VOLUMEN 7 NÚMERO 1 ENERO A JUNIO DE 2003 ISSN: 1870-6525
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VOLUMEN 7 NÚMERO 1 ENERO A JUNIO DE 2003 ISSN: 1870-6525
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Contenido The Lagrange approach to constrained Markov control processes: a survey and extension of results Raquiel R. L´ opez-Mart´ınez and On´esimo Hern´ andez-Lerma . . . . . . . . . . . . . . . 1
Representaciones discretas en tiempo-frecuencia y el problema de la selecci´on de frecuencias Alin Andrei Cˆ arsteanu Manit¸iu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Generalized tilings with height functions Olivier Bodini and Matthieu Latapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
On Anosov energy levels that are of contact type Osvaldo Osuna-Castro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Morfismos, Vol. 7, No. 1, 2003, pp. 1–26
The Lagrange approach to constrained Markov control processes: a survey and extension of results ∗ Raquiel R. L´opez–Mart´ınez
1
On´esimo Hern´andez–Lerma
2
Abstract This paper considers constrained Markov control processes in Borel spaces, with unbounded costs. The criterion to be minimized is the expected total discounted cost and the constraints are imposed on similar criteria. Conditions are given for the constrained problem to be equivalent to a convex program. We present a saddle-point theorem for the Lagrange function associated with the convex program, which is used to obtain the existence of an optimal solution to the constrained problem. In addition, we show that there exists an optimal policy for the constrained problem which is also Pareto optimal for a certain multiobjective Markov control processes.
2000 Mathematics Subject Classification: 90C40, 93E20, 90C25. Keywords and phrases: Constrained Markov control processes, convex problems, saddle point, Pareto policies.
1
Introduction
This paper gives a unified, self–contained presentation of constrained Markov control processes (MCPs) in Borel spaces with unbounded costs. The criterion to be minimized is an expected discounted cost and the constraints are imposed on similar discounted cost functionals. The paper has two main objectives. First, it is a survey of several techniques to ∗
Invited Article. Research partially supported by a PROMEP grant. 2 Research partially supported by CONACyT Grant 37355–E. 1
1
2
R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
analyze constrained MCPs, with emphasis on the Lagrange approach. Second, it extends to constrained MCPs in general (i.e. nondenumerable, noncompact) Borel spaces some results on the existence of optimal policies and it also studies the relation between the Lagrange and the Pareto approaches. In particular, we show the existence of an optimal policy which is also Pareto optimal for a certain multiobjective MCP. The constrained problem (CP) we are concerned with is of the following form: given performance criteria V0 , V1 , . . . , Vq and constants k 1 , . . . , kq , Minimize V0 (π) over the set of control policies π that satisfy the constraints Vi (π) ≤ ki
∀ i = 1, . . . , q.
Control problems of this form appear in many areas — see, for instance, [1–6, 8, 10–14, 19–25, 28–34]. The easiest way to analyze CP is using the so–called direct method. In this method, which of course is also applicable to unconstrained MCPs (e.g. [15], §5.7), the idea is to use occupation measures to transform CP into a “static” optimization problem, say CP’; see [13, 14] and §3 below. If one identifies the set of occupation measures with a convex subset of a suitable linear space of (signed) measures, then one can express CP’ in an obvious manner as either a linear program or a convex program. The linear programming formulation has been done for constrained MCPs in finite [8, 21, 22] or countable [1–3, 20] or even Borel [13, 14] spaces. On the other hand, the convex programming approach, which is the one we are interested in this paper, was originally introduced by Beutler and Ross [4] for MCPs with a countable state space and a single constraint, but it has been extended in many directions, for instance, countable state spaces with compact action sets [1, 3, 5, 6, 31, 32] and Borel state spaces [23, 25, 27, 28, 33]. (For the dynamic programming approach, which is not discussed in this paper, see [29].) As already mentioned above, in this paper we are mainly concerned with the convex programming formulation of constrained MCPs with general Borel state space and unbounded costs. We begin in §2 by introducing some basic terminology and notation. In §3 we define the associated discounted occupation measures and state Lemma 3.3, which ensures that we can consider CP as a convex programming problem. In §4 we study the convex problem. In particular, we obtain a saddle-point theorem for the associated Lagrange function,
Constrained Markov control processes
3
which gives an optimal solution for CP. (A similar result for average cost problems appears in [25].) In §5 we establish some connections between the Lagrange approach and the Pareto optimality of a certain multiobjective MCP. Conditions are given under which an optimal policy for CP is Pareto optimal for the multiobjective problem. To illustrate the results in §4 and §5, in §6 we study the so–called stochastic stabilization problem, from [9] and [27]. In particular, we show a saddle point for the Lagrangean associated with this problem. In §7 and §8 we give the proof of Theorems 4.4, 4.5, and 5.4, which require lengthy preliminaries.
2
Constrained MCPs
Constrained MCPs are rather standard and will be introduced only briefly. (If necessary, see for instance [1, 13, 14, 28, 31, 32] for further details.) The constrained Markov control model is of the form (2.1)
(X, A, {A(x) | x ∈ X}, Q, c, d, k),
where X and A are the state space and the control space, respectively. We shall assume that X and A are Borel spaces, endowed with the corresponding Borel σ-algebras B(X), B(A). For each x ∈ X, the nonempty set A(x) in B(A) consists of the feasible controls or actions when the system is in state x ∈ X. We suppose that the set (2.2)
IK := {(x, a) | x ∈ X, a ∈ A(x)}
of feasible state-action pairs is a Borel subset of X × A. Moreover, Q stands for the transition law, and c : IK → IR is a measurable function that denotes the cost-per-stage . Finally, d = (d1 , . . . , dq ) : IK → IRq is a given function and k = (k1 , . . . , kq ) is a given vector in IRq , which are used to define the constrained problem (CP) in (2.5) and (2.6), below. Let Π be the set of all (randomized, history-dependent) admisible control policies. Let Φ be the set of all the stochastic kernels ϕ on A given X such that ϕ(A(x)| x) = 1 for all x ∈ X, and let IF be the family of measurable functions f : X → A for which f (x) ∈ A(x) for all x ∈ X . As usual, we will identify Φ with the family of randomized stationary policies, and IF with the subfamily of deterministic stationary policies. Throughout the following, we consider a fixed discount factor δ ∈ (0,1), and a fixed initial distribution γ0 ∈ IP(X), where IP(X) denotes the set of probability measures on X. Given the functions c and d =
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R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
(d1 , . . . , dq ) as in (2.1), for each policy π ∈ Π, consider the expected δ-discounted cost functions # !∞ " (2.3) V0 (π, γ0 ) := (1 − δ)Eγπ0 δ t c(xt , at ) , t=0
(2.4)
Vi (π, γ0 ) := (1 −
δ)Eγπ0
!
∞ "
t
δ di (xt , at )
t=0
#
for i = 1, . . . , q.
Furthermore, letting k = (k1 , . . . , kq ) be the q-vector in (2.1), define a subset ∆ of Π as (2.5)
∆ := {π | V0 (π, γ0 ) < ∞ and Vi (π, γ0 ) ≤ ki (i = 1, . . . , q)}.
With this notation, we may then define the constrained problem (CP) we are concerned with as follows: (2.6)
CP : Minimize V0 (π, γ0 ) subject to π ∈ ∆.
If there exists a policy π ∗ in ∆ that solves CP, that is, (2.7)
V0 (π ∗ , γ0 ) = inf{V0 (π, γ0 ) | π ∈ ∆} =: V ∗ (γ0 ),
then π ∗ is said to be an optimal policy for CP, and V ∗ (γ0 ) is called the optimal value of CP.
3
CP as a “static” optimization problem
The following conditions are used, in particular, to express CP as an optimization problem on a certain set of occupation measures —see Lemma 3.3. Assumption 3.1 (a) The set IK (defined in (2.2)) is closed. (b) c(x, a) is nonnegative and inf-compact, which means that for each r ∈ IR the set {(x, a) ∈ IK | c(x, a) ≤ r} is compact.
Constrained Markov control processes
5
(c) di (x, a) is nonnegative and lower semicontinuous (l.s.c.) for i = 1, . . . , q. (d) The transition law Q is weakly continuous, that is (denoting by Cb (S) the space of continuous ! bounded functions on a topological spaces S), Q is such that X u(y)Q(dy|· ) belongs to Cb (IK) for each function u in Cb (X). (e) CP is consistent, that is, the set ∆ in (2.5) is nonempty. Observe that Assumption 3.1(b) yields, in particular, that c is l.s.c. Assumptions 3.1(b) and (c) can be replaced with the following: The cost functions c and d1 , . . . , dq are nonnegative and l.s.c., and at least one of them is inf-compact. On the other hand, the “nonnegativity” condition on c and di may be replaced with “boundedness from below”. Occupation measures. For each policy π ∈ Π, we define the occupation measure µπ = µπγ0 as µπ (Γ) := (1 − δ)
(3.1) µπ
∞ " t=0
δ t Pγπ0 [(xt , at ) ∈ Γ] ∀Γ ∈ B(X × A).
is a probability measure (p.m.) on X ×A, which is concentrated Then on IK, that is, µπ (IKc ) = 0, where IKc stands for the complement of IK. Moreover, using the notation # ⟨µ, h⟩ := hdµ, we can write (2.3) and (2.4) as (3.2)
V0 (π, γ0 ) = ⟨µπ , c⟩ and Vi (π, γ0 ) = ⟨µπ , di ⟩ (i = 1, . . . , q),
respectively. We shall denote by IP(IK) the set of p.m.’s on X × A that are concentrated on IK, and by IPOδ (IK) the subset of occupation measures. Further, for a p.m. µ in IP(IK), we denote by µ $ its marginal on X, that is, µ $(B) := µ(B × A) for all B in B(X).
Remark 3.2 (See Remark 6.3.1 and Theorem 6.3.7 in [15].) For each policy π ∈ Π, the occupation measure µπ ∈ IPOδ (IK) satisfies the following: # %π (B) = (1 − δ)γ0 (B) + δ Q(B|x, a)µπ (d(x, a)) ∀B ∈ B(X). (3.3) µ
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R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
Conversely, if µ is a p.m. in IP(IK) that satisfies (3.3), i.e., " (3.4) µ !(B) = (1 − δ)γ0 (B) + δ Q(B|x, a)µ(d(x, a)) ∀B ∈ B(X),
then µ is in IPOδ (IK). In other words, there is a policy π for which µ is the associated occupation measure, that is, µ = µπ . Therefore, IPOδ (IK) = {µ ∈ IP(IK) | µ satisfies (3.4)}.
We define the following subsets of IPOδ (IK): (3.5) IPδ (IK) := {µ ∈ IPOδ (IK)|⟨µ, c⟩ < ∞, and ⟨µ, di ⟩ < ∞, i = 1, . . . q}, and (3.6)
∆δ := {µ ∈ IPδ (IK)| ⟨µ, di ⟩ ≤ ki , i = 1, . . . q}.
With this notation we can then state the following key fact. Lemma 3.3 CP is equivalent to the problem: CP′ :
Minimize ⟨µ, c⟩
subject to : µ ∈ ∆δ .
Proof: The lemma is a consequence of (3.2) and Remark 3.2. !
4
CP as a convex program
In Lemma 3.3 we already transformed CP into the “static” optimization problem CP′ . We next use CP′ to restate CP as a convex program. Let f and G be the functions on IPδ (IK) defined as f (µ) := ⟨µ, c⟩
and
G(µ) := (G1 (µ), . . . , Gq (µ)),
with Gi (µ) := ⟨µ, di ⟩−ki for i = 1, . . . , q. Obviously, f and G are convex functions. It is just as obvious that IPδ (IK) is a convex set. Thus, by Lemma 3.3 we can represent CP as the convex problem (4.1)
Minimize subject to :
f (µ) µ ∈ IPδ (IK) and
G(µ) ≤ θ,
Constrained Markov control processes
7
where θ is the vector zero in IRq , and G(µ) ≤ θ means that Gi (µ) ≤ 0 for all i = 1, . . . , q. Observe that the constraint in (4.1) can also be written as µ ∈ ∆δ . The Lagrangean L : IPδ (IK)×IRq+ → IR associated with problem (4.1) is given by (4.2)
L(µ, α) := f (µ) + G(µ) · α,
where α = (α1 , . . . , αq ) is in IRq+ , and “·” denotes the inner product in IRq . Remark 4.1 (a) ( See, for instance, [9, p. 88,89] or [18, p. 89]). If µ is in IP(IK), then there exists ϕ ∈ Φ such that µ can be “disintegrated” as ! (4.3) µ(B × C) = ϕ(C|x)" µ(dx) ∀ B ∈ B(X), C ∈ B(A), B
where µ " is the marginal of µ on X. In abbreviated form we write (4.3) as µ = µ "· ϕ. (b) If µ = µ "· ϕ is in IPOδ (IK), then it follows from (3.4) that µ is the occupation measure of the policy ϕ ∈ Φ, that is, µ = µϕ .
The following saddle-point result gives conditions for problem (4.1) to have a solution. Theorem 4.2 Suppose that there exists (µ∗ , α∗ ) ∈ IPδ (IK) × IRq+ such that the Lagrangean L has a saddle point at (µ∗ , α∗ ), i.e., (4.4)
L(µ∗ , α) ≤ L(µ∗ , α∗ ) ≤ L(µ, α∗ )
for all (µ, α) in IPδ (IK) × IRq+ . Then (a) µ∗ solves problem (4.1), and #∗ · ϕ∗ of µ∗ satisfies that ϕ∗ is an optimal (b) the disintegration µ∗ = µ policy for CP.
Proof: The proof of part (a) is similar to that of Theorem 2 in [26, p. 221], and, therefore, is omitted. Part (b) follows from (a), the Remark 4.1(b), and the equivalence of CP and problem (4.1). ! In view of Theorem 4.2, to prove that the problem (4.1) is solvable it suffices to show the existence of a saddle point for L. This is true, in particular, if the following condition holds.
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R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
Assumption 4.3 (Slater condition) There exists µ1 ∈ IPδ (IK) such that G(µ1 ) < θ, that is, Gi (µ1 ) < 0 for i = 1, . . . , q. Theorem 4.4 Under Assumptions 3.1 and 4.3 , there exists a saddle point (µ∗ , α∗ ) for the Lagrangean L, and, therefore, CP is solvable. Proof: See §7. ! To summarize, Theorem 4.4 gives the existence of a saddle point (µ∗ , α∗ ) for L, which, by Theorem 4.2 yields an optimal policy ϕ∗ for CP. It turns out that the converse is also true, as shown in the following result. Theorem 4.5 Suppose that Assumptions 3.1 and 4.3 hold. If µ∗ = !∗ · ϕ∗ ∈ ∆δ is such that ϕ∗ is an optimal policy for CP, then the µ Lagrangean L has a saddle point. Proof: See §7. ! Remark 4.6 (See Remark 4.2.5, p. 51 in [7].) In our present context, Assumption 4.3 is equivalent to the so-called Karlin condition (or constraint qualification), according to which there is no nonzero vector α ∈ IRq+ for which G(µ) · α ≥ 0 for all µ ∈ IPδ (IK).
5
The Lagrange approach vs Pareto optimality
In this section we compare the Lagrange approach to CP with the Pareto optimality of a certain multiobjective MCP. With this in mind, we first briefly introduce multiobjective MCPs (for more information see, for instance [17] or [28]). Let V0 (π, γ0 ) and Vi (π, γ0 ) be as in (2.3) and (2.4), and let V (π, γ0 ) ∈ IRq+1 be the cost vector (5.1)
V (π, γ0 ) := (V0 (π, γ0 ), . . . , Vq (π, γ0 )).
The multiobjective control problem we are concerned with is to find a policy π ∗ that “minimizes” V (·, γ0 ) in the sense of Pareto. To state this in precise form, we first simplify the notation by writing V (π, γ0 ) simply as V (π).
Constrained Markov control processes
9
Definition 5.1 Let Γ(Π) ⊂ IRq+1 be the set of cost vectors in (5.1), i.e., Γ(Π) := {V (π) | π ∈ Π}, which is sometimes called the performance set of the multiobjective MCP. Then a policy π ∗ is said to be Pareto optimal (or a Pareto policy) if there is no π ∈ Π such that V (π) ̸= V (π ∗ ) and Vi (π) ≤ Vi (π ∗ ) for all i = 0, . . . , q. The set of cost vectors in Γ(Π) corresponding to Pareto policies is called the Pareto set of Γ(Π), and it is denoted by Par(Γ(Π)). q+1 Let IRq+1 with strictly positive components. ++ be set of vectors in IR q+1 Let β ∈ IR++ , and consider the scalar (or real-valued) cost-per -stage function
(5.2)
C β (x, a) := β0 c(x, a) +
q !
βi di (x, a),
i=1
and the δ-discounted cost V β (π) = V β (π, γ0 ) with # "∞ ! β π t β (5.3) V (π) := (1 − δ)Eγ0 δ C (xt , at ) . t=0
Using (5.1) and (5.2) we may write V β (π) as (5.4)
V β (π) = β · V (π) =
q !
βi Vi (π).
i=0
Let (5.5)
Λ := {β ∈ IRq+1 ++ |
q !
βi = 1}.
i=0
We may then obtain the existence of Pareto policies by the standard “scalarization” approach, as follows. Theorem 5.2 Choose an arbitrary vector β ∈ Λ. If π ∗ ∈ Π is an optimal policy for the scalar criterion (5.3), that is, (5.6)
V β (π ∗ ) ≤ V β (π) ∀ π ∈ Π,
then π ∗ is Pareto optimal.
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R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
For a proof of Theorem 5.2 see, for instance, Theorem 3.2(a) in [17]. In general, the constrained problem CP in (2.6) can have optimal policies that are not Pareto optimal. On the other hand, if CP has a unique optimal policy π ∗ , then it is easily seen (directly from the Definition 5.1) that π ∗ is a Pareto policy. The following two theorems give other cases in which an optimal policy for CP is in fact a Pareto policy. Theorem 5.3 Let (µ∗ , α∗ ) ∈ IPδ (IK) × IRq++ be a saddle point for the !∗ ·ϕ∗ . Then ϕ∗ is Pareto optimal. Lagrangean L, and disintegrate µ∗ as µ Proof: From the definition (4.4) of a saddle point, we have that L(µ∗ , α∗ ) ≤ L(µ, α∗ ) ∀ µ ∈ IPδ (IK).
(5.7)
On the other hand, from (3.2) and the definition (4.2) of L it follows that L(µ, α) = V0 (π) +
(5.8)
q " i=1
αi (Vi (π) − ki ),
where π is a policy associated to the occupation measure µ. Hence, from (5.7) and (5.8) we have that V0 (ϕ∗ ) +
q " i=1
αi∗ (Vi (ϕ∗ ) − ki ) ≤ V0 (π) +
q " i=1
αi∗ (Vi (π) − ki )
∀ π ∈ Π.
Equivalently, defining β∗ := (1, α∗ ) ∈ IRq+1 ++ , we have
β∗ · V (ϕ∗ ) − α∗ · k ≤ β∗ · V (π) − α∗ · k ∀ π ∈ Π, and so
β∗ · V (ϕ∗ ) ≤ β∗ · V (π) ∀ π ∈ Π. # Finally, let P = 1 + qi=1 αi∗ . Then, multiplying both sides of (5.9) by 1/P , it follows from Theorem 5.2 that ϕ∗ is Pareto optimal. ! (5.9)
Now consider the following subset of Γ(Π) (5.10)
Γ∗ (Π) := {V (π) | π an optimal policy for CP}.
Let Par(Γ∗ (Π)) be the Pareto set of Γ∗ (Π).
Constrained Markov control processes
11
Theorem 5.4 Under Assumption 3.1, the Pareto set Par (Γ∗ (Π)) of Γ∗ (Π) is nonempty. Proof: See §8. ! It turns out that the nonemptiness of Par (Γ∗ (Π)) in Theorem 5.4 ensures the existence of a Pareto policy that is optimal for CP. Theorem 5.5 Under Assumptions 3.1 and 4.3, there exists an optimal policy π ∗ for CP, which is also Pareto optimal. Proof: From Theorem 5.4 there exists a policy π ∗ such that V (π ∗ ) is in Par(Γ∗ (Π)). By (5.10), π ∗ is an optimal policy for CP. We now claim that π ∗ is Pareto optimal, that is, V (π ∗ ) is in Par(Γ(Π)). Indeed, if π ∗ is not Pareto optimal, then there exists a policy π1 ∈ Π such that V (π1 ) ̸= V (π ∗ ) and Vi (π1 ) ≤ Vi (π ∗ ) for i = 0, . . . , q. Hence, V (π1 ) ∈ Γ∗ (Π), which contradicts our assumption on π ∗ . Therefore, π ∗ is Pareto optimal. !
6
Example
To illustrate the results in Sections 4 and 5, we next consider the following problem, which is similar to the stochastic stabilization problem in [9, 27]. First, we show that Assumptions 3.1 and 4.3 hold. Then, we prove that this problem is solvable using the Lagrange approach, that is, we shall obtain a saddle point for the Lagrange function. Finally, we construct the corresponding Pareto set. For notational ease, we shall write the δ-discounted costs in (2.3) and (2.4) without the factor (1−δ). Consider the scalar linear system (6.1)
xt+1 = xt − at + ξt
for t = 0, 1, . . . ,
with state and control spaces X = A = IR. The disturbances ξt are i.i.d. random variables, independent of the initial state x0 , and such that (6.2)
E(ξ0 ) = 0
and
E(ξ02 ) =: σ 2 < ∞.
Let c(x, a) and d(x, a) be the quadratic costs defined as (6.3)
c(x, a) = x2 + a2 , d(x, a) = (x − a)2 ,
12
R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
and consider the following constrained problem in which k is a given positive constant. # !∞ " δ t (x2t + a2t ) Minimize V0 (π, γ0 ) := Eγπ0 subject to :
V1 (π, γ0 ) :=
Eγπ0
!
t=0 ∞ " t=0
t
δ (xt − at )
2
#
≤ k.
It is clear that the Assumptions 3.1(a), (b), (c) are satisfied in this example. Moreover, by the continuity of the right-hand side of (6.1) with respect to xt and at for every ξt it follows that also Assumption 3.1(d) holds. On the other hand, if we take π = f0 ∈ IF as the “identity”policy f0 (x) := x for all x ∈ X, we see that V1 (f0 , x) = 0, and, therefore, Assumptions 3.1(e) and 4.3 are both satisfied. Summarizing, Assumptions 3.1 and 4.3 hold for this problem. Now, from (3.2) and (4.2) the corresponding Lagrange function is L(π, α) = V0 (π, γ0 ) + (V1 (π, γ0 ) − k) · α
(6.4)
with α ≥ 0. Let L1 (α) := inf L(π, α).
(6.5)
π∈Π
Note that defining the new cost per-stage function C α (x, a) := c(x, a) + α · d(x, a) = x2 + a2 + α(x − a)2 and denoting by V α (π, γ0 ) the corresponding δ-discounted cost, we may express (6.4) as L(π, α) = V α (π, γ0 ) − α · k. Therefore, finding a policy that attains the minimun in (6.5) becomes a linear-quadratic problem; see, for instance, p. 162 in [9], p. 70 in [15], or p. 253 in [28]. From any of these references we have inf V α (π, x) − k · α = z(α)v(x) − k · α
π∈Π
∀ x ∈ X,
with v(x) := x2 + (1 − δ)−1 δσ 2 , and z(α) is the maximal solution of the quadratic equation (6.6)
δz 2 + (1 + α − 2δ)z − 1 − 2α = 0.
Constrained Markov control processes
13
Therefore, assuming that the initial distribution γ0 satisfies that ! (6.7) γ¯0 := v(x)γ0 (dx) < ∞, we can express (6.5) as L1 (α) = z(α)γ¯0 − k · α.
(6.8)
Moreover, the deterministic stationary policy fα ∈ IF given by (6.9)
fα (x) =
α + δz(α) x 1 + α + δz(α)
is optimal for V α (π, x) for all x ∈ IR, and so we also have L1 (α) = L(fα , α) for each α ≥ 0. Now, to obtain a saddle point for the Lagrangean in (6.4) we first prove the following, which can be seen as an “explicit” form of Lemma 7.2, below. Proposition 6.1 If the constraint constant k satisfies the inequality (6.10)
0 < k < K, √ √ where K := γ¯0 (1 + 2δ − 1 + 4δ 2 )/2δ 1 + 4δ 2 , then there exists a unique α∗ > 0 such that L1 (α∗ ) = max L1 (α). α≥0
Proof: We differentiate the function L1 in (6.8) with respect to α, to get L′1 (α) = z ′ (α)γ¯0 − k. Let us now show that L′1 (α) = 0 has a unique positive solution. With this in mind, first note that the positive solution of (6.6) is " −(1 + α − 2δ) + (1 + α − 2δ)2 + 4δ(1 + 2α) z(α) = . 2δ Hence 1 + α + 2δ 1 , z ′ (α) = − + " 2δ 2δ (1 + α − 2δ)2 + 4δ(1 + 2α)
and so (6.11)
L′1 (α)
#
$ 1 1 + α + 2δ = − + " γ¯0 − k. 2δ 2δ (1 + α − 2δ)2 + 4δ(1 + 2α)
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R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
According to (6.10) and (6.11) we have √ γ¯0 (1 + 2δ − 1 + 4δ 2 ) ′ √ − k > 0. L1 (0) = 2δ 1 + 4δ 2 On the other hand, lim L′1 (α) = −k < 0.
α→∞
Hence the equation L′1 (α) = 0 has a positive solution. Moreover, from (6.11), L′1 (α) = 0 becomes (1 + α + 2δ)2 = 4δ 2 (k(γ¯0 )−1 + (2δ)−1 )2 ((1 + α − 2δ)2 + 4δ(1 + 2α)). As this equation is quadratic in α, it has a unique positive solution. ! Let α∗ be as in Proposition 6.1 and define z ∗ = z(α∗ ) and f ∗ := fα∗ as in (6.9), that is, f ∗ (x) := fα∗ (x) = (α∗ + δz ∗ )(1 + α∗ + δz ∗ )−1 x. Then (f ∗ , α∗ ) is a saddle point for L, and, therefore, from Theorem 4.2 it follows f ∗ is an optimal policy for CP. Moreover, as α∗ is positive, from Theorem 5.3 we have that f ∗ is Pareto optimal. Remark 6.2 If α = 0, then f0∗ (x) = δz0 x(1 + δz0 )−1 is optimal for V0 , that is, V0 (f0∗ , γ0 ) = inf V0 (π, γ0 ) π∈Π
where z0 is the positive solution of the quadratic equation (6.12)
δz 2 + (1 − 2δ)z − 1 = 0.
On the other hand, we can see that the “identity” policy f0 (x) = x is optimal for V1 , and obviously, V1 (f0 , γ0 ) = 0, that is, inf V1 (π, γ0 ) = 0.
π∈Π
Proposition 6.3 Let f! be a constant, and f ∈ IF a stationary policy given by f (x) := f!x for all x ∈ X. Let θ := 1 − f!. If |θ| < 1, then 2
(6.13)
V0 (f, γ0 ) =
1 + f! γ¯0 , 1 − δθ 2
(6.14)
V1 (f, γ0 ) =
(1 − f! )2 γ¯0 . 1 − δθ 2
Constrained Markov control processes
In particular, for K and f0∗ as in (6.10) and Remark 6.2, V1 (f0∗ , γ0 ) = K.
(6.15)
Proof: Replacing at in (6.1) with at = f (xt ) = f!xt , we obtain xt = (1 − f!)xt−1 + ξt−1 = θxt−1 + ξt−1
∀ t = 1, 2, . . . .
Hence, for all t = 1, 2, . . .
xt = θt x0 +
t−1 "
θj ξt−1−j ,
j=0
and so Exf (x2t ) = θ2t x2 + This yields that (6.16)
Exf
#
∞ "
δ t x2t
$
Eγf0
#
t=0
=
1 1 − δθ 2
σ 2 (1 − θ2t ) . 1 − θ2 & % v(x) σ2δ = . x2 + 1−δ 1 − δθ 2
Hence, from (6.7), (6.17)
∞ "
δ t x2t
t=0
$
=
γ¯0 . 1 − δθ 2
Now note that using a = f (x) = f!x in (6.3) we get (6.18)
2
c(x, a) = (1 + ! f )x2
and
d(x, a) = (1 − f!)2 x2
for all x. Thus, inserting (6.17) and (6.18) in V0 and V1 we obtain (6.13) and (6.14). Finally, from (6.14) and Remark 6.2 we have (6.19)
V1 (f0∗ , γ0 ) =
γ¯0 . (1 + δz0 )2 − δ
On the other hand, from (6.12) we get √ 2δ − 1 + 1 + 4δ 2 (6.20) z0 = . 2δ Hence, substituting (6.20) in (6.19) we obtain (6.15). !
15
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R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
Remark 6.4 Suppose that instead of (6.10) we have k ≥ K, and let f0∗ (x) = δz0 x(1+δz0 )−1 be the optimal policy for V0 (see Remark 6.2). Then, from (6.15) it follows that V1 (f0∗ , γ0 ) = K ≤ k and, therefore, f0∗ is an optimal policy for the constrained problem. Moreover, f0∗ is the unique optimal policy for CP, and so it is Pareto optimal, that is, (V0 (f0∗ , γ0 ), V1 (f0∗ , γ0 )) belongs to the Pareto set. (See Figure 6.1.) The Pareto set. We next construct the Pareto set in an explicit form. As seen above, f ∗ is an optimal policy for CP which is also Pareto optimal, that is, (V0 (f ∗ , γ0 ), V1 (f ∗ , γ0 )) is in the Pareto set. When the constraint constant k varies in the interval (0, K), with K as in (6.10), then (V0 (f ∗ , γ0 ), V1 (f ∗ , γ0 )) describes the Pareto set. Obviously, V0 (f ∗ , γ0 ) is the optimal value for the constrained problem, that is, V0 (f ∗ , γ0 ) = V ∗ (γ0 ). Now, we wish to find the value of V1 (f ∗ , γ0 ). Proposition 6.5 For each k as in (6.10), !∞ # " ∗ δ t (xt − at )2 = k V1 (f ∗ , γ0 ) := Eγf0 t=0
and so (V0 (f ∗ , γ0 ), V1 (f ∗ , γ0 )) = (V ∗ (γ0 ), k) belongs to the Pareto set. Proof: Since (f ∗ , α∗ ) is a saddle point and f ∗ is an optimal policy for CP we have V ∗ (γ0 ) ≤ L(f ∗ , α∗ ) = V ∗ (γ0 ) + (V1 (f ∗ , γ0 ) − k)α∗ . On the other hand, as (V1 (f ∗ , γ0 ) − k)α∗ ≤ 0, it follows that V ∗ (γ0 ) + (V1 (f ∗ , γ0 ) − k)α∗ ≤ V ∗ (γ0 ) and so we have (V1 (f ∗ , γ0 ) − k)α∗ = 0. This equality together with Proposition 6.1 yields that V1 (f ∗ , γ0 ) = k. ! Proposition 6.5 ensures that (V ∗ (γ0 ), k) belongs to the Pareto set when k varies in (0, K). Furthermore, if α∗ is as in Proposition 6.1,
Constrained Markov control processes
17
it is clear then that V ∗ (γ0 ) = L1 (α∗ ). Now, in connection with the Figure 6.1, let us fix w = k and calculate y = L1 (α∗ ). First, we note the following facts. Proposition 6.3 yields that (6.21)
V1 (f ∗ , γ0 ) =
(1 +
α∗
γ¯0 . + δz ∗ )2 − δ
Further, from (6.6) with α = α∗ we have (6.22)
α∗ =
δ(z ∗ )2 + (1 − 2δ)z ∗ − 1 2 − z∗
and subtituting this value of α∗ in (6.21) it follows that (6.23)
V1 (f ∗ , γ0 ) =
(2 − z ∗ )2 γ¯0 . 1 − δ(2 − z ∗ )2
Hence, from Proposition 6.5 we get (2 − z ∗ )2 γ¯0 = k. 1 − δ(2 − z ∗ )2
(6.24)
Now, substituting 6.22 in L(α∗ ) we obtain δ(z ∗ )2 + (1 − 2δ)z ∗ − 1 k 2 − z∗ −(z ∗ )2 (γ¯0 + δk) + 2z ∗ (γ¯0 + δk) − kz ∗ + k . = 2 − z∗
y = z ∗ γ¯0 − (6.25)
From (6.24) we have that (6.26)
−(z ∗ )2 (γ¯0 + δk) = 4(γ¯0 + δk)(1 − z ∗ ) − k.
The latter equality together with (6.25) gives (6.27)
y = 2(γ¯0 + δk) −
Now let
z∗ k. 2 − z∗
z∗ . 2 − z∗ Solving this equation for z ∗ and substituting the solution in (6.24) we get 4γ¯0 = k, (1 + s)2 − 4δ s :=
18
R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
which yields
(6.28)
w = k = γ¯0
4 (s + 1)2 − 4δ
and so, from (6.25), y = 2(γ¯0 + kδ) −
(6.29)
4δ γ¯0 . (s + 1)2 − 4δ
In (6.28) and (6.29) s is the parameter which varies as k is in (0, K). The graph of (6.28)-(4.29) is the Pareto set, which is represented in Figure 6.1.
w
V1 ( f0 , γ0 ) k Γ ( Π)
0
Par (Γ(Π)) V0 ( f0*, γ 0 ) V(* γ 0) Figure 6.1
2γ 0
y
Constrained Markov control processes
7
19
Proof of Theorems 4.4 and 4.5
The proof of Theorems 4.4 and 4.5 is based on the following preliminary facts. Consider the functions (7.1)
(7.2)
L1 (α) :=
inf
µ∈IPδ (IK)
L(µ, α),
L2 (µ) := sup L(µ, α), α≥θ
and let V ∗ (γ0 ) be as in (2.7). Note that, by Lemma 3.3, V ∗ (γ0 ) = inf{⟨µ, c⟩ | µ ∈ ∆δ }. Remark 7.1 As ∆δ ⊂ IPδ (IK), for each α ∈ IRq+ we have L1 (α) ≤ inf L(µ, α) ≤ inf ⟨µ, c⟩ = V ∗ (γ0 ), µ∈∆δ
µ∈∆δ
that is, L1 (α) ≤ V ∗ (γ0 ) for all α ∈ IRq+ . Similarly, V ∗ (γ0 ) ≤ L2 (µ) for all µ ∈ ∆δ . Hence (7.3)
sup L1 (α) ≤ V ∗ (γ0 ) ≤ inf L2 (µ). µ∈IPδ (IK) α≥θ
The following lemmas show that equality holds throughout (7.3). Lemma 7.2 Under Assumptions 3.1 and 4.3, there exists α∗ in IRq+ such that L1 (α∗ ) = sup L1 (α) = V ∗ (γ0 ). α≥θ Proof: In the space IR × IRq define the sets B1 := {(r, α)| r ≥ f (µ), α ≥ G(µ) for some µ ∈ IPδ (IK) }, B2 := {(r, α)| r ≤ V ∗ (γ0 ), α ≤ θ }.
The set B2 is obviously convex, and so is B1 because f and G are convex. By definition of V ∗ (γ0 ), the set B1 contains no interior points of B2 . On the other hand, it is clear that B2 contains an interior point. Thus, by the Separating Hyperplane Theorem (see, for example, [26], p. 133, Theorem 3), there is a vector (r∗ , α∗ ) ∈ IR × IRq such that r ∗ r1 + α 1 · α ∗ ≥ r ∗ r2 + α 2 · α ∗
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R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
for all (r1 , α1 ) ∈ B1 and all (r2 , α2 ) ∈ B2 . By the definition of B2 it follows that r∗ ≥ 0, α∗ ≥ θ. We next show that in fact r∗ > 0. Indeed, as the vector (V ∗ (γ0 ), θ) is in B2 , we have r∗ r + α · α∗ ≥ r∗ V ∗ (γ0 )
(7.4)
for all (r, α) ∈ B1 . Thus, if r∗ = 0, then α · α∗ ≥ 0 for all α ∈ IRq such that (r, α) ∈ B1 . In particular, taking α = G(µ1 ) with µ1 as in Assumption 4.3, we obtain G(µ1 )· α∗ ≥ 0, which implies that Gi (µ1 ) ≥ 0 for some i = 1, . . . , q. As this contradicts Assumption 4.3, it follows that r∗ > 0 and, without loss of generality, we may assume r∗ = 1. Now, since the point (V ∗ (γ0 ), θ) is in the closure of both B1 and B2 , we have (with r∗ = 1 in (7.4)) V ∗ (γ0 ) = =
inf
[r + α · α∗ ] ≤
inf
L(µ, α∗ ) ≤ inf f (u) = V ∗ (γ0 ).
(r,α)∈B1 µ∈IPδ (IK)
inf
µ∈IPδ (IK)
[f (µ) + G(µ) · α∗ ]
µ∈∆δ
Hence, recalling (7.3) the lemma is proved. ! By (7.1) and (7.2) the following lemma is a “minimax” result. Lemma 7.3 Under Assumptions 3.1 and 4.3, we have max L1 (α) = inf L2 (µ) = V ∗ (γ0 ). α≥θ µ∈IPδ (IK)
(7.5)
Proof: Since G(µ) · α ≤ θ for all µ ∈ ∆δ and α ≥ 0, we see that L2 (µ) = sup L(µ, α) = ⟨µ, c⟩ for all µ ∈ ∆δ . α≥θ Hence inf L2 (µ) = V ∗ (γ0 ).
µ∈∆δ
It follows that inf
µ∈IPδ (IK)
L2 (µ) ≤ V ∗ (γ0 ),
and so, by ( 7.3) and Lemma 7.2, the equality (7.5) holds. ! Lemma 7.4 Under Assumption 3.1, there exists a p.m. µ∗ in IPδ (IK) such that L2 (µ∗ ) = inf L2 (µ) = V ∗ (γ0 ). µ∈IPδ (IK)
Constrained Markov control processes
21
Proof: If µ is in IPδ (IK) but not in ∆δ , then there exists i0 in {1, . . . , q} such that Gi0 (µ) > 0, which implies that L2 (µ) = +∞. Therefore, (7.6)
inf
µ∈IPδ (IK)
L2 (µ) = inf L2 (µ) = V ∗ (γ0 ). µ∈∆δ
On the other hand, for all µ ∈ ∆δ and α ≥ θ, we have G(µ) · α ≤ 0, and so it follows that L2 (µ) = sup L(µ, α) = ⟨µ, c⟩ ∀ µ ∈ ∆δ . α≥θ
(7.7)
Therefore, from the (7.6), (7.7), together with Lemma 3.3 and Theorem 3.2 in [13], the desired conclusion follows. ! We are now ready for the proof of Theorems 4.4 and 4.5. Proof of Theorem 4.4. Let α∗ and µ∗ be as in Lemma 7.2 and Lemma 7.4, respectively. From lemma 7.3 we have that L(µ∗ , α∗ ) = V ∗ (γ0 ). Now, by the latter equality together with Lemmas 7.2, 7.4 and the definition of L1 and L2 it follows that L(µ∗ , α∗ ) = L1 (α∗ ) ≤ L(µ, α∗ ) for all µ ∈ IPδ (IK), and, similarly, L(µ∗ , α∗ ) = L2 (µ∗ ) ≥ L(µ∗ , α) for all α ≥ θ. Therefore, the pair (µ∗ , α∗ ) is a saddle point. ! Proof of Theorem 4.5. Let α∗ be as in Lemma 7.2. As G(µ∗ ) ≤ 0 and f (µ∗ ) = V ∗ (γ0 ), it follows that L(µ∗ , α∗ ) ≤ L(µ, α∗ ) for all µ ∈ IPδ (IK), which gives the second inequality in (4.4). On other hand, since V ∗ (γ0 ) ≤ f (µ∗ ) + G(µ∗ ) · α∗ ≤ f (µ∗ ) = V ∗ (γ0 ), we have G(µ∗ ) · α∗ = 0. Therefore, L(µ∗ , α) − L(µ∗ , α∗ ) = G(µ∗ ) · α − G(µ∗ ) · α∗ = G(µ∗ ) · α ≤ 0, and the first inequality in (4.4) follows. !
22
R. R. L´opez–Mart´ınez and O. Hern´andez–Lerma
8
Proof of Theorem 5.4
For completeness, we first state some well-known results that are needed to prove Theorem 5.4. Lemma 8.1 Let Y be a metric space and M a family of probability measures on Y. If there exists a nonnegative and inf-compact function v on Y such that sup{⟨µ, v⟩, µ ∈ M } < ∞, then M is relatively compact, that is, for each sequence {µn } in M there is a probability measure µ on Y and a subsequence {µm } of {µn } such that µm converges weakly to µ in the sense that (8.1)
⟨µm , v⟩ → ⟨µ, v⟩
∀v ∈ Cb (Y ).
To prove Lemma 8.1, one first shows that the hypothesis implies that M is tight, and then the relative compactness of M follows from Prohorov’s Theorem (see [16]). Lemma 8.2 Let Y a metric space, and v : Y → IR lower semicontinuous and bounded below. If {µn } and µ are probability measures on Y and µn converges weakly to µ (that is, as in (8.1)), then lim inf ⟨µn , v⟩ ≥ ⟨µ, v⟩. n→∞
Lemma 8.2 is well known (and easy to prove): see, for instance, statement (12.3.37) in [16, p. 243] Lemma 8.3 The set IPδ (IK) is closed with respect to the topology of weak convergence. For a proof of Lemma 8.3 see Lemma 5.5 in [17], for instance. Let ∆′δ := {µ ∈ ∆δ | µ is an optimal solution for (4.1)}. Lemma 8.4 Let V β (π) and Γ∗ (Π) be as in (5.3) and (5.10), respectively. Let Π∗ be set of policies π such that V (π) in Γ∗ (Π). Then there exists a policy π ∗ such that (8.2)
V β (π ∗ ) = min∗ V β (π). π∈Π
Constrained Markov control processes
23
Proof: It is clear that minimizing V β (·) on Γ∗ (Π) is equivalent to minimizing ⟨·, C β ⟩ on ∆′δ , with C β as in (5.2). Let ρ∗ := inf{⟨µ, C β ⟩ | µ ∈ ∆′δ } and take a sequence {µn } in ∆′δ such that ⟨µn , C β ⟩ ↓ ρ∗ . Therefore, given ϵ > 0, there exists an integer N such that ρ∗ ≤ ⟨µn , C β ⟩ ≤ ρ∗ + ϵ
(8.3)
∀n ≥ N.
On the other hand, by definition of ∆′δ , it follows that ⟨µn , c⟩ = V ∗ (γ0 ) for all
(8.4)
n≥0
with V ∗ (γ0 ) as in (2.7), which implies that sup⟨µn , c⟩ = V ∗ (γ0 ). n
Since c is inf-compact (Assumption 3.1(b)), from Lemma 8.1 it follows that {µn } is relatively compact, that is, there exists a probability measure µ∗ on IK and a subsequence {µm } of {µn } that converges weakly to µ∗ . The latter convergence, together with (8.3) and Lemma 8.2, yields that ⟨µ∗ , C β ⟩ = ρ∗ . Finally, from Lemma 8.3 we conclude that µ∗ is !∗ · ϕ∗ of µ∗ is such indeed a p.m. in ∆′δ , and so the disintegration µ∗ = µ ∗ ∗ that π := ϕ satisfies (8.2). ! Proof of Theorem 5.4 From Lemma 8.4 and Theorem 5.2, it follows that Par(Γ∗ (Π)) ̸= ∅. ! Raquiel R. L´ opez–Mart´ınez Facultad de Matem´ aticas Universidad Veracruzana A. P. 270 Xalapa, Ver., 91090 M´exico ralopez@uv.mx
On´esimo Hern´ andez–Lerma Departamento de Matem´ aticas CINVESTAV–IPN A. P. 14–740 M´exico D.F., 07000 M´exico ohernand@math.cinvestav.mx
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[2] Altman E., Constrained Markov decision processes with total cost criteria: occupation measures and primal LP, Math. Meth. Oper. Res., 43 (1996), 45-72. [3] Altman E., Constrained Markov decision processes with total cost criteria: Lagrange approach and dual LP, Math. Meth. Oper. Res., 48 (1998), 387-417. [4] Beutler F. J.; Ross K. W., Optimal policies for controlled Markov chains with a constraint, J. Math. Anal. Appl., 112 (1983), 236252. [5] Borkar V.S., A convex analytic approach to Markov decision processes, Prob. Theory Related Fields, 78 (1988), 583-602. [6] Borkar V.S., Ergodic control of Markov chains with constraints— the general case, SIAM J. Control Optim., 32 (1994), 176-186. [7] Craven B. D., Mathematical Programming and Control Theory, Chapman and Hall, London, 1978. [8] Derman B. D.; Veinott A. F. Jr., Constrained Markov decision chains, Management Science, 19 (1972), 389-390. [9] Dynkin E.B.; Yushkevich A. A., Controlled Markov Processes, Springer-Verlag, Berlin, 1979. [10] Feinberg E.; Shwartz A., Constrained discounted dynamic programming, Math. Oper. Res., 21 (1996), 922-945. [11] Feinberg E.; Shwartz A., Constrained dynamic programming with two discount factors: applications and an algorithm, IEEE Trans. Autom. Control, 44 (1999), 628-631. [12] Golabi K.; Kulkarni R.B.; Way G.B., A statewide pavement management system, Interfaces, 12 (1982), 5-21. [13] Hern´andez–Lerma O.; Gonz´alez-Hern´andez J., Constrained Markov control processes in Borel spaces: the discounted case, Math. Meth. Oper. Res., 52 (2000), 271-285. [14] Hern´andez–Lerma O.; Gonz´alez-Hern´andez J.; L´opez-Mart´ınez R.R., Constrained average cost Markov control processes in Borel spaces, SIAM J. Control Optim., (to appear).
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[15] Hern´andez–Lerma O.; Lasserre J.B., Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer-Verlag, New York, 1996. [16] Hern´andez–Lerma O.; Lasserre J.B., Further Topics on DiscreteTime Markov Control Processes, Springer-Verlag, New York, 1999. [17] Hern´andez–Lerma O.; Romera R. , Multiobjective Markov control processes: a linear programing approach, preprint, Departamento de Matem´aticas, CINVESTAV-IPN, M´exico, 2003. (Submitted). [18] Hinderer K., Foundations of Non-stationary Dynamic Programming with Discrete-Time Parameter, Lecture Notes Oper. Res. Math. Syst. 33, Springer-Verlag, Berlin, 1970. [19] Hordijk A.; Spieksma F., Constrained admission control to a queueing system, Adv. Appl. Prob., 21 (1989), 409-431. [20] Huang Y.; Kurano M., The LP approach in average rewards MDPs with multiple cost constraints: The countable state case, J. Inform. Optim. Sci., 18 (1997), 33-47. [21] Kallenberg L. C. M., Linear Programming and Finite Markovian Control Problems, Mathematical Centre Tracts 148, Amsterdam, 1983. [22] Kallenberg L. C. M., Survey of linear programming for standard and nonstandard Markovian control problems, Part I: Theory, ZOR–Math. Methods Oper. Res., 40 (1994), 1-42. [23] Kurano M., Nakagami J.; Y. Huang, Constrained Markov decision processes with compact state and action spaces: the average case, Optimization, 48 (2000), 255-269. [24] A. Lazar, Optimal flow control of a class of queueing networks in equilibrium, IEEE Trans. Autom. Control, 28 (1983), 1001-1007. [25] L´opez–Mart´ınez R.R., A saddle–point theorem for constrained Markov control processes, Morfismos, 3 (1999), 69–79. (Available at http://chucha.math.cinvestav.mx/morfismos/ v3n2/index.html)
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[26] Luenberger D. G., Optimization by Vector Space Methods, Wiley, New York, 1969. [27] Mao X.; Piunovskiy A.B., Strategic measure in optimal control problems for stochastic sequences, Stoch. Anal. Appl., 18 (2000), 755-776. [28] Piunovskiy A.B., Optimal Control of Random Sequences in Problems with Constraints, Kluwer, Boston, 1997. [29] Piunovskiy A.B.; Mao X., Constrained Markovian decision processes: the dynamic programming approach, Oper. Res. Letters, 27 (2000), 119-126. [30] Ross K.; Varadarajan R., Multichain Markov decision processes with a sample path constraint: a decomposition approach, Math. Oper. Res., 16 (1991), 195-207. [31] Sennott L.I., Constrained discounted Markov decision chains, Prob. Eng. Inform. Sci., 5 (1991), 463-475. [32] Sennott L.I., Constrained average cost Markov decision chains, Prob. Eng. Inform. Sci., 7 (1993), 69-83. [33] Tanaka K., On discounted dynamic programming with constraints, J. Math. Anal. Appl., 155 (1991), 264-277. [34] Vakil F.; Lazar A. A., Flow control protocols for integrated networks with partially observed voice traffic, IEEE Trans. Autom. Control, 32 (1987), 2-14.
Morfismos, Vol. 7, No. 1, 2003, pp. 27–45
Representaciones discretas en tiempo–frecuencia y el problema de la selecci´on de frecuencias ∗ Alin Andrei Cˆarsteanu Mani¸tiu
Resumen El art´ıculo presenta la problem´atica general de la obtenci´on de bases en Rn con significado de frecuencia. Tambi´en se presenta el efecto sobre la caracter´ıstica de contraste de una base, de la minimizaci´on de una funci´on objetivo aditiva (tal como lo ser´ıa una funci´on de costo), definida sobre los elementos de aquella base. Se discuten varios algoritmos de optimizaci´on, basados en el paquete de ond´ıculas. Finalmente, se propone una soluci´on novedosa a lo que es el problema del c´alculo de un espectro energ´etico significativo de la representacio ´n de una sen ˜al en una base obtenida a partir del paquete de ond´ıculas.
2000 Mathematics Subject Classification: 42C40, 65T60. Keywords and phrases: ond´ıcula, base ortonormal, representaci´ on tiempo–frecuencia, an´ alisis arm´ onico, minimizaci´ on discreta, contraste.
1
Introducci´ on
El an´alisis en tiempo–frecuencia se dedica a la representacio´n de sen ˜ales, con soporte contiguo o discreto, en t´erminos de un diccionario de formas de onda u ond´ıculas. Otros t´erminos usados en espan ˜ol son “onduleta” y “ondeleta”, imitaciones onomatop´eyicas del franc´es “ondelette”. Preferimos “ond´ıcula”, con el cual se guarda el sentido original de la palabra. Las “sen ˜ales” pueden ser desde funciones en el sentido cl´asico hasta distribuciones y procesos estoc´asticos. Tambi´en llamado an´alisis ∗
Art´ıculo invitado.
27
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Alin Andrei Cˆarsteanu Manit¸iu
arm´onico (por razones hist´oricas, dado que las primeras funciones analizadoras fueron funciones arm´onicas, siendo ´este el caso de las transformadas de Fourier), el an´alisis en tiempo–frecuencia tiene sus ra´ıces en la relaci´on complementaria que guardan el tiempo y la frecuencia. Dicha relaci´on tambi´en llama la atenci´on a la localizaci´on en tiempo– frecuencia de los elementos de la representaci´on, aspecto que se discutir´a a continuaci´on, as´ı como a la posibilidad de obtener algoritmos para el uso de funciones analizadoras de formas arbitrarias (en vista que en la transformada de Fourier, el algoritmo hace uso de las propiedades de la exponencial compleja, de las cuales resulta la simetr´ıa de la transformada inversa con respecto a la directa). As´ı, dado el significado de amplitudes correspondientes a diferentes frecuencias, que adquieren los coeficientes de las representaciones en tiempo–frecuencia, varias formas de las funciones analizadoras tienen un inter´es particular, de acuerdo a su interpretaci´on f´ısica, habi´endose compilado hasta diccionarios de funciones analizadoras. Tambi´en, el an´alisis en tiempo–frecuencia se extiende de manera natural a soportes multidimensionales, que pueden ser asociados, en aplicaciones, con espacios f´ısicos bidimensionales o tridimensionales, espacio–tiempos u otras entidades. El presente art´ıculo se limitar´a, de acuerdo a su t´ıtulo, al soporte unidimensional discreto. Consideremos una funci´on f definida sobre un soporte discreto, formado por n puntos equidistantes: f = [f (τ ) , f (2τ ) , f (3τ ) , . . . , f (nτ )], donde n es una potencia de 2. En las aplicaciones, tal f resulta generalmente de un muestreo. El prop´osito declarado del an´alisis arm´onico es de encontrar una descomposici´on de f de la forma (1)
f=
n !
c k wk ,
k=1
donde las wk = [wk (τ ) , wk (2τ ) , . . . , wk (nτ )] son n ond´ıculas (funciones analizadoras) de una familia y los ck son los coeficientes (o amplitudes) correspondientes a cada ond´ıcula. Si las ond´ıculas se escogen de tal manera que los coeficientes ck sean u ´nicamente determinados, entonces la familia de ond´ıculas constituye una base en Rn (f es un vector real ndimensional). Las ond´ıculas se llaman “´atomos” en tiempo frecuencia, en el sentido de que unas son versiones re–escalonadas y/o trasladadas de las otras, ocupando por convenci´on distintos rect´angulos, de ´areas iguales, en el plano tiempo–frecuencia: " # 1 t − bk wk (t) = √ W (2) , ak ak
Representaciones discretas en tiempo–frecuencia
29
donde W es la ond´ıcula–base, los ak son factores de escala y los bk son factores de posici´on en su argumento. En el presente art´ıculo se usar´an ond´ıculas de Haar-Walsh [3, 9], por sus cualidades de localizaci´on, presentadas en la siguiente secci´on. A continuaci´on se presentar´a el algoritmo de paquetes de ond´ıculas, que permite construir varias bases de ond´ıculas a partir de una ond´ıcula de origen. Finalmente, se mostrar´an los algoritmos que permiten escoger una base ´optima con respecto a una funci´on objetivo definida sobre el conjunto de los coeficientes, as´ı como el contenido en frecuencias de dichas bases y algunas de sus aplicaciones.
2
Paquetes de ond´ıculas
Fieles al prop´osito de descomponer la se˜ nal f en funci´on de las frecuencias presentes en ella, consideremos dos operadores, un operador A de “alta frecuencia” y un operador B de “baja frecuencia”, definidos sobre R2 de la siguiente manera: √ ! A [x, y] = (x − y)/ √2 (3) B [x, y] = (x + y)/ 2. Observemos que al aplicar los operadores A y B a una se˜ nal f resulta [A, B] f = [Af , Bf ] = [A [f1 , f2 ] , . . . , A [fn−1 , fn ] , B [f1 , f2 ] , . . . , B [fn−1 , fn ]] y se obtiene una base en Rn (recordemos que n es divisible por 2, siendo una potencia de 2), dado que los operadores son lineales e invertibles, de manera que √ ! (A [x, y] + B [x, y])/ √2 = x (4) (A [x, y] − B [x, y])/ 2 = y. Es f´acil verificar que esta base es ortonormal. Observemos tambi´en que en el cambio de base, a cada par de coordenadas en la base temporal le corresponde un par de coordenadas en la nueva base. Por lo tanto, las coordenadas de cada par en la nueva base cubren exactamente el doble de la extensi´on temporal de una coordenada en la base temporal (el per´ıodo del muestreo), y difieren entre s´ı u ´nicamente en su contenido de frecuencia. Una representaci´on intuitiva de la posici´on de los elementos de las dos bases en el plano tiempo–frecuencia, para n = 8, se muestra en la figura 1. El sistema de funciones ortogonales (3) fue propuesto por primera vez por Haar [3] y Walsh [9].
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Alin Andrei Cˆarsteanu Manit¸iu
ω
ω Af12 Af34 Af56 Af78 f1 f2 f3 f4 f5 f6 f7 f8 Bf12 Bf34 Bf56 Bf78 t
t
Base temporal
Nueva base
Figura 1: Localizaci´on en el plano tiempo–frecuencia de las coordenadas en la base temporal y en la nueva base obtenida por la aplicaci´on de los operadores [A, B]. Aplicando de nuevo los operadores A y B a cada uno de Af y Bf , obtenemos la siguiente base, representada en la figura 2. A cada uno de BAf , AAf , ABf y BBf se le pueden aplicar una u ´ltima vez los operadores A y B, ya que en cada uno de los vectores antes mencionados quedan dos elementos. El resultado es la base frecuencial (o base de Fourier), representada en la misma figura 2. ω
BAf1−4
BAf5−8
AAf1−4
AAf5−8
ABf1−4
ABf5−8
BBf1−4
BBf5−8
ω
BBAf1−8 ABAf1−8 AAAf1−8 BAAf1−8 BABf1−8 AABf1−8 ABBf1−8 BBBf1−8
t Siguiente base
t Base frecuencial
Figura 2: Localizaci´on en el plano tiempo–frecuencia de las coordenadas en la siguiente base obtenida por la aplicaci´on de los operadores [A, B], as´ı como en la base frecuencial. Las 4 bases representadas formalmente (as´ı como gr´aficamente) para n = 8 en las figuras 1 y 2 tienen en com´ un el hecho de que dentro de cada una de ellas, los operadores A o B aparecen un mismo n´ umero
Representaciones discretas en tiempo–frecuencia
31
de veces en cada elemento. Por lo tanto, llamaremos a las bases con esta propiedad “bases fundamentales”. En el caso m´as general en que n es una potencia cualquiera de 2, hay log2 (n) + 1 bases fundamentales, correspondientes al aplicar los operadores 0, 1, . . . , log2 (n) veces. Notemos que cada base fundamental es exhaustiva con respecto a las 2k permutaciones posibles (con repeticiones) de los dos operadores A y/o B, aplicados k veces, sobre un subconjunto de n/2k puntos del dominio de definici´on de f . Por lo tanto, el conjunto de bases fundamentales contiene todos los elementos de bases que se puedan obtener aplicando los operadores A y/o B; llamamos a este conjunto el “paquete de ond´ıculas”. Una observaci´on importante para la obtenci´on de diferentes bases usando los operadores A y B es el hecho siguiente: debido a que la transformaci´on (3) involucra cada vez solamente 2 elementos de una base, sobre cada subconjunto {m2k + 1, . . . , (m + 1) 2k }, k ∈ {1, . . . , log2 (n)} y m ∈ {0, . . . , n/2k −1}, del dominio de definici´on de f se pueden aplicar los operadores A o B, en cualquier orden, cualquier n´ umero de veces entre 0 y k, y el resultado ser´ıa de nuevo una base en Rn (al ser lineal e invertible la transformaci´on (3), la ortonormalidad de tales bases es menos trivial y ser´a discutida aparte). Un ejemplo de base construida de este modo es la base–ond´ıcula (representada en la figura 3 para n = 8). La propiedad distintiva de esta base es que, siendo sus operadores (de altas a bajas frecuencias) A, AB, ABB, ABBB etc., en cada banda de frecuencia, la escala (el factor ak de la ecuaci´on (2), o equivalentemente el n´ umero de elementos de la base temporal involucrados en la representaci´on del elemento en cuesti´on de la presente base) es inversamente proporcional con la frecuencia de aquella banda. Hist´oricamente, ´esta fue la primera base en tiempo–frecuencia desarrollada a parte de las bases fundamentales. ω Af12 Af34 Af56 Af78
ABf1−4
ABf5−8
ABBf1−8 BBBf1−8 t Figura 3: Localizaci´on en el plano tiempo–frecuencia de las coordenadas en la base–ond´ıcula.
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Alin Andrei Cˆarsteanu Manit¸iu
La base frecuencial resulta, si se toma la base temporal como base est´andar, despu´es de O (n log (n)) operaciones. Este orden de velocidad es similar a la del algoritmo de la transformada r´apida de Fourier [2], dado que siguiendo la definici´on se trata de n convoluciones de la funci´on f , definida en n puntos, con diferentes ond´ıculas de ´este mismo tama˜ no, tales como son representadas en la figura 4. Algoritmo discreto de Fourier ω0
Algoritmo del paquete de ond´ıculas
ω1
ω2
ω3
ω4
Figura 4: Formas de las ondas analizadoras correspondientes a diferentes frecuencias ω para el algoritmo del paquete de ond´ıculas (derecha) comparadas con las del algoritmo discreto de Fourier (izquierda), ambos algoritmos usan la funci´on analizador de Haar. Wickerhauser [10] demuestra la equivalencia, en primer orden (y por lo tanto, tambi´en as´ıntoticamente, para n grande), entre el algoritmo de los paquetes de ond´ıculas y el algoritmo de Fourier. Adem´as, se puede
Representaciones discretas en tiempo–frecuencia
33
construir un paquete similar al de ond´ıculas aplicando el algoritmo de Fourier sucesivamente a los n, n/2 + n/2, n/4 + n/4 + n/4 + n/4, etc. puntos del dominio de definici´on de f . Sin embargo, el algoritmo de Fourier es “r´apido” solamente para bases trigonom´etricas, mientras que el algoritmo presentado en esta secci´on mantiene el mismo O (n log (n)) para cualquier par de operadores [A, B]. Demostraremos a continuaci´on la ortonormalidad de las bases construidas a partir del paquete de ond´ıculas Haar. (a) Por su modo de construcci´on, que consiste en aplicar el conjunto de operadores [A, B] a subconjuntos {m2k + 1, . . . , (m + 1)2k }, para k ∈ {1, . . . , log2 (n)} y m ∈ {0, . . . , n/2k − 1} del dominio de definici´on de f , dos elementos arbitrarios de una base obtenida a partir del paquete de ond´ıculas Haar tienen que ser ajenos o en sus representaciones por elementos de la base temporal, o en sus representaciones por elementos de la base frecuencial (o en ambos casos). Supongamos, sin p´erdida de generalidad, que sea ´este el caso de la base temporal, y consideremos la misma base temporal como base est´andar. (Podemos hacer la suposici´on sin p´erdida de generalidad debido a que el producto de A y B aplicados a un par de vectores unitarios [1k , 1j ] resulta ser A [1k , 1j ] · B [1k , 1j ] = 12 12k − 12 12j = 0.) Entonces los dos elementos son trivialmente ortogonales. (b) La normalidad de las bases est´a garantizada por el hecho que los operadores [A, B] conservan las normas en L2 : (A [fk , fj ])2 + (B [fk , fj ])2 = 12 (fk + fj )2 + 2 1 2 2 2 (fk − fj ) = fk + fj . Notemos que la ortonormalidad de los operadores A, B no es suficiente para la ortonormalidad de toda base as´ı obtenida del paquete de ond´ıculas; sino que es necesario que los elementos involucrados en cada aplicaci´on de los operadores sean los mismos como en el caso de los operadores Haar-Walsh. De lo contrario, al aplicar la transformaci´on de base, por intermedio de tales operadores, a un n´ umero de n elementos, su norma llega a encontrarse, despu´es de la transformaci´on, en un n´ umero de elementos mayor que n, o bien al restringirnos a n elementos, la norma de ´estos disminuir´ıa generalmente con respecto a la norma inicial. Estos efectos son llamados efectos terminales, y si no son muy importantes en magnitud, se dice que las respectivas bases son casi ortonormales. Varios autores (por ejemplo, Herley et al. [5]) proponen para estos casos la ortonormalizaci´on de las bases, por procedimientos tipo Gramm-Schmidt o similares. Notemos que con este proceder se pierde el significado en tiempo–frecuencia de dichas bases. (Mientras que este significado no es importante, por ejemplo, para fines de com-
34
Alin Andrei Cˆarsteanu Manit¸iu
presi´on de se˜ nales, para la mayor´ıa de las aplicaciones de investigaci´on s´ı lo es.)
3 3.1
Optimalidad de las bases construidas del paquete de ond´ıculas Criterios globales y funciones objetivo locales
El prop´osito de esta optimizaci´on es de escoger la base que minimiza una funci´on objetivo, definida sobre los coeficientes de la representaci´on de f en la base, de tal manera que se optimicen ciertas caracter´ısticas cualitativas globales de dicha representaci´on. Tal caracter´ıstica es, por ejemplo, el llamado “contraste”. Intuitivamente, aumentar el contraste quiere decir aumentar los elementos mayores y disminuir los elementos menores. En t´erminos de una base que resultar´ıa en la representaci´on m´as contrastante de un vector dado, se elige de un conjunto de bases aquella que maximiza las coordenadas mayores (en valor absoluto) del vector considerado (la ortonormalidad de las bases asegura la minimizaci´on de las coordenadas menores). En las aplicaciones, maximizar el contraste es u ´til para fines de compresi´on de se˜ nales, haciendo posible la eliminaci´on de un mayor n´ umero de elementos peque˜ nos (compresi´on aproximada) o ceros (compresi´on exacta), as´ı como para fines de investigaci´on en la estructura en tiempo–frecuencia de las se˜ nales, revelando las frecuencias dominantes por un per´ıodo limitado de tiempo. Una manera para definir el contraste en una representaci´on es la de ordenar las coordenadas de manera decreciente y construir la secuencia de normas (usuales, en L2 ) de los vectores formados por las primeras 1, 2, . . . , n coordenadas, lo que es equivalente (para bases ortonormales) con la secuencia de sumas parciales de los cuadrados de las primeras 1, 2, . . . , n coordenadas. Por su construcci´on, esta secuencia representa una funci´on no convexa, llam´emosla σ (k) en funci´on del n´ umero k de coordenadas involucradas, haciendo la convenci´on σ (0) = 0. Podemos definir la superioridad del contraste de una primera representaci´on en comparaci´on con una segunda representaci´on: si σ1 (k) ≥ σ2 (k), ∀k ∈ {1, . . . , n}, y ∃k ∈ {1, . . . , n} tal que σ1 (k) > σ2 (k). Observemos que, en el caso general, esta definici´on no introduce un orden completo en el conjunto de representaciones: las gr´aficas de las funciones σ (k) de dos determinadas representaciones se pueden cruzar, en cuyo caso ninguna de las dos (de acuerdo a esta definici´on) tiene un contraste superior a la
Representaciones discretas en tiempo–frecuencia
35
otra. Un modo sencillo de resolver este problema ser´ıa de fijar un cierto ´nico punto en donde comparamos σ1 (k0 ) y σ2 (k0 ). k0 , para que sea el u Para algunas aplicaciones en particular, fijar un tal k0 podr´ıa ser u ´til, pero aqu´ı preferimos analizar el caso general, aun con la desventaja de no tener un orden completo de las representaciones por medio del contraste. Queda entonces encontrar una funci´on objetivo aditiva, definida sobre los coeficientes de la representaci´on de f en una base, de tal manera que se optimice el contraste. Demostraremos en este contexto el siguiente teorema: Teorema 3.1.1 Sean las representaciones de una se˜ nal f en dos bases ortonormales distintas, con una de las representaciones m´ as contrastada que la otra, en el sentido que σ1 (k) ≥ σ2 (k), ∀k ∈ {1, . . . , n}, y ∃k ∈ {1, . . . , n} tal que σ1 (k) > σ2 (k), y sea s (·) una funci´ on objetivo real, c´ oncava. Entonces n n ! "# $ % "# 2 $ % ! s c2k 2 . s ck 1 < k=1
k=1
Demostraci´ on: Probamos que se puede construir una operaci´on que transforma la secuencia ordenada {|ck |1 } a {|ck |2 } en un n´ umero finito " % & de pasos, tal que en cada paso ocurre una disminuci´on de nk=1 s c2k . Sea δ := min [σ1 (k) − σ2 (k)] k∈{l,...,m−1}
tal que δ > 0, σ1 (l − 1) = σ2 (l − 1) y σ1 (m) = σ2 (m), n ≥ m > l > 0. La existencia de una tal subsecuencia l, . . . , m − 1 est´a garantizada por σ1 (0) = σ2 (0) = 0, σ1 (n) = σ2 (n) por ortonormalidad de las $ y #∃k2 $∈ {1, .#. .2, n} $ tal# que $ σ1 (k) > σ2 (k). Cambiemos entonces #bases, 2 2 cl 2 ← cl 2 +δ y cm 2 ← cm 2 −δ, lo que significa σ2 (k) ← σ2 (k)+δ, ∀k una$disminuci´ "# 2Como $ $ % "# o2n$ de % %previsto, "# 2 encontramos % "# &n∈ {l,"#. . 2. ,$m% − 1}. 2 k=1 s ck 2 : s cl 2 + δ + s cm 2 − δ < s cl 2 + s cm 2 , como δ > 0, l ≥ m ⇐⇒ [cl ]2 ≥ [cm ]2 , y la funci´on s es c´oncava. En este momento, para por lo menos un punto k0 ∈ {l, ..., m − 1} tenemos σ1 (k0 ) = σ2 (k0 ). Podemos repetir este procedimiento para subsecuencias terminando, y respectivamente empezando, en k0 . La no-convexidad y la monotonicidad de σ ser´an respetadas dentro de las subsecuencias, como solamente hay una traslaci´on vertical, y ser´an tambi´en respetadas alrededor de k0 , como σ2 (k0 − 1) ≤ σ1 (k0 − 1) ≤ σ1 (k0 ) =
36
Alin Andrei Cˆarsteanu Manit¸iu
σ2 (k0 ) ≤ σ2 (k0 + 1) ≤ σ1 (k0 + 1). Por lo tanto, la existencia y el orden % a preservado a cada paso, realizando una disminuci´on de "#k ]22 $ser´ !n de [c s c k=1 k 2 al respectivo paso siguiente, hasta que max {σ1 (k) − σ2 (k)} = 0 ⇐⇒ σ1 (k) = σ2 (k), ∀k ∈ {1, . . . , n}
k∈{1,...,n}
⇐⇒ [ck ]1 = [ck ]2 , ∀k ∈ {1, . . . , n}, es decir que las dos representaciones se vuelven id´enticas y el algoritmo se termina (despu´es de un n´ umero de pasos menor o igual a n). !
3.2
Algoritmos de optimizaci´ on
En la secci´on precedente hemos mostrado como escoger una funci´on objetivo, definida sobre las coordenadas de f en una base, y cuya minimizaci´on optimice el contraste de la representaci´on de f en dicha base. Queda entonces dise˜ nar algoritmos para buscar dentro del paquete de ond´ıculas la base ortonormal que minimice una funci´on objetivo dada. La b´ usqueda est´a basada en el hecho que para realizar cambios de base usando los operadores A y B sobre 2 coordenadas temporalmente consecutivas de una base, se puede reemplazar este par de coordenadas con un par de coordenadas vecinas en frecuencia, y rec´ıprocamente, como se mostr´o en la secci´on 2. El primer algoritmo de minimizaci´on de la funci´on objetivo, actuando solamente sobre un subconjunto de bases del paquete de ond´ıculas, se encuentra en Wickerhauser [10], y se conoce como el algoritmo (sencillo) del ´arbol (binario). Para ilustrar el algoritmo, es u ´til poner las representaciones en bases tiempo–frecuencia de las figuras 1 y 2 en forma del ´arbol de la figura 5, en el cual cada nivel es la representaci´on de la se˜ nal en una base, correspondiendo (de abajo hacia arriba) de base temporal hasta la base
BBBf ABBf AABf BABf BBf ABf Bf
BAAf AAAf AAf
ABAf BBAf BAf Af
f ´ Figura 5: Arbol del paquete de ond´ıculas.
Representaciones discretas en tiempo–frecuencia
37
frecuencial. (Observemos que para n > 23 , hay m´as niveles en el ´arbol, hacia arriba, para llegar a la base frecuencial y completar todo el paquete de ond´ıculas.) Para escoger los elementos de la base optimizada en este caso, empezando con la base frecuencial, se compara la funci´on objetivo para cada par de coordenadas adyacentes en frecuencia con el par que les corresponde por la ecuaci´on (3) (es decir que se comparan los elementos correspondientes entre la pen´ ultima y la u ´ltima base – la frecuencial), escogiendo cada vez entre los dos pares aquel que minimice la funci´on objetivo. A continuaci´on, se comparan cuartetos de coordenadas de la misma frecuencia en la antepen´ ultima base con los cuartetos correspondientes de la base obtenida al paso anterior, escogiendo entre los cuartetos correspondientes cada vez aquel que minimice la funci´on objetivo. Despu´es, se comparan octetos de coordenadas de la misma frecuencia en la precedente base con los octetos correspondientes de la base obtenida al paso anterior, y as´ı sucesivamente hasta la base temporal. El algoritmo es f´acil de seguir, de arriba hacia abajo, sobre el ´arbol de la figura 5. Sin embargo, aunque el algoritmo minimice la funci´on objetivo sumada sobre los elementos de la base (con las consecuencias globales discutidas en la secci´on precedente), se observa inmediatamente que este algoritmo no tiene acceso a todas las bases que se puedan construir a partir del paquete de ond´ıculas; adem´as, induce una asimetr´ıa artificial entre tiempo y frecuencia. El mismo algoritmo puede ser empleado empezando con la base temporal (llam´emoslo algoritmo inverso del ´arbol sencillo). Aunque ´esto no mejora el algoritmo, ha sugerido la idea para otro algoritmo, operando sobre un conjunto sensiblemente m´as grande de bases, y publicado por Herley et al. [4, 5], as´ı como por Ramchandran y Vetterli [7]. El llamado algoritmo doble del ´arbol consta en correr el algoritmo sencillo a partir de cada una de las bases fundamentales (correspondientes a cada ! " uno de los niveles del ´arbol), teniendo en consecuencia O n log2 (n) operaciones. El algoritmo guarda las mismas desventajas que el algoritmo sencillo, en el sentido que todav´ıa no tiene acceso a todas las bases que puedan resultar del paquete de ond´ıculas (lo que resultar´a obvio en la presentaci´on de los siguientes algoritmos), y tambi´en guarda una asimetr´ıa entre tiempo y frecuencia (por correr hacia la base temporal, o posiblemente en el otro sentido, pero no en ambos). Finalmente, algoritmos que escogen entre todas las bases posibles del paquete de ond´ıculas fueron construidos independientemente por Thiele y Villemoes [8], Herley et al. [6] y Cˆarsteanu et al. [1]. Estos algoritmos est´an basados en la misma idea (presentada a continuaci´on),
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Alin Andrei Cˆarsteanu Manit¸iu
diferenci´andose solamente en el manejo de las variables en memoria (para minimizar la memoria ocupada [1] o el tiempo de ejecuci´on del algoritmo [8, 6], respectivamente). Observemos primero que las bases ortonormales que se pueden construir por las relaciones (3) y (4) a partir del paquete de ond´ıculas no pueden contener simult´aneamente elementos de la base temporal y de la base frecuencial (por no ser ortogonales entre s´ı). Esta constataci´on ofrece la clave para construir un algoritmo que tenga acceso a todas las bases ortonormales que se puedan construir a partir del paquete de ond´ıculas, y de ella resulta directamente el siguiente lema: Lema 3.2.1 Sea P ({1, . . . , n}) el conjunto de bases ortonormales en Rn que se pueden construir a partir del respectivo paquete de ond´ıculas, F ({1, . . . , n}) el subconjunto de P ({1, . . . , n}) de bases que no contienen ning´ un elemento de la base temporal, y T ({1, . . . , n}) el subconjunto de P ({1, . . . , n}) de bases que no contienen ning´ un elemento de la base frecuencial. Entonces F ({1, . . . , n}) ∪ T ({1, . . . , n}) = P ({1, . . . , n}) . De este lema desprendemos que cualquier base ortonormal construida a partir del paquete de ond´ıculas (y por lo tanto la base ´optima buscada) pertenece a F ({1, . . . , n}) o a T ({1, . . . , n}) (o a los dos), por lo cual la b´ usqueda algor´ıtmica de la base ´optima dentro de P ({1, . . . , n}) se reduce a dos b´ usquedas dentro de F ({1, . . . , n}) y de T ({1, . . . , n}), respectivamente. Obviamente, para n > 2, tenemos F ({1, . . . , n}) ∩ T ({1, . . . , n}) ̸= ∅, lo que significa que se puede mejorar el algoritmo de b´ usqueda en t´erminos de rapidez (a precio de usar mas memoria) memorizando F ({1, . . . , n}) ∩ T ({1, . . . , n}) y buscando una sola vez en este subconjunto (en lugar de dos veces, al buscar dentro de F ({1, . . . , n}) y dentro de T ({1, . . . , n})). Thiele y Villemoes [8] muestran que memorizando todas las comparaciones hechas en cada etapa, el algoritmo llega a tener O (n log (n)) operaciones. Veremos a continuaci´on dos lemas, que nos permitir´an continuar la b´ usqueda de la base ´optima dentro de T ({1, . . . , n}) y dentro de F ({1, . . . , n}), respectivamente: Lema 3.2.2 Las bases del subconjunto T ({1, . . . , n}) contienen solamente elementos de las primeras n − 1 bases del respectivo paquete. Por lo tanto, tenemos que P ({1, . . . , n/2}) × P ({n/2 + 1, . . . , n}) = T ({1, . . . , n}). Es decir que T ({1, . . . , n}) consta del paquete construido de los primeros n/2 elementos de la base temporal y del paquete construido de los u ´ltimos n/2 elementos de dicha base.
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Lema 3.2.3 Las bases del subconjunto F ({1, . . . , n}) contienen solamente elementos de las u ´ltimas n−1 bases del respectivo paquete. Por lo tanto, F ({1, . . . , n}) consta del paquete construido de los primeros n/2 elementos de la base temporal y del paquete construido de los u ´ltimos n/2 elementos de dicha base. Hemos entonces reducido, en 2 pasos intermedios, la b´ usqueda de la base ´optima dentro de P ({1, . . . , n}) a la b´ usqueda dentro de 4 paquetes de bases de dimensi´on n/2. Como el conjunto generado por los 4 paquetes es id´entico a P ({1, . . . , n}) (cualquier base que exista en P ({1, . . . , n}) se encuentra tambi´en en el conjunto), concluimos que el algoritmo toma en cuenta todas las bases ortonormales que se puedan construir del paquete de bases de dimensi´on n y,! por " inducci´on, que el 2 algoritmo encuentra el resultado despu´es de O n pasos (sin tomar en cuenta ning´ un modo de eliminar las redundancias entre F (·) y T (·) mencionadas anteriormente). Observemos sin embargo que # (P ({1, . . . , n})), el n´ umero de bases contenidas en P ({1, . . . , n}), no crece con n de manera polinomial, sino exponencial, lo que har´ıa un procedimiento por comparaciones directas pr´acticamente imposible de usar. Efectivamente, como por razones de simetr´ıa tenemos # (P ({1, . . . , n})) = 2#2 (P ({1, . . . , n/2}))− #4 (P ({1, . . . , n/4})), y como # (P ({1})) = 1 y # (P ({1, !√ 2})) = "#2, tenemos que ∃β tal que limn→∞ # (P ({1, . . . , n}))/ β n = 5 − 1 2, con β ≈ 1.84454757 . . . [1]. Para tener una idea del n´ umero de bases que pueden tomar en cuenta los algoritmos antes presentados, mencionemos que para el algoritmo del ´arbol doble, β ≈ 1.7148445 . . ., mientras que para el algoritmo del ´arbol sencillo, β ≈ 1.5028368 . . . (el l´ımite es igual a 1 en los dos u ´ltimos casos) [8]. Para precisar esta comparaci´on visualizando gr´aficamente ciertas diferencias entre los algoritmos, en la figura 6 se presentan (para n = 8) las localizaciones en tiempo–frecuencia de: (A) una base accesible a los tres algoritmos; (B) una base que no puede ser tomada en cuenta por el algoritmo del ´arbol sencillo; y (C) una base a la cual no tienen acceso ni el algoritmo del ´arbol sencillo, ni el algoritmo del ´arbol doble.
4
Aplicaciones
Como ya discutido, maximizar el contraste de una se˜ nal es u ´til para fines de compresi´on, haciendo posible la eliminaci´on de un mayor n´ umero de
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ω
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Figura 6: Representaciones en tiempo–frecuencia de: (A) una base accesible a los tres algoritmos; (B) una base que no puede ser tomada en cuenta por el algoritmo del ´arbol sencillo; y (C) una base a la cual no tienen acceso ni el algoritmo del ´arbol sencillo, ni el algoritmo del ´arbol doble.
elementos peque˜ nos para compresi´on aproximada, o ceros para compresi´on exacta. En la figura 7, presentamos la base temporal, as´ı como la base ´optima de una se˜ nal de 256 muestras. La funci´on objetivo escogida es la entrop´ıa, la se˜ nal llegando de una entrop´ıa de 5.1444 . . . en la base temporal a una entrop´ıa de 1.1858 . . . en la base ´optima. Dado que la entrop´ıa es una funci´on c´oncava, se realiza tambi´en un contraste mas fuerte en la base ´optima (figura 8). Este contraste nos permite reconstruir aproximadamente la se˜ nal usando solamente 20 coordenadas de la base ´optima (figura 9). Para fines de investigaci´on en la estructura en tiempo–frecuencia de una se˜ nal, una representaci´on ´optima puede revelar las frecuencias dominantes por un per´ıodo limitado de tiempo, lo que no es realizable ni en la representaci´on temporal, ni en la representaci´on frecuencial. La figura 10 nos muestra la representaci´on en su base ´optima de un ejemplo de una tal se˜ nal con frecuencia variable en el tiempo, un chirrido.
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5
El problema de las frecuencias para el espectro energ´ etico
En las secciones precedentes hemos visto c´omo, a partir del paquete de ond´ıculas, se pueden construir bases en Rn , con significado en tiempo– frecuencia, que minimicen una funci´on aditiva sobre los elementos de la base, y en consecuencia adquieren ciertas caracter´ısticas de contraste. Sin embargo, una vista m´as cuidadosa al paquete de ond´ıculas revela que, igual que en el caso del algoritmo (real) discreto de Fourier, los pares de elementos consecutivos en frecuencia (con excepci´on del primero y del u ´ltimo elemento) de cualquier base construida, son caracterizados en realidad por la misma frecuencia, y difieren solamente por una traslaci´on (bk de la ecuaci´on (2)). Se presenta entonces el problema de c´omo obtener la norma de las coordenadas correspondientes a cada frecuencia, el conjunto de dichas normas constituye el llamado “espectro energ´etico”. Dicho espectro, interpretado como “contenido” de energ´ıa en cada frecuencia (tambi´en “densidad” de energ´ıa en el caso de espectros continuos) tiene varias aplicaciones en procesos f´ısicos caracterizados por frecuencias. Para ejemplificar, el espectro energ´etico de una se˜ nal que representa la variaci´ on en el tiempo de la velocidad en un punto de un fluido turbulento es gobernado por las leyes de escalamiento que rigen el mismo fluido, y son el resultado de la transferencia de energ´ıa dentro del rango inercial de escalas del fluido, desde las escalas de
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Figura 8: Fracci´on de la norma L2 de la se˜ nal reconstruida, en funci´on del n´ umero de las coordenadas mayores usadas en la reconstrucci´on: notemos la superioridad del contraste de la base ´optima (l´ınea interrumpida) comparado con el contraste de la base temporal (l´ınea continua). inyecci´on de energ´ıa hac´ıa el rango viscoso. El estudio del respectivo espectro energ´etico ha permitido explicar, aunque todav´ıa no completamente, el fen´omeno de movimiento turbulento. Por lo tanto, estimamos como muy importante la posibilidad de extraer el espectro energ´etico de una se˜ nal a partir de las bases construidas en tiempo–frecuencia. Para poder evaluar todas las normas correspondientes a frecuencias en una base, en ella cada elemento de la base temporal debe entrar en el c´alculo de elementos de una sola representaci´on fundamental. El conjunto de bases que satisfacen esta condici´on es precisamente el conjunto de bases accesibles por el algoritmo inverso del ´arbol sencillo (recordemos que por “inverso” hemos entendido el intercambio de tiempo con frecuencia en el algoritmo usual). Esto, porque en el algoritmo del ´arbol sencillo, la descomposici´on frecuencial es constante en el tiempo [10], y por lo tanto, en el algoritmo inverso, la descomposici´on temporal es la misma para todas las frecuencias. Lo que quiere tambi´en decir que para
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Figura 9: La se˜ nal original de 256 muestras (izquierda) y la se˜ nal reconstruida de 20 coordenadas de la base ´optima (derecha). las bases obtenidas por los dem´as algoritmos, en general no se puede calcular un espectro energ´etico con significado. Proponemos aqu´ı una visi´on diferente acerca de este asunto: calcular el espectro energ´etico con solamente n/2 + 2 en lugar de n elementos. El siguiente lema nos ayuda a hacerlo: Lema 5.1.1 Usando el algoritmo para obtener la base ´ optima, si paramos la descomposici´ on en frecuencia a 4 elementos en lugar de 2, y si anulamos para prop´ ositos de comparaci´ on los 2 elementos extremos de los 4 (aparte del caso que son los mismos extremos de una base fundamental), procuramos comparar todas las bases construidas de esta manera, a partir del paquete de ond´ıculas. Para probarlo, es suficiente observar que el algoritmo es equivalente al aplicar el algoritmo usual al conjunto F descrito en la secci´on anterior. La ventaja de este acercamiento radica en el hecho que para la mayor´ıa de las se˜ nales, las n/2 + 2 coordenadas mayores de la base ´optima contienen casi toda la energ´ıa de la se˜ nal, mientras que la base ´optima tiene en la mayor´ıa de los casos un σ (k) superior a lo de la base obtenida por el algoritmo inverso del ´arbol sencillo, para todo k ∈ {1, . . . , n/2 + 2}.
6
Conclusiones
Hemos presentado aqu´ı una revisi´on del estado actual de los algoritmos de optimizaci´on discreta para representaciones en tiempo–frecuencia,
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Figura 10: Un chirrido y su representaci´on en la base ´optima (donde las coordenadas est´an representadas por intensidad de gris): observemos c´omo las frecuencias dominantes son delimitadas durante diversos intervalos de tiempo. enfocando el trabajo en ejemplificar las caracter´ısticas especiales y los logros de cada algoritmo. Una atenci´on particular fue extendida al problema de la selecci´on de frecuencias, la cual juega un papel importante en la definici´on del espectro energ´etico de una se˜ nal. En este contexto, observamos que una versi´on nueva de un algoritmo existente puede resolver el problema, y tambi´en proponemos un algoritmo nuevo, basado en un acercamiento diferente, que surge de los lemas comprobados en el presente art´ıculo. Si bien este algoritmo resulta superior en t´erminos de contraste, la cuesti´on de la optimizaci´on del contraste entre todas las bases que permiten la definici´on de un espectro energ´etico queda como problema abierto para el futuro. Agradecimientos El autor quiere agradecer a los editores de Morfismos la invitaci´on a escribir el presente art´ıculo, as´ı como a los tres revisores an´onimos quienes, con sus comentarios, contribuyeron a mejorar el texto, y a Jes´ us Gonz´alez Espino Barros por su revisi´on final. Alin Cˆ arsteanu Manit¸iu Departamento de Matem´ aticas, CINVESTAV-IPN A.P. 14-740 M´exico D.F. 07000 M´exico alin@math.cinvestav.mx
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Referencias [1] Cˆarsteanu A.; Sapozhnikov V. B.; Venugopal V.; FoufoulaGeorgiou E., Absolute optimal time-frequency basis – a research tool, J. Phys. A: Math. Gen., 30 (1997), 7133-7146. [2] Cooley J. W.; Tukey J. W., An algorithm for machine calculation of complex Fourier series, Math. Computation, 19 (1965), 297-301. [3] Haar A., Zur Theorie der orthogonalen Funktionensysteme, Math. Ann., 69 (1910), 331-371. [4] Herley C.; Kovaˇcevi´c J.; Ramchandran K.; Vetterli M., Arbitrary orthogonal tilings of the time-frequency plane (Int. Symp. on TimeFrequency and Time-Scale Analysis, 11-14), Victoria, BC 1992. [5] Herley C.; Kovaˇcevi´c J.; Ramchandran K.; Vetterli M., Tilings of the time-frequency plane: construction of arbitrary orthogonal bases and fast tiling algorithms, IEEE Trans. Signal Processing, 41 (1993), 3341-3359. [6] Herley C.; Xiong Z.; Ramchandran K.; Orchard M. T., Joint spacefrequency segmentation using balanced wavelet packet trees for least cost image representation, IEEE Trans. Image Processing, 6 (1997), 1213-1230. [7] Ramchandran K.; Vetterli M., Best wavelet packet bases in a ratedistortion sense, IEEE Trans. Image Processing, 2 (1993), 160-173. [8] Thiele C. M.; Villemoes L. F., A Fast Algorithm for Adapted Time– Frequency Tilings, Applied and Computational Harmonic Analysis, 3 (1996), 91-99. [9] Walsh J. L., A Closed Set of Normal Orthogonal Functions, Amer. J. Math., 45 (1923), 5-24. [10] Wickerhauser M. V., Lectures on wavelet packet algorithms, Preprint, Department of Mathematics, Washington University, 1991.
Morfismos, Vol. 7, No. 1, 2003, pp. 47â&#x20AC;&#x201C;68
Generalized tilings with height functions Olivier Bodini
â&#x2C6;&#x2014;
Matthieu Latapy
Abstract In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be defined (for example, the famous tilings of polyominoes with dominoes). We show that many properties of these tilings can be seen as the consequences of properties of the generalized tilings we introduce. In particular, we show that any tiling problem which can be modelized in our generalized framework has the following properties: the tilability of a region can be constructively decided in polynomial time, the number of connected components in the undirected flip-accessibility graph can be determined, and the directed flip-accessibility graph induces a distributive lattice structure. Finally, we give a few examples of known tiling problems which can be viewed as particular cases of the new notions we introduce.
2000 Mathematics Subject Classification: Primary: 05B45; Secondary: 52C20 Keywords and phrases: Tilings, Height Functions, Tilability, Distributive Lattices, Random Sampling, Potentials, Flows.
1
Preliminaries
Given a finite set of elementary shapes, called tiles, a tiling of a given region is a set of translated tiles such that the union of the tiles covers exactly the region, and such that there is no overlapping between any tiles. See for example Figure 1 for a tiling of a polyomino (set of â&#x2C6;&#x2014;
This work arose from ideas develop by the first author while he was completing his Ph.D. studies at the Mathematics Department of the University of P. and M. Curie at Paris.
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Olivier Bodini and Matthieu Latapy
squares on a two-dimensional grid) with dominoes (1 × 2 and 2 × 1 rectangles). Tilings are widely used in physics to modelize natural objects and phenomena. For example, quasicrystals are modelized by Penrose tilings [4] and dimers on a lattice are modelized by domino tilings [10]. Tilings appeared in computer science with the famous undecidability of the question of whether the plane is tilable or not using a given finite set of tiles [2]. Since then, many studies appeared concerning these objects, which are also strongly related to many important combinatorial problems [11]. A local transformation is often defined over tilings. This
Figure 1: From left to right: the two possible tiles (called dominoes), a polyomino (i.e. a set of squares) to tile, and a possible tiling of the polyomino with dominoes. transformation, called flip, is a local rearrangement of some tiles which makes it possible to obtain a new tiling from a given one. One then defines the (undirected) flip-accessibility graph of the tilings of a region R, denoted by AR , as follows: the vertices of AR are all the tilings of R, and {t, t′ } is an (undirected) edge of AR if and only if there is a flip between t and t′ . See Figure 2 for an example. The flip notion is a key element for the generation and enumeration of the tilings of a given region, and for many algorithmical questions. For example, we will see in the following that the structure of AR may give a way to sample randomly a tiling of R with the uniform distribution, which is crucial for physicists. This notion is also a key element to study the entropy of the physical objects [13], and to examine some of their properties like frozen areas, weaknesses, and others [6]. On some classes of tilings which can be drawn on a regular grid, it is possible to define a height function which associates an integer to any node of the grid (it is called the height of the node). For example, one can define such a function over domino tilings as follows. As already noticed, a domino tiling can be drawn on a two dimensional square grid. We can draw the squares of the grid in black and white like on
Generalized tilings with height functions
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Figure 2: From left to right: the flip operation over dominoes, and two examples of tilings which can be obtained from the one shown in Figure 1 by one flip. In these tilings, we shaded the tiles which moved during the flip. a chessboard. Let us consider a polyomino P and a domino tiling T of P , and let us distinguish a particular node p on the boundary of P , say the one with smaller coordinates. We say that p is of height 0, and that the height of any other node p′ of P is computed as follows: initialize a counter to zero, and go from p to p′ using a path composed only of edges of dominoes in T , increasing the counter when the square on the right is black and decreasing it when the square is white. The height of p′ is the value of the counter when one reaches p′ . One can prove that this definition is consistent. This can be used as the height function for domino tilings [18]. See Figure 3 for an example. These height functions make it possible to define AR , the directed flip-accessibility graph of the tilings of a region R: the vertices of AR are the tilings of R and there is a directed edge (t, t′ ) if and only if t can be transformed into t′ by a flip which decreases the sum of the heights of all the vertices (a flip always changes the height function). See Figure 3 for an example with domino tilings. The generalized tilings we introduce in this paper are based on these height functions, and most of our results are induced by them. These notions of height functions are related to classical notions of flow theory in graphs. Let G = (V, E) be a directed graph. A flow on G is a map from E into C (actually, we will only use flows with values in Z). Given two vertices v and v ′ of G, a travel from s to s′ is a set of edges of G such that, if one forgets their orientations, then one obtains a path from s to s′ . Given a flow C, the flux of C on the travel T is ! ! C(e) − C(e) FT (C) = e∈T +
e∈T −
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Figure 3: The directed flip-accessibility graph of the tilings of a polyomino by dominoes. The height of each node of the polyomino is shown for each tiling. The set of all the tilings of this polyomino is ordered by the flip relation directed with respect to the height functions. right direction when one goes from s to s′ , and T − is the set of oriented edges traveled in the reverse direction. One can easily notice that the flux is additive by concatenation of travels: if T1 and T2 are two travels such that the ending point of T1 is equal to the starting point of T2 , then FT1 ·T2 (C) = FT1 (C) + FT2 (C). See [1] for more details about flow theory in graphs. Then we notice that a height function is a flux. Since there is no circuit in the graph AR (there exists no nonempty sequence of directed flips which transforms a tiling into itself), it induces an order relation over all the tilings of R: t ≤ t′ if and only if t′ can be obtained from t by a sequence of (directed) flips. In Section 3, we will study AR under the order theory point of view, and we will meet some special classes of orders, which we introduce now. A lattice is an order L such that any two elements x and y of L have a greatest lower bound, called the infimum of x and y and denoted by x ∧ y, and a lowest greater bound, called the supremum of x and y and denoted
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by x ∨ y. The infimum of x and y is nothing but the greatest element among the ones which are lower than both x and y. The supremum is defined dually. A lattice L is distributive if for all x, y and z in L, x∨(y∧z) = (x∨y)∧(x∨z) and x∧(y∨z) = (x∧y)∨(x∧z). For example, it is known that the flip-accessibility graph of the domino tilings of a polyomino without holes is always a distributive lattice [17]. Therefore, this is the case of the flip-accessibility graph shown in Figure 3 (notice that the maximal element of the order is at the bottom, and the minimal one is at the top of the diagram since we used the discrete dynamical models convention: the flips go from top to bottom). Lattices (and especially distributive lattices) are strongly structured sets. Their study is an important part of order theory, and many results about them exist. In particular, various codings and algorithms are known about lattices and distributive lattices. For example, there exists a generic algorithm to sample randomly an element of any distributive lattice with the uniform distribution [16]. For more details about orders and lattices, we refer to [8]. Finally, let us introduce a useful notation about graphs. Given a directed graph G = (V, E), the undirected graph G = (V , E) is the graph obtained from G by removing the orientations of the edges. In other words, V = V , and E is the set of undirected edges {v, v ′ } such that (v, v ′ ) ∈ E. We will also call G the undirected version of G. Notice that this is consistent with our definitions of AR and AR . In this paper, we introduce a generalization of tilings on which a height function can be defined, and show how some known results may be understood in this more general context. All along this paper, like we did in the present section, we will use the tilings with dominoes as a reference to illustrate our definitions and results. We used this unique example because it is very famous and simple, and permits to give clear figures. We emphasize however on the fact that our definitions and results are much more general, as explained in the last section of the paper.
2
Generalized tilings.
In this section, we give all the definitions of the generalized notions we introduce, starting from the objects we tile to the notions of tilings, height functions, and flips. The first definitions are very general, therefore we will only consider some classes of the obtained objects, in order
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to make the more specific notions (mainly height functions and flips) relevant in this context. However, the general objects introduced may be useful in other cases. Let G be a simple directed graph (G has no multiple edges, no loops, and if (v, v ′ ) is an edge then (v ′ , v) can not be an edge). We consider a set Θ of elementary circuits of G, which we will call cells. Then, a polycell is any set of cells in Θ. Given a polycell P , we call the edges of cells in P the edges of P , and their vertices the vertices of P . A polycell P is k-regular if and only if there exists an integer k such that each cell of P is a circuit of length k. Given a polycell P , we fix an arbitrarily not empty partial subgraph of P that we denote by ∂P . We call it the boundary of P . Notice that, it is not a topological definition. In the following, a polycell will be always considered with its fixed boundary, that is to say (P, ∂P ). A polycell P is full if ∂P is connected. Given an edge e of P which is not on the boundary, we call the set of all the cells in P which have e in common a tile. A set of edges of P \∂P such that the associated tiles constitute a partition of the cells of P is called a tiling Q. An edge in Q is by definition a tiling edge. A polycell P which admits at least a tiling Q is tilable. See Figure 4 and Figure 5 for some examples. Notice that if we distinguish exactly one edge of each cell of a polycell P , in such a way that none of them is on the boundary of P , then the distinguished edges can be viewed as the tiling edges of a tiling of P . Indeed, each edge induces a tile (the set of cells which have this edge in common), and each cell is in exactly one tile.
Figure 4: From left to right: a polycell P (the boundary ∂P being the partial subgraph constituted by the edges belonging to a unique elementary circuit and Θ being the set of all the elementary circuits), the three tiles of P , and a tiling of P represented by its tiling edges (the dotted edges). This polycell is full, tilable, and is not k-regular for any k. Notice that there are two tiles composed of two cells, and another one composed of three cells. Notice also that the tiling given in this figure is the only possible one.
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Figure 5: Left: a 4-regular polycell P (the cells in Θ are the circuits of length 4), the boundary of which is composed of those edges belonging to a unique elementary circuit. Right: a tiling of P represented by its tiling edges (the dotted edges). Notice that this figure is very similar to Figure 1. Let P be a k-regular tilable polycell and Q be a tiling of P . We associate to Q a flow CQ on Θ (seen as a graph): CQ (e) =
!
1 − k if the edge e is a tiling edge of Q 1 otherwise.
For each cell c, we define Tc as the travel which contains exactly the edges of c (in other words, it consists in turning around c). Notice that the flux of CQ on the travel Tc is always null: FTc (CQ ) = 0 since each cell contains exactly a tiling edge, valued 1 − k, and k − 1 other edges, valued 1. Moreover, for each edge e ∈ ∂P , we have CQ (e) = 1 since from the definition e cannot be a tiling edge. Let us consider a polycell P and a flow C on the edges of P . If for every closed travel T (i.e. a cycle when one forgets the orientation of each edge) in P we have FT (C) = 0, then the flow C is called a tension. A polycell P is contractible if the fact that FTc (C) = 0 for every cell c implies that C is a tension. Since the converse is always true, we then have that C is a contractible if the following is true: C is a tension only if for every cell c, FTc (C) = 0. Notice that if P is a contractible k-regular polycell and Q is a tiling of P , then the flow CQ is a tension, since for every cell c, FTc (CQ ) = 0. Now, if we (arbitrarily) distinguish a vertex ν on the boundary of P , we can associate to the tension CQ a potential ϕQ , defined over the vertices of P :
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• ϕQ (ν) = 0. • for all vertices x and y of P , ϕQ (y) − ϕQ (x) = FTx,y (CQ ) where Tx,y is a travel from x to y. The distinguished vertex is needed otherwise ϕQ would only be defined except for a constant, but one can choose any vertex on the boundary. Notice that this potential can be viewed as a height function associated to Q, and we will see that it indeed plays this role in the following. Therefore, we will call the potential ϕQ the height function of Q. See Figure 6 for an example. 1
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Figure 6: From left to right: a tiling Q of a polycell (represented by its tiling edges, the dotted ones), the tension CQ and the height function (or potential) ϕQ it induces. Again, this figure may be compared to Figure 3 (topmost tiling). We now have all the main notions we need about tilings of polycells, including height functions, except the notion of flips. In order to introduce it, we need to prove the following: Theorem 2.0.1 Let P be a k-regular contractible polycell. There is a bijection between the tilings of P and the tensions C on P which verify: • for every edge e in ∂P , C(e) = 1, • and for every edge e of P , C(e) ∈ {1 − k, 1}. Proof: For every tiling Q of P , we have defined above a flow CQ which verifies the property in the claim, and such that for every cell c, FTc (CQ ) = 0. Since P is contractible, this last point implies that CQ is a tension. Conversely, let us consider a tension C which satisfies the hypotheses. Since each cell is of length k, and since C(e) ∈ {1 − k, 1}, the fact that FTc (C) = 0 implies that each cell has exactly one negative edge. These negative edges can be considered as the tiling edges of a tiling of P , which ends the proof. ! Given a k-regular contractible polycell P defined over a graph G, this theorem allows us to identify a tiling Q and the associated tension CQ .
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This makes it possible to define the notion of flip as follows. Suppose there is a vertex x in P which is not on the boundary and such that its height, with respect to the height function of Q, is greater than the height of each of its neighbors in G. We will call such a vertex a maximal vertex. The neighbors of x in G have a smaller height than x, therefore the outgoing edges of x in G are tiling edges of Q and the incoming edges of x in G are not. Let us consider function CQ′ defined as follows: ⎧ if e is an outgoing edge of x ⎨ 1 1 − k if e is an incoming edge of x CQ′ (e) = ⎩ CQ (e) else.
Each cell c which contains x contains exactly one outgoing edge of x and one incoming edge of x, therefore we still have FTc (CQ′ ) = 0. Therefore, CQ′ is a tension, and so it induces from Theorem 2.0.1 a tiling Q′ . We say that Q′ is obtained from Q by a flip around x, or simply by a flip. Notice that Q′ can also be defined as the tiling associated to the height function obtained from the one of Q by decreasing the height of x by k, and without changing anything else. This corresponds to what happens with classical tilings (see for example [17]). See Figure 7 for an example. 1 1
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Figure 7: A flip which transforms a tiling Q of a polycell P into another tiling Q′ of P . From left to right, the flip is represented between the tilings, then between the associated tensions, and finally between the associated height functions. We now can define and study AP , the (directed) flip-accessibility graph of the tilings of P : AP = (VP , EP ) is the directed graph where VP is the set of all the tilings of P and (Q, Q′ ) is an edge in EP if
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Q can be transformed into Q′ by a flip. We will also study the undirected flip-accessibility graph AP . The properties of these graphs are crucial for many questions about tilings, like enumeration, generation and sampling.
3
Structure of the flip-accessibility graph.
Let us consider a k-regular contractible polycell P and a tiling Q of P . Let h be the minimal value among the heights of all the vertices with respect to the height function of Q. If Q is such that all the vertices of height h are on the boundary of P , then it is said to be a maximal tiling. For a given P , we denote by TmaxP the set of the maximal tilings of P . We will see that these tilings play a particular role in the graph AP . In particular, we will give an explicit relation between them and the number of connected components of AP . Recall that we defined the minimal vertices of Q as the vertices which have a height least than the height of each of their neighbors, with respect to the height function of Q (they are local minimums). Lemma 3.0.2 Let P be a k-regular tilable contractible polycell (P is not necessarily full). There exists a maximal tiling Q of P . Proof: Let V be the set of vertices of P , and let Q be a tiling of P such that for every tiling Q′ of P , we have: ! ! ϕQ (x) ≥ ϕQ′ (x). x∈V
x∈V
We will prove that Q is a maximal tiling. Suppose there is a minimal vertex xm which is not on the boundary. " Therefore, one " can transform Q into Q′ by a flip around xm . Then x∈V ϕQ′ (x) = x∈V ϕQ (x) + k, which is in contradiction with the hypothesis. ! Lemma 3.0.3 For every tiling Q of a k-regular contractible polycell P , there exists a unique tiling in TmaxP reachable from Q by a sequence of flips. Proof: It is clear that at least one tiling in TmaxP can be reached from Q by a sequence of flips, since the flip operation decreases the sum of the heights, and since we know from the proof of Lemma 3.0.2 that a tiling such that this sum is maximal is always in TmaxP . We now have to prove that the tiling in TmaxP we obtain does not depend on the order
Generalized tilings with height functions
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in which we flip around the successive minimal vertices. Since making a flip around a minimal vertex x is nothing but increasing its height by k and keeping the other values, if we have two minimal vertices x and x′ then it is equivalent to make first the flip around x and after the flip around x′ or the converse. ! Lemma 3.0.4 Let P be a k-regular contractible and tilable polycell. A tiling Q in TmaxP is totally determined by the values of ϕQ on ∂P . Proof: The proof is by induction over the number of cells in P . Let x be a minimal vertex for ϕQ in ∂P . For all ingoing edges e of x, CQ (e) = 1 − k (otherwise ϕ(x) would not be minimal). Therefore, these edges can be considered as tiling edges, and determine some tiles of a tiling Q of P . Iterating this process, one finally obtains Q. See Figure 8 for an example. ! Theorem 3.0.5 Let P be a k-regular contractible and tilable polycell. The number of connected components in AP is equal to the cardinal of TmaxP . Proof: Immediate from Lemma 3.0.3. ! This theorem is very general and can explain many results which appeared in previous papers. We obtain for example the following corollary, which generalizes the one saying that any domino tiling of a polyomino can be transformed into any other one by a sequence of flips [3]. Corollary 3.0.6 Let P be a full k-regular contractible and tilable polycell. There is a unique element in TmaxP , which implies that AP is connected. Proof: Since ∂P is connected, the heights of the vertices in ∂P are totally determined by the orientation of the edges of ∂P and do not depend on any tiling Q. Therefore, from Lemma 3.0.4, there is a unique tiling in TmaxP . ! As a consequence, if P is a full tilable and contractible polycell, the height of a vertex x on the boundary of P is independent of the considered tiling. In the case of full polyominoes, this restriction of ϕQ to the boundary of P is called height on the boundary [9] and has been introduced in [18]. Notice also that the proof of Lemma 3.0.4 gives an algorithm to build the unique maximal tiling of any k-regular contractible and tilable full polycell P , since the height function on the
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boundary of P can be computed without knowing any tiling of P . See Algorithm 1 and Figure 8. This algorithm gives in polynomial time a tiling of P if it is tilable. It can also be used to decide whether P is tilable or not. Therefore, it generalizes the result of Thurston [18] saying that it can be decided in polynomial time if a given polyomino is tilable with dominoes.
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Figure 8: An example of execution of Algorithm 1. The polycell we want to tile is drawn on the top. Its boundary is composed of the vertices which belong to at most three edges. Below, from left to right, we show the result of each iteration of the algorithm (computation of the tension on the first line, and of the height function on the second line). In this example, the first iteration of the algorithm gives one vertical tile, and the second (and last) iteration gives four horizontal tiles. With these results, we obtained much information concerning a central question of tilings: the connectivity of the undirected flip-accessibility graph. We did not only give a condition under which this graph is connected, but we also gave a relation between the number of its connected components and some special tilings. We will now deepen the study of the structure induced by the flip relation by studying the directed flipaccessibility graph, and in particular the partial order it induces over the tilings: t ≤ t′ if and only if t′ can be obtained from t by a sequence of (directed) flips.
Generalized tilings with height functions
Algorithm 1 Computation of the maximal tiling of a full kregular contractible polycell. Input: A full k-regular contractible polycell P , its boundary ∂P and a distinguished vertex ν on this boundary. Output: An array tension on integers indexed by the edges of P and another one height indexed by the vertices of P . The first gives the tension associated to the maximal tiling, and the second gives its height function. begin P′ ← P; height[ν] ← 0; for each edge e = (v, v ′ ) on the boundary of P ′ do tension[e] ← 1; for each vertex v in ∂P ′ do Compute height[v] using the values in tension;
repeat for each vertex v in ∂P ′ which has the maximal height among the heights of all the vertices in ∂P ′ do for each incoming edge e of v do tension[e] ← 1 − k; for each edge e′ in a cell containing e do tension[e′ ] ← 1; for each edge e = (v, v ′ ) such that tension[e] has newly been computed do Compute height[v] and height[v ′ ] using the values in tension; Remove in P ′ the cells which contain a negative edge; Compute the boundary of P ′ : it is composed of all the vertices of P ′ which have a computed height; until P ′ is empty; end
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Lemma 3.0.7 Let Q and Q′ be two tilings in the same connected component of AP of a polycell P . By definition, we put: X̸= := {x such that ϕQ (x) ̸= ϕQ′ (x) }, we take xm ∈ X̸= such that: max(ϕQ (xm ), ϕQ′ (xm )) = max{ϕQ (x), ϕQ′ (x); x ∈ X̸= } (in other words, xm is the maximum among the vertices x where ϕQ (x) ̸= ϕQ′ (x) ), We can suppose that ϕQ (xm ) > ϕQ′ (xm ) - else we exchange Q and Q′ -, then the potential ϕ defined by : ϕ(x) := ϕQ (xm ) − k, when x := xm and by ϕ(x) := ϕQ (x), otherwise is associated to a tiling of P . Proof: Let us consider any cell which contains xm . Therefore, it contains an incoming edge (xp , xm ) and an outgoing edge (xm , xs ) of xm . We will prove that ϕQ (xp ) = ϕQ (xm ) − 1 and ϕQ (xs ) = ϕQ (xm ) − k + 1, which will prove the claim since it proves that xm is a maximal vertex for ϕQ and so ϕ defines a tiling of P (which is therefore obtained from Q by a flip around xm ). The couple (ϕQ (xp ), ϕQ (xs )) can have three values: (ϕQ (xm ) − 1, ϕQ (xm ) +1), (ϕQ (xm )−1, ϕQ (xm )−k+1), or (ϕQ (xm )+k−1, ϕQ (xm )+ 1). But, if ϕQ (xs ) = ϕQ (xm ) + 1 then ϕQ′ (xs ) = ϕQ (xm ) + 1 (xs ∈ / X̸= ), and so ϕQ′ (xm ) = ϕQ (xm ) + k (xm ∈ X̸= ), which is in contradiction with ϕQ (xm ) > ϕQ′ (xm ). Therefore, (ϕQ (xp ), ϕQ (xs )) must be equal to (ϕQ (xm ) − 1, ϕQ (xm ) − k + 1) for every cell which contain xm , which is what we needed to prove. ! Let us now consider two tilings Q and Q′ of a k-regular contractible polycell P . Let us define max(ϕQ , ϕQ′ ) as the height function such that its value at each vertex is the maximal between the values of ϕQ and ϕQ′ at this vertex. Let us define min(ϕQ , ϕQ′ ) dually. Then, we have the following result: Lemma 3.0.8 Given two tilings Q and Q′ in the same connected component of AP for a k-regular contractible polycell P , max(ϕQ , ϕQ′ ) and min(ϕQ , ϕQ′ ) are the height functions of tilings of P . , ϕQ Proof: We can see that max(ϕQ! " ′ ) is the height "function of a tiling "ϕQ (x) − ϕQ′ (x)" can be decreased of P by iterating Lemma 3.0.7: x
until we reach max(ϕQ , ϕQ′ ). The proof for min(ϕQ , ϕQ′ ) is symmetric. !
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Theorem 3.0.9 If P is a k-regular contractible polycell, then each connected component of AP induces a distributive lattice structure over the tilings of P . Proof: Given two tilings Q and Q′ in the same connected component of AP , let us define the following binary operations: ϕQ ∧ ϕQ′ = min(ϕQ , ϕQ′ ) and ϕQ ∨ ϕQ′ = max(ϕQ , ϕQ′ ). It is clear from the previous results that this defines the infimum and supremum of Q and Q′ . To show that the obtained lattice is distributive, it suffices now to verify that these relations are distributive together. ! As already discussed, this last theorem gives much information on the structure of the flip-accessibility graphs of tilings of polycells. It also gives the possibility to use in the context of tilings the numerous results known about distributive lattices, in particular the generic random sampling algorithm described in [16]. To finish this section, we give another proof of Theorem 3.0.9 using only discrete dynamical models notions. This proof is very simple and has the advantage of putting two combinatorial object in a relation which may help understanding them. However, the reader not interested in discrete dynamical models may skip the end of this section. An Edge Firing Game (EFG) is defined by a connected undirected graph G with a distinguished vertex ν, and an orientation O of G. In other words, O = G. We then consider the set of obtainable orientations when we iterate the following rule: if a vertex v ̸= ν only has incoming edges (it is a sink ) then one can reverse all these edges. This set of orientations is ordered by the reflexive and transitive closure of the evolution rule, and it is proved in [15] that it is a distributive lattice. We will show that the set of tilings of any k-regular contractible polycell P is isomorphic to configuration space of an EFG, which implies Theorem 3.0.9. Let us consider a k-regular contractible polycell P defined over a graph G. Let G′ be the sub-graph of G which contains exactly the vertices and edges in P plus a new vertex ν and a edge (v, ν) for every v in ∂P . This vertex will be the distinguished vertex of our EFG. Let us now consider the height function ϕQ of a tiling Q of P , and let us define the orientation π(Q) of G′ as follows: the edges involving ν are directed towards ν, and each other undirected edge {v, v ′ } in G′ is directed from v to v ′ in π(Q) if ϕQ (v ′ ) > ϕQ (v). Then, the maximal vertices of Q are exactly the ones which have only incoming edges in π(Q), and applying the EFG rule to a vertex of π(Q) is clearly equivalent to making a flip
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around this vertex in Q. Moreover, one can never apply the EFG rule to a vertex in ∂P , since it always has an outgoing edge to ν, which can never be reversed. Finally, the configuration space of the EFG is isomorphic to the connected component of AP which contains Q, which proves Theorem 3.0.9 again. An example is given in Figure 9.
Figure 9: The configuration space of the EFG obtained from Figure 3 (the distinguished vertex ν is not represented: there is an additional outgoing edge from each vertex on the boundary to ν). The two orders are isomorphic.
4
Some applications.
In this section, we present some examples which appear in the literature, and we show how these tiling problems can be seen as special cases of kregular contractible polycells tilings. We therefore obtain as corollaries some known results about these problems, as well as some new results.
4.1
Polycell drawn on the plane or the sphere.
Let us consider a set of vertices V and a set Θ of elementary (undirected) cycles of length k, with vertices in V , such that any couple of cycles in Θ have at most one edge in common. Now let us consider the undirected
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graph G = (V, E) such that e is an edge of G if and only if it is an edge of a cycle in Θ. Moreover, let us restrict ourselves to the case where G is a planar graph which can be drawn in such a way that no cycle of Θ is drawn inside another one. G is 2-dual-colorable if one can color in black and white each bounded face in such a way that two faces which have an edge in common have different colors. See for example Figure 10. The
Figure 10: Two examples of graphs which satisfy all the properties given in the text. The leftmost is composed of cycles of length 3 and has a hole. The rightmost one is composed of cycles of length 4.
Figure 11: A tiling of each of the objects shown in Figure 10, obtained using the polycells formalism. fact that G has the properties above, including being 2-dual-colorable, makes it possible to encode tilings with bifaces (the tiles are two adjacent faces) as tilings of polycells. This includes tilings with dominoes, and tilings with calissons. Following Thurston [18], let us define an oriented version of G as follows: the edges which constitute the white cycles boundaries are directed to travel the cycle in the clockwise orientation, and the edges which constitute the black cycles boundaries are directed counterclockwise. If for every closed travel (its origin and its extremity coincide) on the boundary of polycell P we have FT (C) = 0 where C is a flow such that C(e) = 1 for every e ∈ ∂P , then we say that P has a balanced boundary. One can verify that a polycell with a balanced boundary defined in this way is always contractible. Therefore, our results can be applied, which generalizes some results of Chaboud [5] and Thurston [18].
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4.2
Rhombus tiling in higher dimension.
Let us consider the canonical basis {e1 ,! . . . , ed } of the d-dimensional d affine space R , and let us define ed+1 = di=1 ei . For every α between α as the following set of points: 1 and d+1, let us define the zonotope Zd,d α Zd,d
d
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d+1 "
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α is the zonotope defined by all the vectors e In other words, the Zd,d i except the α-th. We are interested in the tilability of a given solid S α , 1 ≤ α ≤ d + 1}. These tilings are when the set of allowed tiles is {Zd,d called codimension one rhombus tilings, and they are very important as a physical model of quasicristals [7]. If d = 2, they are nothing but the tilings of regions of the plane with three parallelograms which tile an hexagon, which have been widely studied. See Figure 12 for an example in dimension 2, and Figure 13 for an example in dimension 3. In order to encode this problem by a problem over polycells, let us consider the directed graph G with vertices in Zd and such that e = (x, y) is an edge if and only if y = x+ej for an integer j between 1 and d or y = x−ed+1 . We will call diagonal edges the edges which correspond to the second case. This graph can be viewed as a d-dimensional directed grid to which we add a diagonal edge in the reverse direction, at each point of the grid. An example in dimension 3 is given in Figure 14. Each edge
Figure 12: If one forgets the orientations and removes the dotted edges, then the rightmost object is a classical codimension one rhombus tiling of a part of the plane (d = 2). From the polycells point of view, the leftmost object represents the underlying graph G, the middle object represents a polycell P (the boundary of which is the set of the edges which belong to only one cell), and the rightmost object represents a tiling of P (the dotted edges are the tiling edges). α translated by an is in a one-to-one correspondence with a copy of a Zd,d integer vector, namely the one of which it is the diagonal edge. The set Θ of the cells we will consider is the set of all the circuits of length d + 1
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Figure 13: A codimension one rhombus tiling with d = 3 (first line, rightmost object). It is composed of four different three dimensional tiles, and the first line shows how it can be constructing by adding successive tiles. The second line shows the position of each tile with respect to the cube.
Figure 14: Top: the 3-dimensional grid is obtained by a concatenation of cubes with reverse diagonal edges, like this one. Bottom: the cells in Θ. Each tile is composed of six such cells, since each edge belongs to exactly six cells. which contain exactly one diagonal edge. Therefore, each edge belongs to d! cells, and so the tiles will be themselves composed of d! cells. See Figure 14 for an example in dimension 3. Given a polycell P defined over Θ, we define ∂P as the partial subgraph composed by the edges of P which do not belong to d! circuits of P . First notice that a full polycell defined over G is always contractible. Therefore, our previous results can be applied, which generalizes some results presented in [7] and [12]. We also generalize some results about the 2-dimensional case, which has been widely studied.
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Conclusion and Perspectives.
In conclusion, we gave in this paper a generalized framework to study some tiling problems over which a height function can be defined. This includes the famous tilings of polyominoes with dominoes, as well as various other classes, like codimension one rhombus tilings, tilings with holes, tilings on torus, on spheres, three-dimensional tilings, and others we did not detail here. We gave some results on our generalized tilings which made it possible to obtain a large set of known results as corollaries, as well as to obtain new results on tiling problems which appear in the scientific literature. Many other problems may exist which can be modelized in the general framework we have introduced, and we hope that this paper will help understanding them. Many tiling problems, however, do not lead to the definition of any height function. The key element to make such a function exist is the presence of a strong underlying structure (the k-regularity of the polycell, for example). Some important tiling problems (for example tilings of zonotopes) do not have this property, and so we can not apply our results in this context. Some of these problems do not have the strong properties we obtained on the tilings of k-regular contractible polycells, but may be included in our framework, since our basic definitions of polycells and tilings being very general. This would lead to general results on more complex polycells, for example polycells which are not k-regular, or with cells which have more than one edge in common. Acknowledgement The authors thank Fr´ed´eric Chavanon for useful comments on preliminary versions, which deeply improved the manuscript quality. Olivier Bodini LIRMM, University of Montpellier 2, 161, rue Ada, 34392 Montpellier Cedex 5, FRANCE bodini@lirmm.fr
Matthieu Latapy LIAFA, University of Paris 7, 2, place Jussieu, 75005 Paris, FRANCE latapy@liafa.fr
References [1] Ahura R.K., Network Flows Theory, Algorithms and Applications, Prentice Hall (1976).
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[2] Berger R., The indecidability of the domino problem, Mem. of Amer. Math. Soc., 66 (1966). [3] Beauquier D.; Nivat M.; Remila E.; Robson J.M., Tiling figures of the plane with two bars, a horizontal and a vertical one, Computational Geometry, 5 (1995), 1-25. [4] Caspar D.L.D; Fontano E., Five-fold symmetry in crystalline quasicrystal lattices, Proc. Natl. Acad. Sci.USA, 93 (1996), 1427114278. [5] Chaboud T., Domino Tiling in planar graphs with regular and bipartite dual, TCS, 159 (1996), 137-142. [6] Cohn H.; Propp J.; Shor P., Random domino tilings and the arctic circle theorem, (2001) Preprint. [7] Destainville N.; Mosseri R.; Bailly F., Configurational entropy of codimension-one tilings and directed membranes, Journal of Statistical Physics, 87, (1997), 697-713. [8] Davey B.A.; Priestley H.A., Introduction to lattices and orders, Cambridge University Press, (1990). [9] Fournier J.C., Tiling pictures of the plane with dominoes, Discrete Mathematics, 156 (1997), 313-320. [10] Kenyon R., The planar dimer model with boundary: a survey, AMS-CRM Monogr., (2000), 307-328. [11] Latapy M., Generalized integer partitions, tilings of zonotopes and lattices, preprint. [12] Linde J.; Moore C.; Nordahl M.G., An n-dimensional generalization of the rhombus tiling, Proceedings of DM-CCGâ&#x20AC;&#x2122;01 (2001), 23-42. [13] Lagarias J.C.; Romano D.S., A polyomino tiling ploblem of Thurston and its configuration entropy, Journal of Combinatorial Theory, 63 (1993), 338-358. [14] Moore C.; Robson J.M., Hard tiling problems with simple tiles, preprint to appear in Discrete Mathematics.
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[15] Propp J., Lattice structure of orientations of graphs, preprint (1993). [16] Propp J., Generating random elements of finite distributive lattices, Electronic Journal of Combinatorics, 4 (1998). [17] Remila E., The lattice structure of the set of domino tilings of a polygon, to appear. [18] Thurston W.P., Conwayâ&#x20AC;&#x2122;s tiling groups, Amer. Math. Monthly, 97 (1990), 757-773.
Morfismos, Vol. 7, No. 1, 2003, pp. 69–75
On Anosov energy levels that are of contact type Osvaldo Osuna-Castro
1
Abstract In this work we prove that given an autonomous Lagrangian L on a closed manifold M , if an Anosov energy level k can be reparametrized to make it of contact type, then k > c0 (L), the critical value of L associated with the abelian covering.
2000 Mathematics Subject Classification: 37D40, 53D25. Keywords and phrases: Anosov flows, asymptotic cycle, contact type levels, Man ˜e´s critical values.
1
Introduction
Let M be a closed connected manifold, T M its tangent bundle. An autonomous Lagrangian is a smooth function, L : T M → R convex and superlinear. This means that L restricted to each Tx M has positive definite Hessian and for some Riemannian metric we have L(x, v) = ∞, |v| |v|→∞ lim
uniformly on x. Since M is compact, the Euler-Lagrange equation defines a complete flow ϕt on T M . Recall that the energy E : T M → R is defined by ∂L (x, v)v − L(x, v). E(x, v) := ∂v Since L is autonomous, E is a first integral of the flow ϕt . Let us set e := maxx∈M E(x, 0) = −minx∈M L(x, 0). 1
Ph.D. Student, CIMAT, Guanajuato, Gto., M´exico. Supported by a scholarship from the CONACYT.
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Note that the energy level E −1 {k} projects onto the manifold M if and only if k ≥ e.
We shall denote by L:T M → T ∗ M the Legendre transform which is defined by (x, v) → ∂L ∂v (x, v). Our hypotheses on L assure that L is a diffeomorphism. Let H : T ∗ M → R be the Hamiltonian associated to L: H(x, v) := maxv∈Tx M {pv − L(x, v)}.
If θ denotes the canonical 1-form on T ∗ M , then the Euler-Lagrange flow of L can be also obtained as the Hamiltonian flow of E with respect to the symplectic form on T M given by −L∗ dθ thus, if X denotes the vector field associated with the Euler-Lagrange flow then iX L∗ dθ = −dE. In other words, the energy function satisfies E = H ◦ L, so that energy levels for L are sent to level sets of H. Definition: An energy level Σ = H −1 {k} is of contact type if there exists a 1-form λ on Σ such that dλ = ω(= −dθ) and λ(X) ̸= 0 on every point of Σ. An Anosov energy level, is a regular energy level E −1 {k} on which the flow ϕt is an Anosov flow. In [6] G. Paternain shows that if an Anosov energy level k on a surface can be reparametrized to make it of contact type then k > c0 (L) the critical value of L associated with the abelian covering. Our goal in this note is to generalize this result, we shall prove the following: Theorem A. Given an autonomous Lagrangian L on a closed manifold M with dim M ≥ 2 , If an Anosov energy level k can be reparametrized to make it of contact type then k > c0 (L). This completes the previous result.
2
Preliminaries and proofs
Let M(L) be the set of probabilities on the Borel σ-algebra on T M that have compact support and are invariant under the flow ϕt . Let H1 (M, R) be the first real homology group of M . Given a closed oneform ω on M and ρ ∈ H1 (M, R), let ⟨[w], ρ⟩ denote the integral of ω on any closed curve in the homology class ρ. If µ ∈ M(L), its homology is defined as the unique ρ(µ) ∈ H1 (M, R) such that ⟨[w], ρ(µ)⟩ =
!
TM
ωdµ,
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for all closed 1-form on M . The integral on the right-hand side is with respect to µ with ω considered as the function ω : T M → R.
Let µ be a ϕt -invariant probability supported on the energy level Σ = E −1 {k}. The Schwartzman’s asymptotic cycle S(µ) ∈ H1 (Σ, R) of µ is defined by ! ⟨[Ω], S(µ)⟩ =
Ω(X)dµ,
Σ
for every closed 1-form Ω on Σ, where X is the Lagrangian field on Σ. The homology ρ(µ) of the measure µ is the projection of its asymptotic cycle by π∗ : H1 (Σ, R) → H1 (M, R).
Recall that the L-action of an absolutely continuous curve γ : [a, b] → M is defined by AL (γ) :=
! b
L(γ(t), γ(t))dt. ˙
a
Given two points x1 , x2 ∈ M and some T > 0 denoted by C(x1 , x2 ) the set of absolutely continuous curves γ : [a, b] → M with γ(0) = x1 and γ(T ) = x2 . For each k ∈ R, we define Φk (x1 , x2 ; T ) := inf{AL+k (γ) | γ ∈ C(x1 , x2 ) }. The action potential Φk : M × M → R ∪ ∞ of L is defined by Φk (x1 , x2 ) := infT >0 Φk (x1 , x2 ; T ). Definition (Ma˜ ne): The critical value of L is the real number c = c(L) := inf{k | Φk (x, x) > −∞ for some x ∈ M }. Note that if k > c(L) actually Φk (x, x) > −∞ for all x ∈ M . Since L is convex and superlinear, and M is compact, such a number exists. We can also consider the critical value of the lift of the Lagrangian L to a covering of the compact manifold M . Suppose that p : N → M is a covering space and consider the Lagrangian L : T N → R given by L := L◦dp, for each k ∈ R we can define an action potential Φk in N × N just as above and similarly we obtain a critical value c(L) for L. It can be easily checked that if N1 and N2 are coverings of M such that N1 covers N2 then c(L1 ) ≤ c(L2 ), where L1 and L2 denote the lifts of the Lagrangian L to N1 and N2 respectively.
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Among all possible coverings of M there are two distinguished ones; !, and the abelian covthe universal covering which we shall denote by M ering which we shall denote by M . The latter is defined as the covering of M whose fundamental group is the kernel of the Hurewicz homomorphis π1 (M ) → H1 (M, R) these coverings give rise to the critical values " and ca (L) = c0 (L) := c(L) cu (L) := c(L) " and L denote the lifts of the Lagrangian L to M ! and M respecwhere L tively. Therefore we have cu (L) ≤ c0 (L), but in general the inequality may be strict as it was shown in [5].
2.1
Contact and Anosov energy levels
We begin by introducing some concepts related to Euler-Lagrange flow restricted on energy levels. Definition: An energy level Σ = H −1 {k} is of contact type if there exists a 1-form λ on Σ such that dλ = ω(= −dθ) and λ(X) ̸= 0 on every point of Σ. Equivalently, if there exists a vector field Y based on Σ, such that the Lie derivative LY ω = ω. The correspondence is given by iY ω = λ. The vector field Y must be tranverse to Σ because if it is tangent to Σ one has that λ(X) = ω(Y, X) = dH(Y ) = 0. Lemma 2.2.1 The set {k ∈ R | H −1 {k} is of contact type} is open. Proof: Suppose that Σ = H −1 {k} is of contact type, then k is a regular point of H, for otherwise the Hamiltonian flow contains a singularity on Σ and that violates the condition λ(X) ̸= 0. If λ is a contact form for λ, since dλ = ω then λ = pdx|Σ + τ , where τ is a closed 1-form on Σ. We can extend λ as follows. Let π : U → Σ be the projection of an open neighbourhood U of Σ onto Σ. Let λ := pdx + π ∗ (τ ) then dλ = ω and for m near k dλ|H −1 {m} ̸= 0. ✷ The following criterion for contact type appears in [2]
Proposition 2.2.2 If L is a convex Lagrangian then an energy level # −1 E {k} is of contact type if and only if T M (L+k)dµ > 0 for any invariant measure µ supported E −1 {k} with zero asymptotic cycle S(µ) = 0. We shall need the following result:
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Lemma 2.2.3 Suppose M a closed connected manifold with dim M ≥ 2 and M ̸= T 2 . If k > e then π∗ : H1 (E −1 {k}, R)→ H1 (M, R) is an isomorphism. Proof: Since k > e and dim M ≥ 2 then the energy level E −1 {k} is isomorphic to the unit tangent bundle of M with the canonical projection. Using the Gysin exact sequence of the circle bundle π : E −1 {k} → M one can show that (see [3], lemma 1.45) the lemma follows if M is orientable. If M is not orientable and dim M ≥ 3, using the exact homotopy sequence of the circle bundle π : E −1 {k} → M : π
0 = π1 (S n−1 ) → π1 (E −1 {k}) →∗ π1 (M ) → π0 (S n−1 ) = 0, thus we obtain that π∗ : π1 (E −1 {k}) → π1 (M ) is an isomorphism, which in turn implies that π∗ : H1 (E −1 {k}, R)→ H1 (M, R) is a isomorphism. In the case that M is not orientable and dim M = 2, the proof is a minor modification of the above arguments. ✷ An Anosov energy level, is a regular energy level E −1 {k} on which the flow ϕt is an Anosov flow. In [1] was shown Proposition 2.2.4 If the energy level k is Anosov, then k > cu (L). In [5] G. Paternain and M. Paternain gave examples of Anosov energy levels k with k < c0 (L) on surface of genus greater or equal than two. These examples give a negative answer to a question raised by Ma˜ ne.
2.3
Proof of theorem A
Now we shall prove the theorem A, for this we use the next result of Paternain [4] and following his ideas we shall prove this result Proposition 2.3.1 If cu (L) < k < c0 (L), there exists an invariant measure µ supported in the energy level k, such that ρ(µ) = 0 and !
E −1 {k}
(L + k)dµ ≤ 0.
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Proof of theorem A: It follows from a result of Margulis that the energy levels of T 2 does not support Anosov flows thus in the case of T 2 the theorem is valid trivially. Therefore we can assume that M ̸= T 2 . Now as the flow is Anosov, by the proposition 2.2.4 we have that k > cu (L). But if the energy level k ∈ (cu (L), c0 (L)) then applying the proposition 2.3.1, there exists an invariant measure µ such that ρ(µ) = 0 and !
E −1 {k}
(L + k)dµ ≤ 0.
therefore the lemma 2.2.3 and proposition 2.2.2 implies that, the energy level k is not of contact type. Finally by lemma 2.2.1 the energy k = c0 (L) cannot be of contact type then, we must have that k > c0 (L). ✷ Proof of proposition 2.3.1 : Since k < c0 (L) = ca (L) there exists T > 0 and an absolutely continuous closed curve γ : [0, T ] → M homologous to zero such that (1) AL+k (γ) < 0. For n ≥ 1, let us denote by γ n : [0, nT ] → M the curve γ wrapped up n times. Since k > cu (L), by (1) γ n cannot be homotopic to zero. " → M the covering projection and take y such that p(y) = Let p : M " be the unique lift of γ n with γ(0) = γ(T ), and let γ# n : [0, nT ] → M γ# n (0) = y. As k > cu (L) for each n there exists a solution xn (t) of Euler-Lagrange with energy k and some Tn > 0 such that xn (0) = y and xn (Tn ) = γ# n (nT ).
Let µn denote the probability measure in T M uniformly distributed along p ◦ xn |[0,Tn ] and take µ a point of accumulation of µn , this measure µ has the required properties of the proposition 2.3.1. ✷
Osvaldo Osuna Castro CIMAT, A. Postal 402, ´ 3600 Guanajuato, Gto., MEXICO. osvaldo@cimat.mx
References [1] Contreras, G.; Iturriaga, I.; Paternain, G. P.; Paternain, M., Lagrangian graphs, minimizing measure and Man´es critical values, Geom. Func. Anal., 8 (1998), 788-809.
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[2] McDuff, D., Applications of convex integration to symplectic and contact geometry, Ann. Inst. Fourier, 37 (1987), 107-133. [3] Paternain, G. P., Geodesic flows, Birkhauser Boston Inc, Boston, MA, 1999. [4] Paternain, G. P., Hyperbolic dynamics of Euler-Lagrange flows on prescribed energy levels, S´eminaire de Th´erie spectrale et G´eom´etrie, 1996-97, Grenoble, 15 (1998). [5] Paternain, G. P.; Paternain, M., Critical values of autonomous Lagrangian systems, Comment. Math. Helvet., 72 (1997), 481499. [6] Paternain, G. P., On the regularity of the Anosov splitting for twisted geodesic flows, Math Res. Lett., 4 (1998), 871-888.
MORFISMOS, Comunicaciones Estudiantiles del Departamento de Matem´aticas del CINVESTAV, se termin´ o de imprimir en el mes de junio de 2003 en el taller de reproducci´ on del mismo departamento localizado en Av. IPN 2508, Col. San Pedro Zacatenco, M´exico, D.F. 07300. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares en pasta tintoreto color verde.
Apoyo t´ecnico: Omar Hern´ andez Orozco.
Contenido The Lagrange approach to constrained Markov control processes: a survey and extension of results Raquiel R. L´ opez-Mart´ınez and On´esimo Hern´ andez-Lerma . . . . . . . . . . . . . . . 1
Representaciones discretas en tiempo-frecuencia y el problema de la seleccio´n de frecuencias Alin Andrei Ca ˆrsteanu Mani¸tiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Generalized tilings with height functions Olivier Bodini and Matthieu Latapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
On Anosov energy levels that are of contact type Osvaldo Osuna-Castro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69