound on the modulus of conjugates of Ito-Sadahiro numbers; The- orem 7 shows that periodicity of (−β)-expansion of all numbers in the field Q(β) requires β to be a Pisot or Salem number. Statements which we prove, as well as results of other authors that we recall, demonstrate similarities between the behaviour of β-expansions and (−β)-expansions. We mention also phenomena in which the two essentially differ. 2 Preliminaries Let us first recall some number theoretical notions. A complex number β is called an algebraic number, if it is a root of a monic polynomial xn +an−1x n−1+. . . + a1x + a0, with rational coefficients a0, . . . , an−1 ∈ Q. A monic polynomial with rational coefficients and root β of the minimal degree among all polynomials with the same properties is called the minimal poly- nomial of β, and its degree is called the degree of β. The roots of the minimal polynomial are algebraic conjugates. If the minimal polynomial of β has integer coeffi- cients, β is called an algebraic integer. An algebraic integer β > 1 is called a Perron number, if all its con- jugates are in modulus strictly smaller than β. An algebraic integer β > 1 is called a Pisot number, if all its conjugates are in modulus strictly smaller than 1. An algebraic integer β > 1 is called a Salem number, if all its conjugates are in modulus smaller than or equal to 1 and β is not a Pisot number. If β is an algebraic number of degree n, then the minimal subfield of the field of complex numbers con- taining β is denoted by Q(β) and is of the form Q(β) = {c0 + c1β + . . . + cn−1βn−1 | ci ∈ Q} . 60 Acta Polytechnica Vol. 51 No. 4/2011 If γ is a conjugate of an algebraic number β, then the fields Q(β) and Q(γ) are isomorphic. The corre- sponding isomorphism is given by c0+c1β +. . .+cn−1β n−1 �→ c0+c1γ +. . .+cn−1γn−1 . In particular, this means that β is a root of some polynomial f with rational coefficients if and only if γ is a root of the same polynomial f . 3 Ito-Sadahiro polynomial From now on, we shall consider for bases of the numeration system only Ito-Sadahiro numbers, i.e. numbers β such that d−β (lβ ) = d1 . . . dm(dm+1 . . . dm+p) ω . (7) Without loss of generality we shall assume that m ≥ 0, p ≥ 1 are minimal values so that d−β (lβ ) can be written in the above form. Recall that lβ = − β β + 1 . Therefore (7) can be rewritten as − β β + 1 = d1 −β + . . . + dm (−β)m +( dm+1 (−β)m+1 + . . . + dm+p (−β)m+p ) ∞∑ i=0 1 (−β)p i , and after arrangement 0 = −β −β − 1 + d1 −β + . . . + dm (−β)m + (−β)p (−β)p − 1 ·( dm+1 (−β)m+1 + . . . + dm+p (−β)m+p ) . Multiplying by (−β)m ( (−β)p − 1 ) , we obtain the fol- lowing lemma. Lemma 1 Let β be an Ito-Sadahiro number and let d−β (lβ ) be of the form (7). Then β is a root of the polynomial P (x) = (−x)m+1 p−1∑ i=0 (−x)i + ( (−x)p − 1 ) · (8) m∑ i=1 di(−x)m−i + m+p∑ i=m+1 di(−x)m+p−i . Such a polynomial is called the Ito-Sadahiro polyno- mial of β. Corollary 2 An Ito-Sadahiro number is an alge- braic integer of degree smaller than or equal to m + p, where m, p are given by (7). It is useful to mention that the Ito-Sadahiro poly- nomial is not necessarily irreducible over Q. As an example one can take the minimal Pisot num- ber. For such β, we have d−β (lβ ) = 1 001 ω, and thus the Ito-Sadahiro polynomial is equal to P (x) = x4−x3−x2 + 1 = (x−1)(x3−x−1), where x3−x−1 is the minimal polynomial of β. Remark 3 Note that for p = 1 and dm+1 = 0, we have d−β (lβ ) = d1 . . . dm0 ω , and the Ito-Sadahiro polynomial of β is of the form P (x) = (−x)m+1 + d1(−x)m + (d2 − d1)(−x)m−1 + . . . + (dm − dm−1)(−x) − dm , (9) and thus β is an algebraic integer of degree at most m + 1. Theorem 4 Let β be an Ito-Sadahiro number. All roots γ, γ �= β, of the Ito-Sadahiro polynomial (in particular all conjugates of β) satisfy |γ| < 2. Proof. Since β is a root of its Ito-Sadahiro poly- nomial P , there must exist a polynomial Q such that P (x) = (x − β)Q(x). Let us first determine Q and show that it is a monic polynomial with coefficients in modulus not exceeding 1. The coefficients di in the polynomial P in the form (8) are the digits of the (−β)-expansion of lβ , and thus, by (5), they satisfy di = #−βT i−1(lβ ) − lβ$. Relation (4) then implies T i(lβ ) = −βT i−1(lβ )−#−βT i−1(lβ )−lβ$, wherefrom we have di = −T i(lβ ) − βT i−1(lβ) . For simplicity of notation in this proof, denote Ti = T i(lβ ), for i = 0, 1, . . . , m + p. Substituting di = −Ti − βTi−1 into (8), we obtain P (x) = (−x)m+1 p−1∑ i=0 (−x)i + ( (−x)p − 1 ) · m∑ i=1 (−Ti − βTi−1)(−x)m−i + m+p∑ i=m+1 (−Ti − βTi−1)(−x)m+p−i = (−x)m+1 p−1∑ i=0 (−x)i + ( (−x)p − 1 ) (x − β) · m∑ i=2 Ti−1(−x)m−i + (10) (x − β) p∑ i=1 Tm+i−1(−x)p−i −( (−x)p − 1 ) βT0(−x)m−1 + Tm − Tm+p . First realize that Tm − Tm+p = 0, since d−β (lβ) is eventually periodic with a preperiod of length m and a period of length p. As T0 = T 0(lβ ) = − β β + 1 , we can derive that (−x)m+1 p−1∑ i=0 (−x)i − ( (−x)p − 1 ) βT0(−x)m−1 = (−x)m−1(x − β)(x − T0) p−1∑ i=0 (−x)i . 61 Acta Polytechnica Vol. 51 No. 4/2011 Putting back to (10), we obtain that the desired poly- nomial Q defined by P (x) = (x − β)Q(x) is of the form Q(x) = (−x)m−1(x − T0) p−1∑ i=0 (−x)i + ( (−x)p − 1 ) m∑ i=2 Ti−1(−x)m−i + p∑ i=1 Tm+i−1(−x)p−i , which can be rewritten in another form, namely, Q(x) = −(−x)m+p−1 + m+p−2∑ i=m (Tm+p−1−i − T0 − 1)(−x)i + (11) m−1∑ i=0 (Tm+p−1−i − Tm−1−i)(−x)i . Note that the coefficients at individual powers of −x are of two types, namely Tm+p−1−i − T0 − 1 ∈ [−1, 0) , and Tm+p−1−i − Tm−1−i ∈ (−1, 1) . In order to complete the proof, realize that every root γ, γ �= β, of the polynomial P satisfies Q(γ) = 0. We thus have (−γ)m+p−1 = m+p−2∑ i=m (Tm+p−1−i − T0 − 1)(−γ)i + m−1∑ i=0 (Tm+p−1−i − Tm−1−i)(−γ)i , and hence |γ|m+p−1 ≤ m+p−2∑ i=0 |γ|i = |γ|m+p−1 − 1 |γ| − 1 < |γ|m+p−1 |γ| − 1 . From this, we easily derive that |γ| < 2. As a consequence, we can easily deduce the rela- tion between Ito-Sadahiro numbers greater or equal to 2 and Perron numbers. Corollary 5 Every Ito-Sadahiro number β ≥ 2 is a Perron number. In a recent preprint [9], it is shown that also Ito- Sadahiro numbers β < 2 are Perron numbers. 4 Periodic expansions in the Ito-Sadahiro system Representations of numbers in the numeration sys- tem with a negative base from the point of view of dynamical systems have been studied by Frougny and Lai [7]. They have shown the following statement. Theorem 6 If β is a Pisot number, then d−β (x) is eventually periodic for any x ∈ Iβ ∩ Q(β). In particular, their result implies that every Pisot number is an Ito-Sadahiro number. Here, we show a ‘reversed’ statement. Theorem 7 If any x ∈ Iβ ∩ Q(β) has eventually pe- riodic (−β)-expansion, then β is either a Pisot num- ber or a Salem number. Proof. First realize that since l−β ∈ Q(β), by as- sumption, d−β (lβ ) is eventually periodic, and thus β is an Ito-Sadahiro number. Therefore, using Corol- lary 2, β is an algebraic integer. It remains to show that all conjugates of β are in modulus smaller than or equal to 1. Consider a real number x whose (−β)-expansion is of the form d−β (x) = x1x2x3 . . . We now show that x1 = x2 = . . . = xk−1 = 0 and xk �= 0 implies |x| ≥ 1 βk (β + 1) . (12) In order to see this, we estimate the series |x| = ∣∣∣ xk (−β)k + ∞∑ i=1 xk+i (−β)k+i ∣∣∣ ≥ 1 βk − 1 βk ∣∣∣ ∞∑ i=1 xk+i (−β)i ∣∣∣ . Since the set D−β of all (−β)-expansions is shift in- variant, the sum ∞∑ i=1 xk+i (−β)i is a (−β)-expansion of some y ∈ Iβ . Therefore we can write |x| ≥ 1 βk − 1 βk |y| ≥ 1 βk − 1 βk β β + 1 = 1 βk(β + 1) . As β > 1, there exists L ∈ N such that − β β + 1 < 1 (−β)2L+1 . Let M ∈ N satisfy M > 2L + 1. Choose a rational number r such that 1 (−β)2L+1 < r < 1 (−β)2L+1 + 1 βM (β + 1) . (13) According to the auxiliary statement (12), the (−β)- expansion of r must be of the form r = 1 (−β)2L+1 + ∞∑ i=M+1 ri (−β)i . (14) 62 Acta Polytechnica Vol. 51 No. 4/2011 As r is rational, by assumption, the infinite word rM+1rM+2 . . . is eventually periodic and by sum- ming a geometric series, the sum ∞∑ i=M+1 ri (−β)i can be rewritten as ∞∑ i=M+1 ri (−β)i = c0 + c1β + . . . + cn−1β n−1 ∈ Q(β) , where n is the degree of β. In order to prove the theorem by contradiction, assume that a conjugate γ �= β is in modulus greater than 1. By application of the isomorphism between Q(β) and Q(γ), we get c0 + c1γ + . . . + cn−1γ n−1 = ∞∑ i=M+1 ri (−γ)i , and thus r = 1 (−γ)2L+1 + ∞∑ i=M+1 ri (−γ)i . (15) Subtracting (15) from (14), we obtain 0 < ∣∣∣ 1 (−β)2L+1 − 1 (−γ)2L+1 ∣∣∣ ≤ (16) ∞∑ i=M+1 ri ∣∣(−β)−i − (−γ)−i∣∣ ≤ 2#β$ηM+1 1 − η , where η = max{|β|−1, |γ|−1} < 1. Obviously, for any M > 2L + 1, we can find a rational r satisfy- ing (13) and thus derive the inequality (16). However, the left-hand side of (16) is a fixed positive number, whereas the right-hand side decreases to zero with increasing M , which is a contradiction. In order to stress the analogy of the Ito-Sadahiro numeration system with Rényi β-expansions of num- bers, recall that already Schmidt in [13] has shown that for a Pisot number β, any x ∈ [0, 1) ∩ Q(β) has an eventually periodic β-expansion and also, con- versely, that every x ∈ [0, 1) ∩ Q(β) having an even- tually periodic β-expansion force β is either a Pisot number or a Salem number. In fact, the proof of The- orem 6 given by Frougny and Lai, as well as our proof of Theorem 7 are using the ideas presented in [13]. A special case of numbers with periodic (−β)- expansion is given by those numbers x for which the infinite word d−β (x) has suffix 0 ω. We then say that the expansion d−β (x) is finite. An example of such a number is x = 0 with (−β)-expansion d−β (x) = 0ω. As is shown in [10], if β < 1 2 (1 + √ 5), then x = 0 is the only number with finite (−β)-expansion. This property of the Ito-Sadahiro numeration system has no analogue in Rényi β-expansions; for positive base, the set of finite β-expansions is always dense in [0, 1). Just as in the numeration system with a positive base, we can extend the definition of (−β)-expansions of x to all real numbers x, and define the notion of a (−β)-integer as a real number y such that y = yk(−β)k + . . . + y1(−β) + y0 , where yk . . . y1y00 ω is the (−β)-expansion of some number in Iβ . The set of (−β)-integers is denoted by Z−β. With this notation, we can write the set of all numbers with finite (−β)-expansions as Fin(−β) = ∞⋃ k=0 1 (−β)k Z−β . It is not surprising that the arithmetical properties of β-expansions and (−β)-expansions depend on the choice of the base β. It can be shown that both Zβ and Z−β is closed under addition and multiplication if and only if β ∈ N. On the other hand, Fin(β) and Fin(−β) can have a ring structure even if β is not an integer. Frougny and Solomyak [8] have shown that if Fin(β) is a ring, then β is a Pisot number. A similar result is given in [10] for a negative base: Fin(−β) being a ring implies that β is either a Pisot number or a Salem number. In [10] we also prove the conjecture of Ito and Sadahiro that in the case of quadratic Pisot base β the set Fin(−β) is a ring if and only if the conjugate of β is negative. 5 Comments and open questions • Every Pisot number is a Parry number and every Parry number is a Perron number, and neither of these statements can be reversed. The for- mer is a consequence of the mentioned result of Schmidt, the latter statement follows for exam- ple from the fact that every Perron number has an associated canonical substitution ϕβ , see [4]. The substitution is primitive, and its incidence matrix has β as its eigenvalue. The fixed point of ϕβ is an infinite word which codes the sequence of distances between consecutive β-integers. • For the negative base numeration system, we can derive from Theorem 6 that every Pisot number is an Ito-Sadahiro number. From Corollary 5 we know that an Ito-Sadahiro number β ≥ 2 is a Perron number. Based on our investigation, we conjecture that for any Ito-Sadahiro number β ≥ 1 2 (1 + √ 5), the sequence of distances be- tween consecutive (−β)-integers can be coded by a fixed point of a ‘canonical’ substitution which is primitive and its incidence matrix has β2 for its dominant eigenvalue. Thus we expect that 63 Acta Polytechnica Vol. 51 No. 4/2011 every Ito-Sadahiro number β ≥ 1 2 (1 + √ 5) is also a Perron number. In the case that β < 1 2 (1 + √ 5), we have Z−β = {0} and so the situ- ation is not at all obvious. • In [14], Solomyak has explicitly described the set of conjugates of all Parry numbers. In particu- lar, he has shown that this set is included in the complex disc of radius 1 2 (1 + √ 5), and that this radius cannot be diminished. For his proof it was important that all conjugates of a Parry number are roots of a polynomial with real coefficients in the interval [0, 1). In the proof of Theorem 4 we show that conjugates of an Ito-Sadahiro number are roots of a polynomial (11) with coefficients in [−1, 1]. From this, we derive that conjugates of Ito-Sadahiro numbers lie in the complex disc of radius ≤ 2. We do not know whether this value can be diminished. Acknowledgement We acknowledge financial support from Czech Sci- ence Foundation grant 201/09/0584 and from grants MSM6840770039 and LC06002 of the Ministry of Ed- ucation, Youth, and Sports of the Czech Republic. References [1] Ambrož, P., Dombek, D., Masáková, Z., Pelan- tová, E.: Numbers with integer expansion in the numeration system with negative base, preprint 2009, 13pp. http://arxiv.org/abs/0912.4597 [2] Bassino, F.: β-expansions for cubic Pisot num- bers, 5th Latin American Theoretical INformat- ics Symposium (LATIN’02), 2286 LNCS. Can- cun, Mexico. April, 2002. pp. 141–152. Springer- Verlag. [3] Burd́ık, Č., Frougny, Ch., Gazeau, J. P., Krej- car, R.: Beta-Integers as Natural Counting Sys- tems for Quasicrystals, J. Phys. A: Math. Gen. 31 (1998), 6 449–6 472. [4] Fabre, S.: Substitutions et β-systèmes de numération, Theoret. Comput. Sci. 137 (1995), 219–236. [5] Ito, S., Sadahiro, T.: (−β)-expansions of real numbers, Integers 9 (2009), 239–259. [6] Ito, S., Takahashi, Y.: Markov subshifts and re- alization of β-expansions, J. Math. Soc. Japan 26 (1974), 33–55. [7] Frougny, Ch., Lai, A. C.: Negative bases and automata, Discr. Math. Theor. Comp. Sci. 13, No 1 (2011), 75–94. [8] Frougny, Ch., Solomyak, B.: Finite β-expan- sions, Ergodic Theory Dynamical Systems 12 (1994), 713–723. [9] Liao, L., Steiner, W.: Dynamical properties of the negative beta transformation, preprint 2011, 18pp. http://arxiv.org/abs/1101.2366 [10] Masáková, Z., Pelantová, E., Vávra, T.: Arith- metics in number systems with negative base, Theor. Comp. Sci. 412 (2011), 835–845. [11] Parry, W.: On the β-expansions of real num- bers, Acta Math. Acad. Sci. Hung. 11 (1960), 401–416. [12] Rényi, A.: Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957), 477–493. [13] Schmidt, K.: On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269–278. [14] Solomyak, B.: Conjugates of beta-numbers and the zero-free domain for a class of analytic functions, Proc. London Math. Soc. 68 (1994), 477–498. [15] Thurston, W. P.: Groups, tilings, and fi- nite state automata, AMS Colloquium Lecture Notes, American Mathematical Society, Boul- der, 1989. Zuzana Masáková E-mail: zuzana.masakova@fjfi.cvut.cz Edita Pelantová E-mail: edita.pelantova@fjfi.cvut.cz Department of Mathematics FNSPE Czech Technical University in Prague Trojanova 13, 120 00 Praha 2, Czech Republic 64