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

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Page 1: Morfismos, Vol 7, No 2, 2003

VOLUMEN 7

NÚMERO 2

JULIO A DICIEMBRE DE 2003

ISSN: 1870-6525

Page 2: Morfismos, Vol 7, No 2, 2003

MORFISMOSComunicaciones EstudiantilesDepartamento de Matematicas

Cinvestav

Editores Responsables

• Isidoro Gitler • Jesus Gonzalez

Consejo Editorial

• Felipe Gayosso • Samuel Gitler• Onesimo Hernandez-Lerma • Raul Quiroga Barranco

• Enrique Ramırez de Arellano • Francisco Ramırez Reyes• Jose Rosales Ortega • Mario Villalobos Arias

Editores Asociados

• Ricardo Berlanga • Emilio Lluis Puebla• Isaıas Lopez • Guillermo Pastor

• Vıctor Perez Abreu • Carlos Prieto• Carlos Renterıa • Luis Verde

Secretarias Tecnicas

• Roxana Martınez • Laura Valencia

Morfismos puede ser consultada electronicamente en “Revista Morfismos”de la direccion http://www.math.cinvestav.mx. Para mayores informes dirigirseal telefono 50 61 38 71.Toda correspondencia debe ir dirigida a la Sra. Laura Valencia, Departamentode Matematicas del Cinvestav, Apartado Postal 14-740, Mexico, D.F. 07000 opor correo electronico: [email protected].

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VOLUMEN 7

NÚMERO 2

JULIO A DICIEMBRE DE 2003

ISSN: 1870-6525

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Informacion para Autores

El Consejo Editorial de MORFISMOS, Comunicaciones Estudiantiles del Departamentode Matematicas del CINVESTAV, convoca a estudiantes de licenciatura y posgrado a someterartıculos para ser publicados dentro de esta revista bajo los siguientes lineamientos

• Todos los artıculos seran enviados a especialistas para su arbitraje. No obstante, losartıculos seran considerados solo como versiones preliminares y por tanto pueden serpublicados en otras revistas especializadas.

• Se debe anexar junto con el nombre del autor, su nivel academico y la instituciondonde estudia o labora.

• El artıculo debe empezar con un resumen en el cual se indique de manera breve yconcisa el resultado principal que se comunicara.

• Es recomendable que los artıculos presentados esten escritos en Latex y sean enviadosa traves de un medio electronico. Los autores interesados pueden obtener el for-mato LATEX utilizado por MORFISMOS en “Revista Morfismos” de la direccion webhttp://www.math.cinvestav.mx, o directamente en el Departamento de Matematicasdel CINVESTAV. La utilizacion de dicho formato ayudara en la pronta publicaciondel artıculo.

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• Los artıculos deben ser dirigidos a la Sra. Laura Valencia, Departamento de Matemati-cas del Cinvestav, Apartado Postal 14 - 740, Mexico, D.F. 07000, o a la direccion decorreo electronico [email protected]

Author Information

MORFISMOS, the student journal of the Mathematics Department of Cinvestav, invitesundergraduate and graduate students to submit manuscripts to be published under thefollowing guidelines

• All manuscripts will be refereed by specialists. However, accepted papers will beconsidered to be “preliminary versions” in that authors may republish their papers inother journals, in the same or similar form.

• In addition to his/her affiliation, the author must state his/her academic status (stu-dent, professor,...).

• Each manuscript should begin with an abstract summarizing the main results.

• Morfismos encourages electronically submitted manuscripts prepared in Latex. Au-thors may retrieve the LATEX macros used for MORFISMOS through the web sitehttp://www.math.cinvestav.mx, at “Revista Morfismos”, or by direct request to theMathematics Department of Cinvestav. The use of these macros will help in theproduction process and also to minimize publishing costs.

• All illustrations must be of professional quality.

• 15 offprints of each article will be provided free of charge.

• Manuscripts submitted for publication should be sent to Mrs. Laura Valencia, De-partamento de Matematicas del Cinvestav, Apartado Postal 14 - 740, Mexico, D.F.07000, or to the e-mail address: [email protected]

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Lineamientos Editoriales

“Morfismos” es la revista semestral de los estudiantes del Departamento deMatematicas del CINVESTAV, que tiene entre sus principales objetivos el que losestudiantes adquieran experiencia en la escritura de resultados matematicos.

La publicacion de trabajos no estara restringida a estudiantes del CINVESTAV;deseamos fomentar tambien la participacion de estudiantes en Mexico y en el extran-jero, ası como la contribucion por invitacion de investigadores.

Los reportes de investigacion matematica o resumenes de tesis de licenciatura,maestrıa o doctorado pueden ser publicados en MORFISMOS. Los artıculos queapareceran seran originales, ya sea en los resultados o en los metodos. Para juzgaresto, el Consejo Editorial designara revisores de reconocido prestigio y con experienciaen la comunicacion clara de ideas y conceptos matematicos.

Aunque MORFISMOS es una revista con arbitraje, los trabajos seconsideraran como versiones preliminares que luego podran aparecer pu-blicados en otras revistas especializadas.

Si tienes alguna sugerencia sobre la revista hazlo saber a los editores y con gustoestudiaremos la posibilidad de implementarla. Esperamos que esta publicacion pro-picie, como una primera experiencia, el desarrollo de un estilo correcto de escribirmatematicas.

Morfismos

Editorial Guidelines

“Morfismos” is the journal of the students of the Mathematics Department ofCINVESTAV. One of its main objectives is for students to acquire experience inwriting mathematics. MORFISMOS appears twice a year.

Publication of papers is not restricted to students of CINVESTAV; we want toencourage students in Mexico and abroad to submit papers. Mathematics researchreports or summaries of bachelor, master and Ph.D. theses will be considered forpublication, as well as invited contributed papers by researchers. Papers submittedshould be original, either in the results or in the methods. The Editors will assignas referees well–established mathematicians.

Even though MORFISMOS is a refereed journal, the papers will beconsidered as preliminary versions which could later appear in othermathematical journals.

If you have any suggestions about the journal, let the Editors know and we willgladly study the possibility of implementing them. We expect this journal to foster, asa preliminary experience, the development of a correct style of writing mathematics.

Morfismos

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Contenido

When does a manifold admit a metric with positive scalar curvature?

Egidio Barrera-Yanez and Jose 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

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

When does a manifold admit a metricwith positive scalar curvature? ∗

zenaY-arerraBoidigE 1 Jose Luis Cisneros-Molina 2

Abstract

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

2000 Mathematics Subject Classification: 53C21, 55N15, 55N22, 34L40Keywords and phrases: Positive scalar curvature, Dirac operator, con-nective K-theory.

1 Introduction

Riemannian Geometry is devoted to the study of Riemannian manifolds(Mn, g), that is, differentiable manifolds Mm endowed with a Rieman-nian metric g. Since the manifold Mn is also a topological manifold, oneof the most important problems in Riemannian Geometry is to studywhich constrains imposes the topology of Mn on the geometry givenby the Riemannian metric g. More specifically, one would like to studythe relation between some topological invariants of the underlying man-ifold Mn with the curvature of the Riemannian manifold (Mn, g). Inthe present paper we shall only consider closed manifolds, i.e., compactmanifolds without boundary.

∗Invited article1Supported by Proyecto PAPIIT IN110702-22Supported by Proyecto PAPIIT IN110702-2

1

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2 aniloM-sorensiCdnazenaY-arerraB

The curvature tensor can be viewed as a quadratic form Q in thedouble exterior product of the tangent bundle of M , 2 T (M), thepositive definiteness of Q is one of the strongest positivity conditions,for example all compact symmetric spaces have Q ≥ 0 while Q > 0distinguishes spheres and real projective spaces. The restriction of Q tobivectors in 2 T (M) is the sectional curvature K and twice the sum ofthe sectional curvatures over all two planes in a tangent space to a pointgive us the scalar curvature s (see [2]) therefore, we have the followingimplications:

Q > 0 ⇒ K > 0 ⇒ s > 0.

In a Riemannian manifold (Mn, g) the scalar curvature can be builtin a certain way out of the first and second derivatives of g, so we canrecover s from the metric g. Hence it is natural to ask:

• Given a Riemannian manifold M = (Mn, g). When does M admita metric with s > 0 or s = 0 or s < 0?

For the case s ≤ 0, this condition has no topological effect on M bya theorem of Kasdan and Warner [12, 13] which claims the existence ofa metric s ≤ 0 on every manifold of dimension n ≥ 3 and a theoremof Lohkamp [18] that states that the space R−(M) of negative scalarcurvature metrics on M is contractible for every closed manifold Mn ofdimension 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 scalarcurvature?

2. How can I decide if a manifold admit a metric with positive scalarcurvature?

For the existence of a metric with positive scalar curvature, one canprove that if a manifold M has a metric with positive scalar curvaturethen M×N also has a metric with positive scalar curvature since we canshrink the product metric by a positive factor at every point and thenusing 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 haspositive scalar curvature is using obstructions:

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Manifolds with positive scalar curvature 3

1. Index Obstructions. This method is based on the “Bochner-Lichnerowicz-Weitzenbrock formula” which gives a relation be-tween the scalar curvature and the “Dirac operator” (see 2.2) de-fined by Atiyah-Singer on any Riemannian manifold with a spinstructure (see 1.1).

2. Minimal hypersurface method. Schoen and Yau proved that ifM is a manifold of dimension n with positive scalar curvature thenany stable minimal hypersurface N (i.e. N is a local minimum ofthe area functional) also admits positive scalar curvature.

3. Seiberg-Witten invariants. This is an invariant for 4-dimensionalmanifolds which vanishes if the manifold admits a metric with pos-itive 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 thiscase, the scalar curvature coincides with the Gaussian curvature andthe Gauss-Bonnet formula relates it to the Euler-Poincare characteristicχ(M), which is a topological invariant of the 2-manifold M :

χ(M) = (4π)−1M s(x)dvol(x).

Thus if a 2 dimensional manifold M admits a metric of positivescalar curvature, then χ(M) > 0 and by the classification theorem of2-manifolds, this implies that M = S2 or M = RP 2 and indeed, thesemanifolds do admit metrics of positive scalar curvature. Thus χ(M) > 0if and only if M admits a metric of positive scalar curvature.

The situation is very different in higher dimensions. In dimensionn = 3 work of Shoen and Yau [29] with the Thurston conjecture [35](perhaps soon established by Perelman [20, 19]) yields a complete clas-sification of 3-manifolds with positive scalar curvature. For n = 4, seecomment 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 havea metric of positive scalar curvature, the two common methods are“surgery” and “attaching handles” which are related. Let M be a man-ifold with boundary ∂M , we recall that a “handle” is the product oftwo discs Dk×Dn−k, the boundary of this “handle” consist of two parts

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4 aniloM-sorensiCdnazenaY-arerraB

Sk−1×Dn−k and Dk×Sn−k−1, given an embedding of Sk−1×Dn−k intothe boundary ∂M of an n-dimensional manifold M , we can construct anew manifold

M = MSk−1×Dn−k

Dk × Dn−k

by taking the disjoint union of M and the “handle” Dk × Dn−k andidentifying the points in Sk−1 ×Dn−k with their image in ∂M . We saythat M is obtained by attaching a k-handle to M , or M is obtained bysurgery (i.e. by removing Sk×Dn−k and replacing it by Dk+1×Sn−k−1).It is natural to ask whether a metric of positive scalar curvature in Mcan be extended to a metric of positive scalar curvature in M , here themetrics we have in mind are product metrics near the boundary (i.e. aneighborhood of ∂M is isometric to the product of ∂M with an inter-val). Gromov-Lawson [9] and Shoen-Yau [29] showed (independently)that if M admits a metric of positive scalar curvature, and n − k (the codimension of the surgery/handle) is greater than 2, then M alsoadmits such a metric. It is worth giving some of the flavor involved.Let Sk be an embedded k dimensional sphere in M with trivial normalbundle ν. This means that a tubular neighborhood of Sk has the formSk×Dm−k and associated boundary Sk×Sm−k−1. Shrink the size of thetubular neighborhood. It is possible to deform the original metric onM to a metric which is greater than 0 in a neighborhood the boundarySk ×Sm−k−1 in such a way that the new metric still has positive scalarcurvature. It is at this point that the assumption that m−k ≥ 3 is cru-cial to ensure that the standard metric of the fiber spheres Sm−k−1 haspositive scalar curvature and this dominates as the size of these spheresis shrunk by taking an adiabatic limit. The surgery can be performed;one cuts out the Sk × intDm−k and glues in a Dk+1 × Sm−k−1 and pre-serves the positivity of the scalar curvature, later Gajer [5] extend theresult to

Theorem 1.1 Let M be a manifold with boundary and let g be a metricof positive scalar curvature on M . Assume that M is obtained from Mby attaching a handle of codimension ≥ 3. Then g extends to a metricof positive scalar curvature in M

Gromov and Lawson [9] made the important observation that if a man-ifold M belongs to certain class of manifolds, called spin manifolds,whether it admits a metric of positive scalar curvature depends only onthe bordism class of M in a suitable bordism group called MSpinn(Bπ)

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Manifolds with positive scalar curvature 5

with π be the fundamental group of the manifold M . Recall that twomanifolds M and N of dimension n are bordant if there exists a mani-fold W of dimension n + 1 such that ∂W is the disjoint union M N .For the group MSpinn(Bπ) the extra structure we need is called spinstructure 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 univer-sal unital algebra generated by Rn subject to the Clifford commutationrelations

v ∗ w + w ∗ v = ±(v, w)1.

Let Cliff c(n) := Cliff −(n) ⊗R C be the complexification. Note thatCliff −(n) ⊗R C and Cliff +(n) ⊗R C are isomorphic. Let Pin±(n) ⊂Cliff ±(n) be the multiplicative subgroup generated by the unit sphereof Rn; i.e.

Pin±(n) = {x = v1 ∗ ... ∗ vk : |vi| = 1 for some k}.

Define the following groups and representations

• Let Pinc(n) := Pin−(n) ×Z2 S1 where we identify (g,λ) and(−g,−λ),

• det : Pinc(n) → S1 by det(g, λ) = λ2,

• χ : Pin±(n) → Z2 by χ(v1 ∗ ... ∗ vk) = (−1)k, and

• Ψ : Pin±(n) → O(n) by Ψ(x) : w χ(x)x ∗ w ∗ x−1.

• Spin(n) = ker(χ) ∩ Pin−(n) ≈ ker(χ) ∩ Pin+(n), and

• Spinc(n) = Spin(n) ×Z2 S1.

Let n ≥ 3. Then Ψ defines a surjective group homomorphism fromSpin(n) to the orthogonal group SO(n). Since Spin(n) is connectedwe have that π1(SO(n)) = Z2, and ker(Ψ) = {±1} ⊂ Spin(n), we haveSpin(n) is the universal covering group of SO(n).

Note that Ψ defines a surjective group homomorphism from Pin±(n)to the orthogonal group O(n); this exhibits Pin±(n) as a universalcovering groups of O(n). Since O(n) is not connected, the universal

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6 aniloM-sorensiCdnazenaY-arerraB

cover is not uniquely defined as a group, one must decide how to multiplythe arc components and Pin±(n) are the two possible universal coveringgroups. We extend χ and Ψ to Pinc(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 thetransition functions of ξ from the orthogonal group O(k) to the groupPin±(k) or Pinc(k). We say that ξ admits a spin or a spinc structureif 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 thetangent 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. Werefer to Giambalvo [6] for the proof of the following results. It showsthat we can stabilize; a bundle ξ admits a suitable structure if and onlyif 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 ifw2(ξ) lifts from H2(M ; Z2) to H2(M ; Z).

• The bundle ξ admits a pin− structure ⇐⇒ w2(ξ) = 0.

• The bundle ξ admits a pinc structure ⇐⇒ w2(ξ) lifts fromH2(M ; Z2) to H2(M ; Z).

For examples of manifolds with these structures, consider RP l, the realprojective manifold of dimension l, since T (RP l) ⊕ 1 = (l + 1)L, whereL 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.

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Manifolds with positive scalar curvature 7

• RP 4l+3 and (4l + 4)L admit spin structures.

• Other examples of manifolds that admit spin structures are: Sn

for n ≥ 2, CPn for n odd.

Now with the notion of spin structure we can define the spin bordismgroups MSpinn(Bπ). Two spin manifolds M and N of dimension n arespin bordant if there exist a spin manifold W of dimension n + 1 suchthat its boundary is the disjoint union of M and N and the restrictionof the spin structure on W coincides with the spin structures on M andN .

2 Non existence of metrics of positive scalarcurvature

We saw that the Euler-Poincare characteristic is the invariant thattell us when a 2-dimensional manifold admits a metric with positivescalar curvature, so we are looking for a generalization of this invariant,the pioneer of the solution for the non-existence of metrics of positivescalar curvature was Lichnerowicz, see [17], his method is based on the

-hciLetatsotredronI.”alumrofkcobneztieW-zciworenhciL-renhcoB“nerowicz Theorem we need to explain the following concepts:

2.1 Spinor Bundle

Let M be a Riemannian manifold of dimension n = 2k, the spinorbundle is a vector bundle S → M .

S = Spin(M) ×Spin(n) ∆

where ∆ is a certain representation of Spin(n) called the spinor rep-resentation, which is constructed as follows: identify Spin(n) with asubgroup of units of the Clifford algebra Cliff (n), and ∆ is a certainCliff c(n)-module considered as a representation of Spin(n) in the unitsof Cliff c(n) = Cliff c(2k) which is the algebra C(2k) = M2k×2k(C) of2k × 2k matrices over C (see [16]).

Let ∆ be C2kwith 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-eigenspaceof the involution given by the multiplication by the complex volume el-ement ωC = ι2ke1 · · · e2k in Cliff c(n), the vectors {e1, . . . e2k} form an

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Manifolds with positive scalar curvature 9

D2 = ∇∗∇+1

4s.

Where the connection Laplacian is the operator

∇∗∇ : C∞(E) → C∞(E) defined by ∇∗∇(ϕ) = −m!

i,j=1

∇ei∇ejϕ.

One can use this formula to compute

|D2ϕ|2L2 = |∇ϕ|2L2 +1

4

"

Ms(ϕ,ϕ)dvol.

Therefore if the metric in question has positive scalar curvature, thenthere are no elements in kerD (harmonic spinors).

2.3 A-genus

We recall that for an oriented vector bundle E → M , A(E) ∈ H∗(M,Q)given by:

A(E) = 1− 1

24p1 +

1

27 · 32 · 5(−4p2 + 7p21) + . . .

where pj = pj(E) ∈ H4j(M,Z) are the Pontryagin classes of E, see [16].The famous Atiyah-Singer Index Theorem (see [16] for details) identifiesIndexD+ with the A-genus.

If M is a manifold of dimension m = 4k the the A-genus is:

A(M) = ⟨A(TM), [M ]⟩ ∈ Q.

Theorem 2.1 (Lichnerowicz) Let M be a closed spin manifold ofdimension M = 4k which admits a metric of positive scalar curvature,then A(M) = 0.

Notice that the assumption of spin is very important, consider the fol-lowing example: CP 2 = S5/S1 is a manifold with positive scalar curva-ture, since it is a Riemannian submersion, and

A(CP 2) = −(1

8)sign(CP 2) = 0

but CP 2 is not a spin manifold, see Lemma 1.2.

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10 Barrera-Yanez and Cisneros-Molina

3 Gromov-Lawson-Rosenberg Conjecture

If g is a Riemannian metric on M , let D(M, s, g) be the associatedDirac operator defined by the spin structure s. We define the A-genusas follows:

1. Ifm ≡ 0 mod 4, decomposeD(M, s, g) = D+(M, s, g)+D−(M, s, g)and let A(M, s, g) := dimker(D+(M, s, g))−dimker(D−(M, s, g)) ∈Z; the D± are the chiral spin operators.

2. If m ≡ 1 mod 8, let A(M, s, g) = dimker(D(M, s, g)) ∈ Z2.

3. If m ≡ 2 mod 8, let A(M, s, g) = 12 dimker(D(M, s, g)) ∈ Z2.

4. If m ≡ 0, 1, 2, 4 mod 8, let A(M, s, g) = 0.

One can use the Atiyah-Singer index theorem to show that A(M, s) =A(M, s, g) is independent of the metric g, also notice that in dimension 2the invariant (Euler-Poincare characteristic χ(M)) is independent of theRiemannian metric and certain index invariant introduced by Hitchinare independent of the Riemannian metric, see [10] for details.If M is simply connected, the spin structure s is unique and we letA(M) = A(M, s).

If M admits a metric of positive scalar curvature, the formula ofLichnerowicz [17] shows there are no harmonic spinors; consequentlyA(M, s) = 0. In other words, if there exists a spin structure s on M sothat A(M, s) = 0, then M does not admit a metric of positive scalarcurvature. Gromov and Lawson conjectured that the A-genus mightbe the only obstruction to the existence of a metric of positive scalarcurvature if the dimension n was at least 5 and if M was a simplyconnected spin manifold. Stolz used deep homotopy theory to identifythe kernel of A(M, s), see [31] for details, he established this conjectureby proving:

Theorem 3.1 If M is a simply connected, closed, spin manifold ofdimension n ≥ 5, then M admits a metric of positive scalar curvatureif and only if A(M) = 0.

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

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Manifolds with positive scalar curvature 11

with fundamental group π and spin universal cover. Rosenberg conjec-tured that M admits a metric of positive scalar curvature if and onlyif a generalized equivariant index απ (see [10, 21]) of the Dirac opera-tor vanishes. For the fundamental groups that we shall be considering,απ can be expressed in terms of the A-genus defined above. In gen-eral Rosenberg’s Index απ lives in the K-theory of a certain C∗-algebraassociated to the fundamental groups of the manifold, but it is not ingeneral a number, see [21, 25, 22] for details.

What about the universal cover of M? Consider a manifold of di-mension 9 which is homotopy equivalent to a sphere, call it Σ9 withα(Σ9) = 0),(see [10]) take the connected sum of RP 7 × S2 and Σ9, no-tice that RP 7×S2 is spin and α(RP 7×S2) = 0, since RP 7×S2 is zerobordant. Since the manifold M = (RP 7 × S2)#Σ9 is spin bordant tothe disjoint union of RP 7 × S2 and Σ9, we have that:

α(M) = α((RP 7 × S2)#Σ9) = α(RP 7 × S2) + α(Σ9) = α(Σ9) = 0.

So M does not admit a metric with positive scalar curvature butits universal cover M = (S7 × S2)#Σ9#Σ9 which is diffeomerphic toS7 × S2 does admit such a metric. So the question whether a spinmanifold with finite fundamental group π admits a metric with positivescalar curvature cannot be reduced to the universal covering.

Kwasik and Schultz [14] showed that the Gromov-Lawson-Rosenbergconjecture holds for a finite group π if and only if the conjecture holdsfor all the Sylow subgroups of π. Thus one can work one prime at atime. The Gromov-Lawson-Rosenberg conjecture has been establishedin 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 includ-ing free groups, free abelian groups and fundamental groups oforientable surfaces (Rosenberg & Stolz [23]).

For more information about results concerning the Gromov-Lawson-Rosenberg conjecture, see the article of Joachim and Shick [11].Note that Schick [26, 27] has shown that this conjecture fails in some

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12 Barrera-Yanez and Cisneros-Molina

instances so it is crucial to investigate the precise conditions under whichthe A-genus carries the full set of obstructions.

The spin bordism groups are too big, so it is useful to reformulatethe Gromov-Lawson-Rosenberg conjecture in terms of more manageablegroups, which are the connective K-theory groups.

3.1 Connective K-theory

Let KO be the periodic real K-theory spectrum and ko the connectivecover of KO. The generalized homology theory associated with ko iscalled the real connective K-theory. We are interested on the connectiveK-theory of the classifying space of a group π, kon(Bπ).

Let HP 2 be the quaternion projective space with the usual homo-geneous metric of positive scalar curvature. Let HP 2 → E → B be afiber bundle where the transition functions are the group of isometriesPSp3 of HP 2. Since HP2 is simply connected, the projection p : E → Binduces an isomorphism on the fundamental group. Let Tn(Bπ) be thesubgroup of MSpinn(Bπ) generated by the total space of geometric fi-brations with fiber HP 2. Using some work of Jung and deep homotopytheory, Stolz [31] has given the following geometrical characterizationof the real connective K-theory groups localized at the special prime 2:

kon(Bπ)(2) = {MSpinn(Bπ)/Tn(Bπ)}(2).

Let MSpin+n (Bπ) be the classes in MSpinn(Bπ) which can be rep-

resented by manifolds which admit metrics of positive scalar curvature.The invariant απ extends to the bordism groups MSpinn(Bπ); theformula of Lichnerowicz [17] show that it vanishes on MSpin+

n (Bπ).One therefore has the following equivalent formulation of the Gromov-Lawson-Rosenberg conjecture, see [31] for details:

Theorem 3.2 Let π be a finite group, if n ≥ 5, then the followingassertions are equivalent:

• Let M be any closed connected spin manifold of dimension n withfundamental group π. Then M admits a metric of positive scalarcurvature if and only if απ(M) = 0.

• MSpin+n (Bπ) = ker(απ) ∩MSpinn(Bπ).

Let ko+n (Bπ) be the image ofMSpin+n (Bπ) in kon(Bπ). The Gromov-

Lawson-Rosenberg conjecture has the following reformulation in termsof connective K theory:

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Manifolds with positive scalar curvature 13

Theorem 3.3 Let π be an Abelian 2 group, if n ≥ 5, then the followingassertions are equivalent:

• Let M be any closed connected spin manifold of dimension n withfundamental group π. Then M admits a metric of positive scalarcurvature if and only if απ(M) = 0.

• ko+n (Bπ) = ker(απ) ∩ kon(Bπ).

Algebraic topology (spectral sequences) give upper bounds of con-nective K-theory and using Spectral invariants of the Dirac operator(eta invariant) give geometric generators and lower bounds of connec-tive K-theory, using this approach the conjecture is valid for certainnon-orientable manifolds with fundamental group an Abelian 2 group,see [1, 7] for details.

AcknowledgmentWe like to thank Dr. Jesus Gonzalez Espino Barros for inviting us

to colaborate with Morfismos journal. The authors acknowledge withgratitude helpful suggestions by the referee which have improved theexposition of the paper.

Egidio Barrera-YanezInstituto de Matematicas, UNAM,Unidad Cuernavaca,Av. Universidad s/n,Col. Lomas de Chamilpa,Cuernavaca, Morelos, [email protected]

Jose Luis Cisneros-MolinaInstituto de Matematicas, UNAM,Unidad Cuernavaca,Av. Universidad s/n,Col. Lomas de Chamilpa,Cuernavaca, Morelos, [email protected]

References

[1] Barrera-Yanez, E., The eta invariant of twisted products of evendimensional 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 NewYork, 1986.

[3] Botvinnik B.; Gilkey P., The Gromov-Lawson-Rosenberg conjec-ture: the twisted case, Houston Math. J., 23 (1997), 143–160.

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[4] Botvinnik B.; Gilkey P.; Stolz S., The Gromov-Lawson-Rosenbergconjecture for groups with periodic cohomology, J. Differential Ge-ometry, 46 (1997), 374–405.

[5] Gajer P., Riemannian metrics of positive scalar curvature on com-pact 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, Rieman-nian 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 pres-ence of a fundamental group, I. Ann. of Math., 111 (1980), 209–230.

[9] Gromov M.; Lawson B. Jr., The classification of simply connectedmanifolds 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 theGromov-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 deformationof metrics with prescribed Gaussian and scalar curvature Ann. ofMath., 101 (1975), 317–331.

[13] Kazdan J. L.; Warner F., Scalar curvature and conformal deforma-tion of Riemannian structure, J. Diff. Geom., 10 (1975), 113–134.

[14] Kuwasik S.; Schultz R., Positive scalar curvature and periodic fun-damental groups, Comment. Math. Helv., 65 (1990), 271–286.

[15] Kuwasik S.; Schultz R., Fake spherical space forms of constant pos-itive 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.

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[17] Lichnerowicz A., Spineurs harmoniques, C. R. Acad. Sci. Paris,Ser. 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 manifoldsarXiv:math.DG/0303109 v1, 10 Mar 2003.

[20] Perelman G., The entropy formula for the Ricci flow ans its geo-metric applications arXiv:math.DG/0211159 v1, 11 Nov 2002.

[21] Rosenberg J., C∗-algebras, positive scalar curvature and theNovikov 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 theNovikov Conjecture, III, Topology, 25 (1986), 319–336.

[23] Rosenberg J.; Stolz S., A “stable” version of the Gromov-Lawson-Rosenberg conjecture, Contemp. Math., 181 (1995), 405–418.

[24] Rosenberg J.; Stolz S., Metrics of positive scalar curvature withsurgery, Surveys on Surgery theory, Annals of Mathematics Stud-ies, 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-Lawson-Rosenbreg 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 funda-mental groups, Proc. Amer. Math. Soc., 125 (1997), 907–915.

[29] Shoen R.; Yau S.-T., On the structure of manifolds with positivescalar curvature, Manuscripta Math. 28 (1979), 159–183.

[30] Stolz S., Concordance classes of positive scalar curvature metrics,in preparation.

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16 Barrera-Yanez 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, Vol1, Princeton University Press.

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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 matricesfor every size n divisible by 4. They are based on the descriptionof several families of modular Golay pairs and quadruples. Highermoduli are also considered, such as 48, 64, 128 and 192. Finally, weexhibit infinite families of circulant modular Hadamard matricesof various types for suitable moduli and sizes.

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

1 Introduction

A square matrix H of size n, with all entries ±1, is a Hadamard matrixif HHT = nI, where HT is the transpose of H and I the identitymatrix of size n.

It is easy to see that the order n of a Hadamard matrix must be1, 2 or else a multiple of 4. There are two fundamental open problemsabout these matrices:

• Hadamard’s conjecture, according to which there should exist aHadamard matrix of every size n divisible by 4. (See [9].)

• Ryser’s conjecture, stating that there probably exists no circulantHadamard 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

17

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18 Eliahou and Kervaire

There are many known constructions of Hadamard matrices. How-ever, Hadamard’s conjecture is widely open. For example, the set of allcurrently known Hadamard matrix sizes (as of 2004) contains no arith-metic progression, and is in fact of density zero in the set of positivemultiples of 4. (See [17].) The cases below 1000 which are currentlyopen are 428, 668, 716, 764 and 892.

As for Ryser’s conjecture, a lot is known, but here again the conjec-ture is widely open. For example, it is known that if n > 4 is the sizeof a circulant Hadamard matrix, then n = 4 · r2 with r odd and not aprime 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 amodular Hadamard matrix. Like in the classical case, this is a squarematrix H, with all entries ±1, but satisfying the above orthogonalitycondition only modulo some given integer m, i.e.

H ·HT ≡ nI mod m.

Of course, the classical Hadamard matrix conjecture has an m-modular counterpart, namely: for every n divisible by 4, there shouldexist an m-modular Hadamard matrix of size n. Even though this m-modular analogue looks much weaker than the classical one, there is asort of converse, which rests on the following

Remark. If H is an m-modular Hadamard matrix of size n, withn < m, then H is an ordinary Hadamard matrix.

The proof is simple enough: the entries of H · HT are at most nin absolute value. Hence, if those outside the diagonal are assumed tovanish mod m, then they must actually be zero.

With the above remark, we see that the classical Hadamard matrixconjecture holds if and only if the modular Hadamard matrix conjecturesimultaneously holds for infinitely many distinct moduli m.

In this sense, the m-modular version of Hadamard’s conjecture canbe considered as an approximation to the classical one, of quality in-creasing with m. Currently, the highest modulus m for which the m-modular analogue of Hadamard’s conjecture has been completely settledis m = 32. We summarize the relevant facts below.

In a series of papers, Marrero and Butson considered modular Hada-mard matrices mainly with respect to moduli m which are either oddor 2 times an odd number. With respect to such moduli, sizes n > 3

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Modular Hadamard matrices 19

not divisible by 4 are no longer excluded in general. For instance, theyshow the existence of a 6-modular Hadamard matrix of size n for everyeven n.

In this survey, we consider only modulim which are divisible by 4, asthis case resembles more the classical one. Indeed, if n > 3 is the sizeof an m-modular Hadamard matrix, with m divisible by 4, then n itselfmust be divisible by 4, as for ordinary Hadamard matrices. The proofis analogous to the one in the classical case, by considering congruencesmod 4 rather than equalities.

As for Ryser’s conjecture, the situation is somewhat different. Thereseems to be a very rich theory of circulant modular Hadamard matri-ces, which ought to be developed for its own sake. Circulant modularHadamard matrices do exist for certain moduli and sizes greater than4 and thus, the conjecture should rather be replaced in the modularcontext by the following question.

Question: For which moduli m and sizes n do there exist m-modularcirculant Hadamard matrices H of size n ?

The question can be enriched by requiring that some entries of thematrix H ·HT be actually zero, not only zero mod m. We will introducetwo such constraints, complementary in some sense, and refer to thecomplying matrices as being of type 1, type 2 respectively.

Informally, H will be of type 1 if any two rows of H with indices atdistance n

2 are orthogonal in Zn, n being the order of H. On the otherhand, H will be of type 2 if any two rows of H with indices at distanceother than 0 and n

2 are orthogonal in Zn.

As we will see, there are nice infinite families of circulant modu-lar Hadamard matrices of either type. These examples all come fromnumber-theoretic constructions.

The complementary nature of types 1 and 2 imply that, if H is acirculant modular Hadamard matrix of both types simultaneously, thenH is actually a true circulant Hadamard matrix.

Hence, investigating the possible moduli and orders of circulantmodular Hadamard matrices of either type, besides being of indepen-dent interest, might shed some light on Ryser’s conjecture itself.

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20 Eliahou and Kervaire

2 Basic Definitions and Lemmas

We shall denote by H(n) the set of Hadamard matrices of size n, and byHm(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 ndivisible by 4. Among other results, we shall see that H32(n) = ∅ forevery n divisible by 4.

There are many constructions for Hadamard matrices. See thequoted surveys [4]. Here, we will mainly use three such constructions.All three use sets of complementary binary sequences, specifically pairsand quadruples. From such sets, Hadamard matrices are obtained byplacing the circulant matrices derived from each sequence into suitablearrays. 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) H ′ = H!"

1 1−1 1

#=

"H H−H H

#

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

#·"

HT −HT

HT HT

#=

"2H ·HT 0

0 2H ·HT

#,

and H ·HT ≡ nI modulo m, i.e. H ·HT = nI +mX for some n× ninteger matrix X. It follows that

H ′ ·H ′T =

"2H ·HT 0

0 2H ·HT

#= 2n

"I 00 I

#+ 2m

"X 00 X

#.

Thus H ′ ·H ′T is congruent 2n times the identity matrix of size 2nmodulo 2m. !

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Modular Hadamard matrices 21

2.2 Complementary sequences

Let A = (a0, . . . , aℓ−1) be a binary sequence of length ℓ, that is asequence with all entries ai = ±1. The Hall polynomial of A, denotedA(z), is defined as A(z) =

!ℓ−1i=0 aiz

i. The kth aperiodic correlation

coefficient ck(A) is defined as ck(A) =!ℓ−1−k

i=0 ai ai+k, for 0 ≤ k ≤ℓ− 1. It is convenient to define ck(A) = 0 if k ≥ ℓ.

Note that the number ck(A) arises as the coefficient of (zk + z−k) inthe product A(z)A(z−1) in the Laurent polynomial ring Z[z, z−1]:

A(z)A(z−1) = c0(A) +ℓ−1"

k=1

ck(A)(zk + z−k).

Here c0(A) = ℓ, the sum of the squares of the ai which are assumedto be binary (i.e. ±1).

A set of r binary sequences A1, . . . , Ar is a set of complementarysequences if for each k ≥ 1, the sum of the kth correlations of thesequences 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 se-

quences A1, . . . , Ar form a set of complementary sequences if and onlyif A1(z)A1(z−1) + · · ·+Ar(z)Ar(z−1) equals a constant in the Laurentpolynomial ring Z[z, z−1]. In this case, the constant will simply be thesum of the respective lengths of A1, . . . , Ar.

Pairs of complementary sequences of the same length are also knownas Golay pairs. Here, as in [6], we shall refer to quadruples of comple-mentary sequences of the same length as Golay quadruples.

We shall denote by GP(n) the set of Golay pairs of length n, andby GQ(n) the set of Golay quadruples of length n. Golay pairs andquadruples may be used to construct Hadamard matrices of appropriatesize. 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 oflength n. Denote by A,B again the circulant matrices derived fromeach sequence respectively. Let

(2) H = H(A,B) =

#A B

−BT AT

$.

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22 Eliahou and Kervaire

Then H is a Hadamard matrix of size 2n.

Proof: A straightforward computation shows that

H ·HT =

(AAT +BBT −AB +BA

−BTAT +ATBT ATA+BTB

).

Now, since A,B are circulant matrices, they commute. Hence,

H ·HT = (A ·AT +B ·BT )⊗(

I 00 I

). !

There is a classical construction, due to Goethals-Seidel, which asso-ciates a Hadamard matrix of size 4n to every Golay quadruple of lengthn.

First we recall what the Goethals-Seidel array is. If A,B,C and Dare matrices of size n, define

(3) GS(A,B,C,D) =

⎜⎜⎝

A −BR −CR −DRBR A −DTR CTRCR DTR A −BTRDR −CTR BTR 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 quadru-ple of length n. Denote by A,B,C,D again the circulant matrices de-rived from each sequence respectively. Let H=GS(A,B,C,D). Then His a Hadamard matrix of size 4n.

The proof of the proposition uses the following properties of thematrix R. Namely, R2 = I,RT = R, and if X, Y are any two circulantmatrices, then XRY T is a symmetric matrix, i.e. XRY T = Y RXT .

Besides the map from GP(n) to H(2n) recalled above, there are otherconstructions associating a Hadamard matrix to a Golay pair, obtainedby associating first a Golay quadruple to a Golay pair, and then usingthe Proposition above.

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Modular Hadamard matrices 23

For example, if (f, g) is a Golay pair of length n, then (f, f, g, g) isa Golay quadruple of the same length n, yielding a Hadamard matrixof size 4n. This yields a map GP(n) −→ H(4n), not as efficient as theone above. There is a subtler classical construction, yielding this timea 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 isstraightforward 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 definingproperty 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 quadru-ples, 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 length2n. 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 lengthsof Golay pairs and Golay quadruples.

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24 Eliahou and Kervaire

For Golay pairs, it is known that GP(2a10b26c) is not empty forevery exponents a, b, c ≥ 0. On the other hand, it is conjectured thatno lengths other than 2a10b26c 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. Atheorem in [8] states that GP(n) is empty if n admits a divisor whichis congruent to 3 mod 4. Computer searches have revealed the absenceof Golay pairs of length 34, 50 and 68, and more recently of length 74and 82 (see [2]). The smallest undecided cases now are n = 106 andn = 116.

As for Golay quadruples, there is the following

Conjecture. (Turyn, [15]) There is a Golay quadruple of length n forevery positive integer n.

Because of the above-mentioned map GQ(n) → GQ(2n), the core ofthe problem is the case where n is odd. Moreover, because of the mapGQ(n) −→ H(4n), the above conjecture implies Hadamard’s conjecture.

Obviously, every Golay pair A,B of length n yields a Golay quadru-ple A,A,B,B of the same length, and a Golay quadruple of length 2n+1by the map GP(n) −→ GQ(2n+ 1).

2.3 Modular complementary sequences

There are modular analogues of the above notions. Let m be a positiveinteger. A set of r binary sequences {A1, . . . , Ar} is a set of m-modularcomplementary sequences if for each k ≥ 1, the sum of the kth correla-tions 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 quadru-

ples. We will denote by GPm(n), GQm(n) the set of m-modular Golaypairs, respectively m-modular Golay quadruples, of length n.

The above constructions, associating Hadamard matrices to suitablesets 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).

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Modular Hadamard matrices 25

Note that, in these maps, the modulus remains unchanged. However,for the third construction GP(n) −→ GQ(2n + 1) −→ H(8n + 4), wehave the happy circumstance that the modulus is multiplied by 4.

Proposition 2.3.2 There is a map GPm(n) −→ GQ4m(2n + 1), andhence a map GPm(n) −→ H4m(8n+ 4).

The multiplication of the modulus m by 4 is apparent in the proofof the last proposition of Section 2.2.

Finally, on the modular level, the doubling of Golay pairs also dou-bles the modulus. That is, there is a map GPm(n) → GP2m(2n), givenby (f, g) #→ ([f ; g], [f ;−g]). This is easily checked using the Hall polyno-mials of the sequences: if A(z) = f(z)+zng(z) and B(z) = f(z)−zng(z),then

A(z)A(z−1) +B(z)B(z−1) = 2(f(z)f(z−1) + g(z)g(z−1)).

3 Modular Hadamard matrices

3.1 The case m = 12

Marrero and Butson have produced 6-modular Hadamard matrices ofsize n for every even positive integer n. (See [11] and [12].) Very simplematrices suffice for this purpose. It turns out that their constructionyields in fact 12-modular Hadamard matrices of every size n divisibleby 4.

For any given size, let I denote the identity matrix, J the con-stant 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

by J , K or

!K K−K K

"depending on whether n ≡ 0, 4 or 8 mod 12

respectively.

Proof: In size n, we have J · JT = nJ and K · KT = nI + (n −4)(J − I). This takes care of the cases n ≡ 0, 4 mod 12. Assume now

n ≡ 8 mod 12, and let H =

!K K−K K

"of size n. Then H · HT =

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26 Eliahou and Kervaire

!2KKT 0

0 2KKT

". Since K is of size n

2 here, we have KKT = n2 I +

(n2 − 4)(J − I), and so 2K ·KT = nI + (n− 8)(J − I).

It follows that H ·HT ≡!

nI 00 nI

"mod 12. !

This solves the 12-modular version of Hadamard’s conjecture. Ob-viously, more elaborate matrices will be needed for higher moduli. Thisis 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 size4ℓ for every positive integer ℓ. By the Doubling Lemma, it is sufficientto 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 quadru-ples of length ℓ. These quadruples yield 32-modular Hadamard matri-ces of size 4ℓ by the map GQm(n) −→ Hm(4n) of Section 2 derivedfrom the Goethals-Seidel array. For ℓ ≡ 3 or 7 mod 8, the descriptionof 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

GPm(r) −→ GQ4m(2r + 1) of Section 2 (with r = ℓ−12 ).

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 quadru-ples of length ℓ. Rather, we shall obtain 32-modular Hadamard matri-ces of size 4ℓ from 32-modular Golay pairs of length 2ℓ and the mapGPm(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 wecan prove that 32-modular Golay pairs do not exist in length congruentto 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 8remains in doubt.)

3.2.1 Modular Golay quadruples of length ℓ ≡ 1 mod 8

Let k = ℓ−18 . 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))

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Modular Hadamard matrices 27

to produce the desired modular Golay quadruples and modular Hadamardmatrices.

Consider an arbitrary binary sequence h = (x0, . . . , xk−1) say, oflength k, with xi = ±1 for all i. Obviously, the pair (h, h) is a 2-modularGolay 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-modularGolay 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−12 .

Corollary 3.2.1 For every ℓ ≡ 1 mod 8, there is a k-parameter familyof 32-modular Golay quadruples of length ℓ and 32-modular Hadamardmatrices of size 4ℓ, where k = ℓ−1

8 .

Proof: Send the above 8-modular Golay pair of length 4k = ℓ−12 to

GQ32(ℓ) and H32(4ℓ) with the maps

GPm(r) −→ GQ4m(2r + 1) −→ H4m(4(2r + 1))

at m = 8 and r = 4k = ℓ−12 . !

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 sequenceF#, defined as

F# = ((−1)kak, . . . , (−1)iai, . . . , a0).

On the level of Hall polynomials, this transformation reads simplyas F#(z) = zkF (−z−1).

Let r = l−34 , and set ε = (−1)r−1. Thus, ε = −1 if r is even, that is

if ℓ ≡ 3 mod 8, while ε = +1 if ℓ ≡ 7 mod 8. Given two (±1)-sequencesH and K of size 2r + 1, we define a quadruple of binary sequences oflength 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#].

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28 Eliahou and Kervaire

For the binary sequencesH,K described below, the associated quadru-ple Q(H,K) turns out to be a 32-modular Golay quadruple. It is con-venient 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] andK = [−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 se-quences of length ℓ−1

2 = 2r+1, that is H = [12r+1] and K = [−1r+1; 1r]

if r is even, H = [1r−1;−1, 1, 1; [−1, 1](r−1)/2] andK = [−1r−1; 1,−1, 1; [−1, 1](r−1)/2] if r is odd. Then the quadruple ofbinary sequences Q(H,K) = (A,B,C,D) as defined above, is a 32-modular Golay quadruple of length ℓ. More precisely, we have the fol-lowing 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) =

4ℓ+ 32

[r/2]!

i=1

([r/2]− i)(z2i + z−2i).

Corollary 3.2.3 There is a 32-modular Hadamard matrix of size 4ℓfor every positive integer ℓ ≡ 3 mod 4.

Proof: Send the above 32-modular Golay quadruple A,B,C,D oflength 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 Hadamardmatrix 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].

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Modular Hadamard matrices 29

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 thekth aperiodic correlation coefficients of A,B,C and D respectively. Wethen 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 matrixH = GS(A,B,C,D) is therefore a 32-modular Hadamardmatrix of size 428 (see Figure 1). It is amusing to observe that amongthe 91378 =

!4282

"entries of the strict upper triangular part of H ·HT ,

there are 86242 entries which are strictly 0, while the remaining 5136non-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 inposition 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 the-orem of Hadamard [9] stating that the determinant of any real matrixM 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 ifand only if M is a Hadamard matrix. Here, in the above example Hof size n = 428, we have nαn < |det(H)| < nβn, with α = 0.347 andβ = 0.348.

3.2.3 The case ℓ ≡ 5 mod 8

We know only one way to obtain 32-modular Hadamard matrices ofsize 4ℓ for ℓ ≡ 5 mod 8. Namely, from 32-modular Golay pairs of length2ℓ ≡ 10 mod 16 and the map GPm(2ℓ) −→ Hm(4ℓ).

The relevant modular Golay pairs are somewhat involved, and arebest described through their Hall polynomials.

Let k = ℓ−58 . Define S(z) =

#k−1i=0 (−1)iz4i. Let x0, x1 be two binary

parameters, and define the pair of polynomials U(z), V (z) as follows:

U(z) = (x0 + x1z + x0z2)S(z) + (−1)k(x0 − x1z − x0z2)z4k

+(−1)k(x0 − x1z + x0z2)S(z)z4(k+1),

V (z) =#2k

i=0(−1)iz4i + z8k+2.

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30 Eliahou and Kervaire

Figure 1: A 32-modular Hadamard matrix of size 428 (white pixels rep-resent +1 and black pixels represent −1)

Finally, let x3 = ±1 be a third free binary parameter, and defineA(z), B(z) as follows:

A(z) = U(z) + x3z3V (z) + z16k+9(U(z−1)− x3z−3V (z−1)),

B(z) = U(z) + x3z3V (z)− z16k+9(U(z−1)− x3z−3V (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 Golaypair A,B of length 2ℓ = 16k + 10.

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Modular Hadamard matrices 31

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 + 32k−1!

i=1

(−1)i(k − i)(z4i + z−4i).

Example 3.2.6 We get true Golay pairs of length 10 for k = 0 andtrue Golay pairs of length 26 for k = 1. Let now k = 2. There areno Golay pairs of length 2ℓ = 16k + 10 = 42, because 42 has a divisorcongruent to 3 mod 4 (see [8]). However, setting x0 = x1 = x3 = 1 forsimplicity in the pair given by the above theorem, we get a 32-modularGolay pair A,B of length 42, namely

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

B = ++++−−−−+−−++−+−−+−+++++−+

−−+−+++−−++++−−− .

Remarkably, this pair is almost a true Golay pair of length 42, as itsatisfies ci(A) + ci(B) = 0 for all 1 ≤ i ≤ 41 with the sole exception ofi = 4, for which c4(A) + c4(B) = −32.

More generally, the formula (4) shows that only (k− 1)/(16k+8) ofthe correlations sums ci(A)+ci(B) are non-zero. On the other hand, weknow that a pair (A,B) with k ≥ 2 as in the above Theorem can neverbe an actual Golay pair even with an arbitrary (binary) polynomialS(z) =

"k−1i=0 uiz4i. (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 4ℓfor every positive integer ℓ ≡ 5 mod 8.

Proof: Send the above 32-modular Golay pair A,B of length 2ℓ toH32(4ℓ) with the map GPm(2ℓ) −→ Hm(4ℓ) of Section 2. !

Corollary 3.2.8 There exist 128-modular Hadamard matrices of size16ℓ+ 4 for every positive integer ℓ ≡ 5 mod 8.

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32 Eliahou and Kervaire

Proof: Send the above 32-modular Golay pair A,B of length 2ℓ toH128(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 knownfor r = 789, 853 and 917. These are the only undecided cases with r ≤1000 and r ≡ 21 mod 32. However, there exist 128-modular Hadamardmatrices 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 corollaryabove.

In the next Section, we shall actually obtain a 192-modular Hadamardmatrix of size 4 · 917.

3.3 Other moduli

We shall exhibit a few more modular Hadamard matrices in sizes forwhich, 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=0 (−1)iz12i. Let x0, x1 be two binary parameters, and define thepair of polynomials U(z), V (z) as follows :

U(z) = {x0(1 + z2 − z4 + z6 − z8 − z10) + x1(z + z5 + z9)}S(z),

V (z) = (1− z4 + z8)S(z) + z12k−2.

Finally, let x3 be a third free binary parameter, and defineA(z), B(z)as follows :

A(z) = U(z) + x3z3V (z) + z24k+1(U(z−1)− x3z−3V (z−1)),

B(z) = U(z) + x3z3V (z)− z24k+1(U(z−1)− x3z−3V (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 Golaypair A,B of length 2ℓ = 24k + 2.

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Modular Hadamard matrices 33

Example 3.3.2 For k = 1, this construction yields true Golay pairsof length 26. Let now k = 2. There are no Golay pairs of length2ℓ = 24k + 2 = 50, as revealed by an exhaustive computer search. (See[1].) However, setting x0 = x1 = x3 = 1 in the pair given by the abovetheorem, 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 whichc12(A) + c12(B) = −48.

Corollary 3.3.3 There exist 48-modular Hadamard matrices of size48k+4 and 192-modular Hadamard matrices of size 192k+20 for everypositive 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. Therealso exist 48-modular Hadamard matrices of size 4 ·721 and 4 ·853 (withk = 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 fortrue Hadamard matrices.

The moduli m = 2t

We have proved above that H32(n) = ∅ for every positive integer ndivisible 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) = ∅ forevery t ≥ 3 and every n divisible by 2t.

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34 Eliahou and Kervaire

However, with further constructions, we shall obtain 64-modularand 128-modular Hadamard matrices of some (but unfortunately notall) sizes n ≡ 4 mod 8.

Recall from Section 3.2.1 that, if h is an arbitrary binary sequenceof length k, then the pair (h, h) is a k-parameter 2-modular Golay pairof length k. In other terms, GP2(k) = ∅ for every positive integer k. Bythe doubling of Golay pairs, that is, by the map GPm(n) −→ GP2m(2n),which doubles both length and modulus, we readily obtain the followingstatements.

Proposition 3.3.5 GP2t(2t−1k) = ∅ for every positive integers t and

k.

Corollary 3.3.6 There exist 2t+2-modular Hadamard matrices of sizen = 4 · (2tk + 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 64-modular Hadamard matrix of size 4·721. (We already had a 48-modularHadamard 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 length8k + 2 for every integer k ≥ 0.

Proof: For k = 0, the pair A(z) = 1+z,B(z) = 1−z will do. Assumenow k ≥ 1. Choose polynomials f(z) =

!k−1i=0 xiz4i, g(z) =

!k−1i=0 yiz4i

with arbitrary xi = ±1, yi = ±1 for i = 0, 1, . . . , k−1. Let also w = ±1be chosen arbitrarily. Further, let F (z) = z−(4k−1)f(z) + z4k−1f(z−1)and G(z) = z−(4k−1)g(z) − z4k−1g(z−1). A 16-modular Golay pair, oflength 8k + 2, is given by

A(z) = {(1 + z3)F (z) + (z + z2)G(z) + w(z − z2)}z4k−1

B(z) = {(1− z3)F (z) + (z − z2)G(z) + w(z + z2)}z4k−1.!

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Modular Hadamard matrices 35

Corollary 3.3.9 There exist 2t+6-modular Hadamard matrices of sizen = 4 · (2t+3k + 2t+1 + 1) for every integers t, k ≥ 0.

Proof: Since GP16(8k + 2) = ∅, it follows by successive doublingthat GP2t+4(2t+3k + 2t+1) = ∅, for every t ≥ 0. Using again the mapsGPm(n) −→ GQ4m(2n+1) −→ H4m(4 · (2n+1)) of Section 2, it followsthat the sets GP2t+6(2t+4k+ 2t+2 + 1) and H2t+6(4 · (2t+4k+ 2t+2 + 1))are both non-empty, for every t, k ≥ 0. !

Example 3.3.10 It is not known whether Hadamard matrices of size4 · ℓ exist for ℓ = 789, 853, 917 and 933. These four values of ℓ arecongruent to 5 mod 16. The above corollary, with t = 0, thereforeyields 64-modular Hadamard matrices of size 4 · 789, 4 · 853, 4 · 917 and4 · 933. Note that only the case 4 · 933 is really of interest here, as wehad already obtained 128-modular Hadamard matrices of size 4 · 789,4 · 853 and 4 · 917 in Section 3.2.3.

4 Circulant modular Hadamard matrices

4.1 Introduction

According to Ryser’s conjecture, there probably exists no circulant Hadamardmatrix of size n > 4. In contrast, the modular level reveals interestingfamilies of examples [5]. These families are all based on the quadraticand 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-modularcirculant Hadamard matrices of size n ?

Definition 4.1.1 Let s = {x0, x1, . . . , xn−1} ∈ {±1}n be a binarysequence of size n. The kth periodic correlation coefficient γk(s) of s,for 0 ≤ k ≤ n− 1, is defined as γk(s) =

!n−1i=0 xixi+k, where the indices

are read modulo n.

Observe that γ0(s) = n, and that γn−k(s) = γk(s) for 1 ≤ k ≤ n−1.Also, setting s(z) =

!n−1i=0 sizi, we have the formula s(z)s(z−1) = n +!n−1

k=1 γk(s)zk in the quotient ring Z[z]/(zn−1). Finally, if H = circ(s)

is the circulant matrix with first row s, then obviously the matrix H ·HT

has γj−i(s) as entry with position i, j. Thus, H will be an m-modular

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36 Eliahou and Kervaire

circulant Hadamard matrix if and only if γk(s) ≡ 0 mod m for all 1 ≤k ≤ n

2 .

The most obvious examples of circulant modular Hadamard matriceswith a large modulus are

J = circ(1, · · · , 1) and K = −2I + J = circ(−1, 1, · · · , 1)

of size n. We have J · JT = nJ and K · KT = 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 pe-riodic correlation coefficients are actually 0, not only 0 mod m. Weintroduce the following definition.

Definition 4.1.2 Let s ∈ {±1}n be a binary sequence of size n, withn even. We say that s is of type 1 if γn

2(s) = 0. We say that s is of type

2if γ1(s) = . . . = γn2−1(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 itsrows is of type i).

Remark 4.1.3 Ryser’s conjecture is equivalent to saying that there areno binary sequences of length greater than 4 which are simultaneouslyof type 1 and of type 2.

Circulant modular Hadamard matrices of type 1 and type 2 wereintroduced in [6]. After finding that “type 2” was equivalent with thenotion of “almost perfect sequence”, we thought of abandoning the term“type 2”, and replacing the term “type 1” by “enhanced”. But we nowchoose to restore the type 1 / type 2 terminology, essentially becauseof the symmetry in the definition. We ask a little indulgence from thereader for these terminological meanderings.

Example 4.1.4 Here is a binary sequence of length 8 and type 2. Lets = (1, 1, 1,−1, 1,−1,−1, 1). Then γ1(s) = γ2(s) = γ3(s) = 0, showingthat s is indeed a sequence of type 2. Additionally, γ4(s) = −4. TakingH to be the circulant matrix with first row s, we have:

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Modular Hadamard matrices 37

H =

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 1 1 −1 1 −1 −1 11 1 1 1 −1 1 −1 −1

−1 1 1 1 1 −1 1 −1−1 −1 1 1 1 1 −1 11 −1 −1 1 1 1 1 −1

−1 1 −1 −1 1 1 1 11 −1 1 −1 −1 1 1 11 1 −1 1 −1 −1 1 1

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

and

H ·HT =

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

8 0 0 0 −4 0 0 00 8 0 0 0 −4 0 00 0 8 0 0 0 −4 00 0 0 8 0 0 0 −4

−4 0 0 0 8 0 0 00 −4 0 0 0 8 0 00 0 −4 0 0 0 8 00 0 0 −4 0 0 0 8

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Another example of a binary sequence of length 8 and type 2 isprovided 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 ourattention to moduli which are powers of 2, we will exhibit 16-modularHadamard 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 aclassical construction from Delsarte, Goethals and Seidel which impliesthat there is a circulant (n−4)-modular Hadamard matrix of type 2 forevery size n of the form n = 2(pr + 1) where p is prime.

4.2 Circulant (p−1)-modular Hadamard matrices of type1 and size 4p

As announced above, we shall construct a circulant (p − 1)-modularHadamard 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 ofits first row.

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38 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 non-squares mod p. Of course, we have |S0| = |S1| = p−1

2 . Let g0(z) denotethe generating function of S0. That is, g0(z) =

!i∈S0

zi.

Similarly, let g1(z) =!

i∈S1zi be the generating function of S1.

Note that, since S = S0"

S1, we have g0(z) + g1(z) =!p−1

i=1 zi.Let x0, x1, x2, x3 ∈ {±1} be four free binary parameters, and con-

sider the polynomial

h(z) = x0(1 + z2p)(1 + g0(z2)) + x1(1 + z2p)zpg0(z

2)+

x2(1− z2p)g1(−z2) + x3(1− z2p)zp(1 + g1(−z2)),

viewed as an element in the quotient ring Z[z]/(z4p − 1).As it turns out, when expressing h(z) in the form

!4p−1i=0 aizi, 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]/(z4p − 1) be the above polynomial,

h(z) = x0(1 + z2p)(1 + g0(z2)) + x1(1 + z2p)zpg0(z

2)+

x2(1− z2p)g1(−z2) + x3(1− z2p)zp(1 + g1(−z2)).

Then h(z) is the Hall polynomial of a 4-parameter binary sequence hof length 4p, with the property that circ(h) is a circulant (p−1)-modularHadamard 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]/(z4p − 1).

We find the following expression:

h(z)h(z−1) = 4p+ (p− 1)R(z),

where R(z) = 2!p−1

i=1 z4i + x0x1(!2p

i=1 z2i−1 + zp + z3p).

Given that h(z)h(z−1) = 4p +!4p−1

j=1 γj(h)zj , the above expressionshows that the gcd of the periodic correlation coefficients γi(h) of h fori = 1, ..., 4p− 1, is equal to p− 1. Note also that γ2p = 0, showing thath 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.

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Modular Hadamard matrices 39

Example 4.2.2 Let p = 5. The non-zero squares mod 5 are 1 and 4.Therefore S0 = {1, 4}, g0(z) = z + z4 and g1(z) = z2 + z3. Finally,

h(z) ≡ x0(1 + z2 + z8 + z10 + z12 + z18) + x1(z3 + z7 + z13 + z17)+

x2(z4−z6−z14+z16)+x3(z+z5+z9−z11−z15−z19) mod (z20−1),

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 arethe following: γ1 = γ3 = γ7 = γ9 = 4x0x1, γ2 = γ6 = γ10 = 0, γ4 = γ8 =8, γ5 = 8x0x1.

4.3 Circulant 16-modular Hadamard matrices of type 1

Our objective is to construct circulant 16-modular Hadamard matricesof type 1 and size 4p, where p is an odd prime. According to the Lemmabelow, 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 4.2already provides us with the desired sort of matrices, as p−1 is divisibleby 16.

In this Section we consider the remaining case p ≡ 9 mod 16. Weshall present a partial solution to our construction problem, which worksin 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 (forexample p = 41 or 137), we do not know how to construct 16-modularcirculant Hadamard matrices of type 1 and size 4p. Quite possibly, noneexists in this case.

We start with the promised result restricting the possible sizes ofcirculant 16-modular Hadamard matrices of type 1.

Lemma 4.3.1 Let r ≥ 1 be a natural number, and assume there existsa circulant 16-modular Hadamard matrix of type 1 and size 4r. Thenr ≡ 0, 1 or 4 mod 8.

Proof: Let h(z) be the Hall polynomial of the first row h of a circulant16-modular Hadamard matrix of type 1 and size 4r. In the quotientring Z[z](z4r − 1), we have the general formula h(z)h(z−1) = 4r +!2r−1

k=1 γk(zk + z−k) + γ2rz2r, where the γk are the periodic correlation

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40 Eliahou and Kervaire

coefficients of the sequence h. Setting z = 1 in the above formula,we get h(1)2 = 4r + 2

!2r−1k=1 γk + γ2r. Now, γ2r = 0 by the type 1

hypothesis, and γk ≡ 0 mod 16 for all 1 ≤ k ≤ 2r − 1. It follows thath(1)2 = 4r+32θ for some integer θ. Thus h(1) is even, and dividing by4 we get r = (h(1)/2)2 +8θ. The conclusion follows as the only squaresmod 8 are 0, 1 and 4. As a side remark, note that the same argumentwould still work under the weaker hypothesis γ2r ≡ 0 mod 32 insteadof γ2r = 0. !

Let p be a prime such that p ≡ 1 mod 8. As in Section 4.2, considerthe set S = {1, . . . , p − 1} and its partition S = S0 ∪ S1, where S0 isthe 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 multi-

plicative 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⟩ generatedby c2, we have S0 = ρ−1(⟨c2⟩) and S1 = ρ−1(c⟨c2⟩), where c⟨c2⟩ is theother coset of ⟨c2⟩ in F∗

p.Consider now the subgroup Γ = ⟨c4⟩ ⊂ F∗

p. Thus, Γ is the onlysubgroup of order (p − 1)/4 in F∗

p. The four cosets of Γ in F∗p are

Γ, cΓ, c2Γ and c3Γ, and of course they partition F∗p into four pieces of

equal size (p − 1)/4. This partition refines the earlier one into squaresand non-squares, as Γ ∪ c2Γ = ⟨c2⟩.

Transporting back the above partition to S by ρ−1, we shall denoteS00 = ρ−1(Γ), S10 = ρ−1(cΓ), S01 = ρ−1(c2Γ) and S11 = ρ−1(c3Γ).

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 allhave cardinality (p− 1)/4.

For u, v = 0, 1, we shall denote by gu,v(z) the generating function ofSu,v, that is gu,v(z) =

!i∈Su,v

zi. Note that g00(z) + g01(z) + g10(z) +

g11(z) =!p−1

i=1 zi.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 inSection 4.2.

Let x0, x1, x2, x3 ∈ {±1} be four free binary parameters, and con-sider the polynomial

h(z) = x0(1+z2p)(1−g00(z2)−g01(z

2))+x1(1+z2p)zp(g00(z2)−g01(z

2))+

x2(1−z2p)(g10(−z2)−g11(−z2))+x3(1−z2p)zp(1−g10(−z2)−g11(−z2)),

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Modular Hadamard matrices 41

viewed as an element in the quotient ring Z[z]/(z4p − 1). As for thecorresponding polynomial in Section 4.2, when expressing h(z) in theform

!4p−1i=0 aizi, 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, leth(z) ∈ Z[z]/(z4p − 1) be the above polynomial

h(z) = x0(1+z2p)(1−g00(z2)−g01(z

2))+x1(1+z2p)zp(g00(z2)−g01(z

2))+

x2(1−z2p)(g10(−z2)−g11(−z2))+x3(1−z2p)zp(1−g10(−z2)−g11(−z2)).

Then h(z) is the Hall polynomial of a 4-parameter binary sequenceh of length 4p, with the property that circ(h) is a circulant 8-modularHadamard 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 16and 2 is a fourth power mod p.

The periodic correlations γk in h(z)h(z−1) = 4p+!2p−1

k=1 γk(zk+z−k)are explicitly determined in [7], using Jacobi sums. They depend on thedecomposition p = a2+b2 with b even, a odd and the sign of a normalizedby 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 showingthat 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 followsthat if p ≡ 9 mod 16 and 2 is a fourth power modulo p, then necessarilya ≡ −3 mod 8 and b ≡ 0 mod 8. Thus, in this case, all the periodiccorrelations 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 afourth 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 binarysequence h, for which circ(h) is a 16-modular Hadamard matrix of type1 and size 292 :

+ + − − − − − + − + − + − − − + − − − − + − + − − − + −+ − + + − − − − − + − + + + + + − − − + − − − + − − − ++ + + + − − + + − + − − + + − + − + − + − + − − + + − +

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42 Eliahou and Kervaire

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

4.4 Circulant modular Hadamard matrices of type 2

We are seeking binary sequences s of even length n with the propertythat γ1(s) = . . . = γn

2−1(s) = 0. In this way, circ(s) will be a circulantm-modular Hadamard matrix of type 2, with m = γn

2(s).

These sequences were first introduced by J.Wolfmann [16] in 1992,and are called almost perfect sequences. See also Langevin [10]. (Recallthat a sequence s of length n ≡ 0 mod 4 is perfect if it satisfies γi(s) = 0for all 1 ≤ i ≤ n/2. This is equivalent to circ(s) being a circulantHadamard matrix. Hence, Ryser’s conjecture amounts to saying thatthere is no perfect sequence of length n ≡ 0 mod 4 with n > 4.)

Almost perfect sequences are known in all lengths n of the formn = 2(q + 1) where q is an odd prime power, and are believed not toexist in other lengths. This follows from a theorem by Delsarte, Goethalsand Seidel, establishing the existence of a negacyclic conference matrixof order q + 1 for every odd prime power q. For convenience of thereader, 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 conditionC · CT=(n− 1)I.

Definition 4.4.2 A negacyclic matrix N is a square matrix of the form

N = NC(u0, u1, . . . , ur) =

⎜⎜⎜⎜⎜⎝

u0 u1 . . . . . . ur−ur u0 u1 . . . ur−1

−ur−1 −ur u0 . . . ur−2...

......

. . ....

−u1 −u2 . . . −ur u0

⎟⎟⎟⎟⎟⎠.

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Modular Hadamard matrices 43

Example 4.4.3 As an illustration of both concepts simultaneously,here is a negacyclic conference matrix of size 6:

C =

⎜⎜⎜⎜⎜⎜⎝

0 1 1 1 −1 1−1 0 1 1 1 −11 −1 0 1 1 1

−1 1 −1 0 1 1−1 −1 1 −1 0 1−1 −1 −1 1 −1 0

⎟⎟⎟⎟⎟⎟⎠.

Theorem 4.4.4 (Delsarte-Goethals-Seidel, [3]) Let q be an odd primepower. 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.

Let A =

(0 −gq+1

1 g + gq

), with entries in the subfield Fq as g · gq and

g + gq are the norm and trace of g, respectively.Let

v =

(10

).

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 ofF∗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 · CT = qI. Finally, let Γ be thealternating diagonal matrix Γ = diag(1,−1, . . . , 1,−1) of size q + 1.As it turns out, the product Γ · C is a negacyclic conference matrix ofsize 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. satisfyingγ1(s) = . . . = γn

2−1(s) = 0. Moreover, γn2(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 easycalculation shows that γn/2(s) = 4 − n, and that γk(s) = 2(ck(s′) −

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44 Eliahou and Kervaire

cq+1−k(s′)) for all 1 ≤ k ≤ q, where ck(s′) =!q−k

j=1 xjxj+k denote

the kth aperiodic correlation coefficient of the sequence s′. Thus, thesequence s will be of type 2 if and only if ck(s′) = cq+1−k(s′) for all1 ≤ k ≤ q.

Now, the latter condition on s′ is equivalent to the negacyclic matrixN = NC(0, x1, x2, . . . , xq) being a conference matrix, as the dot productof the ith row and the (i+k)th row ofN is equal to ck(s′)−cq+1−k(s′). Bythe result of Delsarte, Goethals and Seidel, there exists a negacyclic con-ference matrix C = NC(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 abinary sequence of type 2 and length n, as desired. !

AcknowledgementThe first author gratefully acknowledges partial support from the

Fonds National Suisse de la Recherche Scientifique during the prepara-tion of this paper.

Shalom EliahouDepartement de Mathematiques,Universite du Littoral Coted’Opale,Batiment Poincare, 50 rue Fer-dinand Buisson, B.P. 699, 62228Calais, France,[email protected]

Michel KervaireDepartement de Mathematiques,Universite de Geneve,2 rue du Lievre, B.P. 240, 1211Geneve 24, [email protected]

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 Golaypairs for lengths up to 100, Math. Comput., 47 (2004), 967-985.

[3] Delsarte P.; Goethals J.-M.; Seidel J., Orthogonal matrices withzero diagonal. II, Can. J. Math., XXIII (1971), 816-832.

[4] Dinitz J. H.; Stinton D. R., Contemporary Design Theory, A Col-lection of Surveys, Wiley-Interscience Publication, 1992.

[5] Eliahou S.; Kervaire M., Circulant Modular Hadamard matrices,Ens. Math., 47 (2001), 103-114.

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Modular Hadamard matrices 45

[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 matricesand 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., Resolution d’une question relative aux determinants,Bulletin des Sciences Mathematiques, 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 relateddesigns, J. Comb. Theory, 15 (1973), 257-269.

[12] Marrero O.; Butson A. T., Modular Hadamard matrices and relateddesigns, 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 theAmer. Math. Soc., 12 (1999), 929-952.

[15] Turyn R. J., Hadamard matrices, Baumert-Hall units, four-symbolsequences, pulse compression and surface wave encoding, J. Comb.Theory, 16 (1974), 313-333.

[16] Wolfmann J., Almost perfect autocorrelation sequences, IEEETrans. Inform. Theory, 38 (1992), 1412-1418.

[17] Van Lint J. H.; Wilson R. M., A course in combinatorics, Cam-bridge University Press, 1992.

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Morfismos, Vol. 7, No. 2, 2003, pp. 47–73

Application of modularity to optimal resourceallocation with risk sensitivity

Guadalupe Avila-Godoy 1

Abstract

An optimal allocation problem with a risk-sensitive controller ismodelled by a controlled Markov chain with exponential total costcriterion. Some general results recently obtained are applied toshow that the particular model studied here has a monotone opti-mal policy and monotone optimal value function. Moreover, it isshown that under certain conditions, the allocation problem withboth risk-neutral and risk-sensitive performance criteria has anoptimal policy of the threshold type.

2000 Mathematics Subject Classification: 90C40, 60J05.Keywords and phrases: Controlled Markov Chains, Optimal ResourceAllocation, Exponential Total Cost Criterion, Risk Sensitivity, Modu-larity.

1 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 consideredhas finite state space, compact action space and bounded cost function.Models of dynamic systems that incorporate risk-sensitivity by meansof an exponential utility function have recently received considerableattention in the literature, see for example [2, 3, 6, 7, 8, 9] and refer-ences therein. However, in contrast with the risk-neutral literature (see

1This paper is part of the author’s doctoral research under the direction of Dr.-revinUehtfoscitamehtaMfotnemtrapeDehttadnarehcuaGzednanreFleunammE

sity of Arizona.

47

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48 Avila-Godoy

[11, 13, 14, 15, 16, 17, 18, 19] and references therein), only a few (and re-cent) contributions dealing with structural properties of the CMC havebeen made in the risk-sensitive case; see [5, 9]. We are particularly con-cerned with the contributions made by Avila-Godoy [4], which extendsome results in [13, 14] from the risk-neutral to the risk-sensitive case.It is proved in [4] that appropiate structural conditions (like modularityand/or monotonicity) of certain functions defined in terms of the costand the transition kernel, imply the monotonicity of the optimal expo-nential value function and the existence of monotone optimal policies.Herein, we apply some results in [4] to show that the CMC modeling theoptimal allocation problem has monotone optimal policies and mono-tone optimal value function. See, e.g., [13, 14] for an analysis of thisproblem with risk-neutral total cost.

The paper is organized as follows. Section 2 includes the descriptionof the model and collects the results in [4] needed for our study. InSections 3 and 4, the CMC model for the finite horizon optimal allo-cation 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 thenthe existence of a monotone optimal policy is established (Proposition3.1.12), that is, we show that the decision function of the optimal policyat 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 condi-tions, we prove that the allocation problem can be reduced to a problemwith two actions and that the optimal policy is of the threshold type(Proposition 4.1.13). Finally, we apply those results to a particular ex-ample with a linear final cost. For the purpose of comparing the resultsobtained in Sections 3 and 4, we include an appendix on risk-neutralresource allocation problems. The proof of the result relative to thereduction of the risk-neutral allocation problem to a problem with twoactions and that the optimal policy is of the threshold type is also acontribution of this paper’s author (Section 5.2).

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

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Optimal resource allocation 49

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 Xgiven 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 isnonnegative 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 systemobserved 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 chosenat time t. If X0 = x ∈ X, and A0 = a ∈ A, then (i) a cost C(x, a) isincurred, and (ii) the system moves to a new state X1 according to theprobability distribution P (· | x, a). Once the transition into the newstate has occurred, a new action is chosen, and the process is repeatedfor n times; see [1, 10, 13].

The strategy followed to choose the actions at each stage is calleda policy. The most general set Π of policies considered in the liter-ature includes the admissible, history dependent, randomized policies;see [1, 10, 13]. Herein, we will be concerned only with the subset of Πconsisting of the Markov deterministic policies, denoted by ΠMD. Fora policy π ∈ Π and initial state x ∈ X, Eπ

x will denote the expectationoperator with respect to the probability measure induced by π and x inthe space of trayectories of the chain.

Risk-sensitivity of the controller is modelled by grading the total costwith the exponential (disutility) function Uγ(x) = (sgn γ)eγx, γ = 0,where the parameter γ turns out to be the (constant) risk-sensitivitycoefficient associated to Uγ , see [12, 20]. In this work, only the caseγ > 0, the risk-averse case, will be considered. Thus, the performancecriterion for a policy π when the initial state is x and we proceed for nstages, is given by

(1) Jπn (x, γ) := Eπ

x

#eγ(

!n−1t=0 C(Xt,At)+c(Xn))

$.

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

(2) Jπ∗n (x, γ) = inf

π{Jπ

n (x, γ)} =: Jn(x, γ).

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50 Avila-Godoy

The optimal policy π∗ is called ETC-optimal and Jn(x, γ) is the op-timal ETC. We can interpret Jn(x, γ) as the minimal ETC that canbe obtained starting at state x with risk-sensitivity coefficient γ andproceeding 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] fortheir proofs. The following known assumption will be made in the restof this section (see [10]).

Assumption 2.1

1) C(x, ·) is continuous for each x ∈ X; and2) If v : X −→ R is bounded then the function

a $−→!

y

pxy(a)v(y) is continuous,

for each x ∈ X.

First, we recall a typical forward dynamic programming recursion.

Theorem 2.1.1 (Dynamic Programming Algorithm) The optimalETC, Jn(x, γ), satisfies the following recursion:

J0(x, γ) = eγc(x),

......

Js+1(x, γ) = infa∈A

"eγC(x,a)

!

y

pxy(a)Js(y, γ)#,

(3)

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)

%!

y

pxy (fs(x)) Jn−s−1(y, γ) =

infa∈A

"eγC(x,a)

!

y

pxy(a)Jn−s−1(y, γ)#.

(4)

Then the Markov deterministic policy π∗ = (f0, f1, f2, . . . fn−1) is ETC-optimal.

Next, a lemma that provides sufficient conditions for monotonicityof the optimal value function is stated.

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Optimal resource allocation 51

Lemma 2.1.2 Suppose that

i) C(x, a) is increasing (decreasing) in x for each a, and c(x) is increas-ing (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) inx, for s = 0, 1, · · ·n.

Finally, some standard definitions and notation, and two key theo-rems 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 ∈ Sthen 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 subaddi-tive on S.

We will assume the state and action spaces to be subsets of R withthe usual order and we will consider the product order ! on R2, thatis, ! 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 bemonotone (with respect to x) if all the decision rules ft are monotonefunctions of the state x. In the particular case that the action spacecontains only two actions, say a1 and a2, a monotone policy is called athreshold policy. That is, a threshold policy is a deterministic Markovpolicy π = (f0, f1, . . . , fn−1) such that, for t = 0, 1, . . . , n− 1, the deci-sion rule ft is given by

(5) ft(x) =

"a1 if x # x∗ta2 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 pol-icy exists, because the search for an optimal policy can then be restricted

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52 Avila-Godoy

from the class of Markov deterministic policies to the much smaller sub-class of monotone policies [11, 13]. The following theorem provides suf-ficient conditions for the existence of optimal monotone (with respectto 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)!

y

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

a′{Ht(x, a

′, γ)}#,

and ft(x) := minA∗t (x) (respectively ft(x) := maxA∗

t (x)). Suppose that

logHt(·, ·, γ) is subadditive (respectively superadditive) on (X × A,!),for fixed γ.

Then, (f0, f1, . . . fn−1) is an optimal policy with ft(x) increasing (re-spectively 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 optimalaction ft(x) varies with respect to t for each fixed x. Thus, a Markovdeterministic policy π = (f0, f1, . . . , fn−1) is said to be monotone (withrespect to t) if for each fixed x, the sequence of actions ft(x) is mono-tone in t. The following theorem provides sufficient conditions for theexistence of optimal monotone (with respect to t) policies for CMC’swith 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 logH(·)(x, ·, γ) is superadditive (re-spectively subadditive) on the lattice (A× {0, 1, 2, · · ·n− 1},!). Then,(f0, f1, . . . fn−1) is an optimal policy such that the sequence of actionsft(x) is decreasing (respectively increasing) in t for each x.

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Optimal resource allocation 53

3 An Optimal Allocation Problem.

In this section we follow Ross [14] to model an optimal allocation prob-lem by means of a finite horizon CMC. However, unlike the mentionedreference, we introduce a risk-sensitive performance criterion. The gen-eral results stated in Section 2 are applied to show that, under standardconditions, the optimal value function is increasing in the state and de-creasing in t (Lemma 3.1.6) and that the optimal policy is increasingin x and increasing in t (Proposition 3.1.12). Moreover, under addi-tional conditions, we prove that the allocation problem can be reducedto a problem with two actions and that the optimal policy is of thethreshold-type (Proposition 4.1.13). Finally, we apply those results toa particular example of a linear final cost; see [14] for an analysis of thisproblem with risk-neutral total cost.

The optimal allocation problem can be described as follows. Supposewe have N stages to construct sequentially I successful components . Ateach stage we allocate a certain amount of money for the constructionof a component. If a is the amount allocated, then the component con-structed will be a success with probability P (a), where P is a continuousstrictly increasing function such that P (0) = 0. After each componentis constructed, we are informed whether or not it is successful. If at theend of N stages, we are x components short, then a final penalty costc(x) is incurred, where c(x) is increasing. The problem is to determinehow much money to allocate at each stage to minimize the expectedETC. A CMC (X,A, P,C) which models the described allocation prob-lem can be defined by taking the state space X = {0, 1, . . . I}, the actionspace A = [0,M ], where M is a positive real number, the cost functionC(x, a) = a, and the transition probabilities

(7) pxy(a) =

⎧⎪⎨

⎪⎩

P (a) if y = x− 1

1− P (a) if y = x

0 otherwise.

The state Xt is the number of successful components still needed attime 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 xwith 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∑

y

pxy(a)Jt(y, γ) = P (a)Jt(x− 1, γ) + (1− P (a))Jt(x, γ)

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54 Avila-Godoy

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)

= infa∈[0,M ]

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

}(8)

= infa∈[0,M ]

{eγa[Jt(x, γ)− P (a)(Jt(x, γ)− Jt(x− 1, γ)]}(9)

= infa∈[0,M ]

{eγa[Jt(x− 1, γ) + (1− P (a))(Jt(x, γ)− Jt(x− 1, γ)]} .(10)

First, we will show that the optimal value function Jt(x, γ) is in-creasing 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 xand decreasing in t.

Proof: We will apply Lemma 2.1.2 to prove that Jt(x, γ) is increasingin x. First, we see that this model satisfies (i) of the mentioned lemmasince C(x, a) is constant in x, and the terminal cost c(x) is increasing.Finally, it follows from (7) that

(11)I∑

y=k

pxy(a) =

⎧⎪⎨

⎪⎩

1 if k ! x− 1

1− P (a) if k = x

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 followsfrom (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 optimalpolicies that are increasing in x and increasing in t. To this end, we willprove that

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

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Optimal resource allocation 55

is subadditive on X×A and subadditive on A× {0, 1, 2, . . . N − 1}, sothat the mentioned monotonicity properties will follow from Theorems2.1.3 and 2.1.4 since in this model the function Ht defined in (6) is thefunction 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) := eγagt(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 oflogHt(x, a, γ) we need is equivalent to a structural property of logGt(x,a, γ).

Lemma 3.1.7 a) logHt(x, a, γ) is subadditive on X×A iff logGt(x, a,γ) is subadditive on X × A. b) logHt(x, a, γ) is subadditive on A ×{0, 1, . . . N − 1} iff logGt(x, a, γ) is superadditive on A× {0, 1, . . .N − 1}.

Next, we will see that each of the structural properties of logGt(x, a,γ) we need is equivalent to a structural property of log Jt(x, γ).

Lemma 3.1.8 a) logGt(x, a, γ) is subadditive on X×A iff log Jt(x, γ)is convex in x. b) logGt(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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56 Avila-Godoy

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) ! J2t (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, a)! gt(x+ 1, a′)

gt(x, a′)

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

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

Note that the last step follows from the equality

logGt(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(x, a, γ)" gt+1(x, a′, γ)

gt(x, a′, γ)

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

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Optimal resource allocation 57

Now, we show that log Jt(x, γ) is indeed convex in x for each t, andsubadditive on X× {0, 1, . . . N − 1}. Throughout the rest of this paperwe will assume the following condition, which is reasonable for somesituations.

Assumption 3.1. The terminal cost c(x) is convex.

Lemma 3.1.9 Under Assumption 3.1, the following three statementshold: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

Ax,t :Jt(x+ 2, γ)

Jt(x+ 1, γ)" Jt(x+ 1, γ)

Jt(x, γ)(15)

Bx,t :Jt+2(x, γ)

Jt+1(x, γ)" Jt+1(x, γ)

Jt(x, γ)(16)

Cx,t :Jt+1(x, γ)

Jt(x, γ)" Jt+1(x+ 1, γ)

Jt(x+ 1, γ)(17)

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 theinequalities are true for k = 0. We assume that they are true whenevert + x < k and let k = t + x. Let’s prove Cx,t. It follows from (9) thatfor some a, say a,

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

and hence

(18)Jt+1(x, γ)

Jt(x, γ)= eγa

!1− P (a)

Jt(x, γ)− Jt(x− 1, γ)

Jt(x, γ)

".

On the other hand, it follows from Ax−1,t that

Jt(x, γ)− Jt(x− 1, γ)

Jt(x, γ)# Jt(x+ 1, γ)− Jt(x, γ)

Jt(x+ 1, γ).

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58 Avila-Godoy

Therefore, from (18) we obtain

Jt+1(x, γ)

Jt(x, γ)≥ eγa

!1− P (a)

Jt(x+ 1, γ)− 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 followsfrom (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)

Jt+1(x, γ)= eγa

′!1− P (a′)

Jt+1(x, γ)− Jt+1(x− 1, γ)

Jt+1(x, γ)

".

On the other hand, it follows from Cx−1,t that

Jt+1(x, γ)− Jt+1(x− 1, γ)

Jt+1(x, γ)! Jt(x, γ)− Jt(x− 1, γ)

Jt(x, γ).

Therefore, from (19) we obtain

Jt+2(x, γ)

Jt+1(x, γ)" eγa

′!1− P (a′)

Jt(x, γ)− Jt(x− 1, 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+1(x+ 1, γ)

Jt(x+ 1, γ)" Jt(x+ 1, γ)

Jt−1(x+ 1, γ),

or equivalently,

Jt+1(x+ 1, γ)Jt−1(x+ 1, γ) " J2t (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) = eγa [Jt−1(x+ 1, γ) + (1− P (a)) (Jt−1(x+ 2, γ)

−Jt−1(x+ 1, γ))] ,

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Optimal resource allocation 59

and hence

Jt(x+ 2, γ)

Jt−1(x+ 1, γ)= eγa [1 + (1− P (a))

Jt−1(x+ 2, γ)− Jt−1(x+ 1, γ)

Jt−1(x+ 1, γ)

!.(21)

On the other hand, it follows from Ax,t−1 and Cx,t−1 that

Jt−1(x+ 2, γ)

Jt−1(x+ 1, γ)! Jt(x+ 1, γ)

Jt(x, γ),

and hence

Jt−1(x+ 2, γ)− Jt−1(x+ 1, γ)

Jt−1(x+ 1, γ)! Jt(x+ 1, γ)− Jt(x, γ)

Jt(x, γ).

Thus, from (21) we obtain

Jt(x+ 2, γ)

Jt−1(x+ 1, γ)! eγa

Jt(x, γ)[Jt(x, γ) + (1− P (a))(Jt(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 foreach t.

Proof: Since Jt(x, γ) = exp(log Jt(x, γ)), the claim follows fromLemma 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 existsan optimal policy π = (f0, . . . fN−1) such that ft(x) is increasing in xfor each t, and increasing in t for each x; see [14]. In the followingproposition we show an analogous result for the risk-sensitive case.

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60 Avila-Godoy

Proposition 3.1.12 Under Assumption 3.1, there exists an optimalpolicy π = (f∗

0 , . . . f∗N−1) for the allocation problem with exponential

total cost criterion such that f∗t (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, logHt(x, a, γ) is subadditive on X×A, and subadditive onA× {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 problemwith 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 thatunder the mentioned conditions, the optimal policy obtained in Propo-sition 3.1.12 has further structural properties. Moreover, we comparethose structured optimal policies with those corresponding to the risk-neutral allocation problem (which are obtained in the appendix). Fi-nally, we apply the obtained results to the particular case of a linearterminal cost function, and again we compare the conclusions with thosecorresponding to the risk-neutral problem.

4 Allocation Problem with P(a) Convex andc(x) Strictly Increasing.

Throughout this section, π∗ = (f∗0 , . . . , f

∗t−1) will denote the monotone

optimal policy obtained in Proposition 3.1.12.

Proposition 4.1.13 Assume that P (a) is convex and twice differen-tiable and c(x) strictly increasing. Then, under Assumption 3.1, theoptimal allocation problem can be reduced to a problem with the actionset {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

(22) f∗t (x) =

!0 if x < x∗tM 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 strictlyincreasing in x. Since J0(x, γ) = eγc(x), the result holds for t = 0. Now

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Optimal resource allocation 61

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 havethat

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 continuousin a,

infa∈[0,M ]

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

> infa∈[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 thatfor ax ∈ (0,M),

∂Gt

∂a(x, ax, γ) = 0 =⇒ ∂2Gt

∂2a(x, ax, γ) < 0;

that is, that there are no minimal points in (0,M). Indeed, it followsfrom (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

∂a(x, a, γ) = −eγa[Jt(x, γ)− Jt(x− 1, γ)]P ′(a) + γGt(x, a, γ),

and

∂2Gt

∂2a(x, a, γ) = −eγa[Jt(x, γ)− Jt(x− 1, γ)]P ′′(a)−

γeγa[Jt(x, γ)− Jt(x− 1, γ)]P ′(a) + γ∂Gt

∂a(x, a, γ)

= γ∂Gt

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

[γP ′(a) + P ′′(a)]

If ax ∈ (0,M) is such that ∂Gt∂a (x, ax, γ) = 0, then ∂2Gt

∂2a (x, a, γ) < 0 sinceJt(x− 1, γ)− Jt(x, γ) < 0 and γP ′(a) + P ′′(a) > 0.

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62 Avila-Godoy

Since there are no minimal points in (0,M), then we must havef∗t (x) ∈ {0,M} ∀t, ∀x. Moreover, if we define

x∗t := min{x : f∗t (x) = M},

t = 0, 1, . . . N−1, then (22) follows from the fact that f∗t (x) is increasing

in x. Finally, the sequence {x∗t } is decreasing since f∗t (x) is increasing

in t. ✷

Now, to gain further insight of the consequences of Proposition4.1.13, we apply this proposition to compute the optimal policy in aparticular example with linear final cost.

Example 4.1.14 Take c(x) = 2x, A = [0, 1], and P (a) convex. We startby computing f∗

N−1(x). To do that, by Proposition 4.1.13, we need onlyto compare the values of the function G0(x, a, γ) at the extreme actionsa = 0 and a = 1. We have that

G0(x, a, γ) = eγa[P (a)J0(x− 1, γ) + (1− P (a))J0(x, γ)], x ≥ 1

= eγa[P (a)eγ(2x−2) + (1− P (a))e2γx], 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))

!e2γ − 1

1− P (1)eγ +

P (1)

1− P (1)

"≥ 0

⇐⇒ e2γ − 1

1− P (1)eγ +

P (1)

1− P (1)≥ 0

⇐⇒#eγ − P (1)

1− P (1)

$#eγ − 1

$≥ 0

⇐⇒ γ ≥ logP (1)

1− P (1).(25)

Thus, it follows from (23), (24) and (25) that

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Optimal resource allocation 63

a) if 12 < P (1) < 1 and 0 < γ ≤ log( P (1)

1−P (1)) then

G0(x, 1, γ) ≤ G0(x, 0, γ);

b) if 12 < P (1) < 1 and γ ≥ log( P (1)

1−P (1)) then

G0(x, 0, γ) ≤ G0(x, 1, γ);

c) if P (1) ≤ 12 and γ > 0 then

G0(x, 0, γ) < G0(x, 1, γ);

d) if P (1) = 1 and γ > 0 then

G0(x, 1, γ) < G0(x, 0, γ).

Therefore the optimal decision rule f∗N−1 and the optimal value func-

tion J1 for the cases (a) and (d) are given by

(26) f∗N−1(x) =

!0 if x < 1

1 if x ! 1,

and(27)

J1(x, γ) =

!1 if x = 0

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

and for (b) and (c) by

f∗N−1(x) = 0, ∀x

and

(28) J1(x, γ) = e2γx, x ≥ 0.

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 12 < P (1) < 1 and 0 < γ ≤ log( P (1)

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

Jt(x, γ) =

!1 if x = 0

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

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64 Avila-Godoy

II) if 12 < P (1) < 1 and γ ≥ log( P (1)

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

Jt(x, γ) = J0(x, γ), x ∈ X;

III) if P (1) ≤ 12 and γ > 0 then for t = 1, . . . N − 1,

Jt(x, γ) = J0(x, γ), x ∈ X;

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

Jt(x, γ) =

!1 if x = 0

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

First, let’s prove (I). The validity of assertion (I) for t = 1 followsfrom (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)]}= 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)]}(30)

= eγ [P (1)Jt(x− 1, γ) + (1− P (1))Jt(x, γ)],(31)

where (30) and (31) follow from the induction hypothesis and Lemma3.1.6 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)}= min{Jt(I, γ), eγ [P (1)Jt(I − 1, γ) + (1− P (1))Jt(I, γ)]}(32)

= min{J0(I, γ), eγ [P (1)J0(I − 1, γ) + (1− P (1))J0(I, γ)]}(33)

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

= J0(I)(34)

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Optimal resource allocation 65

where (32), (33) and (34) follow from (29), the induction hypothesis and(25) 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, 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)

follows from (25) since P (1) ≤ 12 =⇒ log P (1)

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 f∗

t (x), t =0, 1, . . . , N − 2, N − 1, are given by

(35) f∗t (x) =

!0 if x < 1

1 if x ! 1

if 12 < P (1) < 1 and 0 < γ ≤ log

"P (1)

1−P (1)

#, or if P (1) = 1 and γ > 0;

andf∗t (x) = 0, ∀x

if 12 < P (1) < 1 and γ ≥ log

"P (1)

1−P (1)

#, or if P (1) ≤ 1

2 and γ > 0.

Remark 4.1.15 Note that

a) if 12 < P (1) < 1 and γ ≥ log( P (1)

1−P (1)) then the preferences of theγ-decision maker differ from those of the risk-neutral decision maker:the γ-decision maker prefers the action a = 0, whereas the risk-neutraldecision 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 ac-tions a = 0 and a = 1; see the appendix.

5 Appendix

For the purpose of comparing the results obtained in Sections 3 and4 about structured optimal policies for an optimal allocation problem

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66 Avila-Godoy

(with ETC criterion), in this appendix we study the corresponding risk-neutral allocation problem. Section 5.1 summarizes some results aboutmonotonicity and convexity properties of the optimal value functionand monotonicity properties of the policies. For the proof of thoseresults we refer the reader to Ross [14]. In Section 5.2 we derive furtherstructural properties of the optimal policies under the assumptions thatthe probability function P (a) is convex and the final cost c(x) is strictlyincreasing.

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 remainto go and the state at time N − t is x. Note that Ft(x, a) is the functionwithin brackets in the (risk-neutral) dynamic programming algorithm

J0(x) = c(x)(37)...

...

Jt+1(x) = infa∈A(x)

!C(x, a) +

"

y

pxy(a)Jt(y)#.(38)

Let

At(x) := {a : Ft(x, a) = infa′{Ft(x, a

′)}}

and

ft(x) := min At(x).

Lemma 5.1.1 The optimal value function Jt(x) is increasing in x anddecreasing in t. Moreover, under Assumption 3.1, Jt(x) is convex in x.

Proposition 5.1.2 Under Assumption 3.1, π = (f0, . . . , fN−1) is anoptimal 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 increasingin t.

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Optimal resource allocation 67

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

Throughout this appendix, the policy π = (f0, . . . , fN−1) will denote themonotone optimal policy obtained in Proposition 5.1.2. In the followingproposition we will show that when the probability function P (a) isstrictly convex and the final cost c(x) is strictly increasing, the allocationmodel is reduced to a problem with two actions: the extreme points ofthe interval [0,M ]. Consequently, there exists an optimal thresholdpolicy.

Proposition 5.2.1 Assume that P (a) is strictly convex and twice dif-ferentiable and c(x) is strictly increasing. Then, under Assumption 3.1,the allocation optimal control problem (with total cost criterion) can bereduced to a problem with two actions: the extreme points of the inter-val [0,M ]. Moreover, the optimal policy π = (f0, f1, . . . fN−1) is of thethreshold-type, that is, there exist states x0, x1, . . . , xN−1 such that

(39) ft(x) =

!0 if x < xt

M if x ! xt,

t = 0, 1, . . . N − 1. Moreover, the sequence of thresholds is decreasing.

Proof: It follows from (36) that

(40) Ft(x, a) = a+ [Jt(x)− Jt(x− 1)](1− P (a)) + Jt(x− 1).

First, we will show by induction on t, that Jt(x) is strictly increasingin x. Since J0(x) = c(x), the result holds for t = 0. Now assume thatJt(x) < Jt(x+1). Then, from Lemma 5.1.1 and by using the inductionhypothesis 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) = infa∈[0,M ]

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

> infa∈[0,M ]

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

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68 Avila-Godoy

It follows from (40) that

∂Ft

∂a(x, a) = 1− P ′(a)[Jt(x)− Jt(x− 1)]

and∂2Ft

∂2a(x, a) = −P ′′(a)[Jt(x)− Jt(x− 1)].

Thus, since P ′′(a) > 0 and Jt(x) is strictly increasing in x we obtain

that ∂2Ft∂2a (x, a) < 0, and therefore Ft(x, a) is concave in a. Consequently,

At(x) = {0,M},

and hence, ft(x) ∈ {0,M}. Moreover, if we define

xt := min{x : ft(x) = M},

then (39) follows from the fact that ft(x) is increasing in x. Finally, thesequence {xt} is decreasing since ft(x) is increasing in t. ✷

Now, we will apply Proposition 5.2.1 to compute the optimal policyfor the example considered in Section 4.

Example 5.2.2 (revisited.) Take the example considered in Section 4with P (a) strictly convex. First we compute fN−1(x). To do that, byProposition 5.2.1, we need only to compare the values of the functionF0(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), x ≥ 1

= a+ P (a)(2x− 2) + (1− P (a))2x, x ≥ 1.

Thus,F0(x, 0) = 2x, x ≥ 1, and

F0(x, 1) = 2x+ (1− 2P (1)), x ≥ 1.

Thus, we obtain

a) if P (1) > 12 then

F0(x, 1) < F0(x, 0), x ≥ 1

b) if P (1) < 12 then

F0(x, 0) < F0(x, 1), x ≥ 1, and

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Optimal resource allocation 69

c) if P (1) = 12 then

F0(x, 1) = F0(x, 0), x ≥ 1.

Therefore the optimal decision rule fN−1 and the optimal value functionJ1 for the case (a) are given by

(41) fN−1(x) =

!0 if x < 1

1 if x ! 1,

and

(42) J1(x) =

!0 if x = 0

2x+ (1− 2P (1)) if x ≥ 1;

for the case (b) byfN−1(x) = 0, ∀x,

and

(43) J1(x) = 2x, x ≥ 0;

and for the case (c) we obtain that both actions a = 0 and a = 1 areoptimal.

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

I) If P (1) > 12 then for t = 1, . . . N − 1,

Jt(1) = 1 + (1− P (1))Jt−1(1);

II) If P (1) ≤ 12 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,

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)}(44)

= 1 + (1− P (1))Jt(1),(45)

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70 Avila-Godoy

where (44) and (45) follow from the induction hypothesis and Lemma5.1.1 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))}= 2I(48)

where (46), (47) and (48) follow from (36), the induction hypothesis andthe hypothesis P (1) < 1

2 respectively. Thus fN−t−1(I) = 0 and sincefN−t−1(x) is increasing in x we obtain that fN−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 ft(x), t = 0, 1, . . . , N−1,

are given by

ft(x) =

!0 if x < 1

1 if x ! 1

if P (1) > 12 ;

ft(x) = 0, ∀x

if P (1) < 12 ; and if P (1) = 1

2 then there are I + 1 threshold optimalpolicies:

fyt (x) =

!0 if x ≤ y

1 if x > y,

y = 0, 1, . . . I.

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Optimal resource allocation 71

AcknowledgementThis paper is part of the author’s doctoral research under the di-

rection of Dr. Emmanuel Fernandez Gaucherand at the Department ofMathematics of the University of Arizona. I want to thank him for hisguidance during the preparation of this work. I am specially grateful toDr. Onesimo Hernandez Lerma for his helpful comments and suggestionsin helping me see this article to its completion.

Guadalupe Avila GodoyDepartamento de Matematicas,Universidad de Sonora,Rosales y Boulevard Luis Encinas,Hermosillo, Sonora, [email protected]

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Optimal resource allocation 73

[19] Weber R. R.; Stidham S. Jr., Optimal Control of Service Rates inNetworks of Queues, Adv. Appl. Prob., 19 (1987), 202-218.

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Page 83: Morfismos, Vol 7, No 2, 2003

MORFISMOS, Comunicaciones Estudiantiles del Departamento de Matema-ticas del CINVESTAV, se termino de imprimir en el mes de noviembre de 2004en el taller de reproduccion del mismo departamento localizado en Av. IPN2508, Col. San Pedro Zacatenco, Mexico, D.F. 07300. El tiraje en papel opalinaimportada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares en pastatintoreto color verde.

Apoyo tecnico: Omar Hernandez Orozco.

Page 84: Morfismos, Vol 7, No 2, 2003

Contenido

When does a manifold admit a metric with positive scalar curvature?

aniloM-sorensiCsiuLesoJdnazenaY-arerraBoidigE . . . . . . . . . . . . . . . . . . . . . 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