Morfismos, Vol 7, No 2, 2003 by Morfismos, Department of Mathematics, Cinvestav

VOLUMEN 7 NÚMERO 2 JULIO A DICIEMBRE DE 2003 ISSN: 1870-6525

MORFISMOS Comunicaciones Estudiantiles Departamento de Matem´aticas Cinvestav Editores Responsables • Isidoro Gitler • Jes´ us Gonz´alez

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VOLUMEN 7 NÚMERO 2 JULIO A DICIEMBRE DE 2003 ISSN: 1870-6525

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Editorial Guidelines “Morfismos” is the journal of the students of the Mathematics Department of CINVESTAV. One of its main objectives is for students to acquire experience in writing mathematics. MORFISMOS appears twice a year. Publication of papers is not restricted to students of CINVESTAV; we want to encourage students in Mexico and abroad to submit papers. Mathematics research reports or summaries of bachelor, master and Ph.D. theses will be considered for publication, as well as invited contributed papers by researchers. Papers submitted should be original, either in the results or in the methods. The Editors will assign as referees well–established mathematicians. Even though MORFISMOS is a refereed journal, the papers will be considered as preliminary versions which could later appear in other mathematical journals. If you have any suggestions about the journal, let the Editors know and we will gladly study the possibility of implementing them. We expect this journal to foster, as a preliminary experience, the development of a correct style of writing mathematics. Morfismos

Contenido When does a manifold admit a metric with positive scalar curvature? Egidio Barrera-YaË&#x153; nez and JosÂ´e Luis Cisneros-Molina . . . . . . . . . . . . . . . . . . . . . 1

A survey on modular Hadamard matrices Shalom Eliahou and Michel Kervaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Application of modularity to optimal resource allocation with risk sensitivity Guadalupe Avila-Godoy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Morfismos, Vol. 7, No. 2, 2003, pp. 1–16

When does a manifold admit a metric with positive scalar curvature? ∗ Egidio Barrera-Yan ˜ez

Jos´e Luis Cisneros-Molina

Abstract The scalar curvature is the weakest geometric invariant in a Riemannian manifold. M. Gromov, B. Lawson Jr. and J. Rosenberg conjectured that a Riemannian manifold admits a metric with positive scalar curvature if and only if certain topological invariant ˆ called A-genus vanishes. This is known as the Gromov-LawsonRosenberg conjecture. In this article we explain this conjecture and give a brief survey of some results related to it.

2000 Mathematics Subject Classification: 53C21, 55N15, 55N22, 34L40 Keywords and phrases: Positive scalar curvature, Dirac operator, connective K-theory.

Introduction

Riemannian Geometry is devoted to the study of Riemannian manifolds (M n , g), that is, diﬀerentiable manifolds M m endowed with a Riemannian metric g. Since the manifold M n is also a topological manifold, one of the most important problems in Riemannian Geometry is to study which constrains imposes the topology of M n on the geometry given by the Riemannian metric g. More specifically, one would like to study the relation between some topological invariants of the underlying manifold M n with the curvature of the Riemannian manifold (M n , g). In the present paper we shall only consider closed manifolds, i.e., compact manifolds without boundary. ∗

Invited article Supported by Proyecto PAPIIT IN110702-2 2 Supported by Proyecto PAPIIT IN110702-2 1

Barrera-Yan ˜ez and Cisneros-Molina

The curvature tensor can be viewed as a quadratic form Q in the double exterior product of the tangent bundle of M , 2 T (M ), the positive definiteness of Q is one of the strongest positivity conditions, for example all compact symmetric spaces have Q ≥ 0 while Q > 0 distinguishes spheres and real projective spaces. The restriction of Q to bivectors in 2 T (M ) is the sectional curvature K and twice the sum of the sectional curvatures over all two planes in a tangent space to a point give us the scalar curvature s (see [2]) therefore, we have the following implications: Q > 0 ⇒ K > 0 ⇒ s > 0. In a Riemannian manifold (M n , g) the scalar curvature can be built in a certain way out of the first and second derivatives of g, so we can recover s from the metric g. Hence it is natural to ask: • Given a Riemannian manifold M = (M n , g). When does M admit a metric with s > 0 or s = 0 or s < 0? For the case s ≤ 0, this condition has no topological eﬀect on M by a theorem of Kasdan and Warner [12, 13] which claims the existence of a metric s ≤ 0 on every manifold of dimension n ≥ 3 and a theorem of Lohkamp [18] that states that the space R− (M ) of negative scalar curvature metrics on M is contractible for every closed manifold M n of dimension n ≥ 3. Going back to the case s > 0, there are two obvious questions: 1. How can I construct a manifold with a metric with positive scalar curvature? 2. How can I decide if a manifold admit a metric with positive scalar curvature? For the existence of a metric with positive scalar curvature, one can prove that if a manifold M has a metric with positive scalar curvature then M ×N also has a metric with positive scalar curvature since we can shrink the product metric by a positive factor at every point and then using the fact that both manifolds are compact find a common factor, there are also generalizations (for vector bundles) of this technique, see [33] for details. Concerning the other question, the way we decide if a metric has positive scalar curvature is using obstructions:

Manifolds with positive scalar curvature

1. Index Obstructions. This method is based on the “BochnerLichnerowicz-Weitzenbrock formula” which gives a relation between the scalar curvature and the “Dirac operator” (see 2.2) defined by Atiyah-Singer on any Riemannian manifold with a spin structure (see 1.1). 2. Minimal hypersurface method. Schoen and Yau proved that if M is a manifold of dimension n with positive scalar curvature then any stable minimal hypersurface N (i.e. N is a local minimum of the area functional) also admits positive scalar curvature. 3. Seiberg-Witten invariants. This is an invariant for 4-dimensional manifolds which vanishes if the manifold admits a metric with positive scalar curvature, see [34] for details. In the present article we shall focus on the Index Obstruction method. For further details on these methods we recommend the survey of Stolz [33]. Let us start considering the dimension of the manifold n = 2, in this case, the scalar curvature coincides with the Gaussian curvature and the Gauss-Bonnet formula relates it to the Euler-Poincar´e characteristic χ(M ), which is a topological invariant of the 2-manifold M : χ(M ) = (4π)−1

s(x)dvol(x).

Thus if a 2 dimensional manifold M admits a metric of positive scalar curvature, then χ(M ) > 0 and by the classification theorem of 2-manifolds, this implies that M = S 2 or M = RP 2 and indeed, these manifolds do admit metrics of positive scalar curvature. Thus χ(M ) > 0 if and only if M admits a metric of positive scalar curvature. The situation is very diﬀerent in higher dimensions. In dimension n = 3 work of Shoen and Yau [29] with the Thurston conjecture [35] (perhaps soon established by Perelman [20, 19]) yields a complete classification of 3-manifolds with positive scalar curvature. For n = 4, see comment about Seiberg–Witten invariants above. We shall concentrate henceforth on the case n ≥ 5. If one deforms (cut and paste) the manifold, one obtains a manifold that will have a metric of positive scalar curvature, the two common methods are “surgery” and “attaching handles” which are related. Let M be a manifold with boundary ∂M , we recall that a “handle” is the product of two discs Dk × Dn−k , the boundary of this “handle” consist of two parts

Barrera-Yan ˜ez and Cisneros-Molina

S k−1 ×Dn−k and Dk ×S n−k−1 , given an embedding of S k−1 ×Dn−k into the boundary ∂M of an n-dimensional manifold M , we can construct a new manifold ˜ =M M Dk × Dn−k S k−1 ×Dn−k

by taking the disjoint union of M and the “handle” Dk × Dn−k and identifying the points in S k−1 × Dn−k with their image in ∂M . We say ˜ is obtained by attaching a k-handle to M , or M ˜ is obtained by that M k n−k surgery (i.e. by removing S ×D and replacing it by Dk+1 ×S n−k−1 ). It is natural to ask whether a metric of positive scalar curvature in M ˜ , here the can be extended to a metric of positive scalar curvature in M metrics we have in mind are product metrics near the boundary (i.e. a neighborhood of ∂M is isometric to the product of ∂M with an interval). Gromov-Lawson [9] and Shoen-Yau [29] showed (independently) that if M admits a metric of positive scalar curvature, and n − k ( ˜ also the codimension of the surgery/handle) is greater than 2, then M admits such a metric. It is worth giving some of the flavor involved. Let S k be an embedded k dimensional sphere in M with trivial normal bundle ν. This means that a tubular neighborhood of S k has the form S k ×Dm−k and associated boundary S k ×S m−k−1 . Shrink the size of the tubular neighborhood. It is possible to deform the original metric on M to a metric which is greater than 0 in a neighborhood the boundary S k × S m−k−1 in such a way that the new metric still has positive scalar curvature. It is at this point that the assumption that m − k ≥ 3 is crucial to ensure that the standard metric of the fiber spheres S m−k−1 has positive scalar curvature and this dominates as the size of these spheres is shrunk by taking an adiabatic limit. The surgery can be performed; one cuts out the S k × intDm−k and glues in a Dk+1 × S m−k−1 and preserves the positivity of the scalar curvature, later Gajer [5] extend the result to Theorem 1.1 Let M be a manifold with boundary and let g be a metric ˜ is obtained from M of positive scalar curvature on M . Assume that M by attaching a handle of codimension ≥ 3. Then g extends to a metric ˜ of positive scalar curvature in M Gromov and Lawson [9] made the important observation that if a manifold M belongs to certain class of manifolds, called spin manifolds, whether it admits a metric of positive scalar curvature depends only on the bordism class of M in a suitable bordism group called M Spinn (Bπ)

Manifolds with positive scalar curvature

with π be the fundamental group of the manifold M . Recall that two manifolds M and N of dimension n are bordant if there exists a manifold W of dimension n + 1 such that ∂W is the disjoint union M N . For the group M Spinn (Bπ) the extra structure we need is called spin structure which we explain in the next section.

1.1

Cliﬀord Algebras and Spin Structures

Let Cliff ± (n) denote the real Clifford algebra on Rn . This is the universal unital algebra generated by Rn subject to the Clifford commutation relations v ∗ w + w ∗ v = ±(v, w)1.

Let Cliff c (n) := Cliff − (n) ⊗R C be the complexification. Note that Cliff − (n) ⊗R C and Cliff + (n) ⊗R C are isomorphic. Let P in± (n) ⊂ Cliff ± (n) be the multiplicative subgroup generated by the unit sphere of Rn ; i.e. P in± (n) = {x = v1 ∗ ... ∗ vk : |vi | = 1

for some

k}.

Define the following groups and representations • Let P inc (n) := P in− (n) ×Z 2 S 1 where we identify (g, λ) and (−g, −λ), • det : P inc (n) → S 1 by det(g, λ) = λ2 , • χ : P in± (n) → Z2 by χ(v1 ∗ ... ∗ vk ) = (−1)k , and • Ψ : P in± (n) → O(n) by Ψ(x) : w

χ(x)x ∗ w ∗ x−1 .

• Spin(n) = ker(χ) ∩ P in− (n) ≈ ker(χ) ∩ P in+ (n), and • Spinc (n) = Spin(n) ×Z 2 S 1 . Let n ≥ 3. Then Ψ defines a surjective group homomorphism from Spin(n) to the orthogonal group SO(n). Since Spin(n) is connected we have that π1 (SO(n)) = Z2 , and ker(Ψ) = {±1} ⊂ Spin(n), we have Spin(n) is the universal covering group of SO(n). Note that Ψ defines a surjective group homomorphism from P in± (n) to the orthogonal group O(n); this exhibits P in± (n) as a universal covering groups of O(n). Since O(n) is not connected, the universal

Barrera-Yan ˜ez and Cisneros-Molina

cover is not uniquely defined as a group, one must decide how to multiply the arc components and P in± (n) are the two possible universal covering groups. We extend χ and Ψ to P inc (n) by defining χ(x, λ) = χ(x)

and

Ψ(x, λ) = Ψ(x).

Let ξ be a real vector bundle of dimension k with an inner product. We say that ξ admits a pin± or a pinc structure if we can lift the transition functions of ξ from the orthogonal group O(k) to the group P in± (k) or P inc (k). We say that ξ admits a spin or a spinc structure if ξ is orientable and if we can lift the transition functions to Spin(k) or Spinc (k). We say that a manifold M admits such a structure if the tangent bundle T (M ) admits this structure. This condition can be expressed in terms of characteristic classes. Let wi (ξ) for i = 1, 2 be the first two Stiefel-Whitney classes of ξ. We refer to Giambalvo [6] for the proof of the following results. It shows that we can stabilize; a bundle ξ admits a suitable structure if and only if ξ ⊕ 1 admits this structure. Lemma 1.2 Let ξ be as before. • The bundle ξ admits a spin structure ⇐⇒ w1 (ξ) = 0 and w2 (ξ) = 0. • The bundle ξ admits a spinc structure ⇐⇒ w1 (ξ) = 0 and if w2 (ξ) lifts from H 2 (M ; Z2 ) to H 2 (M ; Z). • The bundle ξ admits a pin− structure ⇐⇒ w2 (ξ) = 0. • The bundle ξ admits a pinc structure H 2 (M ; Z2 ) to H 2 (M ; Z).

⇐⇒

w2 (ξ) lifts from

For examples of manifolds with these structures, consider RP l , the real projective manifold of dimension l, since T (RP l ) ⊕ 1 = (l + 1)L, where L is the Hopf bundle, we have: • RP 4l and (4l + 1)L admit pin+ structures. • RP 4l+1 and (4l + 2)L admit spinc structures. • RP 4l+2 and (4l + 3)L admit pin− structures.

Manifolds with positive scalar curvature

• RP 4l+3 and (4l + 4)L admit spin structures. • Other examples of manifolds that admit spin structures are: S n for n ≥ 2, CP n for n odd. Now with the notion of spin structure we can define the spin bordism groups M Spinn (Bπ). Two spin manifolds M and N of dimension n are spin bordant if there exist a spin manifold W of dimension n + 1 such that its boundary is the disjoint union of M and N and the restriction of the spin structure on W coincides with the spin structures on M and N.

Non existence of metrics of positive scalar curvature

We saw that the Euler-Poincar´e characteristic is the invariant that tell us when a 2-dimensional manifold admits a metric with positive scalar curvature, so we are looking for a generalization of this invariant, the pioneer of the solution for the non-existence of metrics of positive scalar curvature was Lichnerowicz, see [17], his method is based on the “Bochner-Lichnerowicz-Weitzenbo¨ck formula”. In order to state Lichnerowicz Theorem we need to explain the following concepts:

2.1

Spinor Bundle

Let M be a Riemannian manifold of dimension n = 2k, the spinor bundle is a vector bundle S → M . S = Spin(M ) ×Spin(n) ∆ where ∆ is a certain representation of Spin(n) called the spinor representation, which is constructed as follows: identify Spin(n) with a subgroup of units of the Clifford algebra Cliff (n), and ∆ is a certain Cliff c (n)-module considered as a representation of Spin(n) in the units of Cliff c (n) = Cliff c (2k) which is the algebra C(2k ) = M2k ×2k (C) of 2k × 2k matrices over C (see [16]). k Let ∆ be C2 with the C(2k )-module structure given by multiplying a 2k ×2k -matrix by a 2k -vector. We consider ∆ as a module over Cliff c (2k) and define a Z2 grading, ∆ := ∆+ ⊕∆− where ∆± are the ±1-eigenspace of the involution given by the multiplication by the complex volume element ωC = ι2k e1 · · · e2k in Cliff c (n), the vectors {e1 , . . . e2k } form an

Manifolds with positive scalar curvature

1 D2 = ∇∗ ∇ + s. 4 Where the connection Laplacian is the operator ∗

∞

∗

∇ ∇ : C (E) → C (E) defined by ∇ ∇(ϕ) = −

m !

i,j=1

∇ei ∇ej ϕ.

One can use this formula to compute |D

ϕ|2L2

|∇ϕ|2L2

1 + 4

s(ϕ, ϕ)dvol. M

Therefore if the metric in question has positive scalar curvature, then there are no elements in ker D (harmonic spinors).

2.3

ˆ A-genus

ˆ We recall that for an oriented vector bundle E → M , A(E) ∈ H ∗ (M, Q) given by: 1 1 ˆ A(E) = 1 − p1 + 7 2 (−4p2 + 7p21 ) + . . . 24 2 ·3 ·5 where pj = pj (E) ∈ H 4j (M, Z) are the Pontryagin classes of E, see [16]. The famous Atiyah-Singer Index Theorem (see [16] for details) identifies ˆ IndexD+ with the A-genus. ˆ If M is a manifold of dimension m = 4k the the A-genus is: ˆ ) = ⟨A(T ˆ M ), [M ]⟩ ∈ Q. A(M Theorem 2.1 (Lichnerowicz) Let M be a closed spin manifold of dimension M = 4k which admits a metric of positive scalar curvature, ˆ ) = 0. then A(M Notice that the assumption of spin is very important, consider the following example: CP 2 = S 5 /S 1 is a manifold with positive scalar curvature, since it is a Riemannian submersion, and 1 2 ˆ A(CP ) = −( )sign(CP 2 ) ̸= 0 8 but CP 2 is not a spin manifold, see Lemma 1.2.

Barrera-Ya˜ nez and Cisneros-Molina

Gromov-Lawson-Rosenberg Conjecture

If g is a Riemannian metric on M , let D(M, s, g) be the associated ˆ Dirac operator defined by the spin structure s. We define the A-genus as follows: 1. If m ≡ 0 mod 4, decompose D(M, s, g) = D+ (M, s, g)+D− (M, s, g) ˆ and let A(M, s, g) := dim ker(D+ (M, s, g))−dim ker(D− (M, s, g)) ∈ ± Z; the D are the chiral spin operators. ˆ 2. If m ≡ 1 mod 8, let A(M, s, g) = dim ker(D(M, s, g)) ∈ Z2 . ˆ 3. If m ≡ 2 mod 8, let A(M, s, g) =

1 2

dim ker(D(M, s, g)) ∈ Z2 .

ˆ 4. If m ̸≡ 0, 1, 2, 4 mod 8, let A(M, s, g) = 0. ˆ One can use the Atiyah-Singer index theorem to show that A(M, s) = ˆ A(M, s, g) is independent of the metric g, also notice that in dimension 2 the invariant (Euler-Poincar´e characteristic χ(M )) is independent of the Riemannian metric and certain index invariant introduced by Hitchin are independent of the Riemannian metric, see [10] for details. If M is simply connected, the spin structure s is unique and we let ˆ ) = A(M, ˆ A(M s). If M admits a metric of positive scalar curvature, the formula of Lichnerowicz [17] shows there are no harmonic spinors; consequently ˆ A(M, s) = 0. In other words, if there exists a spin structure s on M so ˆ that A(M, s) ̸= 0, then M does not admit a metric of positive scalar ˆ curvature. Gromov and Lawson conjectured that the A-genus might be the only obstruction to the existence of a metric of positive scalar curvature if the dimension n was at least 5 and if M was a simply connected spin manifold. Stolz used deep homotopy theory to identify ˆ the kernel of A(M, s), see [31] for details, he established this conjecture by proving: Theorem 3.1 If M is a simply connected, closed, spin manifold of dimension n ≥ 5, then M admits a metric of positive scalar curvature ˆ ) = 0. if and only if A(M The situation in the non-simply connected setting is quite diﬀerent. Rosenberg has modified the original conjecture of Lawson and Gromov. Fix a group π. Let M be a connected manifold of dimension n ≥ 5

Manifolds with positive scalar curvature

with fundamental group π and spin universal cover. Rosenberg conjectured that M admits a metric of positive scalar curvature if and only if a generalized equivariant index απ (see [10, 21]) of the Dirac operator vanishes. For the fundamental groups that we shall be considering, ˆ απ can be expressed in terms of the A-genus defined above. In general Rosenberg’s Index απ lives in the K-theory of a certain C ∗ -algebra associated to the fundamental groups of the manifold, but it is not in general a number, see [21, 25, 22] for details. What about the universal cover of M ? Consider a manifold of dimension 9 which is homotopy equivalent to a sphere, call it Σ9 with α(Σ9 ) ̸= 0),(see [10]) take the connected sum of RP 7 × S 2 and Σ9 , notice that RP 7 × S 2 is spin and α(RP 7 × S 2 ) = 0, since RP 7 × S 2 is zero bordant. Since the manifold M = (RP 7 × S 2 )#Σ9 is spin bordant to the disjoint union of RP 7 × S 2 and Σ9 , we have that: α(M ) = α((RP 7 × S 2 )#Σ9 ) = α(RP 7 × S 2 ) + α(Σ9 ) = α(Σ9 ) ̸= 0. So M does not admit a metric with positive scalar curvature but ˜ = (S 7 × S 2 )#Σ9 #Σ9 which is diﬀeomerphic to its universal cover M 7 2 S × S does admit such a metric. So the question whether a spin manifold with finite fundamental group π admits a metric with positive scalar curvature cannot be reduced to the universal covering. Kwasik and Schultz [14] showed that the Gromov-Lawson-Rosenberg conjecture holds for a finite group π if and only if the conjecture holds for all the Sylow subgroups of π. Thus one can work one prime at a time. The Gromov-Lawson-Rosenberg conjecture has been established in the following cases: • If π is a spherical space form group and if M is spin (Botvinnik, Gilkey and Stolz [4]). • If π = Zp ⊕ Zp and if p is an odd prime (Schultz [28]). • If π belongs to a short list of infinite fundamental groups including free groups, free abelian groups and fundamental groups of orientable surfaces (Rosenberg & Stolz [23]). For more information about results concerning the Gromov-LawsonRosenberg conjecture, see the article of Joachim and Shick [11]. Note that Schick [26, 27] has shown that this conjecture fails in some

Barrera-Ya˜ nez and Cisneros-Molina

instances so it is crucial to investigate the precise conditions under which ˆ the A-genus carries the full set of obstructions. The spin bordism groups are too big, so it is useful to reformulate the Gromov-Lawson-Rosenberg conjecture in terms of more manageable groups, which are the connective K-theory groups.

3.1

Connective K-theory

Let KO be the periodic real K-theory spectrum and ko the connective cover of KO. The generalized homology theory associated with ko is called the real connective K-theory. We are interested on the connective K-theory of the classifying space of a group π, kon (Bπ). Let HP 2 be the quaternion projective space with the usual homogeneous metric of positive scalar curvature. Let HP 2 → E → B be a fiber bundle where the transition functions are the group of isometries P Sp3 of HP 2 . Since HP2 is simply connected, the projection p : E → B induces an isomorphism on the fundamental group. Let Tn (Bπ) be the subgroup of M Spinn (Bπ) generated by the total space of geometric fibrations with fiber HP 2 . Using some work of Jung and deep homotopy theory, Stolz [31] has given the following geometrical characterization of the real connective K-theory groups localized at the special prime 2: kon (Bπ)(2) = {M Spinn (Bπ)/Tn (Bπ)}(2) .

Let M Spin+ n (Bπ) be the classes in M Spinn (Bπ) which can be represented by manifolds which admit metrics of positive scalar curvature. The invariant απ extends to the bordism groups M Spinn (Bπ); the formula of Lichnerowicz [17] show that it vanishes on M Spin+ n (Bπ). One therefore has the following equivalent formulation of the GromovLawson-Rosenberg conjecture, see [31] for details: Theorem 3.2 Let π be a finite group, if n ≥ 5, then the following assertions are equivalent: • Let M be any closed connected spin manifold of dimension n with fundamental group π. Then M admits a metric of positive scalar curvature if and only if απ (M ) = 0. • M Spin+ n (Bπ) = ker(απ ) ∩ M Spinn (Bπ). + Let ko+ n (Bπ) be the image of M Spinn (Bπ) in kon (Bπ). The GromovLawson-Rosenberg conjecture has the following reformulation in terms of connective K theory:

Manifolds with positive scalar curvature

Theorem 3.3 Let π be an Abelian 2 group, if n ≥ 5, then the following assertions are equivalent: • Let M be any closed connected spin manifold of dimension n with fundamental group π. Then M admits a metric of positive scalar curvature if and only if απ (M ) = 0. • ko+ n (Bπ) = ker(απ ) ∩ kon (Bπ). Algebraic topology (spectral sequences) give upper bounds of connective K-theory and using Spectral invariants of the Dirac operator (eta invariant) give geometric generators and lower bounds of connective K-theory, using this approach the conjecture is valid for certain non-orientable manifolds with fundamental group an Abelian 2 group, see [1, 7] for details. Acknowledgment We like to thank Dr. Jes´ us Gonz´alez Espino Barros for inviting us to colaborate with Morfismos journal. The authors acknowledge with gratitude helpful suggestions by the referee which have improved the exposition of the paper. Egidio Barrera-Ya˜ nez Instituto de Matem´ aticas, UNAM, Unidad Cuernavaca, Av. Universidad s/n, Col. Lomas de Chamilpa, Cuernavaca, Morelos, M´exico. ebarrera@matcuer.unam.mx

Jos´e Luis Cisneros-Molina Instituto de Matem´ aticas, UNAM, Unidad Cuernavaca, Av. Universidad s/n, Col. Lomas de Chamilpa, Cuernavaca, Morelos, M´exico. jlcm@matcuer.unam.mx

References [1] Barrera-Ya˜ nez, E., The eta invariant of twisted products of even dimensional manifolds whose fundamental group is a cyclic 2 group, Diﬀerential Geometry and its Applications, 11 (1999), 221–235. [2] Besse A. L., Einstein manifolds, Springer Verlag, Berlin and New York, 1986. [3] Botvinnik B.; Gilkey P., The Gromov-Lawson-Rosenberg conjecture: the twisted case, Houston Math. J., 23 (1997), 143–160.

Barrera-Ya˜ nez and Cisneros-Molina

[4] Botvinnik B.; Gilkey P.; Stolz S., The Gromov-Lawson-Rosenberg conjecture for groups with periodic cohomology, J. Diﬀerential Geometry, 46 (1997), 374–405. [5] Gajer P., Riemannian metrics of positive scalar curvature on compact manifolds with boundary, Ann. of Global Anal. Geom., 5 (1987), 179–191. [6] Giambalvo V., Pin and spinc cobordism, Proc. Amer. Math. Soc., 39 (1973), 395–401. [7] Gilkey P. B.; Leahy J. V.; Park J. H., Spectral geometry, Riemannian submersions and the Gromov-Lawson conjecture, CRC Press, Publ 1999, ISBN 0–8493–8277–7. [8] Gromov M.; Lawson B. Jr., Spin and scalar curvature in the presence of a fundamental group, I. Ann. of Math., 111 (1980), 209–230. [9] Gromov M.; Lawson B. Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math., 111 (1980), 423–434. [10] Hitchin N., Harmonic spinors, Adv. in Math., 14 (1974), 1–55. [11] Joachim M., Shick T., Positive and negative results concerning the Gromov-Lawson-Rosenberg conjecture, in: Geometry and topology, (1998), 213–226. Contemp. Math. 258, Amer. Math. Soc., (2000). [12] Kazdan J. L.; Warner F., Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature Ann. of Math., 101 (1975), 317–331. [13] Kazdan J. L.; Warner F., Scalar curvature and conformal deformation of Riemannian structure, J. Diﬀ. Geom., 10 (1975), 113–134. [14] Kuwasik S.; Schultz R., Positive scalar curvature and periodic fundamental groups, Comment. Math. Helv., 65 (1990), 271–286. [15] Kuwasik S.; Schultz R., Fake spherical space forms of constant positive scalar curvature, Comment. Math. Helv., 71 (1996), 1–40. [16] Lawson H. B. Jr.; Michelson M.-L., Spin geometry, Princeton Math. Ser., vol. 38, Princeton Univ. Press, Princeton, 1989.

Manifolds with positive scalar curvature

[17] Lichnerowicz A., Spineurs harmoniques, C. R. Acad. Sci. Paris, S`er. A–B 257, 7–9. [18] Lonhkamp J., The space of negative curvature metrics, Invent. Math. 110, (1992), No. 2, 403–407. [19] Perelman G., Ricci flow with surgery on three manifolds arXiv:math.DG/0303109 v1, 10 Mar 2003. [20] Perelman G., The entropy formula for the Ricci flow ans its geometric applications arXiv:math.DG/0211159 v1, 11 Nov 2002. [21] Rosenberg J., C ∗ -algebras, positive scalar curvature and the Novikov conjecture, II, Geometric Methods in Operator Algebras, Pitmant Research Notes in Math., 123 (1986), 341–374. [22] Rosenberg J., C ∗ -algebras, positive scalar curvature and the Novikov Conjecture, III, Topology, 25 (1986), 319–336. [23] Rosenberg J.; Stolz S., A “stable” version of the Gromov-LawsonRosenberg conjecture, Contemp. Math., 181 (1995), 405–418. [24] Rosenberg J.; Stolz S., Metrics of positive scalar curvature with surgery, Surveys on Surgery theory, Annals of Mathematics Studies, 2 (2001), 353–386. [25] Rosenberg J., The KO-assembly map and positive scalar curvature, Algebraic Topology, Lecture Notes in Math., 1474 (1991), 170–182. [26] Schick T., A counterexample to the (unstable) Gromov-LawsonRosenbreg conjecture, Topology, 37, (1998), 1165–1168. [27] Shick T., Operator Algebras and Topology, ICTP Lect. Notes, 9 (2002), 571–660. [28] Schultz R., Positive scalar curvature and odd order Abelian fundamental groups, Proc. Amer. Math. Soc., 125 (1997), 907–915. [29] Shoen R.; Yau S.-T., On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159–183. [30] Stolz S., Concordance classes of positive scalar curvature metrics, in preparation.

Barrera-Ya˜ nez and Cisneros-Molina

[31] Stolz S.,Simply connected manifolds of positive scalar curvature, Ann. of Math., 136, (1992), 511–540. [32] Stolz S.,Splitting certain MSpin-module spectra, Topology, 33, (1994), 159–180. [33] Stolz S.,Manifolds of positve scalar curvature, ICTP Lect. Notes, 9 (2002), 661–709. [34] Taubes C. H., The Seiberg-Witten invariants and symplectic forms, Math. Res. Let. 1, (1994), 809–822. [35] Thurston W. P., Three-Dimensional Geometry and Topology, Vol 1, Princeton University Press.

Morfismos, Vol. 7, No. 2, 2003, pp. 17–45

A survey on modular Hadamard matrices Shalom Eliahou

∗

Michel Kervaire

Abstract We provide constructions of 32-modular Hadamard matrices for every size n divisible by 4. They are based on the description of several families of modular Golay pairs and quadruples. Higher moduli are also considered, such as 48, 64, 128 and 192. Finally, we exhibit infinite families of circulant modular Hadamard matrices of various types for suitable moduli and sizes.

2000 Mathematics Subject Classification: 05B20, 11L05, 94A99. Keywords and phrases: modular Hadamard matrix, modular Golay pair.

Introduction

A square matrix H of size n, with all entries ±1, is a Hadamard matrix if HH T = nI, where H T is the transpose of H and I the identity matrix of size n. It is easy to see that the order n of a Hadamard matrix must be 1, 2 or else a multiple of 4. There are two fundamental open problems about these matrices: • Hadamard’s conjecture, according to which there should exist a Hadamard matrix of every size n divisible by 4. (See [9].) • Ryser’s conjecture, stating that there probably exists no circulant Hadamard matrix of size greater than 4. (See [13].) Recall that a circulant matrix is a square matrix C = (ci,j )0≤i,j≤n−1 of size n, such that ci,j = c0,j−i for every i, j (with indices read mod n). ∗

Invited article

Eliahou and Kervaire

There are many known constructions of Hadamard matrices. However, Hadamard’s conjecture is widely open. For example, the set of all currently known Hadamard matrix sizes (as of 2004) contains no arithmetic progression, and is in fact of density zero in the set of positive multiples of 4. (See [17].) The cases below 1000 which are currently open are 428, 668, 716, 764 and 892. As for Ryser’s conjecture, a lot is known, but here again the conjecture is widely open. For example, it is known that if n > 4 is the size of a circulant Hadamard matrix, then n = 4 · r2 with r odd and not a prime power. Actually further constraints on r are known, due to R. Turyn and more recently B. Schmidt [14]. In 1972, Marrero and Butson introduced the weaker notion of a modular Hadamard matrix. Like in the classical case, this is a square matrix H, with all entries ±1, but satisfying the above orthogonality condition only modulo some given integer m, i.e. H · H T ≡ nI mod m. Of course, the classical Hadamard matrix conjecture has an mmodular counterpart, namely: for every n divisible by 4, there should exist an m-modular Hadamard matrix of size n. Even though this mmodular analogue looks much weaker than the classical one, there is a sort of converse, which rests on the following Remark. If H is an m-modular Hadamard matrix of size n, with n < m, then H is an ordinary Hadamard matrix. The proof is simple enough: the entries of H · H T are at most n in absolute value. Hence, if those outside the diagonal are assumed to vanish mod m, then they must actually be zero. With the above remark, we see that the classical Hadamard matrix conjecture holds if and only if the modular Hadamard matrix conjecture simultaneously holds for infinitely many distinct moduli m. In this sense, the m-modular version of Hadamard’s conjecture can be considered as an approximation to the classical one, of quality increasing with m. Currently, the highest modulus m for which the mmodular analogue of Hadamard’s conjecture has been completely settled is m = 32. We summarize the relevant facts below. In a series of papers, Marrero and Butson considered modular Hadamard matrices mainly with respect to moduli m which are either odd or 2 times an odd number. With respect to such moduli, sizes n > 3

Modular Hadamard matrices

not divisible by 4 are no longer excluded in general. For instance, they show the existence of a 6-modular Hadamard matrix of size n for every even n. In this survey, we consider only moduli m which are divisible by 4, as this case resembles more the classical one. Indeed, if n > 3 is the size of an m-modular Hadamard matrix, with m divisible by 4, then n itself must be divisible by 4, as for ordinary Hadamard matrices. The proof is analogous to the one in the classical case, by considering congruences mod 4 rather than equalities. As for Ryser’s conjecture, the situation is somewhat diﬀerent. There seems to be a very rich theory of circulant modular Hadamard matrices, which ought to be developed for its own sake. Circulant modular Hadamard matrices do exist for certain moduli and sizes greater than 4 and thus, the conjecture should rather be replaced in the modular context by the following question. Question: For which moduli m and sizes n do there exist m-modular circulant Hadamard matrices H of size n ? The question can be enriched by requiring that some entries of the matrix H · H T be actually zero, not only zero mod m. We will introduce two such constraints, complementary in some sense, and refer to the complying matrices as being of type 1, type 2 respectively. Informally, H will be of type 1 if any two rows of H with indices at distance n2 are orthogonal in Zn , n being the order of H. On the other hand, H will be of type 2 if any two rows of H with indices at distance other than 0 and n2 are orthogonal in Zn . As we will see, there are nice infinite families of circulant modular Hadamard matrices of either type. These examples all come from number-theoretic constructions. The complementary nature of types 1 and 2 imply that, if H is a circulant modular Hadamard matrix of both types simultaneously, then H is actually a true circulant Hadamard matrix. Hence, investigating the possible moduli and orders of circulant modular Hadamard matrices of either type, besides being of independent interest, might shed some light on Ryser’s conjecture itself.

Eliahou and Kervaire

Basic Definitions and Lemmas

We shall denote by H(n) the set of Hadamard matrices of size n, and by Hm (n) the set of m-modular Hadamard matrices of size n. Of course, H(n) ⊂ Hm (n). Hadamard’s conjecture reads H(n) ̸= ∅ for every n divisible by 4. Among other results, we shall see that H32 (n) ̸= ∅ for every n divisible by 4. There are many constructions for Hadamard matrices. See the quoted surveys [4]. Here, we will mainly use three such constructions. All three use sets of complementary binary sequences, specifically pairs and quadruples. From such sets, Hadamard matrices are obtained by placing the circulant matrices derived from each sequence into suitable arrays. For convenience of the reader, this is recalled below.

2.1

The doubling lemma

We start with a very simple result. Lemma 2.1.1 There is a map Hm (n) → H2m (2n). More specifically, if H is an m-modular Hadamard matrix of size n, then the matrix # " !" 1 1 # H H = (1) H′ = H −H H −1 1 is a 2m-modular Hadamard matrix of size 2n. Observe that the modulus has also been doubled in the process. Proof: ′

H ·H

′T

H H −H H

# " T H · HT

−H T HT

2H · H T 0

0 2H · H T

# ,

and H · H T ≡ nI modulo m, i.e. H · H T = nI + mX for some n × n integer matrix X. It follows that # # " " # " X 0 2H · H T I 0 0 ′ ′T . + 2m H ·H = = 2n 0 X 0 I 0 2H · H T Thus H ′ · H ′T is congruent 2n times the identity matrix of size 2n modulo 2m. !

Modular Hadamard matrices

2.2

Complementary sequences

Let A = (a0 , . . . , aℓ−1 ) be a binary sequence of length ℓ, that is a sequence with all entries ai = The Hall polynomial of A, denoted !±1. ℓ−1 i A(z), is defined as A(z) = kth aperiodic correlation i=0 ai z . The !ℓ−1−k ai ai+k , for 0 ≤ k ≤ coeﬃcient ck (A) is defined as ck (A) = i=0 ℓ − 1. It is convenient to define ck (A) = 0 if k ≥ ℓ. Note that the number ck (A) arises as the coeﬃcient of (z k + z −k ) in the product A(z)A(z −1 ) in the Laurent polynomial ring Z[z, z −1 ]: A(z)A(z −1 ) = c0 (A) +

ℓ−1 "

ck (A)(z k + z −k ).

k=1

Here c0 (A) = ℓ, the sum of the squares of the ai which are assumed to be binary (i.e. ±1). A set of r binary sequences A1 , . . . , Ar is a set of complementary sequences if for each k ≥ ! 1, the sum of the kth correlations of the sequences vanishes, that is rj=1 ck (Aj ) = 0 for all k ≥ 1. (Recall our convention ck (A) = 0 if k is not smaller than the length of A.) Equivalently, using Hall polynomials, it is clear that the binary sequences A1 , . . . , Ar form a set of complementary sequences if and only if A1 (z)A1 (z −1 ) + · · · + Ar (z)Ar (z −1 ) equals a constant in the Laurent polynomial ring Z[z, z −1 ]. In this case, the constant will simply be the sum of the respective lengths of A1 , . . . , Ar . Pairs of complementary sequences of the same length are also known as Golay pairs. Here, as in [6], we shall refer to quadruples of complementary sequences of the same length as Golay quadruples. We shall denote by GP(n) the set of Golay pairs of length n, and by GQ(n) the set of Golay quadruples of length n. Golay pairs and quadruples may be used to construct Hadamard matrices of appropriate size. We recall these classical constructions now. Proposition 2.2.1 There is a map GP(n) −→ H(2n) obtained by the following construction. Let A, B be a Golay pair of length n. Denote by A, B again the circulant matrices derived from each sequence respectively. Let # $ A B (2) H = H(A, B) = . −B T AT

Eliahou and Kervaire

Then H is a Hadamard matrix of size 2n. Proof:

A straightforward computation shows that H ·H

AAT + BB T −B T AT + AT B T

−AB + BA AT A + B T B

" .

Now, since A, B are circulant matrices, they commute. Hence, H · H T = (A · AT + B · B T )

#! I 0 " . ! 0 I

There is a classical construction, due to Goethals-Seidel, which associates a Hadamard matrix of size 4n to every Golay quadruple of length n. First we recall what the Goethals-Seidel array is. If A, B, C and D are matrices of size n, define ⎛ ⎞ A −BR −CR −DR ⎜ BR A −DT R C T R ⎟ ⎟, (3) GS(A, B, C, D) = ⎜ ⎝ CR DT R A −B T R ⎠ DR −C T R B T R A where R is the back-circulant matrix of size n defined by R = (Ri,j ) with Ri,j = δi+j,n+1 for 0 ≤ i, j ≤ n − 1. Proposition 2.2.2 There is a map GQ(n) −→ H(4n) obtained by the following construction. Let A, B, C, D be a Golay quadruple of length n. Denote by A, B, C, D again the circulant matrices derived from each sequence respectively. Let H=GS(A, B, C, D). Then H is a Hadamard matrix of size 4n. The proof of the proposition uses the following properties of the matrix R. Namely, R2 = I, RT = R, and if X, Y are any two circulant matrices, then XRY T is a symmetric matrix, i.e. XRY T = Y RX T . Besides the map from GP(n) to H(2n) recalled above, there are other constructions associating a Hadamard matrix to a Golay pair, obtained by associating first a Golay quadruple to a Golay pair, and then using the Proposition above.

Modular Hadamard matrices

For example, if (f, g) is a Golay pair of length n, then (f, f, g, g) is a Golay quadruple of the same length n, yielding a Hadamard matrix of size 4n. This yields a map GP(n) −→ H(4n), not as eﬃcient as the one above. There is a subtler classical construction, yielding this time a map from GP(n) to H(8n + 4). It is obtained as follows. Notation. If f = (f1 , . . . , fℓ ), g = (g1 , . . . , gn ) are (binary) sequences, we denote their concatenation by [f ; g] = (f1 , . . . , fℓ , g1 , . . . , gn ). Note that the length of [f ; g] is the sum of the lengths of f and g. Proposition 2.2.3 There are maps GP(n) −→ GQ(2n + 1) −→ H(8n + 4). The first map associates to the Golay pair (f, g) a Golay quadruple (A, B, C, D), where A = [f ; 1; g], B = [f ; 1; −g], C = [f ; −1; g], D = [f ; −1; −g]. Proof: Using the Hall polynomials of the respective sequences, it is straightforward to check the formula A(z)A(z −1 ) + B(z)B(z −1 ) + C(z)C(z −1 ) + D(z)D(z −1 ) = 4(1 + f (z)f (z −1 ) + g(z)g(z −1 )). Thus, if f (z)f (z −1 ) + g(z)g(z −1 ) is a constant, this being the defining property of a Golay pair, then so will also be the expression A(z)A(z −1 ) + B(z)B(z −1 ) + C(z)C(z −1 ) + D(z)D(z −1 ). ! We now recall doubling constructions for Golay pairs and quadruples, that is, maps GP(n) −→ GP(2n) and GQ(n) −→ GQ(2n). If (f, g) is a Golay pair of length n, then ([f ; g], [f ; −g]) is a Golay pair of length 2n. If A, B, C, D is a Golay quadruple of length n, then [A; B], [A; −B], [C; D], [C; −D] is a Golay quadruple of length 2n. Both statements are easy to verify. We shall close this Section with a few comments about the lengths of Golay pairs and Golay quadruples.

Eliahou and Kervaire

For Golay pairs, it is known that GP(2a 10b 26c ) is not empty for every exponents a, b, c ≥ 0. On the other hand, it is conjectured that no lengths other than 2a 10b 26c may be realized as Golay pair lengths. It is easy to see that GP(n) is empty if n is odd and greater than 1. A theorem in [8] states that GP(n) is empty if n admits a divisor which is congruent to 3 mod 4. Computer searches have revealed the absence of Golay pairs of length 34, 50 and 68, and more recently of length 74 and 82 (see [2]). The smallest undecided cases now are n = 106 and n = 116. As for Golay quadruples, there is the following Conjecture. (Turyn, [15]) There is a Golay quadruple of length n for every positive integer n. Because of the above-mentioned map GQ(n) → GQ(2n), the core of the problem is the case where n is odd. Moreover, because of the map GQ(n) −→ H(4n), the above conjecture implies Hadamard’s conjecture. Obviously, every Golay pair A, B of length n yields a Golay quadruple A, A, B, B of the same length, and a Golay quadruple of length 2n+1 by the map GP(n) −→ GQ(2n + 1).

2.3

Modular complementary sequences

There are modular analogues of the above notions. Let m be a positive integer. A set of r binary sequences {A1 , . . . , Ar } is a set of m-modular complementary sequences if for each k ≥ 1, the!sum of the kth correlations of the sequences vanishes mod m, that is rj=1 ck (Aj ) ≡ 0 mod m for all k ≥ 1. This is equivalent to the statement that A1 (z)A1 (z −1 ) + · · · + Ar (z)Ar (z −1 )

equals a constant in the Laurent polynomial ring (Z/mZ)[z, z −1 ]. In particular, we have the notion of modular Golay pairs and quadruples. We will denote by GPm (n), GQm (n) the set of m-modular Golay pairs, respectively m-modular Golay quadruples, of length n. The above constructions, associating Hadamard matrices to suitable sets of Golay sequences, work as well in the modular context. Proposition 2.3.1 There are maps GPm (n) −→ Hm (2n) and GQm (n) −→ Hm (4n).

Modular Hadamard matrices

Note that, in these maps, the modulus remains unchanged. However, for the third construction GP(n) −→ GQ(2n + 1) −→ H(8n + 4), we have the happy circumstance that the modulus is multiplied by 4. Proposition 2.3.2 There is a map GP m (n) −→ GQ4m (2n + 1), and hence a map GPm (n) −→ H4m (8n + 4). The multiplication of the modulus m by 4 is apparent in the proof of the last proposition of Section 2.2. Finally, on the modular level, the doubling of Golay pairs also doubles the modulus. That is, there is a map GPm (n) → GP2m (2n), given by (f, g) #→ ([f ; g], [f ; −g]). This is easily checked using the Hall polynomials of the sequences: if A(z) = f (z)+z n g(z) and B(z) = f (z)−z n g(z), then A(z)A(z −1 ) + B(z)B(z −1 ) = 2(f (z)f (z −1 ) + g(z)g(z −1 )).

3 3.1

Modular Hadamard matrices The case m = 12

Marrero and Butson have produced 6-modular Hadamard matrices of size n for every even positive integer n. (See [11] and [12].) Very simple matrices suﬃce for this purpose. It turns out that their construction yields in fact 12-modular Hadamard matrices of every size n divisible by 4. For any given size, let I denote the identity matrix, J the constant all-one matrix, and K = −2I + J, the circulant with first row (−1, 1, . . . , 1). Proposition! 3.1.1 A 12-modular Hadamard matrix of size n is given " K K depending on whether n ≡ 0, 4 or 8 mod 12 by J, K or −K K respectively. Proof: In size n, we have J · J T = nJ and K · K T = nI + (n − 4)(J − I). This takes care of !the cases n"≡ 0, 4 mod 12. Assume now K K of size n. Then H · H T = n ≡ 8 mod 12, and let H = −K K

Eliahou and Kervaire

" 2KK T 0 . Since K is of size n2 here, we have KK T = n2 I + 0 2KK T ( n2 − 4)(J − I), and so 2K · K T = nI + (n − 8)(J − I). " ! nI 0 T mod 12. ! It follows that H · H ≡ 0 nI This solves the 12-modular version of Hadamard’s conjecture. Obviously, more elaborate matrices will be needed for higher moduli. This is plainly illustrated in the case m = 32.

3.2

The solution of the 32-modular Hadamard conjecture

We shall prove the existence of a 32-modular Hadamard matrix of size 4ℓ for every positive integer ℓ. By the Doubling Lemma, it is suﬃcient to consider the case where ℓ is odd. Our constructions depend on the class of ℓ mod 8, and, in contrast to [6], are all based in this paper on modular Golay pairs and quadruples. For ℓ ≡ 1, 3 or 7 mod 8, we shall exhibit 32-modular Golay quadruples of length ℓ. These quadruples yield 32-modular Hadamard matrices of size 4ℓ by the map GQm (n) −→ Hm (4n) of Section 2 derived from the Goethals-Seidel array. For ℓ ≡ 3 or 7 mod 8, the description of these quadruples is by direct construction, while for ℓ ≡ 1 mod 8, they derive from 8-modular Golay pairs of length ℓ−1 2 , and the map ℓ−1 GPm (r) −→ GQ4m (2r + 1) of Section 2 (with r = 2 ). In the remaining case ℓ ≡ 5 mod 8, and more specifically for ℓ ≡ 13 mod 16, we are so far unable to produce 32-modular Golay quadruples of length ℓ. Rather, we shall obtain 32-modular Hadamard matrices of size 4ℓ from 32-modular Golay pairs of length 2ℓ and the map GPm (2ℓ) −→ Hm (4ℓ) of Section 2. We observe that this construction, which works for ℓ ≡ 5 mod 8, cannot work for ℓ ≡ 3 or 7 mod 8, as we can prove that 32-modular Golay pairs do not exist in length congruent to 6 or 14 mod 16, as well as in length congruent to 12 mod 16, see [6]. (The existence of 32-modular Golay pairs of length 2ℓ with ℓ ≡ 1 mod 8 remains in doubt.) 3.2.1 Modular Golay quadruples of length ℓ ≡ 1 mod 8 Let k = ℓ−1 8 . We shall construct a family of 8-modular Golay pairs of length 4k with k free binary parameters, and then use the maps GPm (r) −→ GQ4m (2r + 1) −→ H4m (4(2r + 1))

Modular Hadamard matrices

to produce the desired modular Golay quadruples and modular Hadamard matrices. Consider an arbitrary binary sequence h = (x0 , . . . , xk−1 ) say, of length k, with xi = ±1 for all i. Obviously, the pair (h, h) is a 2-modular Golay pair of length k. By the doubling of Golay pairs, the pair (f, g) with f = [h; h] and g = [h; −h] is a 4-modular Golay pair of length 2k, and the pair (A, B) with A = [f ; g] and B = [f ; −g] is an 8-modular Golay pair of length 4k with k free binary parameters, as desired. In summary, with A = [h; h; h; −h] and B = [h; h; −h; h], the pair (A, B) is a k-parameter family of 8-modular Golay pairs of length 4k = ℓ−1 2 . Corollary 3.2.1 For every ℓ ≡ 1 mod 8, there is a k-parameter family of 32-modular Golay quadruples of length ℓ and 32-modular Hadamard matrices of size 4ℓ, where k = ℓ−1 8 . Proof: Send the above 8-modular Golay pair of length 4k = GQ32 (ℓ) and H32 (4ℓ) with the maps

ℓ−1 2

GPm (r) −→ GQ4m (2r + 1) −→ H4m (4(2r + 1)) at m = 8 and r = 4k =

ℓ−1 2 .

3.2.2 Modular Golay quadruples of length ℓ ≡ 3, 7 mod 8 Our objective here is to show that GQ32 (ℓ) ̸= ∅ for ℓ ≡ 3 mod 4. We shall need the following operation on binary sequences. To the sequence F = (a0 , . . . , ak ), we associate the new sequence F # , defined as F # = ((−1)k ak , . . . , (−1)i ai , . . . , a0 ). On the level of Hall polynomials, this transformation reads simply as F # (z) = z k F (−z −1 ). r−1 . Thus, ε = −1 if r is even, that is Let r = l−3 4 , and set ε = (−1) if ℓ ≡ 3 mod 8, while ε = +1 if ℓ ≡ 7 mod 8. Given two (±1)-sequences H and K of size 2r + 1, we define a quadruple of binary sequences of length ℓ = 4r + 3, Q(H, K) = (A, B, C, D), as follows: A = [H; ε; −H # ], B = [H; ε; −K # ] C = [K; ε; −H # ], D = [K; −ε; −K # ].

Eliahou and Kervaire

For the binary sequences H, K described below, the associated quadruple Q(H, K) turns out to be a 32-modular Golay quadruple. It is convenient to separate the cases r even and r odd. For r even, define H = [12r+1 ] and K = [−1r+1 ; 1r ], where [12r+1 ] denotes the constant all 1 sequence of length (2r + 1), and [−1r+1 ] denotes a constant sequence of −1 repeated (r + 1) times. For r odd, let f = [1r−1 ]. Define H = [f ; −1, 1, 1; f # ] and K = [−f ; 1, −1, 1; f # ]. Since f # = [−1, 1](r−1)/2 in the present case, we have in fact H = [1r−1 ; −1, 1, 1; [−1, 1](r−1)/2 ] and K = [−1r−1 ; 1, −1, 1; [−1, 1](r−1)/2 ]. In [6], we established the following result. Theorem 3.2.2 Let ℓ ≡ 3 mod 4, and let H, K be the above binary se2r+1 ] and K = [−1r+1 ; 1r ] quences of length ℓ−1 2 = 2r + 1, that is H = [1 if r is even, H = [1r−1 ; −1, 1, 1; [−1, 1](r−1)/2 ] and K = [−1r−1 ; 1, −1, 1; [−1, 1](r−1)/2 ] if r is odd. Then the quadruple of binary sequences Q(H, K) = (A, B, C, D) as defined above, is a 32modular Golay quadruple of length ℓ. More precisely, we have the following formula in terms of the Hall polynomials of A,B,C,D : A(z)A(z −1 ) + B(z)B(z −1 ) + C(z)C(z −1 ) + D(z)D(z −1 ) = [r/2]

4ℓ + 32

! i=1

([r/2] − i)(z 2i + z −2i ).

Corollary 3.2.3 There is a 32-modular Hadamard matrix of size 4ℓ for every positive integer ℓ ≡ 3 mod 4. Proof: Send the above 32-modular Golay quadruple A, B, C, D of length ℓ to H32 (4ℓ) with the map GQm (ℓ) −→ Hm (4ℓ) of Section 2. ! Example 3.2.4 No true Hadamard matrix is known yet in size n = 428. But the above construction yields the following 32-modular Hadamard matrix of this size n. Let A = [153 ; −1; [−1, 1]26 ; −1], B = [153 ; −1; [−1, 1]13 ; [1, −1]13 ; 1], C = [−127 ; 126 ; −1; [−1, 1]26 ; −1], D = [−127 ; 126 ; 1; [−1, 1]13 ; [1, −1]13 ; 1].

Modular Hadamard matrices

This is a quadruple of binary sequences of length 107. For 1 ≤ k ≤ 106, let αk = ck (A) + ck (B) + ck (C) + ck (D) be the sum of the kth aperiodic correlation coeﬃcients of A, B, C and D respectively. We then find αk = 0 for all k ∈ {1, 2, . . . , 106}\{2, 4, . . . , 24}, and α2k = 32 · (13 − k) for k in the interval 1 ≤ k ≤ 12. Thus, as claimed, (A, B, C, D) is a 32-modular Golay quadruple of length 107. The matrix H = GS(A, B, C, D) is therefore a 32-modular Hadamard matrix of size! 428 " (see Figure 1). It is amusing to observe that among the 91378 = 428 entries of the strict upper triangular part of H · H T , 2 there are 86242 entries which are strictly 0, while the remaining 5136 non-zero ones consist of 428 entries of the form 32k for each 1 ≤ k ≤ 12. Actually, any row in H is orthogonal to exactly 403 other rows in H. For example, the 25 rows not orthogonal to the first row are the rows in position 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104 and 106. One last remark concerning the determinant of H. Recall the theorem of Hadamard [9] stating that the determinant of any real matrix M of size n with entries from the interval [−1, 1] satisfies the inequality |det(M )| ≤ nn/2 . Moreover, the equality |det(M )| = nn/2 holds true if and only if M is a Hadamard matrix. Here, in the above example H of size n = 428, we have nαn < |det(H)| < nβn , with α = 0.347 and β = 0.348.

3.2.3 The case ℓ ≡ 5 mod 8 We know only one way to obtain 32-modular Hadamard matrices of size 4ℓ for ℓ ≡ 5 mod 8. Namely, from 32-modular Golay pairs of length 2ℓ ≡ 10 mod 16 and the map GPm (2ℓ) −→ Hm (4ℓ). The relevant modular Golay pairs are somewhat involved, and are best described through their Hall polynomials. #k−1 i 4i Let k = ℓ−5 i=0 (−1) z . Let x0 , x1 be two binary 8 . Define S(z) = parameters, and define the pair of polynomials U (z), V (z) as follows: U (z) = (x0 + x1 z + x0 z 2 )S(z) + (−1)k (x0 − x1 z − x0 z 2 )z 4k +(−1)k (x0 − x1 z + x0 z 2 )S(z)z 4(k+1) , V (z) =

#2k

i=0 (−1)

i z 4i

+ z 8k+2 .

Eliahou and Kervaire

Figure 1: A 32-modular Hadamard matrix of size 428 (white pixels represent +1 and black pixels represent −1) Finally, let x3 = ±1 be a third free binary parameter, and define A(z), B(z) as follows: A(z) = U (z) + x3 z 3 V (z) + z 16k+9 (U (z −1 ) − x3 z −3 V (z −1 )), B(z) = U (z) + x3 z 3 V (z) − z 16k+9 (U (z −1 ) − x3 z −3 V (z −1 )). In [6], we prove the following result. Theorem 3.2.5 For every ℓ ≡ 5 mod 8, the above polynomials A(z), B(z) are the respective Hall polynomials of a 3-parameter 32-modular Golay pair A, B of length 2ℓ = 16k + 10.

Modular Hadamard matrices

In this theorem, the total correlation A(z)A(z −1 ) + B(z)B(z −1 ), which we abbreviate AA + BB, is given by the formula (4)

AA + BB = 32k + 20 + 32

k−1 ! i=1

(−1)i (k − i)(z 4i + z −4i ).

Example 3.2.6 We get true Golay pairs of length 10 for k = 0 and true Golay pairs of length 26 for k = 1. Let now k = 2. There are no Golay pairs of length 2ℓ = 16k + 10 = 42, because 42 has a divisor congruent to 3 mod 4 (see [8]). However, setting x0 = x1 = x3 = 1 for simplicity in the pair given by the above theorem, we get a 32-modular Golay pair A, B of length 42, namely

A = ++++−−−−+−−++−+−−+−+−+−−+− + + − + − − − + + − − − − + ++,

B = ++++−−−−+−−++−+−−+−+++++−+ −−+−+++−−++++−−−.

Remarkably, this pair is almost a true Golay pair of length 42, as it satisfies ci (A) + ci (B) = 0 for all 1 ≤ i ≤ 41 with the sole exception of i = 4, for which c4 (A) + c4 (B) = −32. More generally, the formula (4) shows that only (k − 1)/(16k + 8) of the correlations sums ci (A)+ ci (B) are non-zero. On the other hand, we know that a pair (A, B) with k ≥ 2 as in the above Theorem can never be an actual Golay pair even with an arbitrary (binary) polynomial " 4i S(z) = k−1 u i=0 i z . (See [8], Lemma 4.7 and the remark at the end of Section 1.2 in [6].) Corollary 3.2.7 There exist 32-modular Hadamard matrices of size 4ℓ for every positive integer ℓ ≡ 5 mod 8. Proof: Send the above 32-modular Golay pair A, B of length 2ℓ to H32 (4ℓ) with the map GPm (2ℓ) −→ Hm (4ℓ) of Section 2. ! Corollary 3.2.8 There exist 128-modular Hadamard matrices of size 16ℓ + 4 for every positive integer ℓ ≡ 5 mod 8.

Eliahou and Kervaire

Proof: Send the above 32-modular Golay pair A, B of length 2ℓ to H128 (16ℓ+4) with the maps GPm (2ℓ) −→ GQ4m (4ℓ+1) −→ H4m (16ℓ+ 4) of Section 2. ! Example 3.2.9 Currently, no Hadamard matrices of size 4r are known for r = 789, 853 and 917. These are the only undecided cases with r ≤ 1000 and r ≡ 21 mod 32. However, there exist 128-modular Hadamard matrices in size 4r for r = 789, 853 and 917. Indeed, let ℓ = r−1 4 . Then ℓ ≡ 5 mod 8, and the conclusion follows from the second corollary above. In the next Section, we shall actually obtain a 192-modular Hadamard matrix of size 4 · 917.

3.3

Other moduli

We shall exhibit a few more modular Hadamard matrices in sizes for which, as above, no true Hadamard matrices are known yet. The modulus m = 48 We start by constructing 48-modular Golay pairs of length 24k + 2!for every positive integer k. (See [6], Section 1.5.) Define S(z) = k−1 i 12i . Let x , x be two binary parameters, and define the 0 1 i=0 (−1) z pair of polynomials U (z), V (z) as follows : U (z) = {x0 (1 + z 2 − z 4 + z 6 − z 8 − z 10 ) + x1 (z + z 5 + z 9 )}S(z), V (z) = (1 − z 4 + z 8 )S(z) + z 12k−2 . Finally, let x3 be a third free binary parameter, and define A(z), B(z) as follows : A(z) = U (z) + x3 z 3 V (z) + z 24k+1 (U (z −1 ) − x3 z −3 V (z −1 )), B(z) = U (z) + x3 z 3 V (z) − z 24k+1 (U (z −1 ) − x3 z −3 V (z −1 )). We prove the following result in [6]. Theorem 3.3.1 For every ℓ ≡ 1 mod 12, the above polynomials A(z), B(z) are the respective Hall polynomials of a 3-parameter 48-modular Golay pair A, B of length 2ℓ = 24k + 2.

Modular Hadamard matrices

Example 3.3.2 For k = 1, this construction yields true Golay pairs of length 26. Let now k = 2. There are no Golay pairs of length 2ℓ = 24k + 2 = 50, as revealed by an exhaustive computer search. (See [1].) However, setting x0 = x1 = x3 = 1 in the pair given by the above theorem, we get a 48-modular Golay pair A, B of length 50, namely A = ++++−++−−+−+−−−−+−−++−+−+− + + − + − − − + + − − − − − + − + + + − − + ++,

B = ++++−++−−+−+−−−−+−−++−+−−− − − + − + + + − − + + + + + − + − − − + + − −−,

where + stands for +1 and − for −1. This pair satisfies ci (A) + ci (B) = 0 for all 1 ≤ i ≤ 49 with the sole exception of i = 12, for which c12 (A) + c12 (B) = −48. Corollary 3.3.3 There exist 48-modular Hadamard matrices of size 48k + 4 and 192-modular Hadamard matrices of size 192k + 20 for every positive integer k. Proof: Send the above 48-modular Golay pair A, B of length 2ℓ = 24k + 2 to H48 (4ℓ) and to H192 (16ℓ + 4) with the maps GPm (2ℓ) −→ Hm (4ℓ) and GPm (2ℓ) −→ GQ4m (4ℓ + 1) −→ H4m (16ℓ + 4) of Section 2, respectively. ! Example 3.3.4 There exist 192-modular Hadamard matrices of size 4· 917. Indeed, take k = 19 in the above 192-modular construction. There also exist 48-modular Hadamard matrices of size 4 · 721 and 4 · 853 (with k = 60 and k = 71 in the above 48-modular construction, respectively.) These three sizes, 4 · 721, 4 · 853 and 4 · 917, are all undecided cases for true Hadamard matrices.

The moduli m = 2t We have proved above that H32 (n) ̸= ∅ for every positive integer n divisible by 4. Using the map Hm (n) −→ H2m (2n), we see that H64 (n) ̸= ∅ for every n divisible by 8, and more generally that H2t+3 (n) ̸= ∅ for every t ≥ 3 and every n divisible by 2t .

Eliahou and Kervaire

However, with further constructions, we shall obtain 64-modular and 128-modular Hadamard matrices of some (but unfortunately not all) sizes n ≡ 4 mod 8. Recall from Section 3.2.1 that, if h is an arbitrary binary sequence of length k, then the pair (h, h) is a k-parameter 2-modular Golay pair of length k. In other terms, GP2 (k) ̸= ∅ for every positive integer k. By the doubling of Golay pairs, that is, by the map GPm (n) −→ GP2m (2n), which doubles both length and modulus, we readily obtain the following statements. Proposition 3.3.5 GP2t (2t−1 k) ̸= ∅ for every positive integers t and k. Corollary 3.3.6 There exist 2t+2 -modular Hadamard matrices of size n = 4 · (2t k + 1) for every positive integers t, k. Proof: Use the maps GPm (n) −→ GQ4m (2n+1) −→ H4m (4·(2n+1)) of Section 2. ! Example 3.3.7 No Hadamard matrices of size 4 · 721 are known yet. Now, 721 = 24 · 45 + 1. Thus, the above result, with t = 4, yields a 64modular Hadamard matrix of size 4·721. (We already had a 48-modular Hadamard matrix of size 4 · 721. See the case m = 48 above.) We recall one last construction of modular Golay pairs. Proposition 3.3.8 ([6]) There are 16-modular Golay pairs of length 8k + 2 for every integer k ≥ 0. Proof: For k = 0, the pair A(z) = 1 + z,!B(z) = 1 − z will do. Assume !k−1 k−1 4i now k ≥ 1. Choose polynomials f (z) = i=0 xi z , g(z) = i=0 yi z 4i with arbitrary xi = ±1, yi = ±1 for i = 0, 1, . . . , k −1. Let also w = ±1 be chosen arbitrarily. Further, let F (z) = z −(4k−1) f (z) + z 4k−1 f (z −1 ) and G(z) = z −(4k−1) g(z) − z 4k−1 g(z −1 ). A 16-modular Golay pair, of length 8k + 2, is given by A(z) = {(1 + z 3 )F (z) + (z + z 2 )G(z) + w(z − z 2 )}z 4k−1 B(z) = {(1 − z 3 )F (z) + (z − z 2 )G(z) + w(z + z 2 )}z 4k−1 .!

Modular Hadamard matrices

Corollary 3.3.9 There exist 2t+6 -modular Hadamard matrices of size n = 4 · (2t+3 k + 2t+1 + 1) for every integers t, k ≥ 0. Proof: Since GP16 (8k + 2) ̸= ∅, it follows by successive doubling that GP2t+4 (2t+3 k + 2t+1 ) ̸= ∅, for every t ≥ 0. Using again the maps GPm (n) −→ GQ4m (2n + 1) −→ H4m (4 · (2n + 1)) of Section 2, it follows that the sets GP2t+6 (2t+4 k + 2t+2 + 1) and H2t+6 (4 · (2t+4 k + 2t+2 + 1)) are both non-empty, for every t, k ≥ 0. ! Example 3.3.10 It is not known whether Hadamard matrices of size 4 · ℓ exist for ℓ = 789, 853, 917 and 933. These four values of ℓ are congruent to 5 mod 16. The above corollary, with t = 0, therefore yields 64-modular Hadamard matrices of size 4 · 789, 4 · 853, 4 · 917 and 4 · 933. Note that only the case 4 · 933 is really of interest here, as we had already obtained 128-modular Hadamard matrices of size 4 · 789, 4 · 853 and 4 · 917 in Section 3.2.3.

4 4.1

Circulant modular Hadamard matrices Introduction

According to Ryser’s conjecture, there probably exists no circulant Hadamard matrix of size n > 4. In contrast, the modular level reveals interesting families of examples [5]. These families are all based on the quadratic and biquadratic characters of finite fields, and will be exhibited below. Thus, it would seem appropriate to rephrase the problem as follows. Question: For what moduli m and sizes n do there exist m-modular circulant Hadamard matrices of size n ? Definition 4.1.1 Let s = {x0 , x1 , . . . , xn−1 } ∈ {±1}n be a binary sequence of size n. The kth periodic correlation coeﬃcient γk (s) of s, !n−1 for 0 ≤ k ≤ n − 1, is defined as γk (s) = i=0 xi xi+k , where the indices are read modulo n. n, and that γn−k (s) = γk (s) for 1 ≤ k ≤ n − 1. Observe that γ0 (s) != n−1 i −1 Also, setting s(z) = i=0 si z , we have the formula s(z)s(z ) = n + !n−1 k n k=1 γk (s)z in the quotient ring Z[z]/(z − 1). Finally, if H = circ(s) is the circulant matrix with first row s, then obviously the matrix H ·H T has γj−i (s) as entry with position i, j. Thus, H will be an m-modular

Eliahou and Kervaire

circulant Hadamard matrix if and only if γk (s) ≡ 0 mod m for all 1 ≤ k ≤ n2 . The most obvious examples of circulant modular Hadamard matrices with a large modulus are J = circ(1, · · · , 1) and K = −2I + J = circ(−1, 1, · · · , 1) of size n. We have J · J T = nJ and K · K T = nI + (n − 4)(J − I). Thus, J is a circulant n-modular Hadamard matrix, and K is a circulant (n − 4)-modular Hadamard matrix, both of size n. More elaborate examples have the property that some of their periodic correlation coeﬃcients are actually 0, not only 0 mod m. We introduce the following definition. Definition 4.1.2 Let s ∈ {±1}n be a binary sequence of size n, with n even. We say that s is of type 1 if γ n2 (s) = 0. We say that s is of type 2if γ1 (s) = . . . = γ n2 −1 (s) = 0. This definition extends quite naturally to circulant binary matrices. A circulant binary matrix H is of type i (with i = 1 or 2) if its first row is of type i (equivalently, if any of its rows is of type i).

Remark 4.1.3 Ryser’s conjecture is equivalent to saying that there are no binary sequences of length greater than 4 which are simultaneously of type 1 and of type 2.

Circulant modular Hadamard matrices of type 1 and type 2 were introduced in [6]. After finding that “type 2” was equivalent with the notion of “almost perfect sequence”, we thought of abandoning the term “type 2”, and replacing the term “type 1” by “enhanced”. But we now choose to restore the type 1 / type 2 terminology, essentially because of the symmetry in the definition. We ask a little indulgence from the reader for these terminological meanderings. Example 4.1.4 Here is a binary sequence of length 8 and type 2. Let s = (1, 1, 1, −1, 1, −1, −1, 1). Then γ1 (s) = γ2 (s) = γ3 (s) = 0, showing that s is indeed a sequence of type 2. Additionally, γ4 (s) = −4. Taking H to be the circulant matrix with first row s, we have:

Modular Hadamard matrices

⎛

⎜ ⎜ ⎜ ⎜ ⎜ H=⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞ 1 1 1 −1 1 −1 −1 1 1 1 1 1 −1 1 −1 −1 ⎟ ⎟ −1 1 1 1 1 −1 1 −1 ⎟ ⎟ −1 −1 1 1 1 1 −1 1 ⎟ ⎟, 1 −1 −1 1 1 1 1 −1 ⎟ ⎟ −1 1 −1 −1 1 1 1 1 ⎟ ⎟ 1 −1 1 −1 −1 1 1 1 ⎠ 1 1 −1 1 −1 −1 1 1

and ⎛

⎜ ⎜ ⎜ ⎜ ⎜ T H ·H =⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞ 8 0 0 0 −4 0 0 0 0 8 0 0 0 −4 0 0 ⎟ ⎟ 0 0 8 0 0 0 −4 0 ⎟ ⎟ 0 0 0 8 0 0 0 −4 ⎟ ⎟. −4 0 0 0 8 0 0 0 ⎟ ⎟ 0 −4 0 0 0 8 0 0 ⎟ ⎟ 0 0 −4 0 0 0 8 0 ⎠ 0 0 0 −4 0 0 0 8

Another example of a binary sequence of length 8 and type 2 is provided by t = (1, 1, 1, −1, 1, 1, 1, −1). In this case we have γ1 (t) = γ2 (t) = γ3 (t) = 0 and γ4 (t) = 8. For every odd prime p ≡ 1 mod 4, we will exhibit circulant (p − 1)modular Hadamard matrices of type 1 and length 4p. Then, turning our attention to moduli which are powers of 2, we will exhibit 16-modular Hadamard matrices of type 1 and length 4p for every odd prime p ≡ 9 mod 16 for which 2 is a fourth power mod p. Finally, we will recall a classical construction from Delsarte, Goethals and Seidel which implies that there is a circulant (n − 4)-modular Hadamard matrix of type 2 for every size n of the form n = 2(pr + 1) where p is prime.

4.2

Circulant (p − 1)-modular Hadamard matrices of type 1 and size 4p

As announced above, we shall construct a circulant (p − 1)-modular Hadamard matrix of type 1 and size 4p, for every prime number p ≡ 1 mod 4. It is convenient to do so by exhibiting the Hall polynomial of its first row.

Eliahou and Kervaire

Consider the set S = {1, . . . , p − 1} and its partition S = S0 ∪ S1 , where S0 is the subset of squares mod p, and S1 the subset of nonsquares mod p. Of course, we have |S0 | = |S1 | =!p−1 2 . Let g0 (z) denote the generating function of S0 . That is, g0 (z) = i∈S0 z i . ! z i be the generating function of S1 . Similarly, let g1 (z) = ! " i∈S1 i Note that, since S = S0 S1 , we have g0 (z) + g1 (z) = p−1 i=1 z . Let x0 , x1 , x2 , x3 ∈ {±1} be four free binary parameters, and consider the polynomial h(z) = x0 (1 + z 2p )(1 + g0 (z 2 )) + x1 (1 + z 2p )z p g0 (z 2 )+ x2 (1 − z 2p )g1 (−z 2 ) + x3 (1 − z 2p )z p (1 + g1 (−z 2 )),

viewed as an element in the quotient ring Z[z]/(z 4p − 1).! i As it turns out, when expressing h(z) in the form 4p−1 i=0 ai z , we have ai = ±1 for all 0 ≤ i ≤ 4p − 1. In [5], we prove the following result. Theorem 4.2.1 Let p ≡ 1 mod 4 be a prime number. Let h(z) ∈ Z[z]/(z 4p − 1) be the above polynomial, h(z) = x0 (1 + z 2p )(1 + g0 (z 2 )) + x1 (1 + z 2p )z p g0 (z 2 )+ x2 (1 − z 2p )g1 (−z 2 ) + x3 (1 − z 2p )z p (1 + g1 (−z 2 )). Then h(z) is the Hall polynomial of a 4-parameter binary sequence h of length 4p, with the property that circ(h) is a circulant (p − 1)-modular Hadamard matrix of type 1 and size 4p. The proof of the theorem in [5] is obtained by computing h(z)h(z −1 ) explicitly in the ring Z[z]/(z 4p − 1). We find the following expression: h(z)h(z −1 ) = 4p + (p − 1)R(z), !2p 2i−1 ! 4i + z p + z 3p ). where R(z) = 2 p−1 i=1 z + x0 x1 ( i=1 z

! j Given that h(z)h(z −1 ) = 4p + 4p−1 j=1 γj (h)z , the above expression shows that the gcd of the periodic correlation coeﬃcients γi (h) of h for i = 1, ..., 4p − 1, is equal to p − 1. Note also that γ2p = 0, showing that h is a binary sequence of type 1. Thus, as stated, circ(h) is a circulant (p − 1)-modular Hadamard matrix of type 1 and size 4p.

Modular Hadamard matrices

Example 4.2.2 Let p = 5. The non-zero squares mod 5 are 1 and 4. Therefore S0 = {1, 4}, g0 (z) = z + z 4 and g1 (z) = z 2 + z 3 . Finally, h(z) ≡ x0 (1 + z 2 + z 8 + z 10 + z 12 + z 18 ) + x1 (z 3 + z 7 + z 13 + z 17 )+ x2 (z 4 −z 6 −z 14 +z 16 )+x3 (z+z 5 +z 9 −z 11 −z 15 −z 19 ) mod (z 20 −1), so h(z) is the Hall polynomial of the binary sequence h = (x0 , x3 , x0 , x1 , x2 , x3 , −x2 , x1 , x0 , x3 , x0 , −x3 , x0 , x1 , −x2 , −x3 , x2 , x1 , x0 , −x3 ).

The periodic correlation coeﬃcients γi = γi (h) for i = 1, ..., 10 are the following: γ1 = γ3 = γ7 = γ9 = 4x0 x1 , γ2 = γ6 = γ10 = 0, γ4 = γ8 = 8, γ5 = 8x0 x1 .

4.3

Circulant 16-modular Hadamard matrices of type 1

Our objective is to construct circulant 16-modular Hadamard matrices of type 1 and size 4p, where p is an odd prime. According to the Lemma below, this is only possible for p ≡ 1 mod 8, that is p ≡ 1 or 9 mod 16. When p ≡ 1 mod 16, the (p − 1)-modular construction of Section 4.2 already provides us with the desired sort of matrices, as p− 1 is divisible by 16. In this Section we consider the remaining case p ≡ 9 mod 16. We shall present a partial solution to our construction problem, which works in the case where 2 is a fourth power mod p (for example p = 73 or 89). For those primes p ≡ 9 mod 16 where 2 is not a fourth power mod p (for example p = 41 or 137), we do not know how to construct 16-modular circulant Hadamard matrices of type 1 and size 4p. Quite possibly, none exists in this case. We start with the promised result restricting the possible sizes of circulant 16-modular Hadamard matrices of type 1. Lemma 4.3.1 Let r ≥ 1 be a natural number, and assume there exists a circulant 16-modular Hadamard matrix of type 1 and size 4r. Then r ≡ 0, 1 or 4 mod 8. Proof: Let h(z) be the Hall polynomial of the first row h of a circulant 16-modular Hadamard matrix of type 1 and size 4r. In the quotient ring Z[z](z 4r − 1), we have the general formula h(z)h(z −1 ) = 4r + ! 2r−1 k −k ) + γ z 2r , where the γ are the periodic correlation 2r k k=1 γk (z + z

Eliahou and Kervaire

coeﬃcients of the sequence ! h. Setting z = 1 in the above formula, we get h(1)2 = 4r + 2 2r−1 k=1 γk + γ2r . Now, γ2r = 0 by the type 1 hypothesis, and γk ≡ 0 mod 16 for all 1 ≤ k ≤ 2r − 1. It follows that h(1)2 = 4r + 32θ for some integer θ. Thus h(1) is even, and dividing by 4 we get r = (h(1)/2)2 + 8θ. The conclusion follows as the only squares mod 8 are 0, 1 and 4. As a side remark, note that the same argument would still work under the weaker hypothesis γ2r ≡ 0 mod 32 instead of γ2r = 0. ! Let p be a prime such that p ≡ 1 mod 8. As in Section 4.2, consider the set S = {1, . . . , p − 1} and its partition S = S0 ∪ S1 , where S0 is the subset of squares mod p, and S1 the subset of non-squares mod p. For our purposes here, we need to refine this partition as follows. Let ρ : S → F∗p denote the natural projection of S into the multiplicative group F∗p of non-zero elements of the finite field Fp . Let c ∈ F∗p denote a generator of that group, that is an element of multiplicative order p − 1. Given that the squares in F∗p consist of the subgroup ⟨c2 ⟩ generated by c2 , we have S0 = ρ−1 (⟨c2 ⟩) and S1 = ρ−1 (c⟨c2 ⟩), where c⟨c2 ⟩ is the other coset of ⟨c2 ⟩ in F∗p . Consider now the subgroup Γ = ⟨c4 ⟩ ⊂ F∗p . Thus, Γ is the only subgroup of order (p − 1)/4 in F∗p . The four cosets of Γ in F∗p are Γ, cΓ, c2 Γ and c3 Γ, and of course they partition F∗p into four pieces of equal size (p − 1)/4. This partition refines the earlier one into squares and non-squares, as Γ ∪ c2 Γ = ⟨c2 ⟩. Transporting back the above partition to S by ρ−1 , we shall denote S00 = ρ−1 (Γ), S10 = ρ−1 (cΓ), S01 = ρ−1 (c2 Γ) and S11 = ρ−1 (c3 Γ). In this way, we obtain the promised refinement of the partition S = S0 ∪ S1 , as S0 = S00 ∪ S01 and S1 = S10 ∪ S11 . The four subsets Su,v all have cardinality (p − 1)/4. For u, v = 0, 1, we shall by gu,v (z) the generating function of ! denote i Su,v , that is gu,v (z) = i∈Su,v z . Note that g00 (z) + g01 (z) + g10 (z) + ! i g11 (z) = p−1 i=1 z . Note also that g0 (z) = g00 (z) + g01 (z) and g1 (z) = g10 (z) + g11 (z), where g0 (z) and g1 (z) are the generating functions defined and used in Section 4.2. Let x0 , x1 , x2 , x3 ∈ {±1} be four free binary parameters, and consider the polynomial h(z) = x0 (1+z 2p )(1−g00 (z 2 )−g01 (z 2 ))+x1 (1+z 2p )z p (g00 (z 2 )−g01 (z 2 ))+ x2 (1−z 2p )(g10 (−z 2 )−g11 (−z 2 ))+x3 (1−z 2p )z p (1−g10 (−z 2 )−g11 (−z 2 )),

Modular Hadamard matrices

viewed as an element in the quotient ring Z[z]/(z 4p − 1). As for the corresponding in Section 4.2, when expressing h(z) in the !4p−1 polynomial i form i=0 ai z , we have ai = ±1 for all 0 ≤ i ≤ 4p − 1. In [7], we prove the following result.

Theorem 4.3.2 Let p ≡ 1 mod 8 be a prime number. Furthermore, let h(z) ∈ Z[z]/(z 4p − 1) be the above polynomial h(z) = x0 (1+z 2p )(1−g00 (z 2 )−g01 (z 2 ))+x1 (1+z 2p )z p (g00 (z 2 )−g01 (z 2 ))+ x2 (1−z 2p )(g10 (−z 2 )−g11 (−z 2 ))+x3 (1−z 2p )z p (1−g10 (−z 2 )−g11 (−z 2 )). Then h(z) is the Hall polynomial of a 4-parameter binary sequence h of length 4p, with the property that circ(h) is a circulant 8-modular Hadamard matrix of type 1 and size 4p. Moreover, the matrix circ(h) is a circulant 16-modular Hadamard matrix if and only if p ≡ 9 mod 16 and 2 is a fourth power mod p. ! k −k ) The periodic correlations γk in h(z)h(z −1 ) = 4p+ 2p−1 k=1 γk (z +z are explicitly determined in [7], using Jacobi sums. They depend on the decomposition p = a2 +b2 with b even, a odd and the sign of a normalized by the requirement a ≡ 1 mod 4. With this normalization, the correlations γk , are all equal to ±(p−9), ±2(a + 3), or ±2b for k = 1, . . . , 2p − 1. Furthermore, γ2p = 0 showing that h is of type 1. By a theorem of Gauss, a prime p ≡ 1 mod 8 is of the form p = a2 +b2 with b divisible by 8 if and only if 2 is a fourth power modulo p. If follows that if p ≡ 9 mod 16 and 2 is a fourth power modulo p, then necessarily a ≡ −3 mod 8 and b ≡ 0 mod 8. Thus, in this case, all the periodic correlations γk (h) for k = 1, . . . , 2p − 1 are divisible by 16, and circ(h) is a circulant 16-modular Hadamard matrix of type 1 and size 4p. Example 4.3.3 The smallest prime p ≡ 9 mod 16 for which 2 is a fourth power mod p is p = 73 = (−3)2 + 82 . Setting x0 = x1 = x2 = x3 = 1 in the above formula for h(z), we get the following binary sequence h, for which circ(h) is a 16-modular Hadamard matrix of type 1 and size 292 : ++−−−−−+−+−+−−−+−−−−+−+−−−+− +−++−−−−−+−+++++−−−+−−−+−−−+ ++++−−++−+−−++−+−+−+−+−−++−+

− + − − − + + −

Eliahou and Kervaire

− − − − − − − −

4.4

+ − − − + − − −

+−+−+−−++− +−−−−−−−−− −−+++−−−+− −−−−+−−−++ +−+−+−−+−− +−−+−−−+++ +−−−+−−+++ + − − − + − − − +.

− − − − − + −

− − − − − − +

+ − + + + + −

− − + − + − −

− − − − − − −

+ − − − − − +

+ + + + + − +

− + + − − − +

+ − − − + − −

− − − + − − −

+ + + + − + +

− + − − − − −

+ + − − + − −

− − − − − + −

− + + − + + +

Circulant modular Hadamard matrices of type 2

We are seeking binary sequences s of even length n with the property that γ1 (s) = . . . = γ n2 −1 (s) = 0. In this way, circ(s) will be a circulant m-modular Hadamard matrix of type 2, with m = γ n2 (s). These sequences were first introduced by J.Wolfmann [16] in 1992, and are called almost perfect sequences. See also Langevin [10]. (Recall that a sequence s of length n ≡ 0 mod 4 is perfect if it satisfies γi (s) = 0 for all 1 ≤ i ≤ n/2. This is equivalent to circ(s) being a circulant Hadamard matrix. Hence, Ryser’s conjecture amounts to saying that there is no perfect sequence of length n ≡ 0 mod 4 with n > 4.) Almost perfect sequences are known in all lengths n of the form n = 2(q + 1) where q is an odd prime power, and are believed not to exist in other lengths. This follows from a theorem by Delsarte, Goethals and Seidel, establishing the existence of a negacyclic conference matrix of order q + 1 for every odd prime power q. For convenience of the reader, this is recalled below. Definition 4.4.1 A conference matrix C is a square matrix of size n, with entries 0 on the diagonal and ±1 elsewhere, satisfying the condition C · C T =(n − 1)I. Definition 4.4.2 A negacyclic matrix N is a square matrix of the form ⎛

⎜ ⎜ ⎜ N = N C(u0 , u1 , . . . , ur ) = ⎜ ⎜ ⎝

u1 . . . . . . ur u0 −ur u0 u1 . . . ur−1 −ur−1 −ur u0 . . . ur−2 .. .. .. .. .. . . . . . −u1 −u2 . . . −ur u0

⎞

⎟ ⎟ ⎟ ⎟. ⎟ ⎠

Modular Hadamard matrices

Example 4.4.3 As an illustration of both concepts simultaneously, here is a negacyclic conference matrix of size 6: ⎛ ⎞ 0 1 1 1 −1 1 ⎜ −1 0 1 1 1 −1 ⎟ ⎜ ⎟ ⎜ 1 −1 0 1 1 1 ⎟ ⎜ ⎟. C=⎜ 1 −1 0 1 1 ⎟ ⎜ −1 ⎟ ⎝ −1 −1 1 −1 0 1 ⎠ −1 −1 −1

1 −1

Theorem 4.4.4 (Delsarte-Goethals-Seidel, [3]) Let q be an odd prime power. Then there exists a negacyclic conference matrix of size q + 1. Proof: (Sketch) Let g be a primitive element of the finite field Fq2 , that is, a generator of the group F∗q2 of non-zero elements. ( ' 0 −g q+1 , with entries in the subfield Fq as g · g q and Let A = 1 g + gq g + g q are the norm and trace of g, respectively. Let ' ( 1 . v= 0 The q + 1 vectors Ai · v, 0 ≤ i ≤ q, are pairwise independent over Fq . Define the matrix C of size q + 1 by Ci,j = χ(det(Ai · v, Aj · v)) for 0 ≤ i, j ≤ q, where χ : Fq → {0, ±1} is the quadratic character of F∗q , extended by χ(0) = 0. Then C has entries 0 on the diagonal, and ±1 elsewhere. Moreover, C is a conference matrix, that is C · C T = qI. Finally, let Γ be the alternating diagonal matrix Γ = d iag(1, −1, . . . , 1, −1) of size q + 1. As it turns out, the product Γ · C is a negacyclic conference matrix of size q + 1, as desired. See [3] for more details. ! Theorem 4.4.5 Let q be an odd prime power, and let n = 2(q + 1). There exists a binary sequence s of length n and of type 2, i.e. satisfying γ1 (s) = . . . = γ n2 −1 (s) = 0. Moreover, γ n2 (s) = 4 − n. Proof: Given a binary sequence s′ = (x1 , x2 , . . . , xq ) of length q, define the sequence s = [1; s′ ; 1; −s′ ] of length n = 2q + 2. An easy calculation shows that γn/2 (s) = 4 − n, and that γk (s) = 2(ck (s′ ) −

Eliahou and Kervaire

!q−k cq+1−k (s′ )) for all 1 ≤ k ≤ q, where ck (s′ ) = j=1 xj xj+k denote th the k aperiodic correlation coefficient of the sequence s′ . Thus, the sequence s will be of type 2 if and only if ck (s′ ) = cq+1−k (s′ ) for all 1 ≤ k ≤ q. Now, the latter condition on s′ is equivalent to the negacyclic matrix N = N C(0, x1 , x2 , . . . , xq ) being a conference matrix, as the dot product of the ith row and the (i+k)th row of N is equal to ck (s′ )−cq+1−k (s′ ). By the result of Delsarte, Goethals and Seidel, there exists a negacyclic conference matrix C = N C(0, y1 , . . . , yq ) of size q + 1. Let s′ = (y1 , . . . , yq ), and s = [1; s′ ; 1; −s′ ]. From the above discussion, it follows that s is a binary sequence of type 2 and length n, as desired. ! Acknowledgement The first author gratefully acknowledges partial support from the Fonds National Suisse de la Recherche Scientifique during the preparation of this paper. Shalom Eliahou Département de Mathématiques, Université du Littoral Cˆ ote d’Opale, Bˆ atiment Poincaré, 50 rue Ferdinand Buisson, B.P. 699, 62228 Calais, France, eliahou@lmpa.univ-littoral.fr

Michel Kervaire Département de Mathématiques, Université de Genève, 2 rue du Lièvre, B.P. 240, 1211 Genève 24, Suisse. Michel.Kervaire@math.unige.ch

References [1] Terry H. A.; Ralph G. S., Golay sequences, Lect. Notes Math., 622 (1977), 44-54. [2] Borwein P. B.; Ferguson R. A., A complete description of Golay pairs for lengths up to 100, Math. Comput., 47 (2004), 967-985. [3] Delsarte P.; Goethals J.-M.; Seidel J., Orthogonal matrices with zero diagonal. II, Can. J. Math., XXIII (1971), 816-832. [4] Dinitz J. H.; Stinton D. R., Contemporary Design Theory, A Collection of Surveys, Wiley-Interscience Publication, 1992. [5] Eliahou S.; Kervaire M., Circulant Modular Hadamard matrices, Ens. Math., 47 (2001), 103-114.

Modular Hadamard matrices

[6] Eliahou S.; Kervaire M., Modular Sequences and Modular matrices, J. of Comb. Designs, 9 (2001), 187-214. [7] Eliahou S.; Kervaire M., Circulant 16-modular Hadamard matrices and Jacobi sums, J. Comb. Theory, 100 (2002), 116-135. [8] Eliahou S.; Kervaire M.; Saffari B., On Golay polynomial pairs, Adv. Applied Math., 12 (1991), 235-292. [9] Hadamard J., Résolution d’une question relative aux déterminants, Bulletin des Sciences Mathématiques, 17 (1893), 240-246. [10] Langevin P., Almost perfect binary functions, App. Alg. Eng. Comm. Comp., 4 (1993), 95-102. [11] Marrero O.; Butson A. T., Modular Hadamard matrices and related designs, J. Comb. Theory, 15 (1973), 257-269. [12] Marrero O.; Butson A. T., Modular Hadamard matrices and related designs, II, Can. J. Math., XXIV (1972), 1100-1109. [13] Ryser H. J., Combinatorial Mathematics, Carus Monograph 14, Math. Assoc. of America, 1963. [14] Schmidt B., Cyclotomic integers and finite geometry, J. of the Amer. Math. Soc., 12 (1999), 929-952. [15] Turyn R. J., Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulse compression and surface wave encoding, J. Comb. Theory, 16 (1974), 313-333. [16] Wolfmann J., Almost perfect autocorrelation sequences, IEEE Trans. Inform. Theory, 38 (1992), 1412-1418. [17] Van Lint J. H.; Wilson R. M., A course in combinatorics, Cambridge University Press, 1992.

Morfismos, Vol. 7, No. 2, 2003, pp. 47â&#x20AC;&#x201C;73

Application of modularity to optimal resource allocation with risk sensitivity Guadalupe Avila-Godoy

Abstract An optimal allocation problem with a risk-sensitive controller is modelled by a controlled Markov chain with exponential total cost criterion. Some general results recently obtained are applied to show that the particular model studied here has a monotone optimal policy and monotone optimal value function. Moreover, it is shown that under certain conditions, the allocation problem with both risk-neutral and risk-sensitive performance criteria has an optimal policy of the threshold type.

2000 Mathematics Subject Classification: 90C40, 60J05. Keywords and phrases: Controlled Markov Chains, Optimal Resource Allocation, Exponential Total Cost Criterion, Risk Sensitivity, Modularity.

Introduction

In this paper we study a finite horizon controlled Markov chain (CMC) modeling an optimal allocation problem with exponential total cost (ETC) as performance risk-sensitive criterion. The CMC considered has finite state space, compact action space and bounded cost function. Models of dynamic systems that incorporate risk-sensitivity by means of an exponential utility function have recently received considerable attention in the literature, see for example [2, 3, 6, 7, 8, 9] and references therein. However, in contrast with the risk-neutral literature (see 1

This paper is part of the authorâ&#x20AC;&#x2122;s doctoral research under the direction of Dr. Emmanuel Ferna Â´ndez Gaucherand at the Department of Mathematics of the University of Arizona.

Avila-Godoy

[11, 13, 14, 15, 16, 17, 18, 19] and references therein), only a few (and recent) contributions dealing with structural properties of the CMC have been made in the risk-sensitive case; see [5, 9]. We are particularly concerned with the contributions made by Avila-Godoy [4], which extend some results in [13, 14] from the risk-neutral to the risk-sensitive case. It is proved in [4] that appropiate structural conditions (like modularity and/or monotonicity) of certain functions defined in terms of the cost and the transition kernel, imply the monotonicity of the optimal exponential value function and the existence of monotone optimal policies. Herein, we apply some results in [4] to show that the CMC modeling the optimal allocation problem has monotone optimal policies and monotone optimal value function. See, e.g., [13, 14] for an analysis of this problem with risk-neutral total cost. The paper is organized as follows. Section 2 includes the description of the model and collects the results in [4] needed for our study. In Sections 3 and 4, the CMC model for the finite horizon optimal allocation problem is given and the main results of the paper are proved. First, it is shown that the optimal value function for this model, Jt (x), is increasing in the state x and decreasing in t (Lemma 3.1.6), and then the existence of a monotone optimal policy is established (Proposition 3.1.12), that is, we show that the decision function of the optimal policy at the t-th stage is increasing (as a function of the state), for t = 0, 1, . . ., and increasing in t, for each x ∈ X. Moreover, under additional conditions, we prove that the allocation problem can be reduced to a problem with two actions and that the optimal policy is of the threshold type (Proposition 4.1.13). Finally, we apply those results to a particular example with a linear final cost. For the purpose of comparing the results obtained in Sections 3 and 4, we include an appendix on risk-neutral resource allocation problems. The proof of the result relative to the reduction of the risk-neutral allocation problem to a problem with two actions and that the optimal policy is of the threshold type is also a contribution of this paper’s author (Section 5.2).

Description of the model and basic results

Let us consider a CMC specified by the four-tuple (X, A, P, C), where: • X = {1, 2, . . .} is the state space, a countable set.

• A, the action (or control) set, is a compact subset of R. The set

Optimal resource allocation

! " K := (x, a) : x ∈ X, a ∈ A is called the set of state-action pairs.

• P, the transition kernel, is a family of transition probabilities on X given K: P = {P (· | x, a) : (x, a) ∈ K}. We will also denote pxx′ (a) := P (x′ | x, a). Finally,

• C : K −→ R is the one-stage cost function. We will assume that C is nonnegative and bounded: 0 ! C(x, a) ! K < ∞ for every (x, a) ∈ K, and c : X −→ R is the final penalty cost. The above defined CMC represents a stochastic dynamical system observed at times t = 0, 1, · · · , n, whose evolution is as follows. Let Xt and At respectively denote the state of the system and the action chosen at time t. If X0 = x ∈ X, and A0 = a ∈ A, then (i) a cost C(x, a) is incurred, and (ii) the system moves to a new state X1 according to the probability distribution P (· | x, a). Once the transition into the new state has occurred, a new action is chosen, and the process is repeated for n times; see [1, 10, 13]. The strategy followed to choose the actions at each stage is called a policy. The most general set Π of policies considered in the literature includes the admissible, history dependent, randomized policies; see [1, 10, 13]. Herein, we will be concerned only with the subset of Π consisting of the Markov deterministic policies, denoted by ΠM D . For a policy π ∈ Π and initial state x ∈ X, Exπ will denote the expectation operator with respect to the probability measure induced by π and x in the space of trayectories of the chain. Risk-sensitivity of the controller is modelled by grading the total cost with the exponential (disutility) function Uγ (x) = (sgn γ)eγx , γ ̸= 0, where the parameter γ turns out to be the (constant) risk-sensitivity coeﬃcient associated to Uγ , see [12, 20]. In this work, only the case γ > 0, the risk-averse case, will be considered. Thus, the performance criterion for a policy π when the initial state is x and we proceed for n stages, is given by # !n−1 $ (1) Jnπ (x, γ) := Exπ eγ( t=0 C(Xt ,At )+c(Xn )) .

The stochastic optimal control problem is to find a policy π ∗ within the class Π such that (1) is minimized, that is, such that (2)

∗

Jnπ (x, γ) = inf {Jnπ (x, γ)} =: Jn (x, γ). π

Avila-Godoy

The optimal policy π ∗ is called ETC-optimal and Jn (x, γ) is the optimal ETC. We can interpret Jn (x, γ) as the minimal ETC that can be obtained starting at state x with risk-sensitivity coeﬃcient γ and proceeding for n stages. For ease of reference, we end the section by stating (without proofs) some general results that will be needed in the next section; see [4] for their proofs. The following known assumption will be made in the rest of this section (see [10]). Assumption 2.1 1) C(x, ·) is continuous for each x ∈ X; and 2) If v : X −→ R is bounded then the function ! a $−→ pxy (a)v(y) is continuous, y

for each x ∈ X. First, we recall a typical forward dynamic programming recursion. Theorem 2.1.1 (Dynamic Programming Algorithm) The optimal ETC, Jn (x, γ), satisfies the following recursion:

(3)

J0 (x, γ) = eγc(x) , .. .. . . ! " # pxy (a)Js (y, γ) , Js+1 (x, γ) = inf eγC(x,a) a∈A

for s = 0, 1, · · · n − 1. For s = 0, 1, 2, . . . n − 1, let fs : X −→ A be a decision rule defined by $ %! eγC x,fs (x) pxy (fs (x)) Jn−s−1 (y, γ) = (4)

inf

a∈A

" γC(x,a) ! # e pxy (a)Jn−s−1 (y, γ) . y

Then the Markov deterministic policy π ∗ = (f0 , f1 , f2 , . . . fn−1 ) is ETCoptimal. Next, a lemma that provides suﬃcient conditions for monotonicity of the optimal value function is stated.

Optimal resource allocation

Lemma 2.1.2 Suppose that i) C(x, a) is increasing (decreasing) in x for each a, and c(x) is increasing (decreasing). ii)

∞ ! y=z

pxy (a) is increasing in x for all z ∈ X and a ∈ A.

Then, the optimal value function Js (x, γ) is increasing (decreasing) in x, for s = 0, 1, · · · n. Finally, some standard definitions and notation, and two key theorems about structural properties of CMC’s are stated (see [4].) Let (S, !S ) be a lattice, i.e., a partially ordered set such that if s, r ∈ S then s ∨ r ∈ S and s ∧ r ∈ S, and let G : S −→ R. We say that a) G(·) is subadditive (or submodular) on S if G(s ∨ r) + G(s ∧ r) " G(s) + G(r) for every s, r ∈ S; b) G(·) is superadditive (or supermodular) on S if −G(·) is subadditive on S. We will assume the state and action spaces to be subsets of R with the usual order and we will consider the product order ! on R2 , that is, ! is defined by (y, z) ! (y ′ , z ′ ) if y " y ′ and z " z ′ . A Markov deterministic policy π = (f0 , f1 , . . . , fn−1 ) is said to be monotone (with respect to x) if all the decision rules ft are monotone functions of the state x. In the particular case that the action space contains only two actions, say a1 and a2 , a monotone policy is called a threshold policy. That is, a threshold policy is a deterministic Markov policy π = (f0 , f1 , . . . , fn−1 ) such that, for t = 0, 1, . . . , n − 1, the decision rule ft is given by " a1 if x # x∗t (5) ft (x) = a2 if x < x∗t , where x∗t is the control limit or threshold. It is clearly useful to know in advance when a monotone optimal policy exists, because the search for an optimal policy can then be restricted

Avila-Godoy

from the class of Markov deterministic policies to the much smaller subclass of monotone policies [11, 13]. The following theorem provides sufficient conditions for the existence of optimal monotone (with respect to x) policies for CMC’s with ETC criterion. Notation. For t = 0, 1, 2, . . . , n − 1, let (6)

Ht (x, a, γ) := eγC(x,a)

pxy (a)Jn−t−1 (y, γ),

i.e., Ht denotes the function within brackets in (4). Theorem 2.1.3 For t = 0, 1, 2, · · · n − 1, set A∗t (x) =

" # ′ a ∈ A : Ht (x, a, γ) = min {H (x, a , γ)} , t ′ a

and ft (x) := min A∗t (x) (respectively ft (x) := max A∗t (x)). Suppose that log Ht (·, ·, γ) is subadditive (respectively superadditive) on (X × A, !), for fixed γ. Then, (f0 , f1 , . . . fn−1 ) is an optimal policy with ft (x) increasing (respectively decreasing) in x for each t.

Additionally, due to the fact that the optimal policy π = (f0 , f1 , . . . fn−1 ) is in general non-stationary, it is natural to ask how the optimal action ft (x) varies with respect to t for each fixed x. Thus, a Markov deterministic policy π = (f0 , f1 , . . . , fn−1 ) is said to be monotone (with respect to t) if for each fixed x, the sequence of actions ft (x) is monotone in t. The following theorem provides suﬃcient conditions for the existence of optimal monotone (with respect to t) policies for CMC’s with ETC criterion. Theorem 2.1.4 Let A∗t (x) and ft (x) be as in Theorem 2.1.3. Assume that for each x ∈ X, the function log H(·) (x, ·, γ) is superadditive (respectively subadditive) on the lattice (A × {0, 1, 2, · · · n − 1}, !). Then, (f0 , f1 , . . . fn−1 ) is an optimal policy such that the sequence of actions ft (x) is decreasing (respectively increasing) in t for each x.

Optimal resource allocation

An Optimal Allocation Problem.

In this section we follow Ross [14] to model an optimal allocation problem by means of a finite horizon CMC. However, unlike the mentioned reference, we introduce a risk-sensitive performance criterion. The general results stated in Section 2 are applied to show that, under standard conditions, the optimal value function is increasing in the state and decreasing in t (Lemma 3.1.6) and that the optimal policy is increasing in x and increasing in t (Proposition 3.1.12). Moreover, under additional conditions, we prove that the allocation problem can be reduced to a problem with two actions and that the optimal policy is of the threshold-type (Proposition 4.1.13). Finally, we apply those results to a particular example of a linear final cost; see [14] for an analysis of this problem with risk-neutral total cost. The optimal allocation problem can be described as follows. Suppose we have N stages to construct sequentially I successful components . At each stage we allocate a certain amount of money for the construction of a component. If a is the amount allocated, then the component constructed will be a success with probability P (a), where P is a continuous strictly increasing function such that P (0) = 0. After each component is constructed, we are informed whether or not it is successful. If at the end of N stages, we are x components short, then a final penalty cost c(x) is incurred, where c(x) is increasing. The problem is to determine how much money to allocate at each stage to minimize the expected ETC. A CMC (X, A, P, C) which models the described allocation problem can be defined by taking the state space X = {0, 1, . . . I}, the action space A = [0, M ], where M is a positive real number, the cost function C(x, a) = a, and the transition probabilities ⎧ ⎪ if y = x − 1 ⎨P (a) (7) pxy (a) = 1 − P (a) if y = x ⎪ ⎩ 0 otherwise. The state Xt is the number of successful components still needed at time t and the action At is the amount of money allocated at time t. We recall that Jt (x, γ) denotes the minimal cost starting at state x with t stages to go, x ∈ X and t = 0, 1, . . . , N . Remark 3.1.5 This model satisfies Assumption 2.1 since C(x, a) and % pxy (a)Jt (y, γ) = P (a)Jt (x − 1, γ) + (1 − P (a))Jt (x, γ) y

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are continuous functions in a, for each x ∈ X. According to (3), J0 (x, γ) = eγc(x) and for t = 0, 1, . . . N − 1, Jt+1 (x, γ) (8)

(9)

inf

a∈[0,M ]

inf

a∈[0,M ]

eγa [P (a)Jt (x − 1, γ) + (1 − P (a))Jt (x, γ)]

{eγa [Jt (x, γ) − P (a)(Jt (x, γ) − Jt (x − 1, γ)]}

(10) =

inf

a∈[0,M ]

{eγa [Jt (x − 1, γ) + (1 − P (a))(Jt (x, γ) − Jt (x − 1, γ)]} .

First, we will show that the optimal value function Jt (x, γ) is increasing in the state x and decreasing in the number t of stages to go. Lemma 3.1.6 The optimal value function Jt (x, γ) is increasing in x and decreasing in t. Proof: We will apply Lemma 2.1.2 to prove that Jt (x, γ) is increasing in x. First, we see that this model satisfies (i) of the mentioned lemma since C(x, a) is constant in x, and the terminal cost c(x) is increasing. Finally, it follows from (7) that ⎧ ⎪ if k ! x − 1 I ⎨1 # (11) pxy (a) = 1 − P (a) if k = x ⎪ ⎩ y=k 0 if k > x,

and hence, (ii) of Lemma 2.1.2 is valid for this model. Therefore, Jt (x, γ) is increasing in x. Now, since a = 0 is an admissible action, it follows from (8) that Jt+1 (x, γ) ! eγ·0 [P (0)Jt (x − 1, γ) + (1 − P (0))Jt (x, γ)], and hence,

Jt+1 (x, γ) ! Jt (x, γ).

Thus, Jt (x, γ) is decreasing in t for each x. ✷ Our next goal is to show that the allocation problem has optimal policies that are increasing in x and increasing in t. To this end, we will prove that (12)

log {eγa [P (a)JN −t−1 (x − 1, γ) + (1 − P (a))JN −t−1 (x, γ)]}

Optimal resource allocation

is subadditive on X × A and subadditive on A × {0, 1, 2, . . . N − 1}, so that the mentioned monotonicity properties will follow from Theorems 2.1.3 and 2.1.4 since in this model the function Ht defined in (6) is the function within brackets in (12), i.e., Ht (x, a, γ) = eγa [P (a)JN −t−1 (x − 1, γ) + (1 − P (a))JN −t−1 (x, γ)]. Set gt (x, a, γ) = P (a)Jt (x − 1, γ) + (1 − P (a))Jt (x, γ) and (13)

Gt (x, a, γ) := eγa gt (x, a, γ),

so that (14)

Ht (x, a, γ) = GN −t−1 (x, a, γ).

First, it follows from (14) that each of the structural properties of log Ht (x, a, γ) we need is equivalent to a structural property of log Gt (x, a, γ).

Lemma 3.1.7 a) log Ht (x, a, γ) is subadditive on X×A iﬀ log Gt (x, a, γ) is subadditive on X × A. b) log Ht (x, a, γ) is subadditive on A × {0, 1, . . . N − 1} iﬀ log Gt (x, a, γ) is superadditive on A × {0, 1, . . . N − 1}. Next, we will see that each of the structural properties of log Gt (x, a, γ) we need is equivalent to a structural property of log Jt (x, γ).

Lemma 3.1.8 a) log Gt (x, a, γ) is subadditive on X×A iﬀ log Jt (x, γ) is convex in x. b) log Gt (x, a, γ) is superadditive on A × {0, 1, . . . N − 1} iﬀ log Jt (x, γ) is subadditive on X × {0, 1, . . . N − 1}. Proof:

a) Let a′ > a and denote by Dt (x) := Jt (x + 1, γ) − Jt (x, γ).

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Then log Jt (x, γ) is convex in x ⇐⇒ log Jt (x + 1, γ) − log Jt (x, γ) ! log Jt (x, γ) − log Jt (x − 1, γ)

⇐⇒ Jt (x + 1, γ)Jt (x − 1, γ) ! Jt2 (x, γ)

⇐⇒ Jt (x, γ)Dt (x) ! Jt (x + 1, γ)Dt (x − 1)

⇐⇒ (P (a′ ) − P (a))Jt (x, γ)Dt (x) ! (P (a′ ) − P (a))Jt (x + 1, γ) Dt (x − 1)

⇐⇒ −P (a)Jt (x, γ)Dt (x) − P (a )Jt (x + 1, γ)Dt (x − 1) ! ′

− P (a′ )Jt (x, γ)Dt (x) − P (a)Jt (x + 1, γ)Dt (x − 1)

⇐⇒ [Jt (x + 1, γ) − P (a)Dt (x)][Jt (x, γ) − P (a′ )Dt (x − 1)] !

[Jt (x + 1, γ) − P (a′ )Dt (x)][Jt (x, γ) − P (a)Dt (x − 1)]

gt (x + 1, a′ ) gt (x + 1, a) ! gt (x, a) gt (x, a′ ) ⇐⇒ log gt (x, a, γ) is subadditive on X × A ⇐⇒

⇐⇒ log Gt (x, a, γ) is subadditive on X × A.

Note that the last step follows from the equality log Gt (x, a, γ) = γa + log gt (x, a, γ). b) Let a′ > a. Then log Jt (x, γ) is subadditive on X × {0, 1, . . . N − 1}

⇐⇒ log Jt+1 (x − 1) − log Jt+1 (x) ! log Jt (x − 1) − log Jt (x)

⇐⇒ Jt+1 (x − 1)Jt (x) ! Jt+1 (x)Jt (x − 1)

⇐⇒ Jt (x)Dt+1 (x − 1) " Jt+1 (x)Dt (x − 1)

⇐⇒ (P (a′ ) − P (a))Jt (x)Dt+1 (x − 1) " (P (a′ ) − P (a))Jt+1 (x) Dt (x − 1)

⇐⇒ −P (a)Jt (x)Dt+1 (x − 1) − P (a )Jt+1 (x)Dt (x − 1) " ′

− P (a′ )Jt (x)Dt+1 (x − 1) − P (a)Jt+1 (x)Dt (x − 1)

⇐⇒ [Jt+1 (x) − P (a)Dt+1 (x − 1)][Jt (x) − P (a′ )Dt (x − 1)] "

[Jt+1 (x) − P (a′ )Dt+1 (x − 1)][Jt (x) − P (a)Dt (x − 1)]

gt+1 (x, a′ , γ) gt+1 (x, a, γ) " gt (x, a, γ) gt (x, a′ , γ) ⇐⇒ log gt (x, a, γ) is superadditive on A × {0, 1, · · · N − 1} ⇐⇒

⇐⇒ log Gt (x, a, γ) is superadditive on A × {0, 1, · · · N − 1}. ✷

Optimal resource allocation

Now, we show that log Jt (x, γ) is indeed convex in x for each t, and subadditive on X × {0, 1, . . . N − 1}. Throughout the rest of this paper we will assume the following condition, which is reasonable for some situations. Assumption 3.1. The terminal cost c(x) is convex. Lemma 3.1.9 Under Assumption 3.1, the following three statements hold: a) log Jt (x, γ) is convex in x for each t. b) log Jt (x, γ) is convex in t for each x. c) log Jt (x, γ) is subadditive on (X × {0, 1, . . . N − 1}, !). Proof:

First, note that (a), (b) and (c) are equivalent to

(15)

Ax,t :

(16)

Bx,t :

(17)

Cx,t :

Jt (x + 1, γ) Jt (x + 2, γ) " Jt (x + 1, γ) Jt (x, γ) Jt+2 (x, γ) Jt+1 (x, γ) " Jt+1 (x, γ) Jt (x, γ) Jt+1 (x + 1, γ) Jt+1 (x, γ) " Jt (x, γ) Jt (x + 1, γ)

respectively. We will show that those inequalities hold for t = 0, 1, . . . N −2 and x = 0, 1, . . . I −2. The proof will be by induction on k = t+x. We have that C0,0 is true since Jt is decreasing in t (Lemma 3.1.6). B0,0 is an obvious equality, and A0,0 follows from Assumption 3.1. Thus the inequalities are true for k = 0. We assume that they are true whenever t + x < k and let k = t + x. Let’s prove Cx,t . It follows from (9) that for some a, say a ¯, a)(Jt (x, γ) − Jt (x − 1, γ))], Jt+1 (x, γ) = eγ¯a [Jt (x, γ) − P (¯ and hence (18)

! " Jt (x, γ) − Jt (x − 1, γ) Jt+1 (x, γ) γ¯ a =e 1 − P (¯ a) . Jt (x, γ) Jt (x, γ)

On the other hand, it follows from Ax−1,t that Jt (x + 1, γ) − Jt (x, γ) Jt (x, γ) − Jt (x − 1, γ) # . Jt (x, γ) Jt (x + 1, γ)

Avila-Godoy

Therefore, from (18) we obtain " ! Jt (x + 1, γ) − Jt (x, γ) Jt+1 (x, γ) γ¯ a ≥e 1 − P (¯ a) Jt (x, γ) Jt (x + 1, γ) Jt+1 (x + 1, γ) , ≥ Jt (x + 1, γ) and Cx,t follows. In a similar way, to prove Bx,t we have that it follows from (9) that for some a, say a′ , ′

Jt+2 (x, γ) = eγa [Jt+1 (x, γ) − P (a′ )(Jt+1 (x, γ) − Jt+1 (x − 1, γ))], and hence (19)

" ! Jt+2 (x, γ) γa′ ′ Jt+1 (x, γ) − Jt+1 (x − 1, γ) =e . 1 − P (a ) Jt+1 (x, γ) Jt+1 (x, γ)

On the other hand, it follows from Cx−1,t that Jt (x, γ) − Jt (x − 1, γ) Jt+1 (x, γ) − Jt+1 (x − 1, γ) ! . Jt+1 (x, γ) Jt (x, γ) Therefore, from (19) we obtain " ! Jt+2 (x, γ) γa′ ′ Jt (x, γ) − Jt (x − 1, γ) "e 1 − P (a ) Jt+1 (x, γ) Jt (x, γ) Jt+1 (x, γ) , " Jt (x, γ) and Bx,t follows. Finally, to prove Ax,t , note that Bx+1,t−1 is just Jt (x + 1, γ) Jt+1 (x + 1, γ) " , Jt (x + 1, γ) Jt−1 (x + 1, γ) or equivalently, Jt+1 (x + 1, γ)Jt−1 (x + 1, γ) " Jt2 (x + 1, γ). Thus to complete the proof of (15) we have to show that (20)

Jt (x + 2, γ)Jt (x, γ) " Jt+1 (x + 1, γ)Jt−1 (x + 1, γ).

It follows from (10) that for some a, say a ˜, Jt (x + 2, γ) = eγ˜a [Jt−1 (x + 1, γ) + (1 − P (˜ a)) (Jt−1 (x + 2, γ) −Jt−1 (x + 1, γ))] ,

Optimal resource allocation

and hence

(21)

Jt (x + 2, γ) = eγ˜a [1 + (1 − P (˜ a)) Jt−1 (x + 1, γ) ! Jt−1 (x + 2, γ) − Jt−1 (x + 1, γ) . Jt−1 (x + 1, γ)

On the other hand, it follows from Ax,t−1 and Cx,t−1 that Jt (x + 1, γ) Jt−1 (x + 2, γ) ! , Jt−1 (x + 1, γ) Jt (x, γ) and hence Jt (x + 1, γ) − Jt (x, γ) Jt−1 (x + 2, γ) − Jt−1 (x + 1, γ) ! . Jt−1 (x + 1, γ) Jt (x, γ) Thus, from (21) we obtain Jt (x + 2, γ) eγ˜a ! [Jt (x, γ) + (1 − P (˜ a))(Jt (x + 1, γ) − Jt (x, γ))] Jt−1 (x + 1, γ) Jt (x, γ) Jt+1 (x + 1, γ) , ≥ Jt (x, γ) and (20) follows. Thus, the proof is complete. ✷ Corollary 3.1.10 Under Assumption 3.1, Jt (x, γ) is convex in x for each t. Proof: Since Jt (x, γ) = exp(log Jt (x, γ)), the claim follows from Lemma 3.1.9 (a). ✷ Lemma 3.1.11 Under Assumption 3.1, a) log[Gt (x, a, γ)] is subadditive on X × A. b) log[Gt (x, a, γ)] is superadditive on A × {0, 1, . . . N − 1}. Proof: ✷

The results in (a) and (b) follow from Lemmas 3.1.8 and 3.1.9.

We know that for the risk-neutral allocation problem there exists an optimal policy π = (f0 , . . . fN −1 ) such that ft (x) is increasing in x for each t, and increasing in t for each x; see [14]. In the following proposition we show an analogous result for the risk-sensitive case.

Avila-Godoy

Proposition 3.1.12 Under Assumption 3.1, there exists an optimal ∗ policy π = (f0∗ , . . . fN −1 ) for the allocation problem with exponential total cost criterion such that ft∗ (x) is increasing in x, for each t, and increasing in t, for each x. Proof: It follows from Lemmas 3.1.7 and 3.1.11 that for t = 0, 1, . . . N − 1, log Ht (x, a, γ) is subadditive on X × A, and subadditive on A × {0, 1, . . . N − 1}. The result follows from Theorems 2.1.3 and 2.1.4. ✷ In the following section we analyze the allocation control problem with ETC criterion for the case in which the probability function P (a) is convex and the final cost c(x) is strictly increasing. We show that under the mentioned conditions, the optimal policy obtained in Proposition 3.1.12 has further structural properties. Moreover, we compare those structured optimal policies with those corresponding to the riskneutral allocation problem (which are obtained in the appendix). Finally, we apply the obtained results to the particular case of a linear terminal cost function, and again we compare the conclusions with those corresponding to the risk-neutral problem.

Allocation Problem with P(a) Convex and c(x) Strictly Increasing.

∗ ) will denote the monotone Throughout this section, π ∗ = (f0∗ , . . . , ft−1 optimal policy obtained in Proposition 3.1.12.

Proposition 4.1.13 Assume that P (a) is convex and twice diﬀerentiable and c(x) strictly increasing. Then, under Assumption 3.1, the optimal allocation problem can be reduced to a problem with the action ∗ set {0, M }. Moreover, the optimal policy π ∗ = (f0∗ , f1∗ , . . . fN −1 ) is of ∗ ∗ ∗ the threshold type, that is, there exist states x0 , x1 , . . . , xN −1 such that ! 0 if x < x∗t (22) ft∗ (x) = M if x ! x∗t , t = 0, 1, . . . N −1. Furthermore, the sequence of thresholds is decreasing. Proof: First, we will show by induction on t, that Jt (x, γ) is strictly increasing in x. Since J0 (x, γ) = eγc(x) , the result holds for t = 0. Now

Optimal resource allocation

assume that Jt (x, γ) is strictly increasing in x for some t ≥ 0. Then, from Corollary 3.1.10 and by using the induction hypothesis we have that eγa [Jt (x, γ)+(1 − P (a))(Jt (x + 1, γ) − Jt (x, γ))]

> eγa [Jt (x − 1, γ) + (1 − P (a))(Jt (x, γ) − Jt (x − 1, γ))]

and since eγa [Jt (x, γ) + (1 − P (a))(Jt (x + 1, γ) − Jt (x, γ))] is continuous in a, inf

a∈[0,M ]

{eγa [Jt (x, γ) + (1 − P (a))(Jt (x + 1, γ) − Jt (x, γ))]}

inf

a∈[0,M ]

{eγa [Jt (x − 1, γ) + (1 − P (a))(Jt (x, γ) − Jt (x − 1, γ))]} .

Thus, from (10), Jt+1 (x + 1, γ) > Jt+1 (x, γ). Next, we will show that for ax ∈ (0, M ), ∂ 2 Gt ∂Gt (x, ax , γ) = 0 =⇒ 2 (x, ax , γ) < 0; ∂a ∂ a that is, that there are no minimal points in (0, M ). Indeed, it follows from (13) that Gt (x, a, γ) = eγa [(Jt (x, γ) − Jt (x − 1, γ))(1 − P (a)) + Jt (x − 1, γ)], which yields by diﬀerentiating both sides two times with respect to a: ∂Gt (x, a, γ) = −eγa [Jt (x, γ) − Jt (x − 1, γ)]P ′ (a) + γGt (x, a, γ), ∂a and ∂ 2 Gt (x, a, γ) = −eγa [Jt (x, γ) − Jt (x − 1, γ)]P ′′ (a)− ∂2a ∂Gt (x, a, γ) γeγa [Jt (x, γ) − Jt (x − 1, γ)]P ′ (a) + γ ∂a ∂Gt (x, a, γ) + eγa [Jt (x − 1, γ) − Jt (x, γ)] =γ ∂a [γP ′ (a) + P ′′ (a)] 2

∂ Gt t If ax ∈ (0, M ) is such that ∂G ∂a (x, ax , γ) = 0, then ∂ 2 a (x, a, γ) < 0 since ′ ′′ Jt (x − 1, γ) − Jt (x, γ) < 0 and γP (a) + P (a) > 0.

Avila-Godoy

Since there are no minimal points in (0, M ), then we must have ft∗ (x) ∈ {0, M } ∀t, ∀x. Moreover, if we define x∗t := min{x : ft∗ (x) = M }, t = 0, 1, . . . N −1, then (22) follows from the fact that ft∗ (x) is increasing in x. Finally, the sequence {x∗t } is decreasing since ft∗ (x) is increasing in t. ✷ Now, to gain further insight of the consequences of Proposition 4.1.13, we apply this proposition to compute the optimal policy in a particular example with linear final cost. Example 4.1.14 Take c(x) = 2x, A = [0, 1], and P (a) convex. We start ∗ by computing fN −1 (x). To do that, by Proposition 4.1.13, we need only to compare the values of the function G0 (x, a, γ) at the extreme actions a = 0 and a = 1. We have that G0 (x, a, γ) = eγa [P (a)J0 (x − 1, γ) + (1 − P (a))J0 (x, γ)], γa

γ(2x−2)

= e [P (a)e

+ (1 − P (a))e

2γx

x ≥ 1.

x≥1

Thus, (23)

G0 (x, 0, γ) = e2γx

and (24)

G0 (x, 1, γ) = e2γx [P (1)e−γ + (1 − P (1))eγ ].

On the other hand, assuming that P (1) ̸= 1, we obtain that 1 ≤ P (1)e−γ + (1 − P (1))eγ ⇐⇒ eγ ≤ P (1) + e2γ (1 − P (1)) " ! 1 P (1) 2γ γ e + ≥0 ⇐⇒ (1 − P (1)) e − 1 − P (1) 1 − P (1) 1 P (1) eγ + ≥0 ⇐⇒ e2γ − 1 − P (1) 1 − P (1) # $ $# P (1) ⇐⇒ eγ − eγ − 1 ≥ 0 1 − P (1) P (1) (25) . ⇐⇒ γ ≥ log 1 − P (1) Thus, it follows from (23), (24) and (25) that

Optimal resource allocation

1 2

P (1) < P (1) < 1 and 0 < γ ≤ log( 1−P (1) ) then

G0 (x, 1, γ) ≤ G0 (x, 0, γ); b)

1 2

P (1) < P (1) < 1 and γ ≥ log( 1−P (1) ) then

G0 (x, 0, γ) ≤ G0 (x, 1, γ); if P (1) ≤

1 2

and γ > 0 then G0 (x, 0, γ) < G0 (x, 1, γ);

if P (1) = 1 and γ > 0 then G0 (x, 1, γ) < G0 (x, 0, γ).

∗ Therefore the optimal decision rule fN −1 and the optimal value function J1 for the cases (a) and (d) are given by ! 0 if x < 1 ∗ (26) fN −1 (x) = 1 if x ! 1,

and (27) J1 (x, γ) =

1 eγ [P (1)J0 (x − 1, γ) + (1 − P (1))J0 (x, γ)]

if x = 0 if x ≥ 1,

and for (b) and (c) by ∗ fN −1 (x) = 0,

∀x

and J1 (x, γ) = e2γx ,

(28)

x ≥ 0.

Now, to compute the optimal decision rules ft∗ , t = 0, . . . , N − 2, we will first prove each one of the following statements by induction on t: I)

1 2

P (1) < P (1) < 1 and 0 < γ ≤ log( 1−P (1) ) then, for t = 1, . . . N − 1,

Jt (x, γ) =

1 eγ [P (1)Jt−1 (x − 1, γ) + (1 − P (1))Jt (x, γ)]

if x = 0 if x ≥ 1,

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II)

1 2

P (1) < P (1) < 1 and γ ≥ log( 1−P (1) ) then, for t = 1, . . . N − 1,

Jt (x, γ) = J0 (x, γ), III)

if P (1) ≤

1 2

and γ > 0 then for t = 1, . . . N − 1, Jt (x, γ) = J0 (x, γ),

IV)

x ∈ X;

if P (1) = 1 and γ > 0 then for t = 1, . . . N − 1,

Jt (x, γ) =

1 eγ [P (1)Jt−1 (x − 1, γ) + (1 − P (1))Jt (x, γ)]

if x = 0 if x ≥ 1,

First, let’s prove (I). The validity of assertion (I) for t = 1 follows from (27). Next, by (8), Jt+1 (x, γ) = min{Gt (x, 0, γ), Gt (x, 1, γ)}, where (29) Gt (x, a, γ) = eγa [P (a)Jt (x − 1, γ) + (1 − P (a))Jt (x, γ)],

x ≥ 1.

Thus, Jt+1 (x, γ) = min{Jt (x, γ), eγ [P (1)Jt (x − 1, γ) + (1 − P (1))Jt (x, γ)]} (30) (31)

= min{eγ [P (1)Jt−1 (x − 1, γ) + (1 − P (1))Jt−1 (x, γ)], eγ [P (1)Jt (x − 1, γ) + (1 − P (1))Jt (x, γ)]}

= eγ [P (1)Jt (x − 1, γ) + (1 − P (1))Jt (x, γ)],

where (30) and (31) follow from the induction hypothesis and Lemma 3.1.6 respectively. Thus, the proof of (I) is complete. Now, let’s prove (II). First, (28) implies that (II) holds for t = 1. Next, similarly as above, Jt+1 (I, γ) = min{Gt (I, 0, γ), Gt (I, 1, γ)} (32) (33) (34)

= min{Jt (I, γ), eγ [P (1)Jt (I − 1, γ) + (1 − P (1))Jt (I, γ)]}

= min{J0 (I, γ), eγ [P (1)J0 (I − 1, γ) + (1 − P (1))J0 (I, γ)]}

= min{J0 (I), J0 (I)[e−γ P (1) + eγ (1 − P (1))], = J0 (I)

Optimal resource allocation

where (32), (33) and (34) follow from (29), the induction hypothesis and ∗ ∗ (25) respectively. Thus fN −t−1 (I) = 0 and since fN −t−1 (x) is increasing ∗ in x, we obtain that fN −t−1 (x) = 0, for all x. Therefore Jt+1 (x, γ) = min{Gt (x, 0, γ), Gt (x, 1, γ)} = Gt (x, 0, γ) = Jt (x, γ) = J0 (x, γ),

∀x ∈ X,

and the proof of (II) is complete. The proof of (III) is similar to the proof of (II) but in this case (34) P (1) follows from (25) since P (1) ≤ 12 =⇒ log 1−P (1) ≤ 0. The proof of (IV) is similar to the proof of (I). Finally, it follows from (I), (II), (III) and (IV) that ft∗ (x), t = 0, 1, . . . , N − 2, N − 1, are given by ! 0 if x < 1 ∗ (35) ft (x) = 1 if x ! 1 # " P (1) if 12 < P (1) < 1 and 0 < γ ≤ log 1−P (1) , or if P (1) = 1 and γ > 0; and ft∗ (x) = 0, ∀x # " P (1) 1 if 12 < P (1) < 1 and γ ≥ log 1−P (1) , or if P (1) ≤ 2 and γ > 0. Remark 4.1.15 Note that

P (1) a) if 12 < P (1) < 1 and γ ≥ log( 1−P (1) ) then the preferences of the γ-decision maker diﬀer from those of the risk-neutral decision maker: the γ-decision maker prefers the action a = 0, whereas the risk-neutral decision maker prefers the action a = 1; see the appendix;

b) if P (1) = 12 then the γ-decision maker prefers the action a = 0, whereas the risk-neutral decision maker is indiﬀerent between the actions a = 0 and a = 1; see the appendix.

Appendix

For the purpose of comparing the results obtained in Sections 3 and 4 about structured optimal policies for an optimal allocation problem

Avila-Godoy

(with ETC criterion), in this appendix we study the corresponding riskneutral allocation problem. Section 5.1 summarizes some results about monotonicity and convexity properties of the optimal value function and monotonicity properties of the policies. For the proof of those results we refer the reader to Ross [14]. In Section 5.2 we derive further structural properties of the optimal policies under the assumptions that the probability function P (a) is convex and the final cost c(x) is strictly increasing.

5.1

Monotone Optimal Policies

For t = 0, 1, . . . , N − 1, denote (36)

Ft (x, a) := a + P (a)Jt (x − 1) + (1 − P (a))Jt (x),

x ≥ 1,

where Jt (x) is the risk-neutral optimal total cost when t stages remain to go and the state at time N − t is x. Note that Ft (x, a) is the function within brackets in the (risk-neutral) dynamic programming algorithm (37)

(38)

J0 (x) = c(x) .. .. . . Jt+1 (x) =

inf

a∈A(x)

C(x, a) +

" y

# pxy (a)Jt (y) .

Let A¯t (x) := {a : Ft (x, a) = inf′ {Ft (x, a′ )}} a

and f¯t (x) := min A¯t (x). Lemma 5.1.1 The optimal value function Jt (x) is increasing in x and decreasing in t. Moreover, under Assumption 3.1, Jt (x) is convex in x.

Proposition 5.1.2 Under Assumption 3.1, π ¯ = (f¯0 , . . . , f¯N −1 ) is an optimal policy for the risk-neutral allocation problem such that for t = 0, . . . N − 1, ft (x) is increasing in x; and for fixed x, ft (x) is increasing in t.

Optimal resource allocation

5.2

Risk-neutral Allocation Problem with P(a) Strictly Convex and c(x) Strictly Increasing.

Throughout this appendix, the policy π ¯ = (f¯0 , . . . , f¯N −1 ) will denote the monotone optimal policy obtained in Proposition 5.1.2. In the following proposition we will show that when the probability function P (a) is strictly convex and the final cost c(x) is strictly increasing, the allocation model is reduced to a problem with two actions: the extreme points of the interval [0, M ]. Consequently, there exists an optimal threshold policy. Proposition 5.2.1 Assume that P (a) is strictly convex and twice differentiable and c(x) is strictly increasing. Then, under Assumption 3.1, the allocation optimal control problem (with total cost criterion) can be reduced to a problem with two actions: the extreme points of the interval [0, M ]. Moreover, the optimal policy π ¯ = (f¯0 , f¯1 , . . . f¯N −1 ) is of the threshold-type, that is, there exist states x¯0 , x¯1 , . . . , x ¯N −1 such that ! 0 if x < x¯t (39) f¯t (x) = M if x ! x¯t , t = 0, 1, . . . N − 1. Moreover, the sequence of thresholds is decreasing. Proof:

It follows from (36) that Ft (x, a) = a + [Jt (x) − Jt (x − 1)](1 − P (a)) + Jt (x − 1).

(40)

First, we will show by induction on t, that Jt (x) is strictly increasing in x. Since J0 (x) = c(x), the result holds for t = 0. Now assume that Jt (x) < Jt (x + 1). Then, from Lemma 5.1.1 and by using the induction hypothesis we have that a + Jt (x)+(1 − P (a))[Jt (x + 1) − Jt (x)]

> a + Jt (x − 1) + (1 − P (a))[Jt (x) − Jt (x − 1)]

and since a + Jt (x) + (1 − P (a))[Jt (x + 1) − Jt (x)] is continuous in a, Jt+1 (x + 1) = >

inf

a∈[0,M ]

inf

a∈[0,M ]

{a + Jt (x) + (1 − P (a))[Jt (x + 1) − Jt (x)]}

{a + Jt (x − 1) + (1 − P (a))[Jt (x) − Jt (x − 1)]} = Jt+1 (x).

Avila-Godoy

It follows from (40) that ∂Ft (x, a) = 1 − P ′ (a)[Jt (x) − Jt (x − 1)] ∂a and

∂ 2 Ft (x, a) = −P ′′ (a)[Jt (x) − Jt (x − 1)]. ∂2a Thus, since P ′′ (a) > 0 and Jt (x) is strictly increasing in x we obtain 2 that ∂∂ 2Fat (x, a) < 0, and therefore Ft (x, a) is concave in a. Consequently, A¯t (x) = {0, M }, and hence, f¯t (x) ∈ {0, M }. Moreover, if we define x ¯t := min{x : f¯t (x) = M },

then (39) follows from the fact that f¯t (x) is increasing in x. Finally, the sequence {¯ xt } is decreasing since f¯t (x) is increasing in t. ✷ Now, we will apply Proposition 5.2.1 to compute the optimal policy for the example considered in Section 4. Example 5.2.2 (revisited.) Take the example considered in Section 4 with P (a) strictly convex. First we compute f¯N −1 (x). To do that, by Proposition 5.2.1, we need only to compare the values of the function F0 (x, a) at the extreme actions a = 0 and a = 1. It follows from (36) that F0 (x, a) = a + P (a)J0 (x − 1) + (1 − P (a))J0 (x), = a + P (a)(2x − 2) + (1 − P (a))2x,

Thus, F0 (x, 0) = 2x,

x ≥ 1,

and

F0 (x, 1) = 2x + (1 − 2P (1)),

x ≥ 1.

Thus, we obtain a)

if P (1) >

1 2

then F0 (x, 1) < F0 (x, 0),

if P (1) <

1 2

x≥1

then F0 (x, 0) < F0 (x, 1),

x ≥ 1,

and

x≥1

x ≥ 1.

Optimal resource allocation

if P (1) =

1 2

then F0 (x, 1) = F0 (x, 0),

x ≥ 1.

Therefore the optimal decision rule f¯N −1 and the optimal value function J1 for the case (a) are given by ! 0 if x < 1 (41) f¯N −1 (x) = 1 if x ! 1, and (42)

J1 (x) =

for the case (b) by

0 if x = 0 2x + (1 − 2P (1)) if x ≥ 1;

f¯N −1 (x) = 0,

∀x,

and J1 (x) = 2x,

(43)

x ≥ 0;

and for the case (c) we obtain that both actions a = 0 and a = 1 are optimal. Now, to compute the optimal decision rules f¯t , t = 0, . . . , N − 2, we will first prove each one of the following statements by induction on t : I)

If P (1) >

1 2

then for t = 1, . . . N − 1, Jt (1) = 1 + (1 − P (1))Jt−1 (1);

II)

If P (1) ≤

1 2

then for t = 1, . . . N − 1, Jt (x) = J0 (x),

x ∈ X.

First, let’s prove (I). The validity of assertion (I) for t = 1 follows from (42). Next, by the dynamic programming algorithm Jt+1 (1) = min{Ft (1, 0), Ft (1, 1)}. Thus,

(44) (45)

Jt+1 (1) = min{Jt (1), 1 + (1 − P (1))Jt (1)}

= min{1 + (1 − P (1))Jt−1 (1), 1 + (1 − P (1))Jt (1)}

= 1 + (1 − P (1))Jt (1),

Avila-Godoy

where (44) and (45) follow from the induction hypothesis and Lemma 5.1.1 respectively. Thus the proof of (I) is complete. Now, let’s prove (II). First, (43) implies that (II) holds for t = 1. Next, similarly as above Jt+1 (I) = min{Ft (I, 0), Ft (I, 1)} = min{Jt (I), 1 + P (1)Jt (I − 1) + (1 − P (1))Jt (I)}

(46)

= min{2I, 1 + P (1)2(I − 1) + (1 − P (1))2I}

(47)

= min{2I, 2I + (1 − 2P (1))}

(48)

= 2I

where (46), (47) and (48) follow from (36), the induction hypothesis and the hypothesis P (1) < 12 respectively. Thus f¯N −t−1 (I) = 0 and since f¯N −t−1 (x) is increasing in x we obtain that f¯N −t−1 (x) = 0, for all x. Therefore Jt+1 (x) = min{Ft (x, 0), Ft (x, 1)} = Ft (x, 0) = Jt (x) = J0 (x),

∀x ∈ X,

and the proof of (II) is complete. Finally, it follows from (I), (II) and (c) that f¯t (x), t = 0, 1, . . . , N −1, are given by ! 0 if x < 1 f¯t (x) = 1 if x ! 1 if P (1) > 12 ; if P (1) < policies:

1 2;

f¯t (x) = 0, and if P (1) =

1 2

then there are I + 1 threshold optimal

fty (x) = y = 0, 1, . . . I.

∀x

0 if x ≤ y 1 if x > y,

Optimal resource allocation

Acknowledgement This paper is part of the author’s doctoral research under the direction of Dr. Emmanuel Fernández Gaucherand at the Department of Mathematics of the University of Arizona. I want to thank him for his guidance during the preparation of this work. I am specially grateful to Dr. Onésimo Hernández Lerma for his helpful comments and suggestions in helping me see this article to its completion. Guadalupe Avila Godoy Departamento de Matem´ aticas, Universidad de Sonora, Rosales y Boulevard Luis Encinas, Hermosillo, Sonora, 83000 gavila@gauss.mat.uson.mx

References [1] Arapostathis A.; Borkar V. S.; Fernández-Gaucherand E.; Ghosh M. K.; Marcus S. I., Discrete-time Controlled Markov Processes with Average Cost Criterion: a Survey, SIAM J. Control Optim., 31 (1993), 282-344. [2] Avila-Godoy G.; Brau A.; Fernández-Gaucherand E., Controlled Markov Chains with Discounted Risk-sensitive Criteria: Applications to Machine Replacement, In Proc. 36th IEEE Conf. Decision Control, (1997), 1115-1120. [3] Avila-Godoy G.; Fernández-Gaucherand E., Exponential Risksensitive Optimal Scheduling, In Proc. 36th IEEE Conf. Decision Control, (1997), 3958-3963. [4] Avila-Godoy G., Controlled Markov Chains with Exponential Risk Sensitive Criteria: Modularity, Structured Policies and Applications, PhD thesis, Department of Mathematics, University of Arizona, (1998). [5] Bouakiz M.; Sobel M. J., Inventory Control with an Exponential Utility Criterion, Math. Oper. Res., 40 (1992), 603-608. [6] Brau A.; Fernández-Gaucherand E., Controlled Markov Chains with Risk Sensitive Exponential Average Cost Criterion, In Proc. 36th IEEE Conf. Decision Control, (1997), 2260-2264.

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[7] Brau A.; Fernández-Gaucherand E., Controlled Markov Chains with Risk-Sensitive Average Cost Criterion: The Non-Irreducible Case, In Proc. 40th IEEE Conf. Decision Control, (2001), 21082109. [8] Brau A.; Fernández-Gaucherand E., On Weak Conditions and Optimality Inequality Solutions in Risk-Sensitive Controlled Markov Processes with Average Criterion, In Proc. 41th IEEE Conf. Decision Control, (2002), 1375-1379. [9] Fernández-Gaucherand E.; Marcus S. I., Risk-sensitive Optimal Control of Hidden Markov Models: Structural Results, IEEE Trans. Autom. Control, 42 (1997), 1418-1442. [10] Hernández-Lerma O.; Lasserre J., Discrete Time Markov Control Processes, Springer, New York, (1996). [11] Hinderer K. F., On the Structure of Solutions of Stochastic Dynamic Programs, In Proc. 7th Conf. Probability Theory, (1982), 173-182. [12] Pratt J. W., Risk Aversion in the Small and in the Large, Econometrica, 32 (1964), 122-136. [13] Puterman M. L., Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, New York, (1994). [14] Ross S. M., Introduction to Stochastic Dynamic Programming, Academic Press, New York, (1983). [15] Serfozo R. F., Monotone Optimal Policies for Markov Decision Processes, Math. Prog. Study, 6 (1976), 202-215. [16] Serfozo R. F., Optimal Control of Random Walks, Birth and Death Processes, and Queues, Adv. Appl. Prob., 13 (1981), 61-83. [17] Stidham S., Jr.; Weber R. R., Monotonic and Insensitive Optimal Policies for Control of Queues with Undiscounted Costs, Oper. Res., 87 (1989), 611-625. [18] Vega-Amaya O.; Montes-de-Oca R., Application of Average Dynamic Programming to Inventory Systems, Math. Met. Oper. Res., 47 (1998), 451-471.

Optimal resource allocation

[19] Weber R. R.; Stidham S. Jr., Optimal Control of Service Rates in Networks of Queues, Adv. Appl. Prob., 19 (1987), 202-218. [20] Whittle P., Risk-sensitive Optimal Control, John Wiley & Sons, Chichester, (1990).

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