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ℵ0 it is easy to construct an example of a κ-dense linear order that is not strongly κ-dense. Moreover there are κ-dense linear orders that are κ-scattered and ones that do not even have a decreasing κ-sequence. See §4. Note that for κ = ℵ0 , the two definitions of κ-density agree. If an order is not κ-scattered for κ = κ<κ then it embeds a copy of Q(κ) so clearly it has a suborder of size κ that is not κ-scattered. For future purposes we note that a similar statement is true about orders that are strongly κ-scattered for any cardinal κ. Claim 2.8. Suppose that P is an order that is not strongly κ-scattered for κ = κ<κ . Then P has a suborder of size κ that is not strongly κ-scattered.

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 Proof of the Claim. We shall define a suborder Q = n<ω Qn of P by defining Qn by induction on n. Let Q0 be any two-element linear suborder of P , which exists by the definition of weak κ-density. Given Qn of size ≤ κ let us choose for any a
we have that Bζ = {x ∈ D : Sζ
i(ζ) for ζ limit. Let i+ = supξ<ζ i(ξ). Hence, either i+ = i∗ , in which case we stop def  def the induction, or i+ < i∗ , in which case we let i(ζ) = i+ and Sζ = ξ<ζ Sξ . Similarly for Tζ . Notice that our induction must stop at some limit stage ζ ∗ < κ as i∗ < κ. Now def  let S = ζ<ζ ∗ Sζ , and similarly for T . By the construction, it follows that S
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Hence, there is x ∈ D with S
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is not [strongly] κ-scattered. Proof of the Subclaim. Let A, B and S be as in the assumptions. Let DA = {c ∈ S : A <S  c <S  B and a ⊥S c for some a ∈ A} DB = {c ∈ S : A <S  c <S  B and b ⊥S c for some b ∈ B}, so that DA ∪ DB = (A, B)S  . Since the union of two [strongly] κ-scattered posets is itself [strongly] κ-scattered by Claim 2.9 [Claim 2.10], then either S   DA or S   DB is not [strongly] κ-scattered. [Let q be the unique element of A or of B, depending on which of the two sets is not strongly κ-scattered, so finishing the proof in this case]. Notice that  def DA = Da , where Da = {c ∈ S : A <S  c <S  B and a ⊥S c}, a∈A

and similarly for DB . Again by Claim 2.9 there is either a ∈ A or b ∈ B such that Da or Db is not κ-scattered. Let q be either a or b, whichever gives us the non-κ-scattered poset. Hence, Dq = (A, B)S   (⊥ q)S is not κ-scattered. 2 Suppose that S ⊆ P is such that S  = P   S is strongly [weakly] κ-dense. Such an S exists because P  embeds a strongly [weakly] κ-dense subset. Since P  S is [strongly] κ-scattered and has size κ there must be A, B ⊆ S with A

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but (An+1 , Bn+1 )Tn+1 = ∅ [|(an+1 , bn+1 )Tn+1 | < κ].   Let Sn+1 = Sn ∪ Tn+1 and hence Sn+1 ⊇ Sn ∪ Tn+1 is P   Sn+1 . We have An+1 , Bn+1 ⊆ Sn+1 and An <Sn An+1 <Sn Bn+1 <Sn Bn . Also Sn+1 as the union of two [strongly] κ-scattered orders is [strongly] κ-scattered by Claim 2.9 [Claim   is not as it includes Tn+1 which is strongly [weakly] κ-dense. Note also 2.10] while Sn+1  Sn that An+1 ∪ Bn+1 ⊆ k≤n (⊥qk ) . At any rate, Subclaim 2.12 applies to An+1 , Bn+1 and Sn+1 in place of A, B and S. Hence we can find qn+1 ∈ An+1 ∪ Bn+1 such that     (⊥qn+1 )Sn+1 = (An+1 , Bn+1 )Sn+1  (⊥qk )Sn+1 (An+1 , Bn+1 )Sn+1 k≤n+1

 (as for k ≤ n we have that (An+1 , Bn+1 )Sn+1 ⊆ (An , Bn )Sn ⊆ (⊥qk )Sk ⊆ (⊥qk )Sn+1 ) is not [strongly] κ-scattered, hence satisfying all the requirements of the induction at this step. Having finished the induction we obtain that if k < n then qn ∈ An ∪ Bn ⊆ (⊥qk )Sn ⊆ (⊥qk )P . Hence qn ⊥P qk . Then the sequence qn : n < ω forms an infinite antichain in P , contradicting the fact that P is FAC. (2) =⇒ (3) Suppose that every augmentation of P is [strongly] κ-scattered but P does not satisfy the λ-AC for λ = κ<κ . (P is automatically [strongly] κ-scattered, since trivially P is an augmentation of itself.) Take a subset S ⊆ P such that |S| = λ and S is a λ-antichain. We can now embed any strongly [weakly] κ-dense set into S, forming an augmentation of P which is not [strongly] κ-scattered. 2

Remark 2.13. For κ = ℵ0 the three conditions in Lemma 2.11 are equivalent, as follows from the lemma. However for κ > ℵ0 the disjoint sum of an ordinal κ with an antichain of size ℵ0 shows that (3) does not imply (1) even for posets of size κ when κ = κ<κ . In Corollary 4.3 we give an example of a linear order L which is weakly κ-scattered and κ-dense. A disjoint sum of this order and an antichain of size ℵ0 shows that (2) does not imply (1). The above proof does not seem to generalise to show that (3) =⇒ (2) and we do not know if this is the case.

3

A Generalisation of the Classification

Here we will generalise the classification of [1] to κ-scattered FAC partial orders for regular κ. From now on we will fix such a cardinal κ. We remind the reader of the notion of the antichain rank of FAC posets, as introduced in Definition 2.1. Definition 3.1. Fix some ρ ≥ 1. By induction on α, an ordinal, we define κ Hαρ as follows: 1. κ H0ρ = {1}. 2. κ H1ρ is the class of all posets P satisfying the FAC with rkA (P ) ≤ ρ such that either P or its inverse, or both, are κ-well founded.

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 3. If α is a limit ordinal, then κ Hαρ = β<α κ Hβρ . 4. If α = β +1 for some β > 0, then κ Hαρ consists of all posets P that are lexicographical  sums of the the form P = i∈I Pi where Pi ∈ κ Hβρ and I is in κ H1ρ . In general, let for all ordinals α and ρ, κ

Hρ =

 α an ordinal

κ

Hαρ

and

κ

H=



κ

Hρ .

ρ an ordinal

We let aug(κ Hρ ) be the set of all augmentations of posets in κ Hρ . Lemma 3.2. (1) The class κ Hρ is the least class that contains the κ-well founded FAC posets with antichain ranks ≤ ρ and is closed under lexicographical sums and inverses. (2) Each κ Hαρ and κ Hρ is closed under restrictions and inverses. (3) If P ∈ κ H then P is κ-scattered and satisfies the FAC. (4) aug(κ Hρ ) is closed under lexicographical sums, restrictions and augmentations. Every poset in aug(κ Hρ ) is κ-scattered. We remind the reader that κ-scattered is used throughout this paper to refer to weakly κ-scattered orders. Proof of the Lemma. (1) It is clear that κ Hρ contains all κ-well founded posets of antichain ranks ≤ ρ and their inverses, as κ H1ρ contains them. The proof that κ Hρ is closed under lexicographical sums is the same as the one in [1] since we fix κ. We will not include it here. It remains to show that κ Hρ is closed under inverses. In fact we shall prove by induction on α that each κ Hαρ is closed under inverses. For P ∈ κ Hρ we shall use the notation α(P ) = min{α : P ∈ κ Hαρ }. Let us commence the induction. At α = 0 the situation is trivial and at α = 1, by definition κ H1ρ contains all inverses of its members. At α = β + 1, if α(P ) < α then this case is covered by the induction hypothesis. So,  assume that α(P ) = α. Then, P = i∈I Pi where Pi ∈ κ Hβρ and I ∈ κ H1ρ . The inverse of  P is P ∗ = i∈I ∗ Pi∗ where I ∗ ∈ κ H1ρ because κ H1ρ is closed under inverses by definition, and Pi∗ is the inverse of Pi . We know that Pi ∈ κ Hβρ , thus Pi∗ is also in κ Hβρ by the induction hypothesis. Hence P ∗ is in κ Hαρ . We know that α(P ) is never a limit because α is a minimum. Therefore, for α a limit ordinal and any P ∈ κ Hαρ , α(P ) is strictly less than α. Thus, this case is covered by the induction hypothesis. Hence κ Hρ has the closure properties as required. We will show that it is the least such class. Suppose that H is another class with such properties. Again by induction on α, we will show that H contains each κ Hαρ . Thus, we will show H ⊇ κ Hρ . The cases of α = 0 and α = 1 are trivial by definition. At α = β + 1, all sets P ∈ κ Hαρ are of the form  P = i∈I Pi where each Pi ∈ κ Hβρ . Since H contains all κ Hβρ by the induction hypothesis and is closed under lexicographical sums, all P ∈ κ Hαρ must be in H. Thus, κ Hαρ ⊆ H.  The case where α a limit is similar since by definition, κ Hαρ = β<α κ Hβρ .

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(2) We already proved the closure under inverses in the proof of (1). By induction on α, we will show that each κ Hαρ is closed under restrictions. The case α = 0 is trivial. For α = 1, if P ∈ κ H1ρ then either P or P ∗ is κ-well founded. Suppose that P is κ-well founded. Thus, if any restriction of P , call it P − , had a κ-decreasing sequence, it would actually be in P , which is a contradiction. The same argument can be used for P ∗ , the inverse of any κ-well founded poset in κ H1ρ .  Let α = β + 1. Suppose we are given P = i∈I Pi where each Pi ∈ κ Hβρ and I ∈ κ H1ρ . By the induction hypothesis, all restrictions of Pi are in κ Hβρ . Any restriction, P − , of P can be expressed as a lexicographical sum of restrictions of the Pi s along a restriction of I. Thus P − is also in κ Hαρ . The limit case is obvious. (3) Fix an ordinal ρ. By induction on α, we will prove that any P ∈ κ Hαρ is κ-scattered. The case α = 0 is trivial. Let α = 1. Notice that since any strongly κ-scattered order has a κ-decreasing sequence by Lemma 2.4, we have that no κ-well founded poset could embed such an order. Similarly, since by the same lemma strongly κ-dense orders have κ-increasing sequences, a poset whose inverse is κ-well founded also cannot embed such an order. The limit case of the induction is taken care of by the induction hypothesis.  Let α = β + 1. By the induction hypothesis, if P ∈ κ Hαρ we can let P = i∈I Pi where each Pi is κ-scattered and I is κ-scattered. We will show that P is κ-scattered. For the sake of contradiction, let Q be a strongly κ-dense order and suppose f : Q → P is an order preserving embedding. Case 1. For every i ∈ I, there is at most one q ∈ Q such that f (q) ∈ Pi . Define g : Q → I by letting g(q) = i iff f (q) ∈ Pi . This is well-defined by the assumptions of Case 1. We also have that q <∗ r implies f (q)

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reiterating the proof. Before we state the theorem, we need to draw attention to an unusual ordinal operation known as Hessenberg based exponentiation. This smoothly extends the Hessenberg product operation which in turn extends the natural sum operation. Since we do not need to know the exact value of the exponent for this paper, we refer the reader to [1] for a more precise definition. We denote the Hessenberg based exponentiation of α and β by αHβ . Theorem 3.3. If P ∈ κ Hαρ then rkA (P ) ≤ ρHα . Hausdorff’s theorem [2] (or see [7]) and the Abraham-Bonnet generalisation in [1] are both characterisations of the class of linear and FAC posets, respectively, which do not embed the rationals. The latter class is exactly ℵ0 H. To prove something like that we would need to know that if P ∈ aug(κ H) is an FAC poset, then Q(κ) embeds into P . Unfortunately we have not been able to prove such a claim for uncountable κ. The question if it is true even if we assume that κ has some large cardinal properties remains open. We shall instead prove a weaker claim, for which we shall fatten up our hierarchy a little. Definition 3.4. Let κ H∗ denote the closure of aug(κ H) under FAC weakenings, that is, the class obtained by taking all FAC orders P for which there is an order P  in κ H such that P is a weakening of P  . We shall show that κ H∗ lies between the classes of strongly and weakly κ-scattered FAC partial orders. Let us first show the easy direction. Claim 3.5. Every poset in κ H∗ is (weakly) κ-scattered and FAC. Proof of the Claim. Let P be in κ H∗ and let P  in aug(κ H) be such that P is an FAC weakening of P  . Clearly P is FAC. If P were not to be weakly κ-scattered then some strongly κ-dense order would embed into P and hence into P  , contradicting Lemma 3.2(3) and Lemma 3.2(4). 2 The heart of our main theorem lies in the following: Claim 3.6. Every strongly κ-scattered FAC partial order belongs to κ H∗ . Proof of the Claim. Suppose for a contradiction that P is a strongly κ-scattered FAC partial order which does not belong to κ H∗ . Let Q be any linear augmentation of P . By Lemma 2.11 Q is strongly κ-scattered, and by the definition of κ H∗ we have that Q ∈ / κH (and even Q ∈ / aug(κ H)).

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We will define an equivalence relation ≡ on elements of Q. For a, b ∈ Q let a ≡ b iff the interval in Q between a and b is in κ H. It is easily seen that this indeed is an equivalence relation. For a ∈ Q let Ca = {b : a ≡ b} be the equivalence class of a. Subclaim 3.7. Each Ca with the order induced from Q is in κ H. Proof of the Subclaim. Given Ca . By induction on γ, an ordinal, pick if possible the elements aγ and bγ in Ca so that a0 = b0 = a, aγ is Q-increasing with γ and bγ is Q-decreasing with γ. Since Ca is a set, there must be ordinals α, the first γ for which we cannot choose aγ and β, the first γ for which we cannot choose bγ . Then Ca is the lexicographic sum Σi<β [bi+1 , bi ) ⊕ {a0 } ⊕ Σj<α (aj , aj+1 ]. Note that each of the intervals mentioned above is κ H, by the definition of Ca and the fact that ≡ is an equivalence relation. Since κ H is closed under lexicographic sums of the 2 above kind, we obtain that Ca ∈ κ H.

Subclaim 3.8. If a, b ∈ Q are not ≡-equivalent and a
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we do not need to start with κ-well-founded FAC posets in the formation of κ H1 . We may start with κ-well founded linear orders and then the FAC posets get picked up when we form κ H∗ . Claim 3.9. Suppose that κ is a regular cardinal. Then κ H∗ is the closure of the class of all κ-well founded linear orders under inversions, lexicographic sums, FAC weakenings and augmentations. Proof of the Claim. Let H denote the closure of of the class of all κ-well founded linear orders under inversions, lexicographic sums, FAC weakenings and augmentations. Since κ ∗ H is the closure of the class of κ-well founded FAC posets under these operations we have that κ H∗ ⊇ H. On the other hand, if P ∈ κ H∗ then let Q ∈ aug(κ H) be such that P is an FAC weakening of Q and let R ∈ κ H be such that Q is an augmentation of R. If R ∈ H then Q ∈ H by the closure of H under augmentations and hence P ∈ H by the closure of H under FAC weakenings. Hence it suffices to show that κ H ⊆ H. Let ρ ≥ 1 be any ordinal. We shall show by induction on α that κ Hαρ ⊆ H. We first need a subclaim. Subclaim 3.10. Every augmentation of a κ-well founded FAC poset is κ-well founded. Proof of the Subclaim. Let P be a κ-well founded FAC poset and Q an augmentation of P . Suppose that aα ; α < κ is a ≤Q -decreasing sequence. For α < β < κ define f (α, β) = 1 if aα and aβ are comparable in P and let f (α, β) = 0 otherwise. We now use the Dushnik-Miller theorem which says that either there is an infinite 0-homogeneous set or a 1-homogeneous set of type κ. Since P is an FAC poset there cannot be an infinite 0-homogeneous set. However, a 1-homogeneous set of type κ would contradict the fact that P is κ-well founded. This contradiction proves the subclaim. 2 We now proceed with the promised inductive proof. If α = 0 the conclusion is clear. If P ∈ κ H1ρ then P is FAC and either P or its inverse (or both) are κ-well founded. In the first case we can use the subclaim to find Q which is a κ-well founded linear augmentation of P . Hence Q ∈ H and as its FAC weakening, P ∈ H. The other case is similar. The case of α a limit ordinal follows from the inductive hypothesis and the case α = β + 1 for β > 0 follows by the closure of H under lexicographic sums. 2 Let us observe the following: Observation 3.11. Suppose that P is a linear order, Q is an FAC weakening of P , and R is an augmentation of Q. Then R is an FAC weakening of P .

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We conclude that the following theorem is true. Main Theorem 3.12. Assume that κ is a regular cardinal. Let κ H∗ denote the closure of the class of all κ-well founded linear orders under inversions, lexicographic sums and FAC weakenings. Equivalently, (1) κ H∗ contains all strongly κ-scattered FAC posets. (2) κ H∗ is contained in the class of all κ-scattered FAC posets. If κ = ℵ0 we obtain an equality between the notions of κ-dense and strongly κ-dense. Hence, applying Theorem 3.12 to κ = ℵ0 we obtain that H∗ is the class of all scattered FAC posets. Since the Abraham-Bonnet theorem already gives that this class of posets is described by H we have as a corollary Corollary 3.13.

ℵ0

H∗ is exactly the Abraham-Bonnet class

ℵ0

H.

In general the two notions of density are not equivalent, as we illustrate in §4. Moreover, example 4.1 shows that for every uncountable κ with κ = κ<κ there are members of κ H which are not strongly κ-scattered. We also do not know for which uncountable κ we obtain that κ H∗ is the same as κ H. Note that it is not to be expected that κ Hρ is closed under FAC weakenings as weakening a partial order generally adds larger antichains and hence increases the antichain rank. When reduced to the class of linear orders the class κ H∗ can be replaced by a simpler class. Theorem 3.14. Assume that κ is a regular cardinal. Let κ L∗ denote the closure of the class of all κ-well founded linear orders under inversions and lexicographic sums. Then: (1) κ L∗ contains all strongly κ-scattered linear orders. (2) κ L∗ is contained in the class of all κ-scattered linear orders.

Proof. Linear orders are FAC posets with antichain rank ≤ 1. By Lemma 3.2(1) the class κ H1 is the least class that contains the κ-well founded linear orders and is closed under inversions and lexicographic sums, hence κ L∗ = κ H1 . Since every order in κ H1 is linear we obtain κ H1 = aug(κ H1 ), and hence Lemma 3.2(4) gives part (2) of the theorem. To prove (1) we use the proof of Claim 3.6. We start with a strongly κ-scattered linear order Q that does not belong to κ L∗ = κ H1 and obtain a contradiction literally as in the proof of that claim.  With κ = ℵ0 Theorem 3.14 gives Hausdorff’s theorem. The above theorems and remarks raise the following questions

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Question 3.15. (1) For which uncountable κ is aug(κ Hρ ) exactly the class of all κscattered FAC posets with antichain rank ≤ ρ? (2) For which κ is it true that any FAC poset all of whose subposets (or even just chains) of size κ belong to κ H, is itself an element of κ H? (3) For which κ > ℵ0 is it true that every augmentation of a [strongly] κ-scattered κ-AC poset is [strongly] κ-scattered? (4) For which κ > ℵ0 is κ H closed under FAC weakenings? We comment that one may generalise Theorem 3.12 to the case of λ < κ where both λ and κ are equal to their weak powers, and consider the situation of posets of size κ that satisfy (strong) λ-density, obtaining the expected results.

4

Appendix

For the sake of completeness we include some examples that illustrate the difference between weak κ-density and strong κ-density. We shall assume that κ is an uncountable regular cardinal. An easy example of a linear order that is κ-dense but not strongly κ-dense is the lexicographic sum along ω + ω ∗ of any strongly κ-dense order. This order is clearly not strongly κ-dense. We give an example of a κ-dense linear order which is weakly κ-scattered and moreover does not have a κ-decreasing sequence. Let L0 be the lexicographic sum along ω ∗ of copies of κ. By induction on n < ω define Ln by letting Ln+1 be the lexicographic sum along Ln of copies of L0 . We denote the  order of Ln by ≤n . Let L = n<ω Ln be ordered by letting p ≤ q iff p ≤n q for the first n that contains both p and q. Claim 4.1. No Ln for n < ω is κ-dense. L is κ-dense. Proof. The first statements can easily be proven by induction. For the second one, let p < q and let n be the first such that p, q ∈ Ln . By the definition of Ln+1 there is a copy of L0 in {x ∈ Ln+1 : p < x < q}, so clearly the size of this set is κ.  Claim 4.2. L does not have a decreasing sequence of size κ.  Proof. Suppose it had such a decreasing sequence, call it S. Then S = n<ω S ∩ (Ln+1 \ Ln ). By the regularity of κ > ℵ0 there has to be n for which the size of S ∩ (Ln+1 \ Ln ) is κ. Hence it suffices for us to show that no Ln can have a decreasing sequence of size κ. This can be done by induction on n. Hence by Lemma 2.4 we have Corollary 4.3. L does not embed any strongly κ-dense order and L is in κ Hρ for any

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ρ ≥ 1. This shows that the boundary of κ Hρ is somewhere in between weakly κ-scattered and strongly κ-scattered. We conjecture that κ Hρ contains all strongly κ-scattered FAC posets.

References [1] U. Abraham and R. Bonnet: “Hausdorff’s Theorem for Posets That Satisfy the Finite Antichain Property”, Fundamenta Mathematica, Vol. 159(1), (1999), pp. 51–69. [2] F. Hausdorff: “Grundz¨ uge einer Theorie der geordneten Mengenlehre” (in German), Mathematische Annalen, Vol. 65, (1908), pp. 435–505. [3] R. Bonnet and M. Pouzet: “Linear Extensions of Ordered Sets”, In: Ordered Sets, D. Reidel Publishing Company, 1982, pp. 125–170. [4] R. Bonnet and M. Pouzet: “Extension et stratification d’ensembles dispers´es” (in French), C.R.A.S., Paris, S´erie A, Vol. 168, (1969), pp. 1512–1515. [5] S. Shelah: Nonstructure Theory, to appear. [6] S. Shelah: Classification Theory, Revised ed., Studies in Logic and Foundations of Mathematics, Vol. 92, North-Holland, 1990. [7] G. Asser, J. Flachsmeyer and W. Rinow: Theory of Sets and Topology; In honour of Felix Hausdorff, Deutscher Verlag der Wissenschaften, 1972. [8] H.J. Kiesler and C.C. Chang: Model Theory, 3rd ed., Studies in Logic and Foundations of Mathematics, Vol. 73, Elsevier Science B.V., 1990. [9] R. Fra¨ıss´e: Theory of Relations, Revised ed., Studies in Logic and Foundations of Mathematics, Vol. 145, Elsevier Science, B.V., 2000. [10] J. Rosenstein: Linear Orderings, Pure and Applied Mathematics, Academic Press, 1982. [11] E. Mendelson: “On a class of universal ordered sets”, Proc. Amer. Math. Soc., Vol. 9, (1958), pp. 712–713. [12] M. Kojman and S. Shelah: “Non-existence of Universal Orders in Many Cardinals”, J. Symbolic Logic, Vol. 57(3), (1992), pp. 875–891.

DOI: 10.2478/s11533-006-0004-y Research article CEJM 4(2) 2006 242–249

Character formulae for classical groups∗ P´eter E. Frenkel† Institute of Mathematics, Budapest University of Technology and Economics, 1111 Budapest, Hungary

Received 21 November 2005; accepted 20 January 2006 Abstract: We give formulae relating the value χλ (g) of an irreducible character of a classical group G to entries of powers of the matrix g ∈ G. This yields a far-reaching generalization of a result of J.L. Cisneros-Molina concerning the GL2 case [1]. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Character formula, irreducible character, classical group, powers of a matrix MSC (2000): 20G05

Introduction The Weyl character formula [2–5] tells us how to compute the character χλ of an irreducible finite dimensional representation Vλ with highest weight λ of a (complex, semisimple, connected) Lie group G:   (−1)σ z σ(λ+ρ) = χλ · (−1)σ z σρ . σ∈W

σ∈W

˜ → C∗ is Here, for each weight  ∈ h∗ of the Lie algebra g of G, the exponential z  : H ˜ of a maximal torus H ≤ G the corresponding multiplicative character of the preimage H ˜ → G. The weight ρ ∈ h∗ is the half-sum of the positive roots in a universal covering G and W is the Weyl group. Note that both sides of the formula are W–antisymmetric ˜ but χλ is well-defined as a W–symmetric character of H. Note also characters of H,  that Δ = σ∈W (−1)σ z σρ is not identically zero, so χλ is expressed as a ratio of Laurent ˜ (We know a priori that χλ is itself a Laurent polynomials in the coordinates on H. ∗ †

Partially supported by OTKA grants T 042769 and T 046365 E-mail: [email protected]

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polynomial in the coordinates on H, but with many more terms in general than the numerator and denominator.) The formula expresses the value of the character χλ at a group element g ∈ G in terms of a conjugate of the semisimple part of g in the maximal torus H, i.e., in the case of the classical matrix groups, in terms of the eigenvalues of g. There are equally explicit expressions, called determinantal identities or Giambelli formulae [2, Section A.1, formulae (A.5), (A.6) and Section A.3], in terms of the elementary resp. the complete symmetric polynomials in the eigenvalues. In the present paper, we consider the connected classical groups and we prove variants of the Weyl character formula that express the value χλ (g) in many different, explicit rational ways in terms of the entries of powers of the generic matrix g ∈ G. It seems likely that these formulae provide the fastest and most straightforward way of calculating χλ (g) for generic g. This paper was motivated by J. L. Cisneros-Molina’s paper [1] whose main result is the following. Let ω = 0 be a linear function on the space M2 of 2 × 2 matrices, such that ω (1) = 0. For example, ω could be one of the two off-diagonal entries. Then, for   λ = 0, 1, . . . , we have ω g λ+1 /ω (g) = tr S λ g, the trace of the action of g on the λ-th symmetric power of the standard vector representation. Our results can be considered as far-reaching generalizations of this fact. In particular, for g ∈ Mr+1 , and λ = 0, 1, . . . , the trace tr S λ g equals the ratio of the r–dimensional volumes of the two parallelepipeds spanned in Mr+1 /M1 by the images of g, g 2 , . . . , g r−1 , g λ+r and of g, g 2 , . . . , g r−1 , g r , respectively (except when both volumes are zero, i.e., g has a minimal polynomial of degree < r + 1). This is a particular case of Corollary 1.3 below. Our proofs are motivated by the first of the four proofs given in [1], which is due to Jeremy Rickard.

1

General linear group

Let G = GLr+1 (C) with H = (C∗ )r+1 the maximal torus consisting of all invertible diagonal matrices, Hom (H, C∗ ) = Zr+1 the weight lattice, and W = Sr+1 the Weyl group. Write   2 − r −r r r−2 , ,..., , ρ= 2 2 2 2 for the half-sum of the positive roots. Set ρt = ρ + (t, t, . . . , t, t) for t ∈ C. For λ = (λ0 , . . . , λr ) ∈ Zr+1 , write z λ : H → C∗ for the corresponding multiplicative character of the torus H, and, when λ is dominant, i.e. λ0 ≥ · · · ≥ λr , write χλ : G → C for the character of the irreducible representation with highest weight λ. The Weyl character formula   (−1)σ z σ(λ+ρt ) = χλ · (−1)σ z σρt σ∈W

σ∈W

holds for any t ∈ C. The freedom in the choice of t comes from the central C∗ in G. ˜ When Both sides of the formula are W–antisymmetric characters of the infinite cover H.

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ρt ∈ Zr+1 , both sides descend to H. For  ∈ Cr+1 and g ∈ G, define g =

r 

g i ∈ Mr+1 (C)⊗(r+1) .

i=0

This is multi-valued since it depends on a choice of the value of log g ∈ g = glr+1 (C). When  ∈ Zr+1 , it is single-valued. When  ∈ Zr+1 ≥0 , we may allow g ∈ Mr+1 (C) rather than g ∈ G. (Recall that, for g ∈ G, a value of log g is defined to be any matrix X ∈ g such that def  n exp X = ∞ n=0 X /n! = g. Such values X exist for any g ∈ G; see e.g. [3, Section 1.3]. Then we define g i = exp(i log g). When i ∈ Z, this does not depend on the chosen value of log g and coincides with the elementary definition of the matrix power.) We have r  (−1)σ g σ = g i ∈ Mr+1 (C)∧(r+1) . i=0

σ∈W

Theorem 1.1. Let λ = (λ0 ≥ · · · ≥ λr ) ∈ Zr+1 and g ∈ GLr+1 (C). Then, for any t ∈ C,   (−1)σ g σ(λ+ρt ) = χλ (g) · (−1)σ g σρt ; σ∈W

equivalently,

σ∈W r

g

λi +r/2−i+t

= χλ (g) ·

i=0

r

g r/2−i+t ,

i=0

where the powers are defined using any (but always the same) value of log g. When ρt ∈ Zr+1 , the powers are single-valued. In particular, for t = r/2, we get r

g λi +r−i = χλ (g) ·

i=0

r

g r−i .

i=0

When ρt and λ are both in Zr+1 ≥0 , we may allow g ∈ Mr+1 (C). Proof. The set of diagonalizable invertible matrices is dense in Mr+1 (C), so we may assume that g is diagonalizable. The statement of the theorem is invariant under conjugation, so we may assume that g = diag (z0 , . . . , zr ) ∈ H. Then  σ∈W

σ

(−1) g

σ

=

r i=0

r

 i

g = zj · ejj , i

j=0

where ejj is the diagonal matrix with a single 1 at the j-th position. The theorem now follows from the Weyl character formula.  Corollary 1.2. Let Ω be an alternating (r + 1)–linear form on the space Mr+1 (C). Then, for λ = (λ0 ≥ · · · ≥ λr ) ∈ Zr+1 , we have     Ω g λ0 +r , g λ1 +r−1 , . . . , g λr = χλ (g) · Ω g r , g r−1 , . . . , 1 .

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To express χλ (g) as a rational function in entries of powers of g, we must choose Ω such that the right hand side is not identically zero. For example, Ω(g0 , . . . , gr ) could be the determinant of the matrix formed by the diagonals, or by the first rows, etc. of the argument matrices. To calculate χλ (g) for a numerically given g, we need to choose  Ω such that Ω(g r , g r−1 , . . . , 1) = 0. This is possible if and only if ri=0 g r−i = 0, i.e., the minimal polynomial of g is its characteristic polynomial. When g has a minimal polynomial of lower degree, we can use l’Hospital’s rule. Corollary 1.3. Let ω be an alternating r–linear form on the space Mr+1 (C) such that ω vanishes if an argument is 1. Then, for λ as above and with λr = 0, we have     ω g λ0 +r , g λ1 +r−1 , . . . , g λr−1 = χλ (g) · ω g r , g r−1 , . . . , g . To express χλ (g) as a rational function in entries of powers of g, we must choose ω such that the right hand side is not identically zero. For example, ω(g0 , . . . , gr−1 ) could be the determinant of the r × r matrix formed by the truncated (i.e., leftmost entry omitted) first rows of the argument matrices. To calculate χλ (g) for a numerically given g, we need to choose ω such that ω(g r , g r−1 , . . . , g) = 0. This is possible if and only if the minimal polynomial of g is its characteristic polynomial. Corollary 1.3, for r = 1, is the result of J.L. Cisneros-Molina’s paper [1] mentioned in the Introduction. To derive Corollary 1.3 from Corollary 1.2, simply set Ω = dω, defined as usual by Ω (g0 , . . . , gr ) =

r 

(−1)i ω (g0 , . . . , gi−1 , gi+1 , . . . , gr ) .

i=0

Then Ω (g0 , . . . , gr−1 , 1) = (−1)r ω (g0 , . . . , gr−1 ) and Corollary 1.3 follows.

2

Special linear group

Let G = SLr+1 (C) with H (C∗ )r the maximal torus consisting of all unimodular diagonal matrices, Hom (H, C∗ ) = Zr+1 /Z the weight lattice, and W = Sr+1 the Weyl group. Write ρ = (r, r − 1, . . . , 0) + Z · (1, . . . , 1) ∈ Zr+1 /Z for the half-sum of the positive roots. When λ = (λ0 , . . . , λr ) ∈ Zr+1 /Z, write z λ : H → C∗ for the corresponding multiplicative character of the torus H, and, when λ is dominant, write χλ : G → C for the character of the irreducible representation with highest weight λ. The Weyl character formula is valid as stated in the introduction. Both sides are W–antisymmetric characters of H. For  ∈ Zr+1 /Z and g ∈ G, the antisymmetric tensor r

g i ∈ Mr+1 (C)∧(r+1)

i=0

is well defined because either g has a minimal polynomial of degree < r + 1, in which case the algebra C[g] has dimension < r +1 and the antisymmetric tensor above is zero, or else

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g has its characteristic polynomial as minimal polynomial, in which case dim C[g] = r + 1 and multiplication by g on it has determinant det g = 1, so the tensor is independent of the chosen representative of . Theorem 2.1. Let λ = (λ0 ≥ · · · ≥ λr ) + Z · (1, . . . , 1) ∈ Zr+1 /Z and g ∈ SLr+1 (C). Then r r g i = χλ (g) · g r−i , i=0

i=0

where i = λi + r − i. Proof. The theorem trivially follows from Theorem 1.1.

3



Odd special orthogonal group

Let G = SO2r+1 (C) be the connected group preserving the quadratic form x 1 y1 + · · · + x r yr + z 2 . We take the maximal torus H = (C∗ )r consisting of all special orthogonal diagonal   matrices diag z1 , z1−1 , . . . , zr , zr−1 , 1 . In the weight lattice Hom (H, C∗ ) = Zr , we take λj zj . The Weyl group W is λ = (λ1 , . . . , λr ) to correspond to the monomial z λ = r the semidirect product of Sr and Z2 . Write ρ = (r − 1/2, r − 3/2, . . . , 3/2, 1/2) for the half-sum of the positive roots. The Weyl character formula is valid as stated in the ˜ introduction. Both sides are W–antisymmetric characters of the double cover H.   r For  ∈ Z + 12 and g ∈ G, define 

g =

r 

g i ∈ M2r (C)⊗r .

i=1

√ √ This is multi-valued since it depends on a choice of g. (We may choose any matrix g √ √ whose square is g. Such matrices g ∈ G always exist. Then write g i = g 2i to define the matrix power.) We have 

(−1)σ g σ =

σ∈W

r  i  g − g −i ∈ M2r (C)∧r . i=1

Theorem 3.1. Let λ = (λ1 ≥ · · · ≥ λr ) ∈ Zr≥0 and g ∈ SO2r+1 (C). Then 

(−1)σ g σ = χλ (g) ·

σ∈W

equivalently,

r  i=1



(−1)σ g σρ ;

σ∈W

r   r+1/2−i  g i − g −i = χλ (g) · g − g −(r+1/2−i) , i=1

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where  = λ + ρ, i.e. i = λi + r + 1/2 − i, and the powers are defined using any, but √ always the same value of g ∈ SO2r+1 (C). √ Proof. The set of diagonalizable matrices is dense in G, so we may assume that g is diagonalizable. The statement of the theorem is invariant under conjugation, so we may assume that

 √ 1/2 −1/2 g = diag z1 , z1 , . . . , zr1/2 , zr−1/2 , 1 ∈ H. Then



σ

(−1) g

σ

=

r 

i

g −g

−i



r



−i

i

= zj − zj · (ejj − fjj ) ,

i=1

σ∈W

j=1

where ejj resp. fjj is the diagonal matrix with a single 1 at the position corresponding to the xj resp. yj coordinate. The theorem now follows from the Weyl character formula.

4

Symplectic group

r     Let G = Sp2r (C) be the group preserving the skew bilinear form i=1 (xi yi − yi xi ) r on C2r . We take the maximal torus H = (C∗ ) consisting of all symplectic diagonal   matrices diag z1 , z1−1 , . . . , zr , zr−1 . In the weight lattice Hom (H, C∗ ) = Zr , we take λ λ = (λ1 , . . . , λr ) to correspond to the monomial z λ = zj j . The Weyl group W is the semidirect product of Sr and Z2r . Write ρ = (r, r − 1, . . . , 1) for the half-sum of the positive roots. The Weyl character formula is valid as stated in the introduction. Both sides are W–antisymmetric characters of H. For  ∈ Zr and g ∈ G, define 

g =

r 

g i ∈ M2r (C)⊗r .

i=1

Then



σ

(−1) g

σ

r  i  g − g −i ∈ M2r (C)∧r . = i=1

σ∈W

Theorem 4.1. Let λ = (λ1 ≥ · · · ≥ λr ) ∈ Zr≥0 and g ∈ Sp2r (C). Then 

(−1)σ g σ = χλ (g) ·

σ∈W

equivalently,

r 

i

g −g



(−1)σ g σρ ;

σ∈W

−i



= χλ (g) ·

i=1

r 

 g r+1−i − g −(r+1−i) ,

i=1

where  = λ + ρ, i.e., i = λi + r + 1 − i. Proof. The set of diagonalizable matrices is dense in G, so we may assume that g is diagonalizable. The statement of the theorem is invariant under conjugation, so we may

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P.E. Frenkel / Central European Journal of Mathematics 4(2) 2006 242–249

  assume that g = diag z1 , z1−1 , . . . , zr , zr−1 ∈ H. Then 

σ

(−1) g

σ

=

r 

i

g −g

−i



r



−i

i

= zj − zj · (ejj − fjj ) ,

i=1

σ∈W

j=1

where ejj resp. fjj is the diagonal matrix with a single 1 at the position corresponding to the xj resp. yj coordinate. The theorem now follows from the Weyl character formula. 

5

Even special orthogonal group

Let G = SO2r (C) be the connected group preserving the quadratic form Q = x 1 y1 + · · · + x r yr . We take the maximal torus H = (C∗ )r consisting of all special orthogonal diagonal   matrices diag z1 , z1−1 , . . . , zr , zr−1 . In the weight lattice Hom (H, C∗ ) = Zr , we take λ λ = (λ1 , . . . , λr ) to correspond to the monomial z λ = zj j . The Weyl group W is the semidirect product of Sr and Z2r−1 . It acts by permuting the indices and by performing ˜ > W for the Weyl group in the full orthogonal an even number of sign changes. Write W ˜ we write [σ] for group O2r (C). It is the semidirect product of Sr and Z2r . If σ ∈ W, its image in Sr . Write ρ = (r − 1, r − 2, . . . , 1, 0) for the half-sum of the positive roots. ˜ The Weyl Write  = (1, 1, . . . , 1, 1) so that e =  + ρ = (r, r − 1, . . . , 2, 1) is regular for W. character formula is valid as stated in the introduction. Both sides are W–antisymmetric characters of H. For  ∈ Zr and g ∈ G, define r 



g =

g i ∈ M2r (C)⊗r .

i=1

We have 2



(−1)σ g σ =

σ∈W

=



 (−1)[σ] + (−1)σ g σ =

˜ σ∈W r



i=1

r   i  g i + g −i + g − g −i ∈ M2r (C)∧r . i=1

Note that the second term is zero if any i is zero. ¯ = Theorem 5.1. Let λ = (λ1 , . . . , λr ) ∈ Zr with λ1 ≥ · · · ≥ λr−1 ≥ |λr |. Set λ (λ1 , . . . , λr−1 , −λr ). Let g ∈ SO2r (C). Then 2

 ˜ σ∈W

(−1)[σ] g σ = (χλ + χλ¯ ) (g) ·



(−1)[σ] g σρ ;

˜ σ∈W

equivalently, 2

r r  i   r−i  g + g −i = (χλ + χλ¯ ) (g) · g + g −(r−i) . i=1

i=1

P.E. Frenkel / Central European Journal of Mathematics 4(2) 2006 242–249

Also,



(χ − χ¯) (g) ·

(−1)σ g σ = (χλ − χλ¯ ) (g) ·

˜ σ∈W



249

(−1)σ g σe ;

˜ σ∈W

equivalently, √

r r   i   r+1−i  −i · g −g = (χλ − χλ¯ ) (g) · g −1 Pf g − g − g −(r+1−i) .



r

−1

i=1

i=1

Throughout,  = λ + ρ, i.e., i = λi + r − i. Note that (χ − χ¯) (g) =

√

  r −1 Pf g − g −1 ,

where the sign of the Pfaffian of g − g −1 ∈ so2r (C) is specified by declaring the ordered Q–  √ r orthonormal bases of the standard vector representation C2r with determinant 2 −1 to be of positive orientation. Proof. The set of diagonalizable matrices is dense in G, so we may assume that g is diagonalizable. The statement of the theorem is invariant under conjugation, so we may   assume that g = diag z1 , z1−1 , . . . , zr , zr−1 ∈ H. Then 

(−1)

[σ]

g

σ

=

 ˜ σ∈W

i

g +g

−i



i=1

˜ σ∈W

and

r 

σ

(−1) g

σ

=

r  i=1

r



−i

i

= zj + zj · (ejj + fjj ) j=1

i

g −g

−i



r



−i

i

= zj − zj · (ejj − fjj ) , j=1

where ejj resp. fjj is the diagonal matrix with a single 1 at the position corresponding to the xj resp. yj coordinate. The theorem now follows from the Weyl character formula. 

References [1] J. L. Cisneros-Molina: “An invariant of 2 × 2 matrices”, Electr. J. Linear Algebra, Vol. 13, (2005), pp. 146–152. [2] W. Fulton and J. Harris: Representation theory, GTM, Springer, New York, 1991. [3] R. Goodman and N.R. Wallach: Representations and invariants of the classical groups, Cambridge University Press, Cambridge, 1998. [4] H. Weyl: “Theorie der Darstellung kontinuerlicher halbeinfacher Gruppen durch lineare Transformationen, I, II, III, und Nachtrag”, Math. Zeitschrift, Vol. 23, (1925), pp. 271–309; Vol. 24, (1925), pp. 328–376, 377–395, 789-791; reprinted in Selecta Hermann Weyl, Birkh¨auser, Basel, 1956, pp. 262–366. [5] H. Weyl: The classical groups, Their invariants and representations, Princeton University Press, Princeton, 1946.

DOI: 10.2478/s11533-006-0010-0 Research article CEJM 4(2) 2006 250–259

Squared cycles in monomial relations algebras Brian Jue∗ Department of Mathematics, California State University, Stanislaus, Turlock, California 95382, USA

Received 8 December 2005; accepted 16 February 2006 Abstract: Let K be an algebraically closed field. Consider a finite dimensional monomial relations algebra Λ = KΓ/I of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path algebra KΓ. There are many modules over Λ which may be represented graphically by a tree with respect to a top element, of which the indecomposable projectives are the most natural example. These trees possess branches which correspond to right subpaths of cycles in the quiver. A pattern in the syzygies of a specific factor module of the corresponding indecomposable projective module is found, allowing us to conclude that the square of any cycle must lie in the ideal I. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Representation theory, homological dimension, syzygies MSC (2000): 16E05, 16E10, 16G10, 16G20

1

Introduction

Let Λ be a finite dimensional algebra over an algebraically closed field K, with Jacobson radical J. Consider a fixed finite sequence S = (S(0), . . . , S()) of simple left modules over Λ. It is then quite natural to study the class of uniserial modules over Λ which possess the sequence of composition factors S, ie. modules U for which J i U/J i+1 U ∼ = S(i) for all 0 ≤ i ≤ . Recall from Gabriel [6] that over an algebraically closed field K, any finite dimensional algebra Λ is Morita equivalent to a path algebra modulo relations of the form KΓ/I, where Γ is a quiver and I an admissible ideal of the path algebra KΓ. Because the ∗

E-mail: [email protected]

B. Jue / Central European Journal of Mathematics 4(2) 2006 250–259

251

module categories over the algebras Λ and KΓ/I are equivalent, we may therefore assume Λ = KΓ/I. Bongartz and Huisgen-Zimmermann [3, 4] have constructed a quasi-projective subvariety G-Uni(S) of a Grassmannian, and a parametrization map ⎧ ⎫ ⎪ ⎨ isomorphism classes of uniserial modules ⎪ ⎬ φS : G-Uni(S) −→ ⎪ ⎩ with sequence of composition factors S ⎪ ⎭ Two important aspects of their construction are that φS is nearly bijective, and although the G-Uni(S) are not tractable in general, each is covered by a collection of open affine sets which are computationally accessible via quiver and relations. We shall focus our discussion on monomial relations algebras (also known as zero relations algebras in the literature), where the ideal I is generated by a finite set of paths in the path algebra KΓ. If all cycles in KΓ lie in the ideal I, then a simple computation using the techniques in [8] will show that either G-Uni(S) is empty, or the affine sets covering the variety G-Uni(S) are all isomorphic to An for some non-negative n. Therefore, more geometrically interesting covering sets can only appear when there exist cycles lying outside the ideal I. We began to look for classes of paths which must always lie in the ideal I, or must always lie outside of I. It was first conjectured that the square of any cycle in the quiver Γ must lie in the ideal I. However, it was soon apparent that our conjecture was false in the situation where the cycle was merely a loop. For a specific counterexample, let Γ be the quiver having a unique vertex and a unique arrow α, with I = α3 . Then α is a cycle, yet α2 does not lie in the ideal I. Now if the quiver has loops and the base field K is algebraically closed, Igusa [11] has shown that the algebra Λ has infinite global dimension. So to avoid such quivers with loops, we then refined our conjecture to If Λ = KΓ/I is a monomial relations algebra of finite global dimension over an algebraically closed field, then the square of any cycle lies in I. The purpose of this article is to prove our assertion. In fact we shall prove a stronger combinatorial result, from which a proof of this conjecture is an immediate corollary. Then in the final section, we will use this result to study the fibres of Bongartz and Huisgen-Zimmermann’s parametrization map φS .

2

Preliminaries

Throughout this paper Λ = KΓ/I will be a finite dimensional path algebra modulo relations over an algebraically closed field K, where Γ is a quiver and I an admissible ideal of the path algebra KΓ. It will be convenient to identify the vertices of Γ with a full set of primitive idempotents of Λ. We denote the Jacobson radical of Λ by J. All modules will be left modules over Λ, and p dim M shall abbreviate the projective dimension of the module M .

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B. Jue / Central European Journal of Mathematics 4(2) 2006 250–259

Our convention for the product of paths p and q in KΓ will be that pq represents the path q followed by p whenever the concatenation is defined, and pq = 0 otherwise. If p, q, and r are paths such that s = pqr, we shall say that r is a right subpath of s and p is a left subpath of s. Moreover, we shall also say that p, q and r are subpaths of s. A cycle is a non-trivial path beginning and ending in the same vertex, while a loop is a cycle of length one. Let us return to our fixed sequence of simple modules S = (S(0), . . . , S()), with U a uniserial module having this sequence of composition factors. Under the correspondence betweeen simple Λ-modules and vertices of the quiver Γ, the sequence S corresponds to a sequence of vertices (e(0), . . . , e()). Let us now consider paths in the quiver which pass through these vertices in this particular order. However, only certain paths are of particular interest. These are called masts of uniserials. Definition 2.1. Suppose U is a uniserial module of length  + 1. Then any path p of length  such that pU = 0 is called a mast of U . Remark 2.2. [8] (1) Every uniserial module has a mast. (2) If the quiver Γ has no double arrows, then masts are unique. Conversely, the uniqueness of masts implies the absence of double arrows. (3) Not every path in KΓ − I must be a mast of some uniserial module. For example, consider the algebra Λ = KΓ/I, where Γ is the quiver q8 2 MMMM qq MMβ MM q MM q q MM q qq & 84 1 MMM q q MM qq MM q q q γ MMM MM qqqq δ & q αqqq

3

and I = βα − δγ. Observe that neither the path βα nor the path δγ is the mast of any uniserial module over Λ. Many modules can be represented graphically as a disjoint union of trees relative to a top element, the uniserials being a natural example here. But the modules which can be represented by a single tree are of particular interest for our discussion. Definition 2.3. A top element of a module M is an element x ∈ M − JM such that ei x = x for some vertex i of the quiver. The top of a module M is the factor module M/JM . Definition 2.4. Suppose the top of a module T is simple, and there exists a top element x of T such that the graph of T relative to x is a tree. We will then refer to T as a tree module. A tree module shall be said to have top e, if T /JT ∼ = Λe/Je for some primitive

B. Jue / Central European Journal of Mathematics 4(2) 2006 250–259

253

idempotent e. Let e be a primitive idempotent of a monomial relations algebra Λ. Then the graph of the indecomposable projective module Λe, with respect to the top element e, is a tree. In fact, numerous factor modules of the indecomposable projectives are tree modules as well. We record these and some other basic properties of monomial relations algebras in the following remark. Remark 2.5. Let Λ be a monomial relations algebra, and p a path beginning in a vertex e. Then: (1) Λp is a tree module. (2) Jp is a direct sum of tree modules. (3) Λe/Λp and Λe/Jp are tree modules. Let us illustrate this remark with a concrete example. Example 2.6. Let Λ = KΓ/I, where Γ is the quiver

α1

O1O β1

β2

/2 α2

γ





4o

α3

3

and I = β2 α3 , α1 β1 α3 , γα1 β1 , α2 α1 β2 , α3 α2 α1 β1 , β1 γα1 β2 , β2 γα1 β2 . Then the module Λα1 is a tree module, and its graph with respect to the top element α1 is

2=

===γ
==

=

3

4 === β1

α3

==β2

==





α2

4 β1

1

1

α1

2 α2

3

1

α1

2 γ

4

Jα1 is a direct sum of the tree modules Λα2 α1 and Λγα1 , which have graphs

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B. Jue / Central European Journal of Mathematics 4(2) 2006 250–259

3 α3

4

4=

===β2
==

=

β1

1

and

α1

β1

1

2 α2

3

1

α1

2 γ

4

with respect to the top elements α2 α1 and γα1 , respectively. The factor module Λe1 /Λγα1 is also a tree module. Its graph with respect to the top element e1 + (Λγα1 ) is 1 α1

2 α2

3 α3

4 β1

1 Observe that Λe1 /Λγα1 is also a uniserial Λ-module, with mast β1 α3 α2 α1 .

3

c-branches and Main Results

Let T be a tree module, so that its graph relative to a top element is a tree. Beginning at the top of the tree and reading down, one can construct corresponding paths in the quiver. Our main concern are the paths which are right subpaths of cycles. Definition 3.1. Let c be a cycle. Suppose that T is a tree module, such that its graph relative to a top element x is a tree. Then a path q is called a c-branch of T if there exists an arrow α such that: (1) For some n ∈ N, the path αq is a right subpath of cn . (2) αqx = 0. (3) qx = 0. Remark 3.2. (1) We allow the path q in Definition 3.1 to be trivial, which ensures that c-branches always exist and are unique. (2) Intuitively, the c-branch of a tree module is the maximal right subpath of a sufficiently high power of c appearing in the graph relative to x.

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Example 3.3. Referring to the algebra Λ from Example 2.6, let c be the cycle α1 β2 α3 α2 . Then clearly α3 α2 is a c-branch of the tree module Λα1 . If we consider the cycle d = β1 γα1 , then α1 is a d-branch of the tree module Λe1 /Λγα1 . In order to prove our main result, two propositions are needed. We shall utilize a fixed cycle to construct a class of factor modules of the indecomposable projectives. The goal of both propositions is to find a pattern in the syzygies of at least one of these factor modules, which proves that the factor module has infinite projective dimension. But let us first make a notational note. Notation. If p is a nontrivial path, then p∗ shall denote the maximal proper right subpath of p, ie. the right subpath of p which is of length one less than p. Proposition 3.4. Let Λ be a monomial relations algebra, and c a cycle which begins and ends at a vertex e but is not a loop. If the projective module Λe has c-branch c∗ ck with k > 0, then p dim (Λe/Jc∗ ) = ∞. Proof. Observe that Λe/Jc∗ is a tree module, with c-branch c∗ . Then there exists a direct summand T1 of Ω1 (Λe/Jc∗ ) which is a tree module with top e and c-branch c∗ ck−1 . There also exists a direct summand T2 of Ω2 (Λe/Jc∗ ) which is again a tree module, with top e and c-branch c∗ . Now we see by an obvious induction that every odd syzygy of Λe/Jc∗ has a direct summand which is a tree module with top e and c-branch c∗ ck−1 , and every even syzygy of Λe/Jc∗ has a direct summand which is a tree module with top e and c-branch c∗ . Clearly no syzygy of Λe/Jc∗ is projective, and therefore p dim (Λe/Jc∗ ) = ∞.  Now consider a cycle c of length  ≥ 2: u 1 fMMMM uu MMM u u MMM uu u MMMα α1 uu MMM uu u MMM uu u MMM uu MMM u u zu / / −1 2 3 α2

α−1

Observe that by starting at each vertex e1 , e2 , . . . e , one can follow the arrows to construct  cycles of length . Moreover, each of these cycles “shifted” from c is a subpath of the cycle c2 . With this idea in hand, we are now ready to prove the second proposition. Proposition 3.5. Let Λ be a monomial relations algebra, and c a cycle which is not a loop. Suppose that every cycle which is of the same length as c and is also a subpath of c2 lies outside of the ideal I. Then there exists such a cycle d shifted from c, beginning and ending in a vertex e , with the property that p dim (Λe /Jd∗ ) = ∞. Proof. Suppose the cycle c begins and ends at the vertex e. If we express the c-branch

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of the projective module Λe in the form uck with u a proper right subpath of c, our hypothesis guarantees that k > 0. If u = c∗ , then Proposition 3.4 allows us to select d = c and e = e to obtain the desired factor module of infinite projective dimension. Therefore, we may assume u = c∗ . As stated in the proof of Proposition 3.4, Λe/Jc∗ is a tree module with c-branch c∗ . Again there exists a direct summand T1 of Ω1 (Λe/Jc∗ ) which is a tree module with top e and c-branch uck−1 , which implies that T1 cannot be projective. Now let α1 , α2 , . . . , α be arrows such that c = α · · · α2 α1 and  ≥ 2. Since u = c∗ but u is still a proper right subpath of c, we may express u = αr · · · α1 for some 0 ≤ r ≤  − 2. Let b be the cycle αr+1 · · · α1 α · · · αr+2 shifted from c which begins and ends in a vertex e . Consider the projective module Λe . Since b ∈ / I by hypothesis, Λe has b-branch vbj with j > 0 and v a proper right subpath of b. If v = b∗ , then let d = b and e = e to obtain the desired factor module by Proposition 3.4 again. Otherwise v = b∗ , and there exists a direct summand T2 of Ω2 (Λe/Jc∗ ) which is a tree module with top e and b-branch b∗ . Clearly T2 cannot be projective, but a simple inductive argument proves that a pattern develops as we progress to higher syzygies of Λe/Jc∗ . For every n > 2, there exists a vertex eˆ with ˆb a cycle shifted from c beginning and ending in eˆ, such that Λˆ e has ˆb-branch vˆˆbm with vˆ a proper right subpath of ˆb and m > 0. Moreover, Ωn (Λe/Jc∗ ) has a direct summand Tn which is a tree module with top eˆ and ˆb-branch (1) ˆb∗ , or (2) vˆˆbm−1 In both of these cases, the summand Tn cannot be projective. If vˆ = ˆb∗ in the second case ever occurs, then Λˆ e/J ˆb∗ is the desired factor module of infinite projective dimension  by Proposition 3.4. Otherwise, p dim (Λe/Jc∗ ) = ∞. Recall in the introduction that our conjecture on squared cycles in monomial relations algebras also includes the assumption of finite global dimension. This assumption was to prevent loops from apprearing in the quiver. Here now is our main theorem. Theorem 3.6. Let Λ be a monomial relations algebra of finite global dimension. Suppose α1 , α2 , . . . , α are arrows such that c = α · · · α2 α1 is a cycle. Then at least one of the  cycles α α−1 · · · α2 α1 , α1 α α−1 · · · α3 α2 , .. . α−1 · · · α2 α1 α shifted from c lies in the ideal I. Proof. If  = 1, then the cycle c would be a loop. But a loop in the quiver implies the algebra Λ has infinite global dimension [11], which is contrary to our hypothesis. We may

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therefore assume  ≥ 2. Let us assume to the contrary that none of the  shifted cycles lies in I. Observe that these cycles are precisely the cycles which have the same length as c, and are also subpaths of c2 . Then by Proposition 3.5, there exists a shifted cycle d beginning and ending in a vertex e such that p dim (Λe /Jd∗ ) = ∞. Therefore Λ has infinite global dimension, which leads to a contradiction.  Before continuing with the discussion, let us pause to compare our main theorem with a result of Burgess [5]: Let Λ be a monomial relations algebra of finite global dimension. If the quiver Γ contains a cycle c = α2 α1 of length 2, then exactly one of α1 α2 or α2 α1 lies in I. 1o

α1 α2

/

2

Our theorem cannot be sharpened to state that precisely one of the shifted cycles lies in the ideal if the length of the cycle is more than two. This is apparent from our next example. Example 3.7. Let Λ = KΓ/I, where Γ is the quiver 1O

α1

/2

α4

4o

α2 α3



3

and I = α2 α1 . One can verify by inspection that p dim Λe1 = 2, and p dim Λe2 = p dim Λe3 = p dim Λe4 = 1. Therefore Λ has global dimension 2, but the three shifted cycles α4 α3 α2 α1 , α2 α1 α4 α3 , and α3 α2 α1 α4 all lie in I. Now with the theorem proven, a proof of our conjecture follows immediately. Theorem 3.8. Let Λ be a monomial relations algebra of finite global dimension, and c a cycle. Then c2 ∈ I. Proof. The hypothesis of finite global dimension again implies that the length of the cycle c is at least two. It remains to observe that every cycle shifted from c in the statement of Theorem 3.6 is a subpath of c2 . The theorem follows immediately.  After proving this result, we learned from D. Zacharia that it can also be derived from [7] using universal covers of monomial relations algebras.

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Applications of the Theorem

We return to our fixed sequence S = (S(0), . . . , S()) of simple modules. The prohibition on squared cycles in the ideal I also places constraints on the fibres of Bongartz and Huisgen-Zimmermann’s parametrization map φS . This section shall be a study of these fibres. Let us first introduce a class of cycles and a class of monomial relations algebras which will be used in our next result. Definition 4.1. (1) A cycle c shall be called basic if no proper right subpath of c is a cycle. (2) If every cycle in KΓ which does not belong to the ideal I is the power of a basic cycle, then we shall say that the algebra Λ = KΓ/I has basic cycle structure. With the needed definitions in hand, let us proceed to our results. Corollary 4.2. Suppose that Λ is a monomial relations algebra of finite global dimension, with basic cycle structure. If the quasi-projective variety G-Uni(S) is non-empty, then the fibres of the parametrization map φS are isomorphic to either A0 or A1 . Proof. Suppose G-Uni(S) is non-empty. Then because the parametrization map φS is surjective, there exists a uniserial module U having the sequence of composition factors S, with some mast p. In fact, let us consider the collection of isomorphism classes of all uniserial modules with mast p. Denote by G-Uni(p) the inverse image of this collection under φS , and φp the restriction of the map φS to this subvariety G-Uni(p). The variety G-Uni(S) and map φS can be identified (see [3] and [4]) with an affine variety Vp and parametrization map Φp of the uniserial modules with mast p. This identification implies −1 φ−1 S (U ) = Φp (U ), so it suffices to show that the fibres of Φp over U are isomorphic to either A0 or A1 . Bongartz and Huisgen-Zimmermann have shown (Theorem A , [4]) that the fibre of Φp over U is isomorphic to Am , where m = t + 1 − dimK EndΛ (U ) and t is the number of non-trivial right subpaths of p which are cycles. If t > 1, then there would exist a cycle c such that c2 is a right subpath of p. But since the mast p cannot lie in the ideal I, our hypothesis forces the right subpath c2 outside of I, which violates Corollary 3.8. Therefore t ≤ 1, and the corollary follows.  Let us now consider a situation in which two distinct basic cycles are glued together at a vertex to create a generalized “figure-eight.” Corollary 4.3. Let Λ be a monomial relations algebra of finite global dimension. Suppose the first simple module S(0) of the sequence S corresponds to the vertex e, and that at most two distinct basic cycles begin and end at e. If G-Uni(S) is non-empty, then the

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fibres of the map φS are isomorphic to Am , where m is at most three. Proof. Let U be a corresponding uniserial module with mast p, as in the proof of Corollary 4.2. Again the fibre of Φp over U is isomorphic to Am , where m = t + 1 − dimK EndΛ (U ) and t is the number of non-trivial right subpaths of p which are cycles. If there is a unique basic cycle beginning and ending at e, then our result follows from Corollary 4.2. Otherwise, suppose c1 and c2 are distinct basic cycles beginning and ending in the vertex e. Then by Corollary 3.8, the longest cycles which begin and end at e and are formed by concatenating the cycles c1 and c2 , yet still lying outside of the ideal I, are c1 c2 c1 and c2 c1 c2 . Clearly this implies t ≤ 3, and the corollary follows. 

References [1] D. Anick and E. Green: “On the homology of quotients of path algebras”, Comm. Algebra, Vol. 15(1,2), (1987), pp. 309–341. [2] M. Auslander, I. Reiten and S. Smalø: Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, Cambridge, 1995. [3] K. Bongartz and B. Huisgen-Zimmermann: “The geometry of uniserial representations of algebras II. Alternate viewpoints and uniqueness”, J. Pure Appl. Algebra, Vol. 157, (2001), pp. 23–32. [4] K. Bongartz and B. Huisgen-Zimmermann: “Varieties of uniserial representations IV. Kinship to geometric quotients”, Trans. Am. Math. Soc., Vol. 353, (2001), pp. 2091–2113. [5] W. D. Burgess: “The graded Cartan matrix and global dimension of 0-relations Algebras”, Proc. Edinburgh Math. Soc., Vol. 30(3), (1987), pp. 351–362. [6] P. Gabriel: Auslander-Reiten seuquence and representation-finite algebras, Lect. Notes Math. 831, Springer-Verlag, New York, 1980, pp. 1–71. [7] E. Green, D. Happel and D. Zacharia: “Projective resolutions over Artin algebras with zero relations”, Illnois J. Math., Vol. 29(1), (1985), pp. 180–190. [8] B. Huisgen-Zimmermann: “The geometry of uniserial representations of finite dimensional algebras I”, J. Pure Appl. Algebra, Vol. 127, (1998), pp. 39–72. [9] B. Huisgen-Zimmermann: “The geometry of uniserial representations of finite dimensional algebras III”, Trans. Am. Math. Soc., Vol. 348(12), (1996), pp. 4775–4812. [10] B. Huisgen-Zimmermann: “Predicting syzygies of monomial relations algebras”, Manuscr. Math., Vol. 70, (1991), pp. 157–182. [11] K. Igusa: “Notes on the no loops conjecture”, J. Pure Appl. Algebra, Vol. 69, (1990), pp. 161–176. [12] B. Jue: The uniserial geometry and homology of finite dimensional algebras, Thesis (Ph.D), University of California, Santa Barbara, 1999.

DOI: 10.1007/s11533-006-0002-0 Research article CEJM 4(2) 2006 260–269

Full discretization of some reaction diffusion equation with blow up Genevi`eve Barro1∗ , Benjamin Mampassi2† , Longin Some1‡ , Jean Marie Ntaganda2§ , Ouss´eni So1¶ 1

2

D´epartement de Math´ematiques et Informatique, UFR/SEA, Universit´e de Ouagadougou 03 B.P.: 7021 Ouagadougou 03 Burkina Faso

D´epartement de Math´ematiques et Informatique, Facult´e des Sciences et Techniques, Universit´e Cheikh Anta Diop, B.P.: 5005 Dakar, S´en´egal

Received 16 September 2005; accepted 5 December 2005 Abstract: This paper aims at the development of numerical schemes for nonlinear reaction diffusion problems with a convection that blows up in a finite time. A full discretization of this problem that preserves the blow - up property is presented as well as a numerical simulation. Efficiency of the method is derived via a numerical comparison with a classical scheme based on the Runge Kutta scheme. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Full discretization, Runge Kutta scheme, reaction diffusion equation, nonlinear parabolic problems, blow-up MSC (2000): 65M06, 65N22

1

Introduction

We are interested here on the computation of the solution of the system → − ut − Δu1+δ + γ V .∇u1+δ = αup in Ω×]0, ∞[ ∗ † ‡ § ¶

E-mail: E-mail: E-mail: E-mail: E-mail:

[email protected] [email protected] hsome@univ ouaga.bf [email protected] [email protected]

(1)

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u = 0 on ∂Ω×]0, ∞[

(2)

u(x, 0) = u0 (x) x ∈ Ω

(3)

where Ω is a bounded domain in Rd (d ≥ 1). In this paper, we specifically consider the case δ > 0 which corresponds to the slow diffusion case, and we shall take p > 1 because otherwise there can be no blow up. It should be noticed that a great number of processes of applied sciences can be modeled by means of the above reaction diffusion equations. One of the important problems described by these equations are porous media. It is well known [2, 3], under some assumptions on initial conditions that any non null positive solution of (1)-(3) blows up in a finite time if 2 1
(4)

The semi-discretization in time of this problem was studied in [4–6] in the case where γ = 0. As in [4–6], we have constructed a full discretization scheme that has same properties as the exact solution. Contrary to the work of M.N. Roux [5], we present here a complete study that takes into account a numerical simulation aspects. On her previously work [5], the numerical study was focused only on the time semi discretization level. The numerical solution we propose here is compared with a classical scheme based on the 4-5th order Runge Kutta time semi-discretization (RKTS). The first part of this work is devoted to a modified M.N. Roux numerical scheme (MRNS) and its blow up properties. The second part (section three) presents the numerical simulation. In particular the efficiency of the scheme is shown and a comparison with a classical ones is performed.

2

The numerical scheme

2.1 The time semi-discretization Let v = u1+δ , m =

1 , r = pm, (m ∈]0, 1[, r ∈ [1, ∞[). Then v satisfies the following 1+δ mv m−1 vt + Av = αv r , x ∈ Ω; t > 0

v(x, 0) = v0 (x), x ∈ Ω → − where A is the operator −Δ + γ V .∇ of domain D(A) = H01 (Ω) ∩ H 2 (Ω). Let us denote vn as an approximation of v at the time tn with

(5) (6)

tn = tn−1 + Δt n = 1, 2, ... here Δt is a fixed step time. As in [5] we define the time semi-discretization as   m m−1 + tAvn+1 = αtvn+1 vnr−1 . vn+1 vnm−1 − vn+1 1−m

(7)

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We easily see that the existence and the uniqueness of the solution vn+1 ∈ H01 (Ω) for a given vn in the equation (7) are closed to the properties of the operator A. However, it is not difficult to see that for appropriate values of the parameter γ, the maximum principle can be used to establish the existence of positive solution of (7). Thus, considering ellipticity properties of the Laplace operator we can obtain, as in [5], a necessary condition for the uniqueness of positive solution of the equation (7). More precisely the following result can be derived from above remarks. Theorem 2.1. There exists a time T ∗ such that if (n + 1)Δt < T ∗ and vn ∈ H01 (Ω) satisfies m vn r−m < (8) ∞ (1 − m)αΔt then the equation (7) admits a unique solution vn+1 ∈ H01 (Ω) that also satisfies (8). Another version of this theorem for γ = 0 can be found in [5]. A complete proof of this theorem can be adapted from the paper of A. Amann [1]. From this result, it will be possible to provide an algorithm that determines a good estimation of the time blow up. The step time Δt is calculated such that (8) is satisfied for the initial solution v0 . So that we must have m Δt < (9) . v0 m−r ∞ (1 − m)α However, since the equation (7) is non linear, it is difficult to derive in the straight way vn+1 . Considering the key idea in [5], one can take the sequence (vn,j )j≥0 such that  Avn+1,j+1 +

 m m m−1 r−1 vn − αvn (vn+1,j )m vn+1,j+1 = (1 − m) Δt (1 − m) Δt

(10)

m Then, we can prove as in [5] that if vn+1,0 = vnm , the sequence (vn+1,j )j converges uniformly to vn+1 and the convergence is very fast.

2.2 The spatial discretization h Assume that vn+1 is the approximation of the solution in some functional space. Equation (10) is written in the approximation form as h (x) = Bnh (x)vn+1,j+1

m  h m vn+1,j (x) (1 − m) Δt

x∈Ω

(11)

with h vn+1,0 = vnh

where

 Bnh (x)

=A+

in Ω

 h m−1  r−1 m vn (x) − α vnh (x) (1 − m) Δt

(12)  (13)

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Proposition 2.2. There exists a constant c > 0 such that if vnh satisfies (8) and the condition   − −  −1  −1 →2 →2 − 2 c + sup  V  < γ < 2 c + sup  V  (14) Ω

then the matrix

Bnh (x)

Ω

is positively defined.

Proof. We must prove that Bnh φ, φ > 0 ∀ φ ∈ H01 (Ω) and φ = 0. We have   

Bnh φ, φ = Ω Bnh φ φdx    h r−1  h m−1   m φ2 dx − α vn (x) = Ω (Aφ) φdx + Ω v (x) (1 − m) Δt n

(15)

∀ φ ∈ H01 (Ω) According to relation (8), we have  r−m  h m−1  r−1  h m−1 m vn (x) > α vnh (x) = α vnh (x) . vn (x) (1 − m) Δt Hence,

  Ω

  h r−1  h m−1 m φ2 dx > 0. − α vn (x) vn (x) (1 − m) Δt

This proves the strict positivity of the second term of the right hand side of (15). It now  remains to show that Ω (Aφ) φdx > 0 if the condition (14) is satisfied. We have  Ω

 − → − (Δφ) φdx + γ Ω V φφdx   − → = Ω |φ|2 dx + γ Ω V φφdx.

(Aφ) φdx =



Ω

By Young inequality, it is easy to establish −  −   →2  →− →  1 2 1 2 |φ| sup + V φ V  φ ≤   |φ|   2 2 Ω

(16)

Let us now distinguish two cases. Case 1: γ ≥ 0. From (16) we have   −   →− − γ γ →2 → 2 γ φV  φ ≥ − |φ| − sup  V  |φ|2 2 Ω 2 Ω Ω Ω hence

  −   γ γ →2 2 (Aφ)φ ≥ 1 − sup  V  |φ| − |φ|2 2 2 Ω Ω Ω Ω



Since φ ∈ H01 (Ω), it is well known (Poincar´e inequality) that there exists a constant c > 0 depending only of the domain Ω such that   2 |φ| < c |φ|2 Ω

Ω

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 −  γ →2 γc (Aφ)φ > 1 − sup  V  − |∇φ|2 2 2 Ω Ω Ω



Thus

and the positivity of the first term of the right hand side of (15) is then obtained if  −  −1 →2 0 ≤ γ < 2 c + sup  V  Ω

Case 2: γ ≤ 0 By (16) we have 

  −   γ γ →2 2 (Aφ)φ > 1 + sup  V  |∇φ| + |φ|2 2 Ω 2 Ω Ω Ω

Then, applying again the previous Poincar´e inequality yields

  −  γ →2 γc (Aφ)φ > 1 + sup  V  + |∇φ|2 2 2 Ω Ω Ω and the positivity of the first term of the right hand side of (15) is obtained by setting  −  −1 →2 −2 c + sup  V  <γ≤0 Ω

We have proved the proposition.



By this proposition, one can see that for any x ∈ Ω, Bnh (x) is invertible. Consequently,   h (x) j≥0 is completely computed in the approximation space of spatial the sequence vn+1,j variables. Therefore, any spatial discretization holds to fully discretized system (11)(12). Next, we discuss the one dimensional case. Both finite differences method and finite elements method can be used in this case.

2.3 The one-dimensional space case For our numerical simulation we shall consider the case d = 1. Let Ω =]a, b[. Consider the spatial uniform grid (17) Ωh = {x0 = a, x1 , ..., xN = b} where xk = xk−1 + h, k = 1, ..., N. Let set  T vnh = vnh (x1 ), vnh (x2 ), ..., vnh (xN −1 ) .

(18)

Using the finite differences approximation in the spatial direction, one gets the following formula h Bnh vn+1 (xk ) =

1 h h h (−vn+1 (xk+1 ) + 2vn+1 (xk ) − vn+1 (xk−1 )) 2 h   γ h (xk−1 ) + v h (xk+1 ) − vn+1 2h   n+1 m−1  h r−1 h  h m vn+1 (xk ) − α vn (xk ) v (xk ) + (1 − m) Δt n

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for k = 1, ..., N − 1. Hence, the equations (11)-(12) can be rewritten as h h h ah vn+1,j+1 (xk−1 ) + wnh (xk )vn+1,j+1 (xk ) + bh vn+1,j+1 (xk+1 ) =

m  h m vn+1,j (xk ) (1 − m) Δt

with the initial condition h vn+1,0 (xk ) = vnh (xk )

where we have set ⎧ 1 γ ⎪ ⎪ ah = −( 2 + ) ⎪ ⎪ ⎪ h  2h  ⎨  h m−1  h r−1 2 m h v (xk ) wn (xk ) = 2 + − α vn (xk ) ⎪ h (1 − m) Δt n ⎪ ⎪ ⎪ 1 γ ⎪ ⎩ bh = (− 2 + ) h 2h

(19)

According to the notation of (11) and taking into account boundary conditions (2), the scheme (11) can be rewritten in the compact form as Ahn (v hn+1,j+1 ) =

m F(v hn+1,j ) (1 − m) Δt

(20)

where the initial value of the sequence is v hn+1,0 = v hn and where Ahn is the (N −1)×(N −1) matrix whose the (k, l) components are defined by ⎧ ⎪ h ⎪ if l = k + 1 ⎪b ⎪ ⎪ ⎪ ⎪ ⎨ wh (xk ) if l = k n h An (k, l) = (21) ⎪ h ⎪ a if l = k − 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0 otherwise F(v hn+1,j ) is a vector whose components are (v hn+1,j )m . The fully discretization process is then summarized by the following algorithm. Algorithm (1) Read nmax, jmax (2) Compute Δt such that (9) is satisfied (3) Set vold = v0 in Ω; n = 0 m (4) While vold r−m < and n < nmax ∞ (1 − m)αΔt n=n+1 vn = vold in Ω for j = 1 : jmax vn (x) = A−1 vold (x) in Ω end for vold = vn in Ω end while. Clearly, the above algorithm aims to estimate the numerically time blow up.

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Numerical simulations

To gain insight into the method developed in the previous section, we consider, as a test problem, the one dimensional spatial system     ut − u1+δ xx + γ u1+δ x = αup in ] − 1, 1[×]0, ∞[

(22)

u(−1, t) = u(1, t) = 0 t > 0

(23)

u(x, 0) = 5(1 − x2 ) x ∈] − 1, 1[

(24)

To this end, we shall compare the method with a classical one which is a combination of the 4-5th order Runge Kutta method in time and the finite differences in the spatial direction. Numerical experiments are done using a new version of Matlab environment which integrates a programming language and a simulation environment. We should notice that the algorithm presented in the previous section is easily implemented in Matlab in a few instructions lines, and the 4-5th order Runge Kutta is already integrated in the Matlab code ”ode45.m”. First, we have to note that, relatively to the previous section, if δ = 1 the solution of the test problem (22)-(23) blows up if the following holds 1
(25)

The aim of this numerical simulation will be on the investigation of stabilities and convergence properties as well as the estimation of the time blow up. Figures 1, 2 and 3 represent the numerical solution for various values of the parameter p. In each of theses figures, the picture on the left represents the Runge Kutta time semi discretization solution while on the right, is represented the numerical solution from the scheme developed in this paper (M RN S). In Figure 1, we can see that both of the two schemes are stable and coincide. In this case, related to the prediction, since (25) is not satisfied, the solution does not blow up. In Figure 2, the value of p corresponds exactly to the bow up condition (25). We can see that our scheme is well adapted and the time blow up is well estimated while the Runge Kutta scheme presents a very instable picture from which it is unable to determine the time blow up. In the other hand, although the solution is global, the Runge Kutta picture is unstable while our scheme presents a stable picture. In Figure 4, we have plotted various cases of the value γ. It appears that the gradient term does not affect the time blow up.

G. Barro et al. / Central European Journal of Mathematics 4(2) 2006 260–269

5

5

4

4

3

3

2

2

1

1

0

0

1

1

267

1

1

0.5

0.5

0.5

0.5

0

0

Ŧ0.5 t

0

Ŧ0.5 t

x

Ŧ1

0

x

Ŧ1

Fig. 1 A global case: p = 1.5, γ = −0.5, α = 0.5, δ = 1. On the left is represented the RKT S solution, and on the right, the RN S solution.

7

x 10 7

40

6

35 30

5

25 4 20 3 15 2

10

1

5

0

0

0.06

0.1 1

0.04 0.5 0

0.02

1 0.5 0.05

0

Ŧ0.5 t

0

Ŧ1

Ŧ0.5 x

t

0

Ŧ1

x

Fig. 2 A blow up case: p = 3, α = 0.95, γ = −0.9, δ = 1. The estimated time blow up is T ∗ = 0.1067. On the left is represented the RKT S solution, and on the right, the M RN S solution.

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Fig. 3 A global case: p = 1.005, γ = 0.99, α = 3.25, δ = 5. On the left is represented the RKT S solution with instability, and on the right, the M RN S solution.

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Concluding remarks

We have developed in this paper a numerical scheme which in comparison with the Runge Kutta scheme - one of most stable. It is very efficient to solve a class of nonlinear evolution problems with possible blow up. One of its particularities is the estimation of the time blow up while a great number of schemes present instabilities in presence of blow up. The numerical simulation presented in this work could be easily extended to a higher space dimension.

References [1] H. Amann: “On the existence of positive solutions of nonlinear elliptic boundary value problems”, Indiana Univ. Math. J., Vol. 21, (1971), p. 125. [2] M. Chlebik and M. Fila: “Blow-up of positive solutions of a semilinear parabolic equation with a gradient term.”, Dyn. Contin. Discrete Impulsive Syst., Vol. 10, (2003), pp. 525–537. [3] V.A. Galaktionov and J.L.V`azquez: “The problem of blow-up in nonlinear parabolic equations”, Discrete Cont. Dyn. S., Vol 8(2), (2002). [4] M.N. Le Roux: “Numerical solution of fast or slow diffusion equations”, J. Comput. Appl. Math., Vol. 97, (1998), pp. 121–136. [5] M.N. Le Roux: “Semidiscretization in time of nonlinear parabolic equations with blow up of the solution”, Siam J. Numer. Anal., Vol. 31, (1994), pp. 170–195. [6] M.N. Le Roux and H. Wilhelmsson: “Simultaneous diffusion, reaction and radiative loss processes in plasmas: numerical analysis with application to the dynamics of a fusion reactor plasma”, Phys. Scripta, Vol. 45, (1992), pp. 188–192.

DOI: 10.2478/s11533-006-0006-9 Research article CEJM 4(2) 2006 270–293

On hyperbolic virtual polytopes and hyperbolic fans Gaiane Panina∗ Institute for Informatics and Automation, St.Petersburg, 199178, Russia

Received 11 March 20005; accepted 30 December 2005 Abstract: Hyperbolic virtual polytopes arose originally as polytopal versions of counterexamples to the following A.D.Alexandrov’s uniqueness conjecture: Let K ⊂ R3 be a smooth convex body. If for a constant C, at every point of ∂K, we have R1 ≤ C ≤ R2 then K is a ball. (R1 and R2 stand for the principal curvature radii of ∂K.) This paper gives a new (in comparison with the previous construction by Y.Martinez-Maure and by G.Panina) series of counterexamples to the conjecture. In particular, a hyperbolic virtual polytope (and therefore, a hyperbolic h´erisson) with odd an number of horns is constructed. Moreover, various properties of hyperbolic virtual polytopes and their fans are discussed. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Virtual polytope, saddle surface, h´erisson MSC (2000): 52A15, 52B70, 52B10

1

Introduction

In this paper, we study hyperbolic virtual polytopes. Figuratively speaking, hyperbolic virtual polytopes relate to the convex ones in the same way as convex surfaces relate to saddle ones. As is known, there exists no closed saddle polytopal surface. Still, non-trivial hyperbolic virtual polytopes do exist and this is probably the most remarkable fact known about them. Non-trivial hyperbolic virtual polytopes appeared originally as an auxillary construction for various counterexamples to the following A.D.Alexandrov’s uniqueness hypothesis: Let K ⊂ R3 be a smooth convex body. If for a constant C, in each point of ∂K, we have R1 ≤ C ≤ R2 , then K is a ball. (R1 and R2 stand for the principal curvature radii of ∂K). ∗

E-mail: [email protected]

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For a long time mathematicians were certain about the correctness of the hypothesis, but obtained only some partial results. Recently, Y.Martinez-Maure [5] has given a counterexample. First, he demonstrated that each smooth hyperbolic h´erisson (see Section 2) generates a desired counterexample. Next, he presented such an example (see Fig. 1). It is a smooth hyperbolic surface with four horns (i.e., points where the surface is neither hyperbolic no smooth), given by an explicit formula. Surprisingly, this counterexample proved to be not unique: a series of counterexamples was given by the author of the paper (see [9]). Using a different technique, she constructed smooth hyperbolic h´erissons with any even number of horns greater than 4. The present paper continues this study and demonstrates that they are even more various. The paper is organized as follows. Sections 2 and 3 give necessary definitions and examples of virtual polytopes (which are, roughly speaking, Minkowski differences of convex polytopes). In addition, Section 2 recalls briefly the notion of h´erissons (i.e., Minkowski differences of smooth convex bodies). The definition of hyperbolic virtual polytopes (i.e., virtual polytopes such that the graph of the support function is a saddle surface) are presented in Section 4. Convex poytopes and hyperbolic polytopes are compared in Theorem 4.4. Section 5 studies the fans of simplicial hyperbolic virtual polytopes. The edges of such a fan admit a proper coloring, which encodes important properties of the virtual polytope. For instance, a cell of the fan corresponds to a horn of the polytope, if and only if the color changes twice as while going around the cell (Theorem 5.3). Theorem 5.7 demonstrates a way of adding a new horn to a hyperbolic polytope. This is called a C-operation (or a S-operation) and arises from some special refinement of the fan of the original polytope. This trick is used in Section 6, which gives advanced examples of hyperbolic virtual polytopes, in particular, with an odd number of horns. Note that it is impossible to add a horn to any hyperbolic polytope known before ([6] and [9]), so we have to construct new ones. Hyperbolic virtual polytopes can be classified in a reasonable way by the number of horns. However, there exists a finer classification since each hyperbolic polytope generates in a natural way an arrangement of oriented great semicircles (each horn gives a semicircle). We are bound by the case when the semicircles (and therefore, the horns) admit a natural ordering. This allows us to assign to a polytope K a necklace (i.e., a circular sequence) consisting of N signs ” + ” and ” − ” (N stands for the number of horns). By Theorem 6.1, each necklace with more than three changes of sign is realizable as a hyperbolic polytope. Much room is left here for further study: applying C- and S-operations one can obtain further types of hyperbolic polytopes, in particular, with new combinatorial type of semicircle arrangement. However, we leave this beyond the paper. In Section 7, we show that each polytope constructed in Section 6 admits a hyperbolic smoothing and therefore yields a counterexample to the A.D. Alexandrov’s hypothesis. As in [9], we smooth not the surface of the virtual polytope, but the collection of graphs

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of its support function.

2

Virtual polytopes and h´ erissons: basic notations.

Virtual polytopes (introduced by A.Pukhlikov, A.Khovanskii [4], appeared also in a natural way in P.McMullen’s polytope algebras [7]) and can be represented in four different ways. • Virtual polytopes are elements of the Grothendieck group of the semigroup of convex polytopes P in Rn equipped with the Minkowski addition ⊗. I.e., they are formal expressions of type K ⊗ L−1 , where K, L ∈ P. • Virtual polytopes are polytopal functions (Definition 2.2), i.e., finite linear combinations of characteristic functions of convex polytopes. So it makes sense to speak of the value of a polytope K at a point X ∈ Rn . • Virtual polytopes are defined by their support functions, i.e., piecewise linear positively homogeneous functions defined on Rn (Definition 2.3). • A virtual polytope is a pair of type (a closed polytopal surface in Rn with cooriented facets; a spherical fan) (see Theorem 2.10). We now give a detailed explanation of the items (restricting ourselves to dimension n = 3). Denote by P the set of all compact convex polytopes in Rn (degenerate polytopes are also included). It is a semigroup with respect to the Minkowski addition ⊗. Denote by P ∗ the Grothendieck group of P. The element of P ∗ that is inverse to K is denoted by K −1 . A function F : R3 → Z is polytopal if it admits a representation of the form  F = ai IKi , i

where ai ∈ Z, Ki ∈ P, and IKi is the indicator function of the polytope Ki :  1 if x ∈ Ki , IKi (x) = 0 otherwise. The set of all polytopal functions M is endowed with two ring operations. The role of addition is played by the pointwise addition, denoted by +. The multiplication is generated by ⊗ and is denoted by the same symbol. The unit element of the ring M is obviously the function E = I{O} . Identifying convex compact polytopes with their indicator functions, we get an inclusion π : P ⊂ M. Keeping this identification in mind, we write for convenience K instead of IK . Due to the following theorem, all elements of the semigroup π(P) are invertible in M.

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Theorem 2.1. (On Minkowski inversion) [4] For any convex polytope K, we have (−1)dim K IRelint(sK) ⊗ K = E, where s is the central symmetry mapping (with respect to the origin O) , Relint(sK) is the relative interior of the polytope sK (i.e., the interior taken in the affine hull of K). Hence the inclusion P ⊂ M induces an inclusion P ∗ ⊂ M. Definition 2.2. The image of the latter inclusion is called the group of virtual polytopes. For convenience we denote it by the same letter P ∗ . Definition 2.3. Let K be a virtual polytope. Then there exists convex polytopes L and M such that K = L ⊗ M −1 . The support function hK of the virtual polytope K is defined to be the pointwise difference of support functions of L and M : hK = hL − hM . Remark 2.4. Recall that the support function of a convex polytope is piecewise linear with respect to a fan. By a fan we mean a splitting of Rn in a union of polytopal cones with a common apex at O. In the sequel, we sometimes speak of (and draw) the intersection of the fan with the unite sphere S n−1 centered at O. Thus, the cones correspond to spherical polytopes (spherical cells).  Definition 2.5. [8] Let K = i ai Ki with Ki ∈ P. Let ei (ξ) be the support hyperplane to Ki with the outer normal vector ξ. The polytope Kiξ = Ki ∩ ei (ξ) is called the face of  the polytope Ki with the normal vector ξ, whereas the polytopal function K ξ = i ai Kiξ is called the face of the polytopal function K with the normal vector ξ. A face of a virtual polytope is a virtual polytope as well. The 0-dimensional, 1dimensional and 2-dimensional faces are called vertices, edges and facets respectively. (By the dimension of a virtual polytope we mean the dimension of the affine hull of its support.) Similarly to faces of convex polytopes, virtual faces behave linearly with respect to the Minkowski addition: Theorem 2.6. [8] In the above notation, K1ξ ⊗ K2ξ = (K1 ⊗ K2 )ξ . Definition 2.7. A point X is called a boundary point of a polytope K, if x ∈ cl(supp(K ξ )) for some ξ ∈ S 2 such that ξ is not orthogonal to af f (K). (cl denotes the closure.)

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Definition 2.8. A fan Σ is a finite collection of compact spherical polytopes on the unit sphere S 2 (possibly nonconvex ones) such that • U, V ∈ Σ ⇒ U ∩ V ∈ Σ;  • Σ U = S 2; • U = V ∈ Σ ⇒ RelintU ∩ RelintV = ∅. The fan of a virtual polytope is defined below analogously to the classical definition of the outer normal fan. Definition 2.9. For a virtual polytope K ∈ P ∗ , its fan ΣK is the collection of spherically polytopal sets {ΣK (ν)}, where ν ranges over the set of faces of K, and ΣK (ν) = cl({ξ|K ξ = ν}) (cl denotes the closure.) These polytopal sets are called the cells of the fan. Similarly to the convex case, the support function of K is linear on each cell of ΣK . And similarly to convex polytopes, the fan of a virtual polytope K can be defined as the minimal fan for which hK is linear on each cell. In addition, we have the usual duality: k-dimensional cells of ΣK correspond to (3 − k − 1)-dimensional faces of K. The 0-dimensional cells are called the vertices of the fan. From now on, we assume that n = 3, and deal with 3-dimensional virtual polytopes. A virtual polytope is said to be simplicial if each of its facets is a virtual triangle (see Section 3). Each simplicial virtual polytope K yields in a natural way a spherehomeomorphic simplicial complex CK which is generated by the collection of triangles {cl(supp(K ξ ))| ξ ∈ S 2 ; dim(K ξ ) = 2}. Alternatively, given a simplicial complex C, it is sometimes possible to associate with C a virtual polytope. Moreover, sometimes it is possible to associate many different virtual polytopes (see Figure 5). The general construction is given in the following theorem. Theorem 2.10. Construction of a virtual polytope related to an immersed simplicial complex [8, 9]. Let C be a closed sphere-homeomorphic immersed (with possible self-intersections) in 3 R simplicial complex generated by a set of triangles {Ti }. Suppose there exists a collection of normal vectors ξi of the triangles Ti and a spherical fan Σ with vertices in {ξi } satisfying the two conditions: Main condition. The combinatorics of Σ is dual to that of C . (In particular, ξi and ξj are connected by an edge of Σ if and only if Ti and Tj share an edge in C.) Condition for complexes with parallel adjacent facets. If two adjacent facets Ti and Tj of the complex are parallel (and therefore, have opposite normal vectors ξi and ξj ), then the points ξi and ξj are connected by an edge (respectively, edges) of ΣK , which is orthogonal to the mutual edge (respectively, mutual edges) of the facets.

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Then there exists a virtual polytope K such that • the set of {cl(supp(K ξ ))| ξ ∈ S 2 ; dim(K ξ ) = 2} coincides with the set {Ti }, and • ΣK = Σ. Remark 2.11. Given a virtual polytope K in R3 = (x, y, z), the vertices of the associated complex CK can be restored by the support function h = hK as follows. Let a vertex A of CK correspond by duality to a cell α of the fan ΣK . Then A = ((h|α )x , (h|α )y , (h|α )z ), where h|α is the restriction of h to the cell α. Similar geometric realization of Minkowski differences of smooth convex bodies makes sense as well. It can be traced in the early paper [1] by A.D. Alexandrov. Let h : S 2 → R be a smooth function. By the h´erisson H with the support function h (see [11]), we mean the envelope of the family of planes {eH (ξ)}ξ∈S 2 , where the plane eH (ξ) is defined by the equation (x, ξ) = h(ξ). It is a sphere-homeomorphic surface with possible self-intersections and self-overlapings. We say that a h´erisson H is smooth if its support function is smooth. As a set of points, a h´erisson H coincides with the image of the mapping φ : S 2 −→ R3 , (x, y, z) −→ (hx (x, y, z), hy (x, y, z), hz (x, y, z)). Analogously to the classical convex case, the principal curvature radii R1 and R2 of a h´erisson H at a point ξ ∈ S 2 (or at the point φ(ξ) ∈ H) are the eigenvalues of the matrix ⎛ ⎞   ⎜hxx (ξ) hxy (ξ)⎟ ⎝ ⎠. hyx (ξ) hyy (ξ) (ξ is codirected with the z axis.) Although the support function of a h´erisson is smooth, the h´erisson itself (regarded as a surface) may have singular points. They appear whenever R1 R2 = 0. If R1 R2 = 0, the h´erisson B is a smooth surface in a neighbourhood of φ(ξ) and the radii R1 and R2 coincide with the principal curvature radii of the surface H. Martinez-Maure observed that a body K together with a constant C give a counterexample to A.D. Alexandrov’s hypothesis if and only if the h´erisson K ⊗ BC−1 is hyperbolic, i.e., R1 R2 ≤ 0 everywhere (BC stands for a ball of radiuce C). An example of a hyperbolic h´erisson presented by Martinez-Maure [5] is a surface (see Fig. 1) obtained by gluing together graphs of two explicitly given functions.

3

Examples of virtual polytopes

1. One-dimensional virtual polytopes are not various. A virtual segment is either a regular convex segment or an inverse to a convex segment. By Theorem 2.1, the latter is

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Fig. 1 a polytopal function admitting the value −1 strictly inside a segment and admitting the value 0 outside it and at the endpoints. 1

–1

1

1

0 0 inverse to a regular segment

a regular segment

Fig. 2 2. Two-dimensional virtual polytopes are much more various. We list below all types of virtual triangles (i.e., virtual polytopes possessing 3 edges and therefore, 3 vertices). In the figure we indicate the values of the polytopal function and the coorientations of the edges. For instance, the second figure means that the polytopal function admits the value −1 strictly inside the triangle and inside the side edges. At the vertices and on the horizontal edge, the function equals 0. 1

1 1

0

1

1 1

–1 1

0

–1 0

1

0

–1 0

0

0

0

1 –1

0 0

0

1 0

0 0

Fig. 3 The virtual polytope in Fig. 4 is not a triangle but a hexangle though the closure of its support is a triangle (similarly to the above virtual triangles). The point is that each of the three segments that serve as edges is taken twice with both coorientations. 3. Some virtual tetrahedra. A hyperbolic tetrahedron. It would take too much space to list all 3-dimensional virtual tetrahedrons (i.e., virtual polytopes with 3 facets). Instead we draw some of them to further illustrate the theorems. This time we do not indicate the values of the polytopal functions. Instead, we show the coorientations of

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0

–1 0

–1 –2 –1

0

Fig. 4 the facets by normal vectors and draw the fans (keeping in mind Theorem 2.10). It is possible to restore the values of the polytopal function owing the methods of [8].

+

=

convex tetrahedron

+

=

hyperbolic tetrahedron

+

Fig. 5 The above hyperbolic tetrahedron is of particular interest. It is the simplest non-trivial example of a hyperbolic polytope.

4

Hyperbolic virtual polytopes: definition and properties

Let K be a virtual polytope and let h = hK be its support function. For ξ ∈ S 2 , let e(ξ) be the plane defined by the equation (x, ξ) = 1. Consider the restriction of h to

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the plane e(ξ) and denote by F = FK (ξ) the graph of the restriction. The surface F is piecewise linear. Its vertices and edges correspond to those of ΣK intersected with the open hemisphere with the pole at ξ. Note that the virtual polytope K is convex if and only if the surface FK (ξ) is convex for any ξ. This motivates the following definition. Definition 4.1. A virtual polytope K is called hyperbolic if FK (ξ) is a saddle surface for any ξ ∈ S 2 . In the sequel, we call such virtual polytopes simply hyperbolic polytopes. Recall that a piecewise linear (or any other non-smooth surface) F is called a saddle surface if there is no plane cutting a bounded connected component off F (see [3]). The polytopal complex CK generated by a hyperbolic polytope K is not a saddle surface (for there exists no closed saddle surface). Some of its vertices can be cut off the surface by a plane. Such vertices are called horns of the hyperbolic polytope K. This preserves the traditional notation of the theory of smooth narrowing saddle surfaces, which deals with infinite horns (see [3] and [13]). Let Ξ = {ξi } be a collection of points on S 2 such that each open hemisphere contains at least one point from the collection. Proposition 4.2. [9] A virtual polytope K is hyperbolic if and only if FK (ξ) is a saddle surface for any ξ ∈ Ξ. In this case, FK (ξ) is saddle for any other ξ ∈ S 2 as well. Definition 4.3. [9] A vertex ξ of a fan Σ is nonconvex (respectively, convex) if there exists (respectively, doesn’t exist) an adjacent to ξ angle greater than π (Fig. 6).

Fig. 6 Denote by Hyp the set of all hyperbolic virtual polytopes. Also put Conv = {K ∈ P ∗ | either K or K −1 is a convex polytope}. The following theorem compares these sets and demonstrates their contraposition. Theorem 4.4. (1) K ∈ Conv if and only if all non-boundary values of its facets are non-negative; K ∈ Hyp if and only if all non-boundary values of its facets are nonpositive.

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(2) K ∈ Conv ∪ Hyp

279

⇔

the coorientations of the f acets of K generate a global orientation of CK . (3) Let K be a simplicial virtual polytope (i.e., all its facets are virtual triangles). Then K ∈ Conv if and only if every vertex of its fan ΣK is convex; K ∈ Hyp if and only if every vertex of its fan ΣK is nonconvex. The above examples of virtual tetrahedrons (Fig. 5) give a good illustration of the assertions. To prove the theorem, we need two auxiliary lemmas. Lemma 4.5. Let K be a 2-dimensional virtual polytope (embedded in R3 ). K is hyperbolic if and only if all its values at non-boundary points are non-positive. Proof. The fan of K is symmetric and has 2 vertices. Therefore, it suffices to consider the surface F = FK (ξ) for ξ ⊥ af f (K). It is a piecewise linear cone. Let e be a plane containing its apex A. By duality, e corresponds to a point E from the plane k = af f (K). The point E is a non-boundary point of K if and only if the plane e does not contain edges of F. The assertion of the lemma easily follows from the equality K ξ (E) = 1 + χ(e ∩ F ∩ U (A)), where U (A) = {x ∈ R3 | 0 = |x, A| < ε} is a deleted neighbourhood of A, F is the subgraph of the restriction of hK on the plane e, χ stands for the Euler characteristic. Indeed, the surface F is saddle if and only if there exists a plane e such that e ∩ F = {A}, which means K(E) = 1. Now prove the equality. −χ(e ∩ F ∩ U (A)) = by duality, #({l | l ⊂ k is an oriented line, E ∈ l; l is a support line to K})/2 = 1 − K(E). The latter equality is well-known for convex polytopes. Owing to linearity, it also is valid for virtual polytopes.  Lemma 4.6. Let K be a virtual polytope. Suppose a point X is a non-boundary point of its facet K ξ . Suppose in addition that X ∈ / af f (K η ) for all η = ξ. Move somewhat the point X in the direction ξ (respectively, −ξ) and obtain the point X + (respectively, X − ). In this notation, we have K ξ (X) = K(M − ) − K(M + ).

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Proof. The assertion follows directly from [8], Section 1.



Now prove the theorem. The first assertion follows from Lemma 4.5. Indeed, for a virtual polytope K, the surface FK (ξ) coincides in a neighbourhood of its vertex A with the surface FK A (ξ), where K A is the facet of K that corresponds to the vertex A. It remains to observe that if F is non-saddle, there exists a plane cutting off a bounded component containing exactly one vertex of F. 2. A virtual polytope K can be regarded as a cycle (recall that K is a polytopal function, i.e., a linear combination of convex polytopes). The winding number of the polytopal surface CK at a point X coincides with the value K(X) provided that X is a non-boundary point of K. Let η denote the coorientation of CK regarded as a cycle, whereas ξi denote the orientations of the facets of K (according to Definition 2.5). By Lemma 4.6, if η coincides with (respectively, is opposite to) ξi , then all non-boundary values of K ξi are non-negative (respectively, non-positive). 3.The third assertion is trivial. 

5

Hyperbolic fans

In the section, all virtual polytopes (respectively, fans) are assumed to be simplicial (respectively, simple). Let K be a hyperbolic polytope with a fan ΣK and a support function hK . The edges of ΣK admit the following natural coloring: we color an edge of ΣK red (respectively, blue), if the graph FK (ξ) of hK is concave up (respectively, concave down) in a neighbourhood of an inner point of the edge. (We assume that the edge intersects with the hemisphere with the pole ξ.) This construction is correct: the color does not depend on the choice of ξ. Remark 5.1. For a hyperbolic K, the three edges adjacent to a vertex of ΣK can be colored only as is shown in Fig. 7.

Fig. 7

Definition 5.2. A fan Σ is hyperbolic, if each of its vertex is nonconvex. A hyperbolic fan Σ (which possibly is not a fan of a virtual polytope) is proper, if its

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edges admit a coloring such that the adjacent edges to every of its vertices are colored as in Fig. 7. Given a hyperbolic fan, either there is no proper coloring or just two opposite ones. The below figure gives an example of a non-proper hyperbolic fan.

Fig. 8

Theorem 5.3. Let K be a simplicial hyperbolic polytope with a fan ΣK . For a 2dimensional cell α of ΣK , denote by S(α) the number of color changes as going around the boundary of the cell. The following assertions are valid: 1. S(α) = 2 if and only if α corresponds by duality to a horn. 2. S(α) = 2 implies that α contains a great semicircle (not vise versa!). 3. S(α) = 0 implies that K is a virtual triangle or a virtual segment.  4. α [S(α) − 4] = −8. Proof. Denote by h the support function of K. Prove 1 and 2. Let A be a horn of the virtual polytope K. Let a Cartesian coordinate system (x, y, z) be chosen so that the vertex A = (0, 0, 0) of the virtual polytope K can be cut off by the plane z = − for a small  > 0. This means (by Remark 2.11) that hx = hy = hz = 0 on the cell α, whereas hz ≤ 0 on neighbour to α cells. Consider the restriction of h to a plane e(ξ) such that the z axis is orthogonal to ξ. Let α = α ⊂ e(ξ) be the polygon which corresponds to the cell α (i.e., the intersection of e(ξ) with the cone built over α. For a vertex X of α, three cases are possible (see Fig. 9): a. Two red edges adjacent to X admit a one-sheet orthogonal projection on z ⊥ . The polygon α lies beneath the red edges. b. Two blue edges adjacent to X admit a one-sheet orthogonal projection on z ⊥ . The polygon α lies over the blue edges. c. Locally, α lies between a blue and a red edges that have the same projection on ⊥ z . Assume that dimK = 3. Altering the vector ξ, make the plane e(ξ) contain a vertex X of α of the type c. Let s be a ray with the apex at the point X which locally lies in α. Then s is contained in α.

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z

a

a)

X

z X

a

b)

a

z

X c)

Fig. 9

Treating similarly other planes e(ξ) (such that ξ is orthogonal to z), we conclude that the color changes twice as going around α and that α contains a great semicircle. Alternatively, let K be a hyperbolic polytope and let the color change twice as going around a cell α of ΣK . Let α correspond by duality to a vertex A. For the above choice of coordinate system, the polytope α looks locally as shown in Fig. 9. In each of these three cases, we have hz ≤ 0 in a neighbourhood of the cell α. Since hz = 0 on α, there exists a plane cutting a bounded component with the vertex A off the complex CK , i.e., A is a horn. 3. Suppose S(α) = 0. Then each angle of the spherical polygon α is greater than π. This means that either α is a spherical 2-gons or S 2 \ α is contained in an open hemisphere, which is impossible. 4.We follow the proof of the Cauchy Lemma (see [2]). Denote by V and F the number of vertices and the number of 2-dimensional cells of ΣK . Count the total number of color  changes for all cells. Since each vertex gives exactly two changes, we have α S(α) = 2V. Applying the Euler formula 2F = V + 4, we conclude the proof.  Definition 5.4. We say that two proper hyperbolic fans are combinatorially equivalent if there exist proper colorings of both of them together with a combinatorial equivalence

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that preserves the colors of the edges. Definition 5.5. We say that a hyperbolic fan Σ is realizable (respectively, combinatorially realizable) if there exists a hyperbolic polytope K such that ΣK = Σ (respectively, Σ is combinatorially equivalent to ΣK ). Given one realizable fan, we sometimes can obtain many others. The operations described below change the combinatorics of a fan but preserve combinatorial realizability. Proposition 5.6. Let Σ be a realizable hyperbolic fan. Suppose two inner points of some red (respectively, blue) edges can be connected by a geodesic segment avoiding intersections with other edges. Then breaking somewhat these red (respectively, blue) edges, coloring this segment blue (respectively, red), and adding it to Σ, we obtain a combinatorially realizable hyperbolic fan. This is called H-operation.

H-operation

Fig. 10

Proof. Let a virtual polytope K correspond to the fan Σ. For a fixed vector ξ, consider the surface FK (ξ). Let the new segment s belong to a cell α, which corresponds to the facet A of the surface FK (ξ). Put a = aff(A). Now break the plane a along the segment s to obtain a concave down (respectively, concave up) surface. Owing to simplicity of the fan, replacment of a by the broken plane causes no harm to the combinatorics at adjacent vertices. This gives a saddle surface with the desired combinatorics. Alter the surfaces FK (ξ) for other vectors ξ to get a self-consistent collection of surfaces. This corresponds to a hyperbolic polytope of the desired type.  Not all hyperbolic polytopes allow an H-operation. For instance, the hyperbolic tetrahedron (Fig. 5) allows no H-operation. We shall apply H-operations in Section 6 for shortening the edges of hyperbolic fans, which is necessary for further smoothing. Theorem 5.7. Let K be a hyperbolic polytope. Suppose that one of the below described C- or S- configurations of four geodesic segments 1, 2, 3, and 4 can be placed on S 2 (see Fig. 11) such that

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• The endpoints of 1 and 4 of C-configuration (respectively, S-configuration) lie on edges of ΣK of different (respectively, one and the same )color. • Intersections of the configuration with the edges of ΣK are avoided (except for the endpoints of 1 and 4). • Segments 2 and 3 are great semicircles. • The vertices of the configuration are nonconvex. • Segments 1 and 4 lie on one and the same great circle. Then after an appropriate coloring and adding this configuration to ΣK , we obtain a combinatorially realizable fan. This is called C-operation (respectively, S-operation).

1 2

3

adding a C-configuration

4

1

2 adding a S-configuration

3 4

Fig. 11

Proof. Consider the surface FK (ξ) for a fixed vector ξ. Denote by a its face corresponding to the cell of ΣK which contains the S-configuration (respectively, C-configuration). Replace the affine hull of a by a polyhedral surface (consisting of three linear parts) as is shown in Fig. 12. The affine hulls of other faces remain unchanged. Thus we obtain a new saddle surface FK (ξ) for the fixed vector ξ. Altering the surfaces FK (ξ) for other planes ξ, we obtain a self-consistent collection of saddle surfaces, and therefore, a hyperbolic polytope of the desired type.  An easy calculation of color changes (according to Theorem 5.3) proves the following proposition. Proposition 5.8. • For a 3-dimensional hyperbolic polytope, H-operations never change the number of

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FK(x)

F'K(x)

x

x

Fig. 12 horns. • C-operation (respectively, S-operation) doesn’t change the number of horns if the Cconfiguration (respectively, S-configuration) is contained in a cell corresponding to a horn. • Otherwise , C-operation (or S-operation) adds one horn. It is often impossible to add a horn by a C- or a S -operation to a hyperbolic polytope K. Indeed, there must exist a cell of ΣK containing a great semicircle but not corresponding to a horn of K. For instance, neither the hyperbolic polytope from [5], nor those from [9] possess this property.

6

Advanced examples: hyperbolic polytopes with even and odd number of horns

Let K be a simplicial hyperbolic polytope. Let α1 , ...αN be the cells of ΣK such that for each k, the color changes twice as going around the boundary of αk . (Recall that each such cell corresponds by duality to a horn.) By Theorem 5.3, each cell αk contains an oriented great semicircle, say, Sk . Its orientation is generated by the coloring. Arrangements of great semicircles on S 2 may have a complicated structure. Their combinatorial classification is a separate problem (not to be discussed here). In this paper, we bound ourselves by simple combinatorics of {Si }N i=1 . Horns ordering property. Let K be a hyperbolic polytope. K is said to possess the horns ordering property, if. there exists a hemisphere S+2 N and a circular ordering of the set {Si }N i=1 (and therefore, the same ordering of {αi }i=1 ) satisfying the following two conditions.

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1. For each i, the prolongation of the great semicircle Si in S+2 first meets the semicircle Si+1 . 2. For each i, the prolongation of the great semicircle Si in S 2 \ S+2 first meets the semicircle Si−1 . In the section, all hyperbolic polytopes constructed possess this property. 2

S+

S3

S1 S2

Fig. 13 Denote by S the boundary of S+2 . Choose an orientation of S and mark the cells αk by a sign ” + ” or ” − ” as follows: if S first meets a red edge of αK , then assign to αk the sign ” + ”. Otherwise, we assign the sign ” − ” (see Fig. 14). +

_

S _

+

_

Fig. 14 Thus we obtain a necklace (i.e., a circular sequence) of N signs N (K) = (± ± ...±) which is defined up to the order inversion combined with the sign inversion (this combination is motivated by the orientation changing of S). The example from [5] gives the necklace (+ − +−). The examples from [9] give the necklace of type (+ − + − ... + −) for even number N ≥ 4 of signs. However, the set of realizable necklaces is much more diverse: Theorem 6.1. Suppose the sign changes at least 4 times as going around a necklace N = (± ± ...±). Then there exists a hyperbolic virtual polytope K such that N (K) = N . Proof. Step 1. New hyperbolic polytope of type (+ − +−). Assuming that a coordinate system (x, y, z) is fixed, consider the collection of points {Ai , Pi , O}4i=1 as shown in Fig. 15. The x and y coordinates of the points can be read from the grid, whereas the z

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y

P4(–d)

P1(d) A4(–e)

T4

T3

z

A1(e) y

A3(e)

P3(d)

O T2

x T1 A2(–e) x

P2(–d)

a sideview

Fig. 15 coordinates are indicated in the brackets. For instance, A1 = (3, 3, ). The values δ > 0 and  > 0 are chosen below. The collection of oriented triangles Si = (Ai Ai+1 O), Ti = (Ai+1 Ai Pi ), Si = (Ai+1 Ai O), Ti = (Ai Ai+1 Pi ) forms a simplicial complex C. The pairs of triangles Si and SI (as well as Ti and Ti ) differ only by orientation: the normal vectors of Si and Ti look upwards (i.e., form an angle with the z axis less than π/2), whereas the normal vectors of Si and Ti look downwards (i.e., form an angle with the z axis greater than π/2). Remark 6.2. We indicate the orientation of a triangle in two ways: by its normal vector ξ and by the order of its vertices. Remark 6.3. Here and in the sequel, given an ordered set of any objects {Xi }M i=1 , we assume that for any k ∈ Z, we have Xk = Xi if k ≡ i(modM ). Keeping in mind Theorem 2.10, mark on the sphere S 2 the endpoints of the normal vectors of the triangles (we denote them by the same letters). For an appropriate choice of the numbers δ and  (for instance, the values  = 0, 1 and δ = 0, 4 are suitable), the geodesic segments connecting the points (see Fig. 16) do not intersect each other (except for the marked points). The complex C together with the fan obtained gives a hyperbolic polytope K4 . To adjust it for further smoothing (Section 7), apply four H-operations and obtain the fan Σ4 (Fig. 16). By Theorems 4.4 and 5.3, this yields a hyperbolic polytope K4 with

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4 horns (namely, the points Pi ), which differs from that constructed in [5]: their fans are not combinatorially equivalent although they contain the same number of vertices and edges.

T3

S3

S4

T2 S2

T4

S1

T1

upper part of S4

upper part of S'4 H-operation

T'1 T'2

S'2

S'1 S'3

S'4

T'4

T'3

lower part of S4

lower part of S'4

Fig. 16 Step 2. New hyperbolic polytopes of type (+ − ... + −) with even number of horns. On this step, we take a star with N vertices (as defined in [9]) instead of the four points A1 , ..., A4 and follow the pattern of Step 1. Again, the z coordinates of the points are indicated in the brackets. The collection of oriented triangles Si = (Ai Ai+1 O), Ti = (Ai+1 Ai Pi ), Si = (Ai+1 Ai O), Ti = (Ai Ai+1 Pi ) form a simplicial complex C. As on the Step 1, the pairs of triangles Si and SI (as well as Ti and Ti ) differ only by orientation: the normal vectors of Si and Ti look upwards, whereas the normal vectors of Si and Ti look downwards. As above,we mark on the sphere S 2 the endpoints of the normal vectors of the triangles (we denote them by the same letters). For an appropriate choice of the numbers δ and , the geodesic segments connecting the points (see Fig. 17) do not intersect each other (except for the marked points). The complex C together with the fan obtained ΣN gives a hyperbolic polytope KN .

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P1(–d)

A2(–e)

O A3(e) P2(d)

A1(e)

upper part

H-operation

lower part

Fig. 17

After a series of H-operations, we obtain a hyperbolic virtual polytope KN with N

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horns (namely, the points Pi ), which differs combinatorially from that constructed in [8]. (Although it contains the same number of vertices and edges). An advantage of such a polytope is that it admits addition of extra horns. Step 3. Adding an extra horn. A hyperbolic polytope of type (+ + − + −... + −) with odd number of horns. The fan of the above constructed hyperbolic polytope KN allows a S-operation (see Fig. 18). Thus we obtain a desired hyperbolic polytope. This time the necessary conditions for further smoothing H-operations are a bit more complicated than those on the previous steps. Nevertheless, they lead to a smoothable hyperbolic polytope with N + 1 horns. upper part

S-operation

H-operations

lower part

Fig. 18 Step 4. Arbitrary necklace N = (± ± ...±). First construct a hyperbolic polytope of type (+, −, ..., +, −) with N horns, where N equals the number of color changes in the necklace N . Applying necessary S-operations, we insert additional horns. After a series of H-operations (as on Step 3), we obtain the required.  Using the polytopes KN as a base for applying S- and C-operations, one has even more freedom. This technique leads to advanced (in comparison with constructed above) combinatorial types of hyperbolic polytopes. However, we leave them for a later paper.

7

Smoothing techniques

Recall that whenever we have a smooth hyperbolic approximation of a hyperbolic virtual polytope, we obtain a counterexample to A.D. Alexandrov’s hypothesis.

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Theorem 7.1. Each polytope constructed in Section 6 admits a hyperbolic C ∞ -smoothing. In particular, there exist C ∞ -smooth hyperbolic h´erissons with odd number of horns. Proof. We follow the pattern of [9]: to obtain the desired approximation, we find mutually consistent smooth saddle surfaces which approximate the surfaces FK (ξ) for different ξ. The approximating smooth surface FK (ξ) coincides with FK (ξ) at the points lying far from the edges. Along the edges, the surface FK (ξ) is replaced by cylinders and cones. By a cylinder (respectively, by a cone with a vertex A) we mean a set of points that is invariant under all translations parallel to a line l (respectively, under homotheties with center in A). Note that as passing to another vector ξ, cones and cylinders may turn to each other. That is why we need different types of local saddle approximations as given in Lemma 7.2. Lemma 7.2. Let F be a piecewise linear surface given in the coordinate system (u, v, w) by the formula  0 if v < |u|, w(u, v) = |u| − v otherwise. Let L1 , L2 , and L3 be its edges (see Fig. 19); let Ai be a point lying on af f (Li ) \ Li or the point lying on af f (Li ) ”at infinity”. Then F admits a C ∞ -smooth approximation by a saddle surface F  such that • The surface F  coincides with F far from the edges. • Along the edges Li , (i = 1, 2, 3), but far from the vertex O, the surface F  is a cone with the apex at Ai (i.e., a cylinder if Ai lies at infinity). • The cylinders (or the cones) approximating L2 and L1 can be chosen independently. w

O

L1

v

u L3

L2

Fig. 19 Proof. The following two cases are already proven in [9]: 1. L1 , L2 , and L3 are approximated by cylinders. 2. L1 and L3 are approximated by cones with the apex at O, whereas L2 is approximated by a cylinder.

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Clearly case 2 is the tightest and implies the whole lemma.



Continue the proof of the theorem. Step 1. Consider a narrow belt about the equator z = 0 on the sphere S 2 and construct first a local approximation in the belt. The fan of K intersected with the belt splits into disjoint union of figures of two types (see Fig. 20).

type 1

type 2 Z=0

A

Fig. 20 Owing to H-operations, we may assume that all edges of the fan ΣK are shorter than π/2. Consider a collection of hemispheres with poles in some points {ηi } lying on the equator such that the union of the hemispheres covers the belt. Choose mutually consistent approximations of the surfaces F(ηi ) along the belt as follows: • Edges of type 1 coming out of of the belt, are approximated by cones with the apex at the vertex corresponding to A (i.e., at the vertex lying on the equator z = 0. • Edges of type 2 coming out of the belt, are approximated by cylinders. Step 2. Put ξ = (0, 0, ±1) and consider the surface FK (ξ). The surface has already some approximations (arising from Step1) along the edges coming from infinity. Namely, if an edge arises from type 1, it is approximated by a cylinder. If an edge comes from type 2, it is approximated by a cone with an apex lying beyond the edge. The only (mild) condition on the approximations we choose is that the cones and cylinders must be narrow (i.e., F  differs from FK (ξ) in a narrow domain along the edges). Step 3. Using the freedom of choice (Lemma 7.2), one can expand these approxima tions to a global saddle surface F  , approximating FK (ξ).

Acknowledgment The author wishes to express her gratitude to Nikolai Mn¨ev for useful discussions and to Alexandr Khodot for preparing the pictures.

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References [1] A.D. Alexandrov: “On uniqueness theorem for closed surfaces”, Doklady Akad. Nauk SSSR, Vol. 22, (1939), pp. 99–102 (Russian). [2] A.D. Alexandrov: Konvexe Polyeder, Berlin, Akademie-Verlag, 1958. [3] Yu. Burago and S.Z. Shefel: “The geometry of surfaces in Euclidean spaces”, In: Geometry III. Theory of surfaces. Encycl. Math. Sci., Vol. 48, 1992, pp. 1–85 (Russian, English). [4] A. Khovanskii and A. Pukhlikov: “Finitely additive measures of virtual polytopes”, St. Petersburg Math. J., Vol. 4(2), (1993), pp. 337–356. [5] Y. Martinez-Maure: “Contre-exemple a` une caract´erisation conjectur´ee de la sph`ere”, C.R. Acad. Sci. Paris, Vol. 332(1), (2001), pp. 41–44. [6] Y. Martinez-Maure: “Th´eorie des h´erissons et polytopes”, C.R. Acad. Sci. Paris Serie 1, Vol. 336, (2003), pp. 41–44. [7] P. McMullen: “The polytope algebra”, Adv. Math., Vol. 78(1), (1989), pp. 76–130. [8] G. Panina: “Virtual polytopes and some classical problems” St. Petersburg Math. J., Vol. 14(5), (2003), pp. 823–834. [9] G. Panina: “New counterexamples to A.D. Alexandrov’s hypothesis”, Adv. Geometry, Vol. 5, (2005), pp. 301–317. [10] A.V. Pogorelov: “On uniqueness theorem for closed convex surfaces” , Doklady Akad. Nauk SSSR, Vol. 366(5), (1999), pp. 602–604 (Russian). [11] R. Langevin, G. Levitt and H. Rosenberg: “H´erissons et multih´erissons (enveloppes param´etr´ees par leur application de Gauss)”, Singularities, Warsaw, Banach Center Publ., Vol. 20, (1985), pp. 245–253. [12] H. Radstr¨om: “An embedding theorem for spaces of convex sets”, Proc. AMS, Vol. 3(1), (1952), pp. 165–169. ` Rozendorn: “Surfaces of negative curvature”, Current Problems Math., Fund. Dir., [13] E. Vol. 48, (1989), pp. 98–195 (Russian).

DOI: 10.2478/s11533-006-0009-6 Research article CEJM 4(2) 2006 294–303

A topological invariant for pairs of maps Marcelo Polezzi1∗ , Claudemir Aniz2† 1

Universidade Estadual de Mato Grosso do Sul-(UEMS), Rodovia MS 306, 79540-000 Cassilˆ andia, Brasil 2

Departamento de Matem´ atica, Universidade Federal de Mato Grosso do Sul-(UFMS), Caixa Postal 549, 79070-900 Campo Grande, Brasil

Received 15 November 2005; accepted 8 February 2006 Abstract: In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from Rn , showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = P er(f )). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from R intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(R) such that f ◦ h = h ◦ f . For this latter set we obtain a generalization of Sharkovsky’s theorem. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Contact orders for pairs of maps, Sharkovsky’s theorem, discrete dynamical systems MSC (2000): 26A18, 37E05

1

Introduction

Problem 1.1. Given two continuous maps f, g : Rm → Rm , find sufficient conditions for Graph(f ) ∩ Graph(g) = ∅, where Graph(f ) = {(x, f (x)) : x ∈ Rm }. We shall solve this problem in the one-dimensional case and let the general case as an open problem. We hope that our solution may catch the readers’s interest for solving the n-dimensional case. For this purpose, let us first give some definitions, some of them ∗ †

E-mail: [email protected] E-mail: [email protected]

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very well known. Let N = {1, 2, . . .} be the set of natural numbers and f : R → R be a continuous function. For each natural n, let f n be defined by f 1 = f and f n = f ◦ f n−1 when n ≥ 2. We say that x∗ ∈ R is a period-n point of f if n is the least natural number such that f n (x∗ ) = x∗ . If n = 1, we say that x∗ is a fixed point of f and define F ix(f ) = {x ∈ R : x is a fixed point of f } Now, consider the sets P ern (f ) = {x ∈ R : x is a period-n point of f } and P er(f ) = {n ∈ N : P ern (f ) = ∅}. For the case in which m = 1 and g is the identity function, our problem has the following solution:

Proposition 1.2. Let f : R → R be a continuous function. If P ern (f ) = ∅ for some n ≥ 2, then F ix(f ) = ∅. This result, whose proof is a straightforward application of Bolzano’s theorem, can also be seen as a very particular case of Sharkovsky’s theorem. Before we state this theorem, let us give some more definitions: Let α(n) be the exponent of the highest power of 2 which divides n. For example, if n = 40, then α(40) = 3. Now, consider the sets 2N = {2, 4, 6, . . .}, 2N+1 = {3, 5, 7, 9, . . .} and 2N = {2, 4, 8, 16, . . .}, and let us construct a total ordering relation in N, represented by the symbol ≺. Given a = b, the Sharkovsky’s ordering is defined as follows: ⎧ ⎪ ⎪ a ∈ 2N + 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a, b ∈ 2N + 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a ∈ 2N \ 2N ⎪ ⎪ ⎨ a ≺ b ⇔ a, b ∈ 2N \ 2N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a, b ∈ 2N \ 2N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a, b ∈ 2N ⎪ ⎪ ⎪ ⎪ ⎪ ⎩b = 1

and b ∈ 2N,

or

and a < b,

or

and b ∈ 2N ,

or

and α(a) < α(b),

or

and α(a) = α(b), a < b, or and α(a) > α(b),

or

Now, denoting by n p when n ≺ p or n = p, we clearly have that (i) a a, (ii) a b and b a ⇒ a = b and (iii) a b and b c ⇒ a c, for any a, b, c ∈ N. Thus, the set of the natural numbers, ordered by Sharkovsky’s ordering, is given by: 3 ≺ 5 ≺ 7 ≺ 9 ≺ · · · 2.3 ≺ 2.5 ≺ 2.7 ≺ 2.9 ≺ · · · 22 .3 ≺ 22 .5 ≺ 22 .7 ≺ 22 .9 ≺ · · · 23 ≺

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22 ≺ 2 ≺ 1. Sharkovsky’s theorem (see, e.g., [2–4, 7]) is a remarkable result in Discrete Dynamical Systems, which says the following:

Theorem 1.3 (Sharkovsky’s theorem). (a)([9]). Let f : R → R be a continuous function. If P ern (f ) = ∅ and n ≺ p, then also P erp (f ) = ∅. (b)([10]). Consider the set S(n) = {p ∈ N : n p}. Then, for every n ∈ N, there exists a continuous function fn : R → R such that P er(fn ) = S(n). Moreover, there exist continuous functions g, h : R → R such that P er(g) = 2N ∪ {1} and P er(h) = ∅. Let us make some remarks about the meaning and the implications of this beautiful theorem: (a) If P er3 (f ) = ∅, then P er(f ) = N. This corollary is known as Li-Yorke’s theorem, due to Tien-Y. Li and James A. Yorke, which was published in 1975 (see [6]), eleven years after the paper [9] (written in Russian) by Alexander N. Sharkovsky was published. (b) If p ≺ n, then P ern (f ) = ∅ does not imply that P erp (f ) = ∅. A nice example of this (see [6]) is the fact that do exist functions f for which P er5 (f ) = ∅ and nevertheless P er3 (f ) = ∅. In fact, consider the function f : [1, 5] → [1, 5], defined such that f (1) = 3, f (3) = 4, f (4) = 2, f (2) = 5, f (5) = 1 and which is affine on each interval [n, n + 1], 1 ≤ n ≤ 4. Obviously, x = 1 ∈ P er5 (f ). Furthermore, it is easy to check, by analyzing the graphs of f , f 2 and f 3 , that there exists only one point x0 such that f 3 (x0 ) = x0 and it is located on the interval [3, 4]. However, on this interval we have that f (x) = 10 − 2x, and so f (10/3) = 10/3. Thus, x0 = 10/3, implying that P er3 (f ) = ∅. (c) P er(f ) is a finite set if, and only if P er(f ) = ∅ or P er(f ) = {1} or P er(f ) = S(2n ) for some n ∈ N . (d) The best strategy for proving the first part of Sharkovsky’s theorem is the observation that it is sufficient to show that: (i) If for a natural number m > 2 we have m ∈ P er(f ), then 2 ∈ P er(f ); (ii) If for an odd natural number m ≥ 3 we have m ∈ P er(f ), then n ∈ P er(f ) for every n > m;

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(iii) If for an odd natural number m ≥ 3 we have m ∈ P er(f ), then 2N ⊂ P er(f ). The veracity of items (i),(ii) and (iii) was proven by P. D. Straffin, Jr. [11]. However, Bau-Sen Du [3] has given a simpler proof, in which (ii) and (iii) are proved simultaneously and the proof of (i) differs from the traditional ones (see [1] and [11]). (e) For the proof of the second part of Sharkovsky’s theorem, it is necessary to make a study of the periodic points of functions called trapezoidal functions, constructed by means of tent functions. An example of such type of function is given by T (x) = 1−|2x−1| for 0 ≤ x ≤ 1 and T (x) = 0 otherwise. Two nice references for this result are the papers of Saber Elayadi [4] and Bau-Sen Du [3]. At this point one can argue that Sharkovsky’s theorem determines all the possible “contact orders” between the function f and the identity function id. More precisely, let g : R → R be any continuous function and, for each natural n, consider the set Conn (f, g) = {x ∈ R : n is the least natural number such that f n (x) = g n (x)}. Since Conn (f, id) = P ern (f ), then Sharkovsky’s theorem can be rephrased as: If Conm (f, id) = ∅, then Conn (f, id) = ∅ precisely when m ≺ n. Now, consider the set of “all possible contact orders” between f and g, given by Con(f, g) = {n ∈ N : Conn (f, g) = ∅}. By Sharkovsky’s theorem, one sees that the set Con(f, id) can not be equal to an arbitrary given subset of the naturals. Indeed, let S ∈ {{2}, {odd naturals}, {2, 3}}. Then, there is no continuous function f : R → R such that Con(f, id) = S. On the other hand, consider the following examples: (1) Let f (x) = −x and g(x) = −x + 1. One easily sees that Con2 (f, g) = R and Conn (f, g) = ∅ for all n ∈ N \ {2}. Thus, Con(f, g) = {2}. (2) Let f (x) = −x + 1 and g(x) = x + 1. Then, by Figure 1, one easily sees that Con2n−1 (f, g) = {x = 1 − n} and Con2n (f, g) = ∅ for all n ∈ N. Thus, Con(f, g) = {odd naturals}. (3) Let f (x) = −x + 2 and g(x) = −|x| + 1. Then, by Figure 2, one easily sees that Con2 (f, g) = [0, 1], Con3 (f, g) = {x = 3/2} and Conn (f, g) = ∅ for all n ∈ N \ {2, 3}. Thus, Con(f, g) = {2, 3}. These examples motivated us to inquire about all the possibilities for the set Con(f, g). As a surprising result, we found that Con(f, g) may appear as any subset of the naturals. This conclusion allows us to generalize an analogue of Sharkovsky’s theorem for maps from Rn , n ≥ 2 to itself [5]. That result, due to Kannan, Saradhi and Seshasai, says the following: Theorem 1.4. Let m ≥ 2 be a fixed natural number and let S ⊆ N be any given subset of the naturals. Then, there exists a continuous map f : Rm → Rm such that P er(f ) = S

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7 6 5 4 3 f f^2=id g g^2 g^3

2 1 0 -3

-2

-1

0

1

2

3

-1 -2 -3 -4

Fig. 1 Contact orders between f (x) = −x + 1 and g(x) = x + 1. 7 6 5 4 3 2 1 0 -1

-4

-3

-2

-1

0

1

2

3

4

f f^2=id g g^2 g^3

-2 -3 -4 -5

Fig. 2 Contact orders between f (x) = −x + 2 and g(x) = −|x| + 1.

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The result above, among other things, tell us that Proposition 1.2 does not hold for m ≥ 2. This is shown by the following example: 2-dimensional counter-example. Let f (x, y) = (−x, y +x2 −1) be a diffeomorphism of the plane. We easily see that {(1, y), (−1, y) : y ∈ R} = P er2 (f ). However, F ix(f ) = ∅. More than this, we have P er(f ) = {2}. Consider the natural extended definitions of the sets Conn (f, g) and P ern (f ) for maps from Rm to itself, and let id be the identity map on Rm . Then, Theorem 1.4 says that if S ⊆ N is any given subset of the naturals, then there exists a continuous map f : Rm → Rm such that Con(f, id) = S. By Sharkovsky’s theorem, we know that Theorem 1.4 does not hold for m = 1. However, if we consider the set Con(f, g) instead of P er(f ), then it is possible to extend Theorem 1.4 in order to include the one-dimensional case (see Theorem 2.4 below, whose proof can be found in our paper [8]).

2

Results

The set Con(f, g) is a Topological Invariant. Theorem 2.1. Let f, g, f1 , g1 : Rn → Rn be continuous maps such that the pairs (f, g) and (f1 , g1 ) are topologically equivalent. That is, there exists a homeomorphism h : Rn → Rn such that f1 = h◦f ◦h−1 and g1 = h◦g◦h−1 . Then, we have that Con(f, g) = Con(f1 , g1 ). In other words, the set Con(f, g) is a topological invariant. Proof. Suppose that m ∈ Con(f, g). Then, there exists xm ∈ Rn such that m is the least natural number satisfying f m (xm ) = g m (xm ). On the other hand, f1m = (h ◦ f ◦ h−1 )m = h ◦ f m ◦ h−1 and g1m = (h ◦ g ◦ h−1 )m = h ◦ g m ◦ h−1 . Thus, f1m (h(xm )) = (h ◦ f m ◦ h−1 )(h(xm )) = h(f m (xm )) = h(g m (xm )) = (h ◦ g m ◦ h−1 )(h(xm )) = g1m (h(xm )). Moreover, m is the least natural number such that f1m (h(xm )) = g1m (h(xm )). In fact, suppose that there exists p < m such that f1p (h(xm )) = g1p (h(xm )). In this case, we would have h(f p (xm )) = h(g p (xm )), and then f p (xm ) = g p (xm ), which is a contradiction, since xm ∈ Conm (f, g). Therefore, h(xm ) ∈ Conm (f1 , g1 ), which implies m ∈ Con(f1 , g1 ). Since m is arbitrary, we conclude that Con(f, g) ⊆ Con(f1 , g1 ). Conversely, suppose that m ∈ Con(f1 , g1 ). Then, there exists ym ∈ Rn such that m is the least natural number satisfying f1m (ym ) = g1m (ym ). Thus, (h ◦ f m ◦ h−1 )(ym ) = (h ◦ g m ◦ h−1 )(ym ), which implies f m (h−1 (ym )) = g m (h−1 (ym )). Moreover, m is the least natural number satisfying this last equality. In fact, suppose that there exists p < m such that f p (h−1 (ym )) = g p (h−1 (ym )). In this case, we would have (h ◦ f p ◦ h−1 )(ym ) = (h ◦ g p ◦ h−1 )(ym ), and then f1p (ym ) = g1p (ym ), which is a contradiction, since ym ∈ Conm (f1 , g1 ). Therefore, h−1 (ym ) ∈ Conm (f, g), which implies m ∈ Con(f, g). Since m is arbitrary, we conclude that Con(f1 , g1 ) ⊆ Con(f, g). 

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Definition 2.2. Let f : R → R and g : R → R be continuous functions and consider the following sets: (i) P ern (f, g) = {x ∈ R : n is the least natural number such that f n (x) = x = g n (x)}, (ii) P er(f, g) = {n ∈ N : P ern (f, g) = ∅}. Remark 2.3. In general, the sets Conn (f, g) and P ern (f, g) can be quite different. In fact, let f, g : [1, 4] → [1, 4] be such that f (1) = 2, f (2) = 3, f (3) = 4, f (4) = 1, g(1) = 2, g(2) = 4, g(3) = 1, g(4) = 3 and, on each interval [n, n + 1], n = 1, 2, 3, assume that f and g are both affine. We have that {1, 2, 3, 4} ⊂ P er4 (f, g), whereas 1 ∈ Con1 (f, g), 2 ∈ Con3 (f, g) and {3, 4} ⊂ Con4 (f, g). Proof. Given n ∈ {1, 2, 3, 4}, we have that f j (n) = n = g j (n) for j < 4 and f 4 (n) = n = g 4 (n). Thus, {1, 2, 3, 4} ⊂ P er4 (f, g). We also have that f (1) = 2 = g(1), and so 1 ∈ Con1 (f, g). Furthermore, f j (2) = g j (2) for j < 3 and f 3 (2) = 1 = g 3 (2), which implies that 2 ∈ Con3 (f, g). Finally, for n = 3 and n = 4 we have f j (n) = g j (n) if j < 4. However, since f 4 (n) = n = g 4 (n) for n = 1, 2, 3, 4, we conclude that {3, 4} ⊂ Con4 (f, g). Theorem 2.4. Let S ⊆ N be any given subset of the naturals. Then, there exist continuous functions f, g : R → R such that Con(f, g) = S and Conn (f, g) = P ern (f, g) for all n ∈ S. Comment 2.5. The essence of the proof of this theorem lies on the construction of pairs of continuous functions f, g : R → R which possess the desired properties. For this purpose, we make use of auxiliary functions; among them are fn , gn : [1, 2n] → [1, 2n], with n ∈ N, n ≥ 2, given by ⎧ ⎪ ⎪ x + 1, if 1 ≤ x ≤ n − 1 ⎪ ⎪ ⎨ fn (x) = n + |x − n| and gn (x) = (n − x)(n − 1) + 1, if n − 1 ≤ x ≤ n ⎪ ⎪ ⎪ ⎪ ⎩ 1, if n ≤ x ≤ 2n Of course, fn and gn are both continuous for each n ∈ N. Furthermore, it is not difficult to show that Con(fn , gn ) = {n} and Conn (fn , gn ) = {x = n}. Two solutions to the posed problem for the one-dimensional case. First solution. We now show that under a simple condition on one of the functions f and g, it is possible to guarantee that if Conn (f, g) = ∅ for some n ≥ 2, then Graph(f ) ∩ Graph(g) = ∅. By Examples 1 and 3 of Section 1, we know that it does not hold in general. Proposition 2.6. Let f, g : R → R be continuous functions such that g is nondecreasing. If Conn (f, g) = ∅ for some n ≥ 2, then also Con1 (f, g) = ∅.

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Proof. Since g is nondecreasing, then the same holds for all its iterates g i , i ≥ 2. Suppose Con1 (f, g) = ∅. By Bolzano’s theorem, we conclude that either f (x) > g(x) for all x ∈ R or f (x) < g(x) for all x ∈ R. In the first case, we shall have g n (x) ≤ g n−1 (f (x)) = g n−2 (g(f (x))) ≤ g n−2 (f 2 (x)) = g n−3 (g(f 2 (x))) ≤ g n−3 (f 3 (x)) ≤ · · · ≤ g(f n−1 (x)) < f n (x). That is, g n (x) < f n (x), for all x ∈ R, which is a contradiction, since Conn (f, g) = ∅. Analogously, the assumption f (x) < g(x) for all x ∈ R will also lead to a contradiction. Thus, Con1 (f, g) = ∅.  Second solution. The first solution does not give any clue in order to solve our problem in higher dimensions. This is so because there is no suitable analogue to the concept of nondecreasing functions for n-dimensional self-maps. However, we shall now provide a sufficient condition on the pair (f, g) which might, at least in principle, be extended for self-maps from Rn . Proposition 2.7. Let f, g : R → R be continuous functions such that f ◦ g = g ◦ f . If Conn (f, g) = ∅ for some n ≥ 2, then also Con1 (f, g) = ∅. Proof. Suppose Con1 (f, g) = ∅. By Bolzano’s theorem, we conclude that either f (x) > g(x) for all x ∈ R or f (x) < g(x) for all x ∈ R. In the first case, we shall have f n (x) = f (f n−1 (x)) > g(f n−1 (x)) = f (g(f n−2 (x))) > g 2 (f n−2 (x)) = f (g 2 (f n−3 (x))) > g 3 (f n−3 (x)) . . . > g n (x). That is, f n (x) > g n (x), for all x ∈ R, which is a contradiction, since Conn (f, g) = ∅. Similarly, the assumption f (x) < g(x) for all x ∈ R will also lead to a contradiction. Therefore, Con1 (f, g) = ∅.  The ordering of the sets Con(f, 0) and Con(f, h), for the case in which h ∈ Hom(R) and f ◦ h = h ◦ f Definition 2.8. We say that x∗ ∈ R is an order-n zero of f : R → R if n is the least natural number such that f n (x∗ ) = 0. In this section we shall study the sets Con(f, 0) and Con(f, h), when h is a homeomorphism of the real line which commutes with f . The first set can be seen as the set of all possible orders of order-n zeros of f . More precisely, consider the sets Conn (f, 0) = {xn ∈ R : xn is an order-n zero of f } and Con(f, 0) = {n ∈ N : Conn (f, 0) = ∅}. We shall give for this set an analogue of Sharkovsky’s theorem, as follows:

Theorem 2.9. (a) Let f : R → R be a continuous function. If Conn (f, 0) = ∅ and n > p, then also Conp (f, 0) = ∅. (b) Given any n ∈ N, there exists a continuous function g : R → R such that Con(g, 0) =

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{1, . . . , n}. (c) If h : R → R is a surjective continuous function such that h(0) = 0 and Con2 (h, 0) = ∅, then Con(h, 0) = N. Proof. (a) Let a ∈ Conn (f, 0). Then, f n−1 (f (a)) = 0 and f k (f (a)) = 0 for 1 ≤ k ≤ n − 2. Thus, f (a) ∈ Conn−1 (f, 0). Analogously, f 2 (a) ∈ Conn−2 (f, 0), f 3 (a) ∈ Conn−3 (f, 0), . . . , f n−1 (a) ∈ Con1 (f, 0). (b) Consider the function g : R → R given by ⎧ ⎪ ⎨ −n + 1, if x ≤ −n g(x) = ⎪ ⎩ x + 1, if x ≥ −n If n = 1, than we clearly have Con(g, 0) = {1} and Con1 (g, 0) = (−∞, −1]. Otherwise, we have that x = −n ∈ Conn (g, 0), since g n (−n) = 0 and g k (−n) = 0 for 1 ≤ k ≤ n − 1. In fact, g(−n) = −n + 1, g 2 (−n) = −n + 2, . . . , g n (−n) = 0. Furthermore, the only zero of g is x = −1. Now, suppose that Conp (g, 0) = ∅ for some p > n. In this case, there would exist x¯ ∈ R such that g p (¯ x) = 0 and g k (¯ x) = 0 for 1 ≤ k ≤ p − 1. Thus, it p−1 p−2 2 would imply that g (¯ x) = −1, g (¯ x) = −2, . . . , g (¯ x) = g p−(p−2) (¯ x) = 2−p. Hence, we would have g(¯ x) ∈ (−∞, −n], which is impossible, since Im(g) = [−n + 1, ∞). Therefore, Con(g, 0) = {1, 2, . . . , n}. (c) Let h : R → R be surjective and, in addition, h(0) = 0 and Con2 (h, 0) = ∅. In this case, we shall have Con(h, 0) = N. Indeed, since Con2 (h, 0) = ∅, there exists a2 ∈ R such that h2 (a2 ) = 0 and h(a2 ) = 0. This yields by the hypothesis that a2 = 0. Since h is surjective, there exists a3 ∈ R such that h(a3 ) = a2 . Observe that 0 = a3 = a2 and a3 ∈ Con3 (h, 0). Then, reasoning this way we shall obtain a sequence of infinitely many pairwise distinct points 0, a2 , a3 , a4 , . . . , an . . . such that an ∈ Conn (h, 0) for all n ≥ 2.  Example 2.10 (of (c)). The function f : R → R given by f (x) = 8x(x − 1)2 is surjective, f (0) = 0 and x = 1/2 ∈ Con2 (f, 0). Therefore, Con(f, 0) = N. Now, let us investigate the set Con(f, h), when h ∈ Hom(R) and f ◦ h = h ◦ f . We shall prove that for this case one obtains a generalization of Sharkovsky’s theorem. Indeed, we have the following result: Theorem 2.11. Let f, h : R → R be continuous functions such that h ∈ Hom(R) and f ◦ h = h ◦ f . Then, Con(f, h) = P er(f ◦ h−1 ). In other words, exactly one of the three situations will occur: (i) Con(f, h) = ∅, (ii) Con(f, h) = S(n) for some n ∈ N, (iii) Con(f, h) = 2N ∪ {1}. Proof. First, observe that since f ◦ h = h ◦ f , then also f ◦ h−1 = h−1 ◦ f . We

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have the following equivalences: f n (xn ) = hn (xn ) ⇔ (f n ◦ (h−1 )n )(hn (xn )) = hn (xn ) ⇔ (f ◦h−1 )n (hn (xn )) = hn (xn ). Therefore, xn ∈ Conn (f, h) ⇔ hn (xn ) ∈ P ern (f ◦h−1 ). Now, since any yn ∈ R can be given by yn = hn (xn ) for some xn ∈ R, the latter equivalence implies that Con(f, h) = P er(f ◦ h−1 ), and the theorem is proven.  Example 2.12. Consider the functions f, g : [1, 6] → [1, 6] defined such that f (1) = 2, f (2) = 3, f (3) = 6, f (4) = 1, f (5) = 4, f (6) = 5, g(1) = 5, g(2) = 4, g(3) = 1, g(4) = 6, g(5) = 3, g(6) = 2, and which are affine on each interval [n, n + 1], 1 ≤ n ≤ 5. Also, consider the homeomorphism h : [1, 6] → [1, 6] given by h(x) = −x + 7. It is then easy to check that f ◦ h = g = h ◦ f . Therefore, by Theorem 2.11 we have that Con(f, h) = P er(f ◦ h−1 ) = P er(f ◦ h) = P er(g) = N, since P er3 (g) = ∅. Indeed, xn ∈ P er3 (g), for all xn ∈ {1, 2, 3, 4, 5, 6}.

References [1] R. Barton and K. Burns: “A Simple Special Case of Sharkovskii’s Theorem”, Amer. Math. Monthly, Vol. 107(10), (2000), pp. 932–933. [2] N. Bhatia: “New Proof and Extension of Sarkovskii’s Theorem”, Far East J. Math. Sci., Special Volume, Part I, (1996), pp. 53–68. [3] B.-S. Du: “A Simple Proof of Sharkovsky’s Theorem”, Amer. Math. Monthly, Vol. 111(7), (2004), pp. 595–599. [4] S. Elayadi: “On a Converse of Sharkovsky’s Theorem”, Amer. Math. Monthly, Vol. 103, (1996), pp. 386–392. [5] V. Kannan, P.V.S.P. Saradhi and S.P. Seshasai: “A Generalization of Sarkovskii’s Theorem to Higher Dimensions”, J. Nat. Acad. Math. India, Vol. 11, (1997), pp. 69–82. [6] T.-Y. Li and J.A. Yorke: “Period Three Impies Chaos”, Amer. Math. Monthly, Vol. 82(10), (1975), pp. 985–992. [7] V.J. L´opez and L. Snoha: “All Maps of Type 2∞ are Boundary Maps”, Proc. Amer. Math. Soc., Vol. 125(6), (1997), pp. 1667–1673. ˇ [8] M. Polezzi and C. Aniz: “A Sarkovskii-Type Theorem for Pairs of Maps”, Far East J. Dynamical Systems, Vol. 7(1), (2005), pp. 65–75. [9] A.N. Sharkovsky: “Coexistence of cycles of a continuous map of a line into itself”, Ukrain. Mat. Zh., Vol. 16(1), (1964), pp. 61–71 (Russian); Internat. J. Bifur. Chaos Appl. Sci. Engrg., Vol. 5, (1995), pp. 1263–1273 (English). [10] A.N. Sharkovsky: “On cycles and the structure of a continuous map”, Ukrain. Mat. Zh., Vol. 17(3), (1965), pp. 104–111 (Russian). [11] P.D. Straffin, Jr.: “Periodic Points of Continuous Functions”, Math. Mag., Vol. 51(2), (1978), pp. 99–105.

DOI: 10.2478/s11533-006-0008-7 Research article CEJM 4(2) 2006 304–318

On a generalization of duality triads Matthias Schork∗ Alexanderstr. 76, 60489 Frankfurt, Germany

Received 25 March 2005; accepted 21 December 2005 Abstract: Some aspects of duality triads introduced recently are discussed. In particular, the general solution for the triad polynomials is given. Furthermore, a generalization of the notion of duality triad is proposed and some simple properties of these generalized duality triads are derived. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Duality triad, recurrence relation, inversion relation MSC (2000): 05Axx, 11B37, 11B83

1

Introduction

In [11, 12] the notion of duality triad was introduced. By this the following system is meant. Let three sequences i = {ik }k≥0 , q = {qk }k≥0 , and d = {dk }k≥0 of complex numbers with ik = 0 be given. For such a triple (i, q, d) we introduce the following “dynamical system”. The discrete time steps n start with 0 and at every time n we (i,q,d) consider the sequence cn = {cn,k }k≥0 ≡ {cn,k }k≥0 of complex numbers satisfying the recursion relation cn+1,k = ik−1 cn,k−1 + qk cn,k + dk+1 cn,k+1 (1) with the initial values c0,0 = 1 and c0,k = 0 for k > 0. Of course, given the initial values and the triple of sequences, the numbers cn,k are uniquely determined. Let us consider (i,q,d) the polynomial sequence {Φn (x)}n≥0 ≡ {Φn (x)}n≥0 (i.e., deg Φn (x) = n) satisfying the recurrence relation xΦn (x) = dn Φn−1 (x) + qn Φn (x) + in Φn+1 (x)

(2)

with Φ0 (x) = 1 and the convention Φ−1 (x) = 0 (note that this implies Φ1 (x) = (x−q0 )/i0 ). This recurrence relation for the sequence of polynomials Φn (x), called triad polynomials, ∗

E-mail: [email protected]

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is dual to the relation (1) in the sense that the following inversion relation holds [11, 12]: n

x =

n 

cn,k Φk (x).

(3)

k=0

This system of numbers cn,k and polynomials Φn (x) satisfying (1), (2), and (3) for the given triple of sequences (i, q, d) is called duality triad (associated to (i, q, d)) [11, 12]. Further motivation for these systems (with possible applications) and references can be found in [11, 12]. As stressed in [11], these duality triads are dual recurrences satisfying (1) and (2) as used in dynamical data bases theory [7] (see also [8]) which furthermore satisfy a third relation, the inversion relation (3). In general, the coefficients cn,k are thus the expansion coefficients (or connection coefficients) of the monomials xn in the basis of polynomials {Φk (x)}k≥0 . In [4, 11–14] these triads were studied and several well-known sequences of combinatorial numbers cn,k and polynomials Φn (x) were shown to be special cases. Possibly the simplest example is the following. Example 1.1. (Pascal triad) ik = 1, qk = 1, dk = 0. The corresponding cn,k satisfy the recursion relation cn+1,k = cn,k−1 +cn,k and are therefore given by the binomial coefficients,   i.e., cn,k = nk . Equation (6) implies Φn (x) = (x − 1)n so that the inversion relation (3)    becomes in this case xn = nk=0 nk (x − 1)k . In [14] a generalization of duality triads was suggested. This generalization consists in generalizing (1) - and consequently (2), (3) - to the case of higher order recurrences. It is the aim of the present article to make this suggestion more precise and derive first properties of these generalized duality triads. Let us now describe the structure of this article in detail. In Section 2 we give further examples of duality triads where the connection coefficients cn,k are given by well-known combinatorial numbers. In Section 3 we derive for an arbitrary triple (i, q, d) the form of the triad polynomials and discuss some general properties. In particular, we give a reformulation of the definition of duality triads which makes its structure more transparent. In Section 4 a generalization of duality triads is proposed and some properties of these generalized duality triads are derived. Finally, some conclusions are presented in Section 5.

2

Some well-known examples of duality triads

In this section we want to describe - following [4, 11–14] - some duality triads where the connection coefficients cn,k are given by well-known combinatorial numbers. Example 2.1. (Stirling triad) ik = 1, qk = k, dk = 0. The corresponding cn,k satisfy cn+1,k = cn,k−1 + kcn,k and are therefore given by the Stirling numbers of second kind, i.e.,  n−1 k k p k n p cn,k = S(n, k) = (−1) (−1) [15]. Equation (6) implies Φ (x) = n p=0 k=0 (x−k) = k! p n n n k x and (3) becomes x = k=0 S(n, k)x .

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Example 2.2. (q-Gaussian triad) ik = 1, qk = q k , dk = 0. The corresponding cn,k satisfy cn+1,k = cn,k−1 + q k cn,k and are therefore given by the q-deformed binomial coefficients,     m nq ! i.e., cn,k = nk q , where nk q = kq !(n−k) , m ! = q k=1 kq and the basic q-numbers are given q!  1−q m 2 m−1 k by mq = (1 + q + q + · · · q ) = 1−q [1]. Equation (6) implies Φn (x) = n−1 k=0 (x − q )     n k−1 n l and (3) becomes xn = k=0 k q l=0 (x − q ). The special case q = 1 is treated in Example 1.1. Example 2.3. (Bach-Comtet-Voigt-Konvalina triad) ik = 1, qk , dk = 0. This is a generalization of the above examples. Given the sequence q = (q0 , q1 , . . .), the corresponding cn,k satisfy cn+1,k = cn,k−1 + qk cn,k which is precisely the recursion relation of the generalized Stirling numbers of second kind considered in [2, 6, 9, 10, 16], i.e., cn,k = B(n, k) where  B(n, k) = q0d0 q1d1 q2d2 · · · qndn . (4) n

d0 +d1 +···+dn =n−k

The sum contains k summands. In the case qk = 1 each summand equals 1, yielding    B(n, k) = nk and thus reproducing Example 1.1. Equation (6) implies Φn (x) = n−1 k=0 (x− qk ). This duality triad is treated in detail in [4]. Note that one may choose an arbitrary sequence q as “input”, for example the sequence of prime numbers. Example 2.4. (Lah triad) ik = 1, qk = 2k, dk = k(k − 1). The corresponding cn,k satisfy cn+1,k = cn,k−1 + 2kcn,k + k(k + 1)cn,k+1 and are therefore given by the (unsigned) Lah n−1 ([5], p. 156). (Recall that the usual recursion numbers, i.e., cn,k = L(n, k) = n! k! k−1 relation for the Lah numbers is L(n + 1, k) = L(n, k − 1) + (n + k)L(n, k). From the definition one infers that (n − k)L(n, k) = k(k + 1)L(n, k + 1) and hence the above threeterm recursion relation.) Note that dk = 0 so that we cannot simply use (6) to determine Φn (x) but have to use instead the general formula (10). Many more examples of duality triads can be found in [11, 12, 14], corresponding, e.g., to several versions of deformed Stirling numbers or tangential numbers.

3

General properties of duality triads

Equation (2) may be written equivalently as (recall that ik = 0 for all k) Φn+1 (x) =

(x − qn ) dn Φn (x) − Φn−1 (x). in in

In the case dk = 0 for all k one immediately obtains n−1  (x − qk ) (x) = . Φ(i,q,d=0) n ik k=0 Let us define φn (x) by Φn (x) =

n−1  (x − qk ) k=0

ik

φn (x).

(5)

(6)

(7)

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If dk = 0 for all k, then φn (x) = 1 for all n, so - in a certain sense - φn (x) measures the influence of the nonvanishing dk . In general, one obtains for φn (x) from (5) the recursion relation in−1 dn φn+1 (x) = φn (x) − (8) φn−1 (x) (x − qn−1 )(x − qn ) with φ−1 (x) = 0, φ0 (x) = 1. Recalling Φ1 (x) = (x − q0 )/i0 , the first nontrivial function φ1 (x) = 1 obtained from (7) matches the recursion relation (8). For small n one obtains the explicit formulas: φ1 (x) = 1, i0 d1 , (x − q0 )(x − q1 )

 i0 d1 i1 d2 φ3 (x) = 1 − + , (x − q0 )(x − q1 ) (x − q1 )(x − q2 )

 i0 d1 i1 d2 i2 d3 φ4 (x) = 1 − + + (x − q0 )(x − q1 ) (x − q1 )(x − q2 ) (x − q2 )(x − q3 )

 i0 d1 i2 d3 2 +(−1) . (x − q0 )(x − q1 )(x − q2 )(x − q3 )

φ2 (x) = 1 −

One recognizes the pattern. A straightforward but tedious induction shows that φn (x) = 1 +

[ n2 ]  k=1

k

(−1)

n−2k  k−1  r=0 l=0

ir+2l dr+2l+1 . (x − qr+2l )(x − qr+2l+1 )

(9)

Combining this with (7) yields the general form of Φn (x). Proposition 3.1. Let us consider the duality triad associated to (i, q, d). Then the triad polynomials Φn (x) satisfying (2) are given by ⎧ ⎫ n n−1 ⎪ ⎪ [ ] n−2k 2 ⎬   (x − qk ) ⎨  k−1  ir+2l dr+2l+1 k Φn (x) = (−1) 1+ . (10) ⎪ ik (x − qr+2l )(x − qr+2l+1 ) ⎪ ⎩ ⎭ r=0 l=0 k=0 k=1 Recall the interpretation of (1) as a discrete dynamical system cn  cn+1 (this is discussed briefly in [11] and also in [14]). The “state” cn at time n is given by the infinite column vector cn = (cn,0 , cn,1 , . . .)t and the dynamical law is encoded in the “transition matrix” T ≡ T (i,q,d) . This is an ∞ × ∞ tridiagonal matrix with coefficients Tkl = ik−1 δk−1,l + qk δk,l + dk+1 δk+1,l

(11)

where δr,s is the Kronecker delta (and where the upper left corner of T is given by T00 ). The “dynamical law” mentioned above is then given by cn+1 = T cn

(12)

which is the equivalent form of (1). Thus, the state cn at time n is determined completely by the sequences (i, q, d) - encoded in T via (11) - and the “initial conditions” c0 =

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∞ n (c0,0 , c0,1 , . . .)t through cn = T n c0 . It follows that cn,k = l=0 [T ]kl c0,l , so it remains n to determine the entries of T (note that in the sum only finitely many summands are nonvanishing). Before we do this we introduce - following [14] - a concise notation which will also be useful when we discuss a generalization in Section 4. Thus, (−1)

μk

(0)

≡ ik ,

(1)

μk ≡ qk ,

μk ≡ dk .

(13)

The recurrence relation (1) can then be written as cn+1,k =

1 

(l)

μk+l cn,k+l

(14)

l=−1

and the dual recursion relation (2) is given by xΦn (x) =

1 

μ(−l) n Φn+l (x).

(15)

l=−1

Note that the transfer matrix T may be written in the new notation as 1 

Tkl =

(σ)

μk+σ δk+σ,l .

(16)

σ=−1

Proposition 3.2. Let the triple (i, q, d) ≡ (μ(−1) , μ(0) , μ(1) ) of sequences be given. Then  n ∞ 1    (σ ) m cn,i0 = c0,in . μim−1 (17) +σm δim−1 +σm ,im i1 ,...,in =0 σ1 ,...,σn =−1

m=1

In the case at hand we have furthermore the initial values c0,l = δ0,l . This allows a further  (−1) simplification of (17) and implies cn,n = nk=1 μk−1 as well as cn,k = 0 if k > n. The triad polynomials are given in the new notation by ⎧ ⎫ n ⎪ n−1 ⎪ ] [ n−2k (−1) (1) 2 ⎬   (x − μ(0) ) ⎨  k−1  μr+2l μr+2l+1 k k Φn (x) = (−1) 1 + . (−1) (0) (0) ⎪ ⎪ μ (x − μ )(x − μ ) ⎩ ⎭ r=0 l=0 k r+2l r+2l+1 k=0 k=1

(18)

Let us combine the polynomials Φk (x) in the infinite row vector Φ(x) = (Φ0 (x), Φ1 (x), . . .). Equation (15) can then be written as eigenvalue equation Φ(x)x = Φ(x)T (note that we write this equation - as well as (12) - dual to the corresponding ones in [11]). We would now like to interpret (3) also in another fashion. For this we imagine the sum on the right-hand side as a pairing between the row vector Φ(x) = (Φ0 (x), Φ1 (x), . . .) and the  column vector cn = (cn,0 , cn,1 , . . .)t as follows: Φ(x)|cn  := ∞ k=0 Φk (x)cn,k . This implies n that we can write (3) in the equivalent form x = Φ(x)|cn . With the definition of the linear spaces t C∞ f in := {c = (c0 , c1 , . . .) | ck ∈ C, only finitely many ck = 0}, k  akj xj , akj ∈ C, akk = 0}, PC := {Ψ(x) = (Ψ0 (x), Ψ1 (x), . . .) | Ψk (x) = j=0

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309

and the bilinear pairing ·|· : PC × C∞ f in → C[x], given by Ψ(x)|c := can now summarize the above observations in the following theorem.

∞

k=0 ck Ψk (x),

we

Theorem 3.3. (Equivalent definition of duality triad) Let a triple of sequences μ = (−1) (μ(−1) , μ(0) , μ(1) ) with μk = 0 for all k be given. The associated duality triad is defined  (σ) by the transfer matrix T ≡ T (μ) with Tkl = 1σ=−1 μk+σ δk+σ,l , a sequence {cn }n≥0 with cn ∈ C∞ f in and a Φ(x) ∈ PC such that: (i) c0,k = δ0,k , (ii) cn+1 = T cn , (iii) xΦ(x) = Φ(x)T , and (iv) Φ(x)|cn  = xn for every n ≥ 0. These properties imply the normalization Φ0 (x) = 1 and the symmetry of T with respect to the pairing, i.e., Φ(x)T |cn  = Φ(x)|T cn . (19) Proof. Since the above linear spaces and the pairing are defined exactly for this purpose, it is clear that the properties (ii), (iii), and (iv) are a reformulation of the original defining properties (1), (2), and (3), respectively (and (i) makes the initial value explicit). The normalization Φ0 (x) = 1 follows directly from (iii) by considering n = 0 and using (i). (ii)

Thus, only (19) has to be considered. The right-hand side is given by Φ(x)|T cn  = (iv)

(iii)

(iv)

Φ(x)|cn+1  = xn+1 , whereas the left-hand side equals Φ(x)T |cn  = xΦ(x)|cn  = xxn = xn+1 .  Remark 3.4. Note that we can equivalently take as defining properties of a duality triad (i), (ii), (iii) and (19) - with the normalization Φ0 (x) = 1 included - instead of (ii) (19) (iii) (i)-(iv). Property (iv) then follows since Φ(x)|cn  = Φ(x)|T cn−1  = Φ(x)T |cn−1  = xΦ(x)|cn−1 , thus implying Φ(x)|cn  = xn Φ(x)|c0 . Due to (i) one has c0,k = δ0,k , yielding Φ(x)|c0  = Φ0 (x) = 1 and thereby showing (iv). Remark 3.5. In the interpretation as a dynamical system we thus have an action of T on the state space C∞ f in as well as on PC . Note that due to our assumptions on the state space, the initial conditions and the transfer matrix all infinite sums which appear are in fact finite sums so that no discussion of convergence is necessary.

4

A generalization: duality triads of rank r

In Section 3 we have rewritten the definition of a duality triad in such a fashion that a possible generalization - suggested in [14] - is straightforward. The underlying idea is the following. For the “ordinary” duality triads the basic recurrence relation (14) is a three-term relation. In the interpretation from above where we interpret the index n as a discrete time step and consider the evolution of this dynamical system, the value of the state at the “place” k at time n + 1 is determined by the value at time n at the

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same place k and its nearest neighbors k − 1 and k + 1. Thus, we may loosely speak of a “nearest neighbor interaction”. Including an influence of the next-to nearest neighbors would mean to include in addition k − 2 and k + 2, i.e., considering a five-term recurrence relation for cn+1,k . In general, for a given natural number r we include the influence of the places k − r, . . . , k, . . . , k + r, i.e., we consider a (2r + 1)-term recurrence relation for cn+1,k . Now, let r be a natural number and assume that we are given a (2r + 1)-tuple of sequences μ = (μ(−r) , . . . , μ(r) ). Then the straightforward generalization of (14) is the (2r + 1)-term recurrence relation cn+1,k =

r 

(l)

μk+l cn,k+l

(20)

l=−r (0)

with initial condition c0,k = δ0,k . In particular, every s-term relation cn+1,k = μk cn,k + (−s) (l) · · · + μk−s cn,k−s (i.e., μk = 0 for l > 0 and every k) fits into this scheme, giving a (2s + 1)-term recurrence relation. Note that we may interpret (20) again as a dynamical system cn  cn+1 = T cn where the transition matrix T ≡ T (μ) is given by the following (2r + 1)-diagonal matrix r  (σ) Tkl = μk+σ δk+σ,l (21) σ=−r

generalizing (16). This implies the following generalization of Proposition 3.2. Proposition 4.1. Let the (2r + 1)-tuple (μ(−r) , . . . , μ(r) ) of sequences be given. Then the coefficients cn,k satisfying (20) are given by  n ∞ r    (σ ) m cn,i0 = μim−1 (22) c0,in +σm δim−1 +σm ,im i1 ,...,in =0 σ1 ,...,σn =−r

m=1

(where, again, c0,l = δ0,l ). In particular, cn,k = 0 if k > rn. (Note, however, that in general there will not exist a simple formula for cn,rn as in the case r = 1.) A different formula for the coefficients cn,k will be given later in Theorem 4.8. Now, we would like to define a generalization of the duality triad considered before. For this the reformulation given in Theorem 3.3 and Remark 3.4 is particularly suited. Definition 4.2. (Duality triad of rank r) Let a (2r + 1)-tuple of sequences μ = (−r) (μ(−r) , . . . , μ(r) ) with μk = 0 for all k and a polynomial pr (x) of degree r be given. Furthermore, let an r-tuple Ψ(x) = (Ψ0 (x), . . . , Ψr−1 (x)) of polynomials satisfying Ψ0 (x) = 1 and deg Ψk (x) = k be given. The associated duality triad or rank r is defined by the  (σ) transfer matrix T ≡ T (μ) with Tkl = rσ=−r μk+σ δk+σ,l , a sequence {cn }n≥0 with cn ∈ C∞ f in , and a Φ(x) ∈ PC such that: (i) c0,k = δ0,k , (ii) cn+1 = T cn , (iii) pr (x)Φ(x) = Φ(x)T ,

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311

(iv) Φ(x)T |cn  = Φ(x)|T cn  for every n ≥ 0, and (v) Φk (x) = Ψk (x) for 0 ≤ k ≤ r − 1. The connection between duality triads considered in preceding sections and duality triads of rank r is given by the following proposition. Proposition 4.3. Every duality triad is a duality triad of rank one associated to the polynomial p1 (x) = x. Proof. This follows by comparing Theorem 3.3 and Remark 3.4 with Definition 4.2. The properties (i)-(iii) of Theorem 3.3 and Definition 4.2 coincide and property (iv) of Theorem 3.3 is - according to Remark 3.4 - equivalent to the symmetry of T and the normalization Φ0 (x) = 1, which are precisely properties (iv) and (v) of Definition 4.2 in the rank one case.  Let us give some comments about the definition. Clearly, property (ii) is just the recursion relation (20), whereas (iii) gives the dual relation for the polynomial sequence. Writing this explicitly, it reads pr (x)Φn (x) =

r 

μ(−l) n Φn+l (x)

(23)

l=−r

and is the obvious generalization of (15). Making this even more explicit, we obtain the analogue of (5), i.e, Φn+r (x) = −

r−1 (−k)  μn k=1

(−r)

μn

(0)

Φn+k (x) +

pr (x) − μn (−r)

μn

Φn (x) −

r (k)  μn k=1

(−r)

μn

Φn−k (x).

(24)

(−r)

This shows explicitly why it is necessary to assume that μn = 0 in Definition 4.2. In order to use these equations for arbitrary n ≥ 0 we introduce the convention Φ−k (x) = 0 for k = 1, . . . , r (as in the rank one case). Recall that in the rank one case the normalization Φ0 (x) = 1 is given and that the recursion relation determines the first nontrivial polynomial Φ1 (x) (and, of course, all polynomials of higher degree subsequently). In the case of higher rank the first nontrivial relation results by choosing n = 0 in (24), i.e., (−r+1)

Φr (x) = −

μ0

(−r)

μ0

(−1)

Φr−1 (x) − · · · −

μ0

(−r)

μ0

(0)

Φ1 (x) +

pr (x) − μ0 (−r)

μ0

Φ0 (x).

(25)

This relation determines Φr (x) in terms of the polynomials Φk (x) = Ψk (x) with k = 0, . . . , r − 1. In particular, it follows from (24) that if deg Φk (x) = k for 0 ≤ k ≤ r − 1, then deg Φk (x) = k for every k. This is the reason why we have included the otherwise arbitrary polynomials Ψk (x) in the definition of the duality triad of rank r: demanding that the first r components Φ0 (x), . . . , Φr−1 (x) of Φ(x) have the correct degree implies that Φ(x) ∈ PC . Let us collect the explicit formulas satisfied by a duality triad of rank r in the following theorem.

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Theorem 4.4. Let (T, {cn }n≥0 , Φ(x)) be a duality triad of rank r associated to the (2r+1)tuple of sequences μ = (μ(−r) , . . . , μ(r) ) and the polynomial pr (x). Then the coefficients  (l) cn,k satisfy the recursion relation cn+1,k = rl=−r μk+l cn,k+l , the triad polynomials Φn (x)  (−l) satisfy the dual recursion relation pr (x)Φn (x) = rl=−r μn Φn+l (x), and the inversion relation is given by rn  n [pr (x)] = cn,k Φk (x). (26) k=0 (ii)

(iv)

(iii)

Proof. It remains to show (26). One has Φ(x)|cn  = Φ(x)|T cn−1  = Φ(x)T |cn−1  = pr (x)Φ(x)|cn−1 , hence Φ(x)|cn  = [pr (x)]n Φ(x)|c0  (compare Remark 3.4). Due to (i)  one has Φ(x)|c0  = Φ0 (x) = 1 and it follows that [pr (x)]n = Φ(x)|cn  = ∞ k=0 cn,k Φk (x). The assertion follows by recalling that cn,k = 0 for k > rn (see Proposition 4.1).  In the case of higher rank one may choose different polynomials pr (x), yielding different generalized duality triads. Of course, the simplest choice is pr (x) = xr , but pr (x) = xr seems also to be an interesting choice. Note that due to our assumptions all sums which appear are in fact finite sums (as in the case r = 1, see Remark 3.5). Now, we would like to discuss the simplest examples of duality triads of rank r. From the above definition (−r) we have to assume μk = 0, but all other sequences μ(l) are allowed to vanish. Example 4.5. (A trivial example of a duality triad of rank r) Let the (2r +1)-tuple of sequences μ = (μ(−r) , 0, . . . , 0) be given, i.e., all sequences except μ(−r) vanish. Furthermore, a polynomial pr (x) and an r-tuple of polynomials Ψ(x) = (Ψ0 (x), . . . , Ψr−1 (x)) have to (−r) be given. The recursion relation becomes cn+1,k = μk−r cn,k−r . Since c0,k = δ0,k , the coeffi(−r) (−r) (−r) ¯ then cn,kr cients cn,k vanish if k is not a multiple of r and if k = kr μ(k−1)r · · · μ0 . ¯ = μ¯ ¯ kr The polynomials satisfy Φn∗ +r (x) =

pr (x)

(−r)

μn∗

Φn∗ (x) for 0 ≤ n∗ ≤ r − 1 (note that the right-

hand side is given by the initial values Φn∗ (x) = Ψn∗ (x)). In general, we may write   n = r nr + n∗ with 0 ≤ n∗ ≤ r − 1. Iterating the above relation yields Φn (x) =

n pr (x)[ r ]

(−r) (−r)

(−r)

μn−r μn−2r · · · μn∗ (−1)

(−r)

Ψn∗ (x).

(27) (−1)

˜k := μkr yields the relation dn+1,k = μ ˜k−1 dn,k−1 Note that defining dn,k := cn,kr and μ of a duality triad of rank one. However, due to the dual recursion relation and the inversion relation this example cannot be mapped onto a duality triad of rank one. Before we discuss the next example we introduce some notations and terminology which will be used in the rest of the section. Definition 4.6. Let us consider the lattice Z × Z. A straight line connecting (n, k) and (n + 1, l) with |l − k| ≤ r will be called step with initial point (n, k) and endpoint (n+1,l) (n+1,l) (n + 1, l) and will be denoted by S(n,k) . The weight of a step S(n,k) is defined by

M. Schork / Central European Journal of Mathematics 4(2) 2006 304–318 (n+1,l)

(k−l)

313

(n+1,l)

ω(S(n,k) ) := μk and its height by ht(S(n,k) ) := l − k. A path of length m is a sequence of m steps, where the endpoint of the k-th step is equal to the initial point of the (k + 1)-th step. Thus, we can write a path of length m as a concatenation of m (n+m,lm+1 ) (n+2,l ) (n+1,l ) steps, i.e., in the form S(n+m−1,l · · · S(n+1,l23) S(n,l1 ) 2 . Equivalently, such a path can be m) described by its initial point (n, l1 ) and the m heights νj := (lj+1 − lj ) for 1 ≤ j ≤ m. Note that −r ≤ νj ≤ r. The height of a path is defined as the sum of the heights of its steps, i.e., by νm + · · · + ν1 = lm+1 − l1 . Let us denote by Γm,k the set of paths of length m and height k with initial point (0, 0), i.e., Γm,k := {ν = (ν1 , . . . , νm ) ∈ {−r, . . . , r}m |

m 

νj = k}.

(28)

j=1

The weight ω(ν) of a path ν ∈ Γm,k is defined as the product of the weights of all its steps, i.e., −νm−1 −ν2 −ν1 m ω(ν) := μ−ν (29) r(νm−1 +···+ν1 ) μr(νm−2 +···+ν1 ) · · · μrν1 μ0 . The paths we are interested in stay in the upper half (i.e., “above the x-axis”). Thus, we finally define the subset Γ∗m,k ⊂ Γm,k of admissible paths of length m and height k by Γ∗m,k

:= {ν = (ν1 , . . . , νm ) ∈ Γm,k |

l 

νj ≥ 0 for 1 ≤ l ≤ m}.

(30)

j=1

Example 4.7. (A nontrivial example of a duality triad of rank r) Let the (2r + 1)-tuple μ = (μ(−r) , 0, . . . , 0, μ(0) , 0, . . . , 0) of sequences be given. Furthermore, a polynomial pr (x) and a r-tuple of polynomials Ψ(x) = (Ψ0 (x), . . . , Ψr−1 (x)) have to be given. From (25) p (x)−μ

(0)

it follows that Φn∗ +r (x) = r (−r) n∗ Φn∗ (x) for 0 ≤ n∗ ≤ r − 1. In general, we may write μn∗   n = r nr + n∗ with 0 ≤ n∗ ≤ r − 1. Iterating the above relation yields ⎧ ⎫ n −1 ⎪ ⎪ (0) r] ⎨[  pr (x) − μn∗ +kr ⎬ (31) Ψn∗ (x), Φn (x) = (−r) ⎪ ⎪ μn∗ +kr ⎩ k=0 ⎭ which is the direct analogue of (6). Due to the vanishing of most of the sequences μ(l) the dual recursion relation for the triad polynomials implies that the polynomials “decouple” in the sense that there exist r sequences (Φn∗ (x), Φn∗ +r (x), Φn∗ +2r (x), . . .) for 0 ≤ n∗ ≤ r − 1 which are independent. Let us turn to the coefficients cn,k . They satisfy (−r) (0) the recursion relation cn+1,k = μk−r cn,k−r + μk cn,k (with c0,k = δ0,k ). It follows that (−r) (0) c1,k = μk−r δ0,k−r + μk δ0,k and that (−r) (−r)

(−r) (0)

(0) (−r)

(0) (0)

c2,k = μk−r μk−2r δ0,k−2r + {μk−r μk−r + μk μk−r }δ0,k−r + μk μk δ0,k .

(32)

In general, it is clear that cn,k = 0 if k is not a multiple of r. If k is a multiple of r ¯ In (32) the contribution to c2,kr we may write k = kr. ¯ corresponds to the sum of the ¯ weights of all paths of length 2 from (0, 0) to (2, kr). For example, c2,r corresponds to

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two paths (whereas c2,0 as well as c2,2r correspond to one path). The first path consists (1,r) (−r) (2,r) (0) of the step S(0,0) with weight μ0 followed by the step S(1,r) with weight μr , giving the (0) (−r)

weight μr μ0

(1,0)

(0)

for the path. The second path consists of the step S(0,0) with weight μ0 (2,r)

(−r)

(−r) (0)

followed by the step S(1,0) with weight μ0 , giving the weight μ0 μ0 for the path. In n n ¯ ˜ n,kr analogy to (28) we define Γ ¯ := {ν = (ν1 , . . . , νn ) ∈ {0, r} | l=1 νl = kr} and the weight ω(ν) of a path ν = (ν1 , . . . , νn ) as the product of the weights of all its steps, see  (29). Then the coefficients cn,kr ¯ are given by cn,kr ¯ = ˜ ¯ ω(ν). It is clear that this ν∈Γ n,kr reproduces the coefficients c2,k discussed explicitly above. In particular, in the special case n (0) (−r) μk = 1 = μk the weights ω(ν) of all paths ν equal 1 and since there are |Γn,kr ¯ | = k ¯ n summands, one obtains cn,kr . Thus, this example corresponds to some kind of = ¯ ¯ k “embedded” Pascal triad, see Example 1.1. After having discussed explicitly a particular example of a duality triad of rank r, we can now give another expression for the coefficients cn,k in the general case. Theorem 4.8. Let (T, {cn }n≥0 , Φ(x)) be a duality triad of rank r associated to the (2r+1)tuple of sequences μ = (μ(−r) , . . . , μ(r) ). Then the coefficient cn,k is given as the sum of all weights of all admissible paths of length n and height k, i.e., cn,k =



ω(ν).

(33)

ν∈Γ∗n,k (l)

In particular, if μk = 1 for all k, l, then cn,k = |Γ∗n,k |. Proof. The proof is a simple induction in n. Recalling (20) and using (33) for n, one obtains r   (σ) cn+1,k = μk+σ ω(ν). (34) σ=−r ν∈Γ∗n,k+σ

(σ)

(n+1,k)

(σ)

Since μk+σ is the weight of the step S(n,k+σ) the expression μk+σ ω(ν) is the weight of the admissible path ν˜ ∈ Γ∗n+1,k which consists of the admissible path ν ∈ Γ∗n,k+σ followed (n+1,k) by the step S(n,k+σ) . Since the sum on the right-hand side yields all admissible paths  ν˜ ∈ Γ∗n+1,k , the right-hand side equals ν˜∈Γ∗ ω(˜ ν ), proving the assertion.  n+1,k

(l)

Note that in Example 4.7 all paths are admissible since νj ≥ 0. In general, if μk = 0 for l > 0 and all k then Γn,k = Γ∗n,k , i.e., all paths are admissible. However, note that if (l) μk = 0 for some l ≤ 0, then the set of paths contributing to cn,k is only a subset of Γ∗n,k . An example for this is given in Example 4.7. Let us denote by c(s, t, u) the number of compositions of u with exactly t parts, each ≤ s ([1], p.55). Proposition 4.9. Let (T, {cn }n≥0 , Φ(x)) be a duality triad of rank r associated to the (l) (2r + 1)-tuple of sequences μ = (μ(−r) , . . . , μ(0) , 0, . . . , 0), i.e., μk = 0 for l > 0 and all

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315

k. Then the number of admissible paths of length n and height k is given by |Γ∗n,k |

=

n   n s=0

s

c(r, n − s, k).

(l)

In particular, if μk = 1 for l ≤ 0 and all k, then cn,k =

n

(35)

s=0

n s

c(r, n − s, k).

(l)

Proof. Due to μk = 0 for l > 0 and all k, every path ν ∈ Γn,k is admissible, i.e., (0) (1) (n) (s) Γn,k = Γ∗n,k . We may decompose Γn,k = Γn,k ∪ Γn,k ∪ · · · ∪ Γn,k where Γn,k denotes the set of paths ν = (ν1 , . . . , νn ) where exactly s of the components νj vanish. It follows    (s) (s) that |Γ∗n,k | = ns=0 ns |Γn,k |. The assertion follows since |Γn,k | is given by the number of (s)

compositions of k with exactly n − s parts, each ≤ r, i.e., |Γn,k | = c(r, n − s, k).



Example 4.10. Let us consider a duality triad of rank one, i.e., we are given the triple (0) μ = (μ(−1) , μ(0) , μ(1) ) of sequences. Let us first consider the special case where μk = 0 for n, and all k. A path ν ∈ Γ∗n,0 of height k = 0 necessarily requires n to be even, i.e., n = 2¯ ∗ to have an equal number of “up” and “down” steps. This means that Γ2¯n,0 = D2¯n , where D2¯n is the set of Dyck-paths of length 2¯ n. Thus, c2¯n,0 is given as the sum of all weighted  (−1) (1) Dyck-paths of length 2¯ n, i.e., c2¯n,0 = ν∈D2¯n ω(ν). In particular, if μk = 1 = μk   2¯ n 1 is the n ¯ -th Catalan number. for all k, then c2¯n,0 = |D2¯n | = Cn¯ , where Cn¯ = n¯ +1 n ¯ (0)

Now, let us allow μk = 0. If for the paths ν ∈ Γ∗n,l one has l = 0, then ν ∈ Γ∗n,0 is a Motzkin-path of length n, i.e., Γ∗n,0 = Mn , where we have denoted by Mn the set of Motzkin-paths of length n. Thus, cn,0 is given by the sum of all weighted Motzkin-paths,  (l) i.e., cn,0 = ν∈Mn ω(ν). In particular, if μk = 1 for all l, k, then cn,0 = |Mn | = Mn ,    n Ck is the n-th Motzkin number. where Mn = k≥0 2k (r)

Definition 4.11. Let r ∈ N be given. The set Mn of Motzkin-paths of rank r and (r) length n is given by Mn := Γ∗n,0 (where Γ∗n,0 is defined in (30)). Consequently, the n-th (r) (r) Motzkin number of rank r is defined by Mn := |Mn |. As in the rank one case we define (r) the set Dn of Dyck-paths of rank r and length n as the set of those Motzkin-paths of rank r and length n having no horizontal steps, i.e., Dn(r) := {ν ∈ {−r, . . . , −1, 1, . . . , r}n |

n  j=1

(1)

νj = 0,

l 

νj ≥ 0 for 1 ≤ l ≤ m}.

(36)

j=1 (1)

It is clear that Mn = Mn (and consequently that Mn = Mn ). Furthermore, (1) (1) (r) Dn = 0 if n is odd and Dn = Dn if n is even. Note that in the higher rank case Dn = 0 (2) if n is odd. For example, in the case r = 2 the set D3 consists of the paths (1, 1, −2) and (2, −1, −1). Proposition 4.12. Let (T, {cn }n≥0 , Φ(x)) be a duality triad of rank r associated to the (2r + 1)-tuple of sequences μ = (μ(−r) , . . . , μ(r) ).

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(1) The coefficient cn,0 is given as the sum of all weighted Motzkin-paths of rank r and  (l) (r) length n, i.e., cn,0 = ν∈M(r) ω(ν). If μk = 1 for all l, k, then cn,0 = Mn . n (0)

(2) If in addition μk = 0 for all k, then cn,0 is given as the sum of all weighted Dyck (l) paths of rank r and length n, i.e., cn,0 = ν∈Dn(r) ω(ν). If μk = 1 for all l = 0, k, (r)

then cn,0 = |Dn |.

Proof. Combining (33) and Definition 4.11 yields the first assertion. The second assertion follows trivially from this and Definition 4.11.  Let us return to Example 4.10 and introduce a more combinatorial language used, n+1,k+1 n+1,k e.g., in [17]. We call a step of the form Sn,k ↔ (1, 1) an up-step U , Sn,k ↔ (1, 0) n+1,k−1 a level-step L, and Sn,k ↔ (1, −1) a down-step D. Let l and d be positive integers and color the L steps with l colors and the D steps with d colors. Let A(n, k) be the set of colored paths starting in (0, 0) and ending in (n, k) and let M (n, k) be the set of lattice paths in A(n, k) that never go below the x-axis. Let an,k := |A(n, k)| and mn := |M (n, 0)|. The number mn is called (1, l, d)-Motzkin number. Since the number |A(n, k)| of colored paths equals the sum of all weighted paths in Γn,k , i.e., |A(n, k)| =  (1) (0) (−1) := 1 for every k, we find ν∈Γn,k ω(ν) with weights given by μk := d, μk := l, μk that an,k = cn,k and, therefore, that an+1,k = an,k−1 + lan,k + dan,k+1 . This implies  mn = cn,0 = ν∈Γ∗ ω(ν). It is shown in [17] that the mn satisfy the three-term recursion n,0 (n + 2)mn = l(2n + 1)mn−1 + (4d − l2 )(n − 1)mn−2 . In the case of higher rank r > 1 (r) the analogous Motzkin numbers mn = cn,0 (see Definition 4.11) are part of a duality (r) triad whose coeffcients cn,k satisfy the (2r + 1)-term recursion (20). Thus, the mn are expected to satisfy a (2r + 1)-term recursion similar to the one given for mn above. (l) However, to obtain a close analogy we have to restrict to the case where the weights μk are integers which are independent of k (as in the case r = 1 above). Thus, if the weight (0) n+1,k of the level-step L ↔ Sn,k ↔ (1, 0) is given by μk = l, of the r different down-steps (u)

n+1,k−u Du ↔ Sn,k ↔ (1, −u) by μk = du (with 1 ≤ u ≤ r), and of the r different up-steps (−v)

n+1,k+v Uv ↔ Sn,k ↔ (1, v) by μk = uv (with 1 ≤ v ≤ r), the basic recursion relation (20) is in this case given by cn+1,k = ur cn,k−r + · · · + u1 cn,k−1 + lcn,k + d1 cn,k+1 + · · · + dr cn,k+r . The corresponding Motzkin numbers should be called (ur , . . . , u1 , l, d1 , . . . , dr )-Motzkin numbers.

Remark 4.13. We have used several times the interpretation of the property cn  cn+1 = T cn of the duality triad (or its generalization) as a dynamical system. Clearly, there is a very close connection to one-dimensional cellular automata [18]. Representing a sequence cn as an infinite row of boxes (beginning with the zeroth box B0 ) where the contents of the k-th box Bk is given at time n by cn,k , the transfer matrix T thus describes the contents cn+1,k of Bk at time n + 1 in dependence of the contents cn,k of Bk and some of its neighboring boxes (in the case of a duality triad of rank r the contents of the r boxes on both sides of Bk will contribute).

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5

317

Conclusion

In this article we have discussed some general properties of duality triads. In particular, we gave an equivalent definition for duality triads which makes their structure more transparent. Using this reformulation, we introduced a natural generalization of duality triads where the basic relation is allowed to be a recurrence relation of higher order. Some of the simplest properties of the higher rank case were discussed, but many interesting properties have yet to be determined. It was shown that in the simplest cases the resulting connection coefficients have a nice interpretation in terms of paths in the lattice Z × Z, generalizing some well-known combinatorial structures associated to the rank one case (e.g., Dyck-paths, Motzkin-paths and the associated Catalan and Motzkin numbers). As a very brief outlook let us mention that the generalized Stirling numbers Sr,r (n, k) introduced in [3] (with S1,1 (n, k) ≡ S(n, k) from Example 2.1) satisfying  k [xr ]n = rn k=0 Sr,r (n, k)x are the coefficients cn,k of a natural duality triad of rank r. This as well as other examples will be treated in a future publication.

References [1] G.E. Andrews: The Theory of Partitions, Addison Wesley, Reading, 1976. ¨ [2] G. Bach: “Uber eine Verallgemeinerung der Differenzengleichung der Stirlingschen Zahlen 2.Art und einige damit zusammenh¨angende Fragen”, J. Reine Angew. Math., Vol. 233, (1968), pp. 213–220. [3] P. Blasiak, K.A. Penson and A.I. Solomon: “The Boson Normal Ordering Problem and Generalized Bell Numbers”, Ann. Comb., Vol. 7, (2003), pp. 127–139. [4] E. Borak: “A note on special duality triads and their operator valued counterparts”, Preprint: arXiv:math.CO/0411041. [5] L. Comtet: Advanced Combinatorics, Reidel, Dordrecht, 1974. [6] L. Comtet: “Nombres de Stirling g´en´eraux et fonctions sym´etriques”, C. R. Acad. Sc. Paris, Vol. 275, (1972), pp. 747–750. [7] P. Feinsilver and R. Schott: Algebraic structures and operator calculus. Vol. II: Special functions and computer science, Kluwer Academic Publishers, Dordrecht, 1994. [8] I. Jaroszewski and A.K. Kw´asniewski: “On the principal recurrence of data structures organization and orthogonal polynomials”, Integral Transforms Spec. Funct., Vol. 11, (2001), pp. 1–12. [9] J. Konvalina: “Generalized binomial coefficents and the subset-subspace problem”, Adv. Math., Vol. 21, (1998), pp. 228–240. [10] J. Konvalina: “A unified interpretation of the Binomial Coefficients, the Stirling Numbers and Gaussian Coefficents”, Amer. Math. Monthly, Vol. 107, (2000), pp. 901–910. [11] A.K. Kwa´sniewski: “On duality triads”, Bull. Soc. Sci. Lettres L  ´od´z, Vol. A 53, Ser. Rech. D´eform. 42, (2003), pp. 11–25. [12] A.K. Kwa´sniewski: “On Fibonomial and other triangles versus duality triads”, Bull.

318

[13] [14] [15] [16]

[17] [18]

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Soc. Sci. Lettres L  ´od´z, Vol. A 53, Ser. Rech. D´eform. 42, (2003), pp. 27–37. A.K. Kwa´sniewski: “Fibonomial Cumulative Connection Constants”, Bulletin of the ICA, Vol. 44, (2005), pp. 81–92. M. Schork: “Some remarks on duality triads”, Adv. Stud. Contemp. Math., to appear. R.P. Stanley: Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999. B. Voigt: “A common generalization of binomial coefficients, Stirling numbers and Gaussian coefficents”, Publ. I.R.M.A. Strasbourg, Actes 8e S´eminaire Lotharingien, Vol. 229/S-08, (1984), pp. 87–89. W. Woan: “A Recursive Relation for Weighted Motzkin Sequences”, J. Integer Seq., Vol. 8, (2005), art. 05.1.6. S. Wolfram: A new kind of science, Wolfram Media, Champaign, 2002.

DOI: 10.2478/s11533-006-0011-z Communication CEJM 4(2) 2006 319–322

Rational values of the arccosine function∗ Juan L. Varona† Departamento de Matem´ aticas y Computaci´ on, Universidad de La Rioja, Edificio Vives, Calle Luis de Ulloa s/n, 26004 Logro˜ no, Spain

Received 7 November 2005; accepted 8 February 2006 √ Abstract: We give a short proof to characterize the cases when arccos( r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Arcosine, cosine, rational, irrational MSC (2000): 11J72, 33B10

The arithmetic properties of trigonometric functions has been a recurring topic in the mathematical literature. In 1933, D. H. Lehmer [2] proved that if d > 2 and k/d is an irreducible fraction, then 2 cos(2πk/d) is an algebraic integer of degree ϕ(d)/2 (with ϕ(d) being Euler’s totient function); the proof can be also found in [3, Theorem 3.9]. As a consequence, it can be shown that, for t ∈ Q, the only rational values of cos(πt) are cos(πt) = 0, ±1, ±1/2. But this can be proved independently of Lehmer’s result; see also [3, Chapter 3] for historical references. Another nice and self-contained proof appears in [4, § 6.3, Theorem 6.16]. In [1, Chapter 6], as a key step for the construction of Dehn’s counterexamples to Hilbert’s third problem about decomposing polyhedra, it is established that 1 arccos π



1 √ n

 ∈ / Q when n ∈ N,

n odd,

n ≥ 3.

(1)

The aim of this paper is to give a direct and simple proof of a much more general result: ∗ †

Research partially supported by grant BFM2003-06335-C03-03 of the DGI (Spain). E-mail: [email protected]

320

J.L. Varona / Central European Journal of Mathematics 4(2) 2006 319–322

the complete characterization of the r ∈ Q such that √  1 arccos r ∈ Q. π Let us explain the idea of our proof. The elegant proof of (1) given in [1, Chapter 6] is based in the trigonometric identity cos((k + 1)θ) = 2 cos(θ) cos(kθ) − cos((k − 1)θ),

(2)

which is an immediate consequence of ) cos( α−β ). cos(α) + cos(β) = 2 cos( α+β 2 2 For even n, a different method is suggested in that book, distinguishing between the cases √ n = 2j and n not a power of 2. Thus, it is obtained that (1/π) arccos(1/ n) is rational if and only if n ∈ {1, 2, 4}. The relation (2) can be read in term of Chebyshev polynomials of the first kind. These polynomials are well known in the mathematical literature (see, for instance, [6] or [5]), mainly for their importance in approximation theory (they are orthogonal polynomials, and are also used in least squares fit and in quadrature formulas for numerical integration). At least for this author, polynomial relations are easier to handle than trigonometric relations and, thus, the proof given in [1, Chapter 6] seems to be clearer when written in terms of polynomials. Moreover, this allows a useful generalization of (1) whose proof does not lose the simplicity of [1]; concretely, in this way we show that   1 m ∈ /Q arccos √ π 2 nM √ when n, m, M ∈ N, n ≥ 2, gcd(n, m) = 1 and m/(2 nM ) < 1. Finally, √ the square root of every positive rational number r < 1 can be written as √ √ √ r = m/(2 nM ) with gcd(n, m) = 1, with the exceptions of r = 1, 1/2, 1/ 2 and √ √ 3/2. Thus, we conclude that, for r ∈ Q with 0 ≤ r ≤ 1, the number (1/π) arccos( r) is irrational except in the cases arising from these values of the cosine function: √ √ cos(0) = 1, cos(π/6) = 3/2, cos(π/4) = 1/ 2, (3) cos(π/3) = 1/2, and cos(π/2) = 0. Remark. As an easy consequence, for t ∈ Q, the only possible rational values of cos2 (πt) √ √ are given by cos(πt) = ±1, ± 3/2, ±1/ 2, ±1/2 and 0. Actually, this can also be proved by using that the only rational values of cos(πt) are 0, ±1, and ±1/2, and the relation cos2 (θ) = (1 + cos(2θ))/2; or can be derived from Lehmer’s result by searching algebraic integers of degree at most 2 of cos(πt). But it seems that none of these facts about cos2 have been noticed in the literature. In any case, we are giving a direct and new proof of this result. Of course, using the elementary trigonometric relations cos2 (θ) = 1 − sin2 (θ) and cos2 (θ) = 1/(1 + tan2 (θ)), similar results for the rational values of sin2 (πt) and tan2 (πt) can be obtained.

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321

Thus, let us state Theorem. Let r ∈ Q with 0 ≤ r ≤ 1. Then, the number √  1 arccos r π

√ is rational if and only if r is 0, 1/4, 1/2, 3/4, or 1; and the same holds for (1/π) arcsin( r). Proof. Noticing that arccos(x)+arcsin(x) = π/2, it is enough to analyze the case arccos. Even more, the “if” part is clear from the trigonometric values shown in (3), so let us study the “only if” part. We claim that every r ∈ Q \ {0, 1/4, 2/4, 3/4, 1}, r ≥ 0, can be written as m2 r= 4nM

(4)

with conditions n, m, M ∈ N, n ≥ 2 and gcd(n, m) = 1. This is true because given r = p/q, with p and q co-prime and q not a divisor of 4, there are only two possibilities: • if q has an odd divisor n, say q = nM , we can write r = (2p)2 /(4nM ); • if q is a power of 2, then q = 2j with j ≥ 3, p is odd and we have r = p2 /(4 · 2j−2 p) (with n = 2j−2 ≥ 2). In both cases we have found the decomposition (4). Then, we only need to prove that   1 m A(n, m, M ) = arccos √ ∈ /Q π 2 nM √ when n, m, M ∈ N, n ≥ 2, gcd(n, m) = 1 and m/(2 nM ) < 1. For x ∈ [−1, 1] and k ∈ N ∪ {0}, let Tk (x) be Tk (cos(θ)) = cos(kθ),

x = cos(θ).

It is immediate that T0 (x) = 1 and T1 (x) = x. Moreover, the trigonometric relation cos((k + 1)θ) = 2 cos(θ) cos(kθ) − cos((k − 1)θ) proves the recurrence formula Tk+1 (x) = 2xTk (x) − Tk−1 (x),

k ≥ 1.

In particular, this implies that Tk (x) is a polynomial of degree k, and so we get the so-called Chebyshev polynomials of the first kind. Making the substitution gk (x) = 2Tk (mx/2), we get  g0 (x) = 2, g1 (x) = mx, (5) gk+1 (x) = mxgk (x) − gk−1 (x), k ≥ 1. Then, gk (x) is a polynomial of degree k and coefficients in Z, and it verifies 2 cos(kθ) = 2Tk (cos(θ)) = gk (2 cos(θ)/m).

(6)

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Now, let us take  θ = arccos

m √ 2 nM

 ,

1 x= √ , 2 nM

By (6), we have

 2 cos(kθ) = gk (2x) = gk

1 √ nM



cos(θ) = mx.

Bk = √ ( nM )k

(7)

for some Bk ∈ Z. From (5) it is easy to check that B0 = 2, B1 = m, and Bk+1 = mBk − nM Bk−1 . Now, let us recall that gcd(n, m) = 1. Then, by induction on k (starting with k = 1), it follows that n does not divide Bk for any k ≥ 1. To conclude the proof, let us suppose that A(n, m, M ) = (1/π)θ = h/k ∈ Q. Then, kθ = hπ and, by (7), Bk ±2 = 2 cos(hπ) = 2 cos(kθ) = √ , ( nM )k This implies that n divides Bk , which is a contradiction.

Bk ∈ Z. 

Acknowledgment Thanks to Professors G¨ unter M. Ziegler, Jaime Vinuesa, and the referee, for their interest and their valuable comments, that allowed to improve this paper.

References [1] M. Aigner and G.M. Ziegler: Proofs from THE BOOK, 3rd ed., Springer, 2004. [2] D.H. Lehmer: “A note on trigonometric algebraic numbers”, Amer. Math. Monthly, Vol. 40, (1933), pp. 165–166. [3] I. Niven: Irrational numbers, Carus Monographs, Vol. 11, The Mathematical Association of America (distributed by John Wiley and Sons), 1956. [4] I. Niven, H.S. Zuckerman and H.L. Montgomery: An introduction to the theory of numbers, 5th ed., Wiley, 1991. [5] N.M. Temme: Special functions: an introduction to the classical functions of mathematical physics, John Wiley and Sons, 1996. [6] E. W. Weisstein: “Chebyshev polynomial of the first kind”, In: MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/ ChebyshevPolynomialoftheFirstKind.html.