morfismos, vol 8, no 2, 2004

VOLUMEN 8NÚMERO 2

JULIO A DICIEMBRE DE 2004 ISSN: 1870-6525

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VOLUMEN 8NÚMERO 2

JULIO A DICIEMBRE DE 2004ISSN: 1870-6525

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Contenido

Homotopy triangulations of a manifold triple

Rolando Jimenez and Yuri V. Muranov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

On information measures and prior distributions: a synthesis

Francisco Venegas-Martınez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

On a problem of Steinhaus concerning binary sequences

Shalom Eliahou and Delphine Hachez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Errata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Morfismos, Vol. 8, No. 2, 2004, pp. 1–25

Homotopy triangulations of a manifold triple ∗

Rolando Jimenez Yuri V. Muranov

Abstract

The set of homotopy triangulations of a given manifold fits into asurgery exact sequence which is the main tool for the classificationof manifolds. In the present paper we describe relations betweenhomotopy triangulations of different manifolds for a given man-ifold triple and its connection to surgery theory. We introducea group of obstructions to split a homotopy equivalence along apair of submanifolds and study its properties. The main resultsare given by commutative diagrams of exact sequences.

2000 Mathematics Subject Classification: 57R67, 57Q10, 19J25, 19G24,18F25.Keywords and phrases: surgery on manifolds, surgery and splittingobstruction groups, surgery exact sequence, the set of homotopy trian-gulations.

1 Introduction

Let Xn be a closed n–dimensional CAT (CAT = TOP, PL,Diff)manifold with fundamental group π = π1(X). A fundamental problemof geometric topology is to describe all closed n–dimensional CAT–manifolds which are homotopy (simple homotopy) equivalent to X.The main tool for such investigation is the surgery exact sequence (see[23, 20])

· · · → Ln+1(π) → Sn(X) → [X,G/CAT ]σ→ Ln(π) → · · · (1.1)

∗Invited article. Partially supported by Russian Foundation for FundamentalResearch Grant no. 02–01–00014, CONACyT, DGAPA–UNAM, Fulbright–GarcıaRobles and UW–Madison.

1


2 Rolando Jimenez and Yuri V. Muranov

The elements of the set [X,G/CAT ] are called normal invariants [23,§10]. In the present paper we shall work in the category of topo-logical manifolds (CAT = TOP ), and consider the groups L∗(π) =Ls∗(π) which give obstructions to simple homotopy equivalence. The

elements of Sn(X) = Ssn(X) are called homotopical triangulations or

s–triangulations of the manifold X (see [23, §10],[20]). The structureset Ss

n(X) is the set of s–cobordism classes of CAT–manifolds which aresimple homotopy equivalent to Xn (see [23, §10] and [20, p. 542]).

Let Y ⊂ X be a submanifold of codimension q in X. A simplehomotopy equivalence f : M → X splits along the submanifold Y ifit is homotopy equivalent to a map g transversal to Y , such that forN = g−1(Y ) the restrictions

g|N : N → Y, g|(M\N) : M \N → X \ Y

are simple homotopy equivalences. Let ∂U be the boundary of a tubu-lar neighborhood U of the submanifold Y in X. There exists a groupLSn−q(F ) of obstructions to splitting (see [23, 20]) which depends onlyon n− q mod 4 and a pushout square

F =

⎛

⎝π1(∂U) → π1(X \ Y )

↓ ↓π1(U) → π1(X)

⎞

⎠ (1.2)

of fundamental groups with orientations.

We consider the group LPn−q(F ) of obstructions to surgery on pairsof manifolds (X,Y ) as defined in [23, 20]. This group depends as wellonly on n− q mod 4 and the square F . In [20] Ranicki introduced a setSn+1(X,Y, ξ) of homotopy triangulations of a pair of manifolds (X,Y ),where ξ denotes the normal bundle of Y in X. This set consists of con-cordance classes of maps f : (M,N) → (X,Y ) which are splitted alongY and fits into a commutative braid of exact sequences ([20, Proposition7.2.6])

→ Sn+1(X,Y, ξ) −→ Hn(X,L•) −→ Ln(π1(X)) →

Sn+1(X) LPn−q(F )

→ Ln+1(π1(X)) −→ LSn−q(F ) → Sn(X,Y, ξ) →,(1.3)

where Hn(X,L•) ≃ [X,G/TOP ] and the spectrum L• is a one–connec-ted cover of the Ω–spectrum L(Z) with L•0 ≃ G/TOP [21] (see also[23, 20]).

Homotopy triangulations of a manifold triple 3

Consider a triple Zn−q−q′ ⊂ Y n−q ⊂ Xn of closed topological mani-folds. We shall suppose that every submanifold is locally flat in anambient manifold, and equipped with the structure of normal topolog-ical bundle (see [20, p. 562–563]). The groups LTn−q−q′(X,Y, Z) ofobstructions to surgery on manifold triples were recently introduced in[18]. The groups LT∗ are natural generalizations of the obstructiongroups LP∗ to surgery on manifold pairs.

The set Sn+1(X,Y, Z) of homotopy triangulations of the triple (X,Y, Z) is the natural generalization of the structure sets Sn+1(X) andSn+1(X,Y, ξ). This set fits in the exact sequence

· · · → LTk+1(X,Y, Z) → Sn+1(X,Y, Z) → Hn(X,L•) → LTk(X,Y, Z) → · · ·(1.4)

where k = n− q− q′. The relations between S∗(X,Y, Z) and S∗(X,Y, ξ)are given by the following braid of exact sequences [18]

→ Sn+1(X,Y, Z) −→ Hn(X,L•) −→ LPn−q(F ) →

Sn+1(X,Y, ξ) LTk(X,Y, Z)

→ LPn−q+1(F ) −→ LSk(Ψ) −→ Sn(X,Y, Z) →(1.5)

where Ψ is the square of fundamental groups in the splitting problemfor the pair (Y, Z). Remark that the map

Sn+1(X,Y, Z) −→ Sn+1(X,Y, ξ)

in (1.5) is a natural forgetful map.

We have the following topological normal bundles: ξ for the subman-ifold Y in X, η for the submanifold Z in Y , and ν for the submanifoldZ in X. Let Uξ be a space of normal bundle ξ. We shall suppose thatthe space Uν of the normal bundle ν is identified with the space Vξ ofrestriction of bundle ξ on a space Uη of normal bundle η in such waythat ∂Uν = ∂Uξ|Uη ∪ Uξ|∂Uη .

In the present paper we describe various relations between sets ofhomotopy triangulations S∗(X), S∗(Y ), S∗(Z), S∗(X,Y, ξ), S∗(X,Z, ν),S∗(Y, Z, η), and S∗(X,Y, Z) which arise naturally for a triple of embed-ded manifolds. The main results are given by commutative diagramsof exact sequences. We also introduce a group LSP∗ of obstructions tosplit a simple homotopy equivalence f : M → X along a pair of em-bedded submanifolds (Z ⊂ Y ) ⊂ X, and describe its relations to the


classical obstruction groups in surgery theory. The group LSP∗ is anatural straightforward generalization of the group LS∗ if we considerthe pair of submanifolds Z ⊂ Y instead of the submanifold Y .

2 Preliminaries

In this section we recall some definitions and results used in the paper(see [23, 20, 10, 11, 18, 21, 22, 8]).

The definition of a topological normal map (f, b) : M → X is givenin [20, p. 36] (see also [21, 19]). For n ≥ 5 the set of concordanceclasses of topological normal maps into a manifold X coincides withthe set [X,G/TOP ] ∼= Hn(X,L•) (see [20, 21, 19]). We shall use thedefinition of topological manifold pair (X,Y, ξ) as given in [20, §7.2].Here Y is the submanifold of codimension q for which the topologicalnormal bundle (Dq, Sq−1) → (E(ξ), S(ξ)) → Y is defined. A topologicalnormal map

((f, b), (g, c)) : (M,N, ξ1) → (X,Y, ξ)

with N = f−1(Y ) is defined in [20, §7.2]. For this map the restrictions(f, b)|N = (g, c) : N → Y and (f, b)|P = (h, d) : (P, S(ξ1)) → (Z, S(ξ))

are topological normal maps where P = M \ E(ξ1), Z = X \ E(ξ).Additionally, the restriction (h, d)|S(ξ1) : S(ξ1) → S(ξ) coincides with

the induced map (g, c)! : S(ξ1) → S(ξ), and we have (f, b) = (g, c)! ∪(h, d). It follows from [20, Proposition 7.2.3] that the set of concordanceclasses of normal maps to the pair (X,Y, ξ) coincides with the set ofconcordance classes of normal maps to the manifold X.

A normal map ((f, b), (g, c)) : (M,N) → (X,Y ) represents an ele-ment of Sn+1(X,Y, ξ) if the maps

f : M → X, g : N → Y, and h : (P, S(ν)) → (Z, S(ξ))

are s–triangulations (see [20, p. 571]). It follows from the definition ofs–triangulation of the pair (X,Y, ξ) that the forgetful maps

Sn+1(X,Y, ξ) → Sn+1(X), ((f, b), (g, c)) → (f, b);

Sn+1(X,Y, ξ) → Sn−q+1(Y ), ((f, b), (g, c)) → (g, c)

are well defined. In general, the map Sn+1(X,Y, ξ) → Sn+1(X) is notan epimorphism or a monomorphism [20, p. 571].


Recall that if (X,Y ) is a pair of topological spaces, equipped with anorientation homomorphism, then the relative groups S∗(X,Y ) are welldefined [20, p. 560]. These groups fit into the following exact sequences

· · · → Sn(Y ) → Sn(X) → Sn(X,Y ) → Sn−1(X,Y ) → · · · ,· · · → Hn(X,Y ;L•) → Ln(π1(Y ) → π1(X)) → Sn(X,Y ) → · · · .

(2.1)

For a manifold pair (X,Y ) the groups S∗(X,Y ) differ from the groupsS∗(X,Y, ξ).

A topological normal map

((f, b), (g, c), (h, d)) : (M,N,K) → (X,Y, Z)

of a triple of manifolds (X,Y, Z) is given by topological normal maps ofmanifold pairs ((f, b), (g, c)) : (M,N) → (X,Y ) and ((g, c), (h, d)) :(N,K) → (Y, Z). A topological normal map (f, b) ∈ [X,G/TOP ]with f : M → X (see [20, 18]) defines the topological normal map((f, b), (g, c), (h, d)) : (M,N,K) → (X,Y, Z) as follows from topologicaltransversality (see [20, Proposition 7.2.3]). A topological normal map

((f, b), (g, c), (h, d)) : (M,N,K) → (X,Y, Z)

is an s–triangulation of the triple (X,Y, Z) if the constituent normalmaps ((f, b), (g, c)) : (M,N) → (X,Y ), ((g, c), (h, d)) : (N,K) → (Y, Z),and ((f, b), (h, d)) : (M,K) → (X,Z) are s–triangulations. The set ofconcordance classes of s–triangulations of the triple (X,Y, Z) is denotedby Sn+1(X,Y, Z). As follows from [18] this set has a group structureand fits into the commutative braid of exact sequences (1.5).

In [18] the groups LT∗(X,Y, Z) and the map

Θ∗(f, b) : [X,G/TOP ] = Hn(X,L•) → LTn−q−q′(X,Y, Z)

are defined in such a way that the normal map (f, b) is normally bordantto an s–triangulation of the triple (X,Y, Z) if and only if Θ∗(f, b) = 0(for n− q − q′ ≥ 5).

Proposition 2.1 Suppose that a topological normal map

((f, b), (g, c), (h, d)) : (M,N,K) → (X,Y, Z)

gives s–triangulations of manifold pairs

((f, b), (g, c)) : (M,N) → (X,Y )


and((g, c), (h, d)) : (N,K) → (Y, Z).

Then the map((f, b), (h, d)) : (M,K) → (X,Z)

is an s–triangulation of the pair (X,Z).

Proof: The maps f : M → X and h : K → Z are simple homotopyequivalences by definition. Denote by Vξ the space of restriction of thebundle ξ on the space Uη. We can identify the space Vξ with the spaceUν of the bundle ν. The map g already splitted along the submanifoldZ. Hence its restriction is a simple homotopy equivalence on Y \ Zand, therefore, we have a simple homotopy equivalence f |H where His Uξ \ Uν . By definition the restriction of the map f gives a simple

homotopy equivalence on E = X \ Uξ. The map f |H∩E will be a simplehomotopy equivalence since g is a simple homotopy equivalence on Y \Zand on the boundary of the tubular neighborhood. Hence the mapf |H∪E will be a simple homotopy equivalence (see [8, §23]). We canidentify X \ Uν with H ∪ E and the proposition is proved. !

Proposition 2.2 For the triple Z ⊂ Y ⊂ X the natural forgetful mapsfit in the following commutative diagrams

Sn+1(X,Y, Z) → Sn+1(X,Y, ξ)↓ ↓

Sn−q+1(Y, Z, η) → Sn−q+1(Y ),

Sn+1(X,Y, Z) → Sn+1(X,Y, ξ)↓ ↓

Sn+1(X,Z, ν) → Sn+1(X),

andSn+1(X,Y, Z) → Sn+1(X,Z, ν)

↓ ↓Sn−q+1(Y, Z, η) → Sn−q−q′+1(Z).

Proof: The result follows from the definition of s–triangulation of tripleof manifolds. !

In the present paper we shall use realizations of the various groupsand natural maps in surgery theory on the spectra level (see [23, 20, 18,21, 1, 8]). The surgery exact sequence (1.1) in the topological category


is realized on the spectra level [21]. The commutative diagrams (1.3)and (1.5) are realized on the spectra level (see [20, 18]), too. We recallhere that transfer and induced maps of L–groups are realized on thespectra level. A homomorphism of oriented groups f : π → π′ inducesa cofibration of Ω–spectra [8]

L(π) −→ L(π′) −→ L(f),

where πn(L(π)) = Ln(π) and similarly for the other spectra. The ho-motopy long exact sequence of this cofibration gives a relative exactsequence of L∗–groups of the map f

. . . → Ln(π) → Ln(π′) → Ln(f) → Ln−1(π) → . . . .

Let p : E → X be a bundle over an n–dimensional manifold X whosefiber is an m–dimensional manifold Mm. Then the transfer map (see[23, 10, 11]) p∗ : Ln(π1(X)) → Ln+m(π1(E)) is well defined. This mapis realized on the spectra level by a map

p! : L(π1(X)) → Σ−mL(π1(E)).

For the pair of manifolds (X,Y ) consider a homotopy commutative di-agram of spectra (see [23, 20, 1])

L(π1(Y )) → Σ−qL(π1(∂U) → π1(Y )) → Σ−qL(π1(X \ Y ) → π1(X)) ↓ ↓

Σ1−qL(π1(∂U)) → Σ1−qL(π1(X \ Y )),(2.2)

where the left maps are the transfer maps on the spectra level. Thespectrum LS(F ) is defined as a homotopy cofiber of the map

ΣL(π1(Y )) → Σ−q−1L(π1(X \ Y ) → π1(X)),

and the spectrum LP (F ) is defined as a hmotopy cofiber of the map

Σ−1L(π1(Y )) → Σ−qL(π1(X \ Y ))

(see [1, 13, 17, 12, 8]). Then the homotopy groups of these spectracoincide with the splitting obstruction groups πn(LS(F )) ∼= LSn(F )and the surgery obstruction groups for the manifold pair πn(LP (F )) ∼=LPn(F ).

It follows from [20, 18, 21] that the sets of s–triangulations are re-alized on the spectra level. we shall denote by S(X) the corresponding


spectrum for the structure set Sn+1(X), and similarly for the otherstructure sets.

For the triple Zn−q−q′ ⊂ Y n−q ⊂ Xn denote by j the natural in-clusion (X \ Z, Y \ Z) → (X,Y ) of CW–pairs of codimension q (see[20, §7.2]). Let FZ be the square of fundamental groups for the pair(X \ Z, Y \ Z). Let W = (X \ Z). The map j induces the map ofsquares F → FZ and therefore a commutative diagram

......

↓ ↓. . . → LSn−q(FZ) → Ln−q(π1(Y \ Z)) →

↓ ↓. . . → LSn−q(F ) → Ln−q(π1(Y )) →

↓ ↓. . . → LNSk → Ln−q(π1(Y \ Z) → π1(Y ))

trrel→↓ ↓...

...

...↓

→ Ln(π1(X \ Y ) → π1(W )) → · · ·↓

→ Ln(π1(X \ Y ) → π1(X)) → · · ·↓

trrel→ Ln(π1(W ) → π1(X)) → · · ·↓...

(2.3)

where k = n−q−q′. Two right upper horizontal maps induce the map ofthe relative groups of two right upper vertical maps with relative groupsLNS∗ (see [7, 15]). Diagram (2.3) is realized on the spectra level.

3 Homotopy triangulations of a triple of mani-folds

In this section we describe relations between various structure sets whicharise naturally for a triple of manifolds. Denote by Φ the square offundamental groups in the splitting problem for the pair (X,Z). Recallhere that F , FZ , and Ψ denote the similar squares for the pairs (X,Y ),(X \ Z, Y \ Z), and (Y, Z), respectively.


Theorem 3.1 The natural forgetful maps of Proposition 2.2 fit in thefollowing commutative diagrams of exact sequences

→ Sn(X \ Y ) −→ Sn(X,Y, ξ) −→ LSk−1(Ψ) →

Sn(X,Y, Z) Sn−q(Y )

→ LSk(Ψ) −→ Sn−q(Y, Z, η) −→ Sn−1(X \ Y ) →,

(3.1)

......

...↓ ↓ ↓

· · · → LSn−q(FZ) → LSn−q(F ) → LNSk → · · ·↓ ↓ ↓

· · · → Sn(X,Y, Z) → Sn(X,Y, ξ) → LSk−1(Ψ) → · · · ,↓ ↓ ↓

· · · → Sn(X,Z, ν) → Sn(X) → LSk−1(Φ) → · · ·↓ ↓ ↓...

......

(3.2)

and

......

...↓ ↓ ↓

· · · → LSn−q(FZ) → Sn(X,Y, Z) → Sn(X,Z, ν) → · · ·↓ ↓ ↓

· · · → Sn−q(Y \ Z) → Sn−q(Y, Z, η) → Sk(Z) → · · · ,↓ ↓ ↓

· · · → Sn−1(X \ Z,X \ Y ) → Sn−1(X \ Y ) → Sn−1(X \ Z) → · · ·↓ ↓ ↓...

......

(3.3)

where k = n− q − q′. The diagrams are realized on the spectra level.

Proof: Diagram (3.1) was obtained in [18]. The groupHn(X,L•) mapsto the groups of a commutative diagram of forgetful maps

LTn−q−q′(X,Y, Z) → LPn−q(F )↓ ↓

LPn−q−q′(Φ) → Ln(π1(X)),(3.4)

which is realized on the spectra level (see [18, 15]). We obtain a commu-tative diagram in the form of pyramid with the group Hn(X,L•) in thetop. This diagram is realized on the spectra level. The cofibres of the


maps which correspond to the side edges give a homotopy commutativediagram of spectra of structure sets

S(X,Y, Z) → S(X,Y, ξ)↓ ↓

S(X,Z, ν) → S(X),

which realizes the second commutative diagram from Proposition 2.2.It is easy to see that diagram (3.2) follows from the diagram above anddiagrams (1.3), (2.3). Now consider a commutative diagram

Hn(X,L•) = Hn(X,L•)↓ ↓

Hn−q(Y,L•) → Hn−q−q′(Z,L•),(3.5)

in which the vertical maps and the lower horizontal map are given bycompositions of transfer maps and isomorphisms (see [20, p. 579])

Hn−q(Y,L•) ∼= Hn(X,X \ Y ;L•),Hn−q−q′(Z,L•) ∼= Hn(X,X \ Z;L•),Hn−q−q′(Z,L•) ∼= Hn−q(Y, Y \ Z;L•).

Consider a natural map of (3.5) to the commutative diagram of forgetfulmaps (see [18, 15])

LTn−q−q′(X,Y, Z) → LPn−q−q′(Φ)↓ ↓

LPn−q−q′(Ψ) → Ln−q−q′(π1(Z)).(3.6)

Diagrams (3.5) and (3.6) and the obtained maps between them arerealized on the spectra level. Cofibres give a homotopy commutativediagram of spectra of structure sets

S(X,Y, Z) → S(X,Z, ν)↓ ↓

ΣqS(Y, Z, η) → Σq+q′S(Z).(3.7)

Diagram (3.7) realizes on the spectra level commutative diagram (3.3).Now the result follows similarly to the previous case. !

Theorem 3.2 Let Φ and Ψ be the squares of fundamental groups con-cerning with a splitting problem for the manifold pairs (X,Z) and (Y, Z),


respectively. Then there exists a commutative braid of exact sequences

→ LNSk−1 −→ LSk−1(Ψ) → Sk−1(Z) →

Sl(Y, Y \ Z) LSk−1(Φ)

→ Sk(Z) −→ Sn(X,X \ Z) −→ LNSk−2 →,

(3.8)

where k = n− q − q′, l = n− q.

Proof: The transfer maps give isomorphisms

Hn−q−q′(Z,L•)∼= !!

∼= ""

Hn−q(Y, Y \ Z;L•)

∼=##

Hn(X,X \ Z;L•).

(3.9)

Consider a commutative triangle (see [7, 15])

Ln−q−q′(π1(Z)) !!

$$

Ln−q(π1(Y \ Z) → π1(Y ))

trrel

##Ln(π1(X \ Z) → π1(X)).

(3.10)

The right vertical map in (3.9) is a relative transfer map from (2.3)and the two other maps are compositions of transfer maps and mapsinduced by inclusions. For the pair (X,Y ) the corresponding map onthe spectra level is described in (2.2). Taking the obstruction to surgerywe obtain maps from (3.9) into (3.10) for which homotopy cofibers givea homotopy commutative triangle of the spectra of structure sets

Σq+q′S(Z) → ΣqS(Y, Y \ Z) ↓

S(X,X \ Z).(3.11)

The cofiber of horizontal map in (3.11) is Σk+1LS(Ψ) and the cofiber ofthe sloping map is Σk+1LS(Φ) as follows from [20, Proposition 7.2.6,ii].Thus we have a push–out square of spectra

ΣqS(Y, Y \ Z) → Σk+1LS(Ψ)↓ ↓

S(X,X \ Z) → Σk+1LS(Φ),(3.12)


where the right vertical maps fit into (3.2). Now the homotopy longexact sequences of maps in (3.12) give diagram (3.8). !

Theorem 3.3 There exists the following braid of exact sequences

→ Hk(Z,L•) −→ Ln(C → D) −→ LNSk−1 →

Ll(A → B) Sn(X,X \ Z)

→ LNSk −→ Sl(Y, Y \ Z) −→ Hk−1(Z,L•) →,

(3.13)

where k = n− q − q′, l = n− q,π1(Y \Z) = A,π1(Y ) = B,π1(X \Z) =C,π1(X) = D. Diagram (3.13) is realized on the spectra level.

Proof: The proof is similar to that of Theorem 3.2. It is necessary touse the definition of the map trrel : Ll(A → B) → Ln(C → D) from(2.3) and isomorphisms (3.9). !

Theorem 3.4 There exists a commutative diagram of exact sequences

......

...↓ ↓ ↓

· · · → Sl(Y \ Z) → Sn(X \ Z,X \ Y ) → LSl−1(FZ) → · · ·↓ ↓ ↓

· · · → Sl(Y ) → Sn(X,X \ Y ) → LSl−1(F ) → · · ·↓ ↓ ↓

· · · → Sl(Y, Y \ Z) → Sn(X,X \ Z) → LNSk−1 → · · · ,↓ ↓ ↓...

......

(3.14)

where l = n−q and k = n−q−q′. Diagram (3.14) is realized on spectralevel.

Proof: The proof is similar to that of Theorem 3.2. !

Theorem 3.5 There exists a commutative braid of exact sequences

→ Hn(X,L•) −→ LPk(Φ) −→ LSl−1(FZ) →

LTk(X,Y, Z) Sn(X,Z, ν)

→ LSl(FZ) −→ Sn(X,Y, Z) −→ Hn−1(X,L•) →,

(3.15)

where k = n − q − q′ and l = n − q. Diagram (3.15) is realized on thespectra level.


Proof: Consider the maps from Hn(X,L•) to the groups LTk(X,Y, Z)and LPk(Φ) obtained by taking the obstructions to surgery. We obtain acommutative triangle in which the third map is the natural forgetful mapLTk(X,Y, Z) → LPk(Φ). Now the result follows from the definitions ofthe groups Sn(X,Z, ν) and Sn(X,Y, Z) and from commutative diagram(3.2). !

Theorem 3.6 There exists a commutative diagram of exact sequences

......

...↓ ↓ ↓

· · · → Hn(X \ Y,L•) → Hn(X,L•) → Hn−q(Y,L•) → · · ·↓ ↓ ↓

· · · → Ln(π1(X \ Y )) → LTk(X,Y, Z) → LPk(Ψ) → · · ·↓ ↓ ↓

· · · → Sn(X \ Y ) → Sn(X,Y, Z) → Sl(Y, Z, η) → · · · ,↓ ↓ ↓...

......

(3.16)

where l = n − q and k = n − q − q′. Diagram (3.16) is realized on thespectra level.

Proof: The right upper square in (3.16) is commutative and it is re-alized on the spectra level by [20, 18]. Now diagram (3.16) is obtainedby considering the homotopy long exact sequences of maps from thissquare. !

4 Splitting a homotopy equivalence alonga submanifold pair

In this section we introduce the obstruction groups LSP∗ = LSP∗(X,Y, Z) for a triple of embedded manifolds Z ⊂ Y ⊂ Z. These groups fitinto an exact sequence

. . . → LSPn−q−q′ → LTn−q−q′(X,Y, Z) →

→ Ln(π1(X)) → LSPn−q−q′−1 → . . . .(4.1)

The groups LSP∗(X,Y, Z) are a natural straightforward generalizationof the splitting obstruction groups LS∗(F ) for the manifold X with a


submanifold Y to the case when the manifold X contains a pair of em-bedded submanifolds (Z ⊂ Y ) ⊂ X. Hence the groups LSP∗(X,Y, Z)are obstruction groups for doing surgery on the manifold pair (Y,Z)inside the manifold X. In particular, there is a natural forgetful mapLSP∗(X,Y, Z) → LP∗(Ψ) forgetting the ambient manifold X. We alsodescribe relations between the introduced groups and the sets of homo-topy triangulations which arise for the triple of manifolds.

Recall here that in [18] the spectrum LT (X,Y, Z) with homotopygroups LTn(X,Y, Z) = πn(LT (X,Y, Z)) is defined as a homotopy cofi-ber of the map

Σ−q′−1v : Σ−q′−1LP (F ) → LS(Ψ),

where Σ denotes the suspension functor. The map of homotopy groupsinduced by v coincides with the composition

LPn−q+1(F ) → Sn+1(X,Y, ξ) → Sn−q+1(Y ) → LSn−q−q′(Ψ), (4.2)

where the middle map is the natural forgetful map and the other mapsare described in (1.3). Hence, from the cofibration sequence of themap v, we obtain the map t : Σq+q′LT (X,Y, Z) → ΣqLP (F ). Thecomposition of the map t with the natural forgetful map LPn−q(F ) →Ln(π1(X)) on spectra level provides a map s : Σq+q′LT (X,Y, Z) →L(π1(X)). We define a spectrum LSP (X,Y, Z) as the spectrum fittingin the cofibration

LSP (X,Y, Z) → LT (X,Y, Z) → Σ−q−q′L(π1(X)). (4.3)

Let LSPn = LSPn(X,Y, Z) denote the homotopy group πn(LSP (X,Y,Z)). As follows from the definition, these groups fit into the long exactsequence in (4.1).

Theorem 4.1 There exists a commutative braid of exact sequences

→ Sn+1(X,Y, Z) −→ Hn(X,L•) −→ Ln(π1(X)) →

Sn+1(X) LTn−q−q′

α → Ln+1(π1(X)) −→ LSPn−q−q′ −→ Sn(X,Y, Z) →

(4.4)

which is realized on the spectra level.


Proof: The definition of the spectrum LT yields a homotopy commu-tative right square of a homotopy commutative diagram of spectra

Σ−1S(X) → X ∧ L• → L(π1(X))↓ ↓ ↓=

Σq+q′LSP → Σq+q′LT → L(π1(X))(4.5)

where the existence of the left vertical map follows from [22]. Thecofibers of the two horizontal maps of the left square in (4.5) coincide.Hence the left square is a pull–back square and the homotopy long exactsequences of this square give diagram (4.4). !

Commutative diagram (4.4) is a natural generalization of diagram(1.3) in the case of a triple of embedded manifolds. The left vertical mapin (4.5) induces a map α : Sn+1(X) → LSPn−q−q′(X,Y, Z) in (4.4). Thegeometric meaning of this map is explained in the following theorem.

Theorem 4.2 Let f : M → X be a simple homotopy equivalence whichrepresents an element of Sn+1(X). Then α(f) = 0 if and only if thehomotopy class of the map f contains an s–triangulation of the triple(X,Y, Z) (f splits along the pair Z ⊂ Y ).

Proof: Let the homotopy class of the map f contain an s–trian-gulation of the triple (X,Y, Z). Then the element f lies in the im-age of the forgetful map Sn+1(X,Y, Z) → Sn+1(X). The compositionSn+1(X,Y, Z) → Sn+1(X) → LSPn−q−q′ is trivial as follows from (4.4).Hence α(f) = 0. Conversely, let α(f) = 0. Then the same exact se-quence shows that f lies in the image Sn+1(X,Y, Z) → Sn+1(X), andthe result follows. !

Suppose that the pairs of manifolds (X,Y ) and (Y, Z) are Browder–Livesay pairs (see [3]). In this case the spectrum Σ2LT (X,Y, Z) co-incides with the third member of the filtration in the constructionof the surgery spectral sequence of Hambleton and Kharshiladze (see[18, 6]). Then the map rp : Ln(π1(X)) → LSPn−q−q′−1(X,Y, Z) from(4.1) is a natural generalization of the Browder–Livesay invariant r :Ln(π1(X)) → LSn−q−1(F ) = LNn−q−1(π1(X \Y ) → π1(X)) (see [3, 5]).Recall here (see [3]), that if r(x) = 0, then the element x ∈ Ln(π1(X))is not realized by a normal map of closed manifolds. In fact, the map rpgives an information which is equivalent to consider the first and secondBrowder–Livesay invariants (see [15, 6, 5]).


Proposition 4.3 For a triple of manifolds (X,Y, Z), let (X,Y ) and(Y, Z) be Browder–Livesay pairs. If rp(x) = 0, then the element x ∈Ln(π1(X)) is not realized by a normal map of closed manifolds.

Proof: The result follows from [15, Proposition 3] and [5]. !Recall here the diagram (see [18])

→ Ln(π1(X \ Y )) −→ LPn−q(F ) −→ LSk−1(Ψ) →

LTk Ln−q(π1(Y ))

→ LSk(Ψ) −→ LPk(Ψ) −→ Ln−1(π1(X \ Y )) →,

(4.6)

that gives the relations of LT∗–groups to splitting obstruction groupsand to surgery obstruction groups for manifold pairs. Diagram (4.6) isrealized on spectra level (see [18]).

The relations of LSP∗ to classical surgery obstruction groups for thetriple (X,Y, Z) is given by the following result.

Theorem 4.4 There exist braids of exact sequences

→ Ln(C) −→ Ln(D) −→ LSPk−1 →

LTk(X,Y, Z) Ln(C → D)

→ LSPk −→ LPk(Ψ) −→ Ln−1(C) →,

(4.7)

→ LSn−q(FZ) −→ LTk(X,Y, Z) −→ Ln(D) →

LSPk LPk(Φ)

→ Ln+1(D) −→ LSk(Φ) −→ LSn−q−1(FZ) →,

(4.8)

→ LSn−q(FZ) −→ LSn−q(F ) −→ LSk−1(Ψ) →

LSPk LNSk

→ LSk(Ψ) −→ LSk(Φ) −→ LSn−q−1(FZ) →,

(4.9)

and→ LSn−q+1(F ) −→ LSk(Ψ) −→ LTk →

LPn−q+1(F ) LSPk

→ LTk+1 −→ Ln+1(D) −→ LSn−q(F ) →,

(4.10)


where C = π1(X \Y ), D = π1(X), k = n−q−q′. Diagrams (4.7)–(4.10)are realized on the spectra level.

Proof: Consider a homotopy commutative diagram

LPn−q−q′(Ψ) → Ln(π1(X \ Y ) → π1(X))↓ ↓

Ln−1(π1(X \ Y ))=→ Ln−1(π1(X \ Y ))

in which the upper horizontal map and the left map are compositionsof the natural forgetful map LPn−q−q′(Ψ) → Ln−q(π1(Y )) and maps ofL–groups induced from (2.2). This diagram is realized on spectra leveland by [22] we obtain a map of cofibration sequences

LT (X,Y, Z) → LP (Ψ) →Σ−q−q′+1L(π1(X \ Y ))↓ ↓ ↓=

Σ−q−q′L(π1(X))→Σ−q−q′L(π1(X \ Y )→π1(X))→Σ−q−q′+1L(π1(X \ Y ))

where the left square is a pull–back square of spectra. The homotopylong exact sequences of this square provide the commutative braid ofexact sequences (4.7) if we use the definition of LSP∗–groups.

The natural forgetful maps (see [18, 15]) LTn−q−q′ → LPn−q−q′(Φ)

→ Ln(π1(X)) provide a map of cofibration sequence

LSP (X,Y, Z) → LT (X,Y, Z) → Σ−q−q′L(π1(X))↓ ↓ ↓=

LS(Φ) → LP (Φ) → Σ−q−q′L(π1(X)).

Now, similarly to the previous result, we obtain diagram (4.8), since themap LT∗ → LP∗(Φ) fits into (3.15).

Transfer maps and diagram (2.3) provide a map of the commutativediagram

LPn−q−q′(Ψ) → Ln−q(π1(Y ))↓ ↓

Ln−q−q′(π1(Z)) → Ln−q(π1(Y \ Z) → π1(Y ))(4.11)

to the commutative diagram

Ln(π1(X \ Y ) → π1(X))=→ Ln(π1(X \ Y ) → π1(X))

↓ ↓Ln(π1(X \ Z) → π1(X)) → Ln(π1(X \ Z) → π1(X)).

(4.12)


All these maps are realized on the spectra level. Hence, on the spectralevel, the cofibers of the maps from (4.11) to (4.12) provide a homotopycommutative diagram of spectra

LSP → Σ−q′LS(F )↓ ↓

LS(Φ) → LNS(4.13)

as follows from (2.3) and (4.6). The realizations on spectra level ofdiagrams (4.11) and (4.12) give pull–back squares. Hence the homotopycommutative square (4.13) is a pull–back and diagram (4.9) is obtainedfrom the homotopy long exact sequence of (4.13).

The natural forgetful maps LTn−q−q′ → LPn−q(F ) → Ln(π1(X))from (4.6) provide a homotopy commutative diagram of spectra

LT (X,Y, Z) → Σ−q′LP (F ) → ΣLS(Ψ)↓= ↓ ↓

LT (X,Y, Z) → Σ−q−q′L(π1(X)) → ΣLSP (X,Y, Z),(4.14)

where the rows are cofibrations and the right vertical map is defined by[22]. Hence the right square in (4.14) is a pull–back and its homotopylong exact sequences give diagram (4.10). !

Corollary 4.5 There exist exact sequences

· · · → LSPk → LSn−q(F ) → LSk−1(Ψ) → · · · ,

· · · → LSPk → LSk(Φ) → LSn−q−1(FZ) → · · · ,and

· · · → LSPk → LPk(Ψ) → Ln−1(π1(X \ Y ) → π1(X)) → · · · ,

in which the left maps are natural forgetful maps.

Now we describe some relations between the introduced groups LSP∗and various structure sets which arise for the triple (X,Y, Z).

Theorem 4.6 There exist braids of exact sequences

→ Sn(X) −→ LSPk−1 −→ Sl−1(Y, Z, η) →

Sn(X,X \ Y ) Sn−1(X,Y, Z)

→ Sl(Y, Z, η) −→ Sn−1(X \ Y ) −→ Sn−1(X) →,

(4.15)


→Hl(Y,L•) −→ Ln(π1(X \ Y ) → π1(X)) −→ LSPk−1 →

LPk(Ψ) Sn(X,X \ Y )

→ LSPk −→ Sl(Y, Z, η) −→ Hl−1(Y,L•) →,(4.16)

→ LSl(FZ) −→ Sn(X,Y, Z) −→ Sn(X) →

LSPk Sn(X,Z, ν)

→ Sn+1(X) −→ LSk(Φ) −→ LSl−1(FZ) →,

(4.17)

and

→ LSl+1(F ) −→ LSk(Ψ) −→ Sn(X,Y, Z) →

Sn+1(X,Y, ξ) LSPk

→ Sn+1(X,Y, Z) −→ Sn+1(X) −→ LSl(F ) →,

(4.18)

where l = n− q, k = n− q − q′. Diagrams (4.15)–(4.18) are realized onthe spectra level.

Proof: Transfer maps give a commutative diagram (see [20])

Hn−q(Y,L•)∼=→ Hn(X,X \ Y ;L•) ↓

Hn−1(X \ Y ;L•).

(4.19)

Consider the commutative triangle

LPn−q−q′(Ψ) → Ln(π1(X \ Y ) → π1(X)) ↓

Ln−1(π1(X \ Y ))(4.20)

which follows from the commutative diagram obtained in the proof ofTheorem 4.4.

The results of [20, Proposition 7.2.6] provide the map from (4.19)to (4.20). On the spectra level cofibres of this map give a homotopycommutative triangle of spectra of structure sets

S(Y, Z, η) → Σ−qS(X,X \ Y ) ↓

Σ−q+1S(X \ Y ).(4.21)


By [22] diagram (4.21) induces a map of cofibration sequences

S(Y, Z, η) → Σ−qS(X,X \ Y ) → Σq′+1LSP↓= ↓ ↓

S(Y, Z, η) → Σ−q+1S(X \ Y ) → Σ−q+1S(X,Y, Z).

where the left square is a pull–back square of spectra. The homotopylong exact sequences of this square provide the commutative braid ofexact sequences in (4.15). In a similar way the map from (4.19) to (4.20)provides a pull–back square

Σq′LP (Ψ) → Σ−qL(π1(X \ Y ) → π1(X))↓ ↓

S(Y, Z, η) → Σ−qS(X,X \ Y )

where the cofibers of the vertical maps are homotopy equivalent to thespectrum Y+∧L•. From this the braid of exact sequences (4.16) follows.The diagram (4.17) is obtained in a similar way if we consider on thespectra level the homotopy commutative triangle of the cofibers of themap from Hn(X,L•) to the triangle of natural forgetful maps

LTn−q−q′ → LPn−q−q′(Φ) ↓

Ln(π1(X)).(4.22)

We obtain diagram (4.18) in a similar way to the construction of diagram(4.17). To do this we have to consider the commutative triangle

LTn−q−q′ → LPn−q(F ) ↓

Ln(π1(X)).

So the proof is complete. !

5 Examples

Now we give examples how to compute some LSP–groups.

Consider the triple

(Z ⊂ Y ⊂ X) = (RPn ⊂ RPn+1 ⊂ RPn+2)


of real projective spaces with n ≥ 5.

The orientation homomorphism

w : π1(RPk) = Z/2 → ±1

is trivial for k odd and nontrivial for k even.

We have the following table for surgery obstruction groups (see [23,9])

n = 0 n = 1 n = 2 n = 3Ln(1) Z 0 Z/2 0

Ln(Z/2+) Z⊕ Z 0 Z/2 Z/2Ln(Z/2−) Z/2 0 Z/2 0

where superscript ”+” denotes the trivial orientation of the correspond-ing group and superscript ”−” denotes the nontrivial orientation.

We have two squares for codimension one splitting problems whichappear for different pairs RPk ⊂ RPk+1 of the considered triple.

We denote by

F± =

⎛

⎝1 → 1↓ ↓

Z/2∓ → Z/2±

⎞

⎠

the oriented square F of fundamental groups in accordance with theorientation ± of the ambient manifold.

We have the following isomorphisms (see [9, p. 15] and [23])

LSn(F+) = LNn(1 → Z/2+) = BLn+1(+) = Ln+2(1)

andLSn(F

−) = LNn(1 → Z/2−) = BLn+1(−) = Ln(1).

Now we recall intermediate computations of obstruction groupsLP∗(F±) and LT ∗(X,Y, Z) from [18].

The computation of LP∗–groups for a pair Y ⊂ X is based on thefollowing braid of exact sequences [23]

→ Ln+1(C) −→ Ln+1(D)∂−→ LSn−q(F ) →

LPn−q+1(F ) Ln+1(C→D)

→ LSn−q+1(F ) −→ Ln−q+1(B) −→ Ln(C) →

(4.23)


where A = π1(∂U), B = π1(Y ), C = π1(X \ Y ), and D = π1(X).

In the cases of squares F± we have q = 1, and the natural map thatforget the ambient manifold

LSn(F±) → Ln(Z/2∓)

coincides with the map

ln : BLn(±) → Ln−1(Z/2∓)

which is described in [9, p. 35].

Using this result and a diagram chasing in diagram (4.23) we obtainsurgery obstruction groups (see also [16])

LPn(F+) = LPn−1(F

−) = Z/2,Z/2,Z/2,Z

for n = 0, 1, 2, 3 (mod 4), respectively.

Now a diagram chasing in diagram (4.6) provides the following re-sults.

Proposition 5.1 [18] Let Mn−k be a closed simply connected topolog-ical manifold. For the triple of manifolds

(Zn ⊂ Y n+1 ⊂ Xn+2) = (Mn−k × RPk ⊂ Mn−k × RPk+1 ⊂ Mn−k × RPk+2)

with n ≥ 5 we have the following results.

For k odd the groups LTn are isomorphic to

Z⊕ Z/2,Z/2,Z⊕ Z/2,Z/2


For k even LT0∼= Z/2⊕ Z/2 and LT1

∼= Z/2. The groups LT3 andLT2 fit into the exact sequence

0 → LT3 → Z → Z → LT2 → Z/2 → 0.

Now we apply these results to compute the LSP∗–groups in the consid-ered cases.

Theorem 5.2 Under assumptions of Proposition 5.1 we have the fol-lowing:


For k odd the groups LSPn are isomorphic to

Z,Z,Z/2,Z/2


For k even we have isomorphisms LSP0∼= LSP1

∼= Z/2. The groupsLSP3 and LSP2 fit into the exact sequence

0 → LSP3 → Z → Z → LSP2 → 0.

Proof: Consider the case when k is odd. From diagram (4.6) in theconsidered case we conclude that all maps LTn → LPn+1(F+) areepimorphisms (see also [18]). Now it is easy to describe the mapsLPn(F+) → Ln+1(Z/2+) from diagram (4.23). For n = 1 mod 4 andn = 2 mod 4 these maps are isomorphisms Z/2 → Z/2 as follows con-sidering exact sequences lying in diagram (4.23). For n = 0 mod 4 themap is trivial since the group L1(Z/2+) is trivial.

The mapZ = LP3(F

+) → L0(Z/2+) = Z⊕ Z

is an inclusion on a direct summand. The image of this map coincideswith the image of the map L0(1) → L0(Z/2+) that is induced by theinclusion 1 → Z/2+. This follows from the commutative triangle

Z||

LP3(F+)∼=

L0(1)mono−→ L0(Z/2+)

|| ||Z Z⊕ Z

which lies in diagram (4.23). From diagram (4.7) we obtain an exactsequence

· · · → LTnτ→ Ln+2(Z/2+) → LSPn−1 → LTn−1 → · · · (4.24)

The map τ in (4.24) is a composition

LTn → LPn+1(F+) → Ln+2(Z/2+)

of maps that we already know.


From this we obtain that τ is trivial for n = 3, an isomorphismZ/2 → Z/2 for n = 1, an epimorphism Z⊕ Z/2 → Z/2 with a kernel Zfor n = 0, and a map Z⊕Z/2 → Z⊕Z with kernel Z/2 and cokernel Zfor n = 2. Now considering the exact sequence (4.24) we get the resultof the theorem for k odd. We get the result for k the even case in asimilar way. !

Dr. Rolando Jimenez,Instituto de Matematicas, UNAM,Unidad Cuernavaca,Av. Universidad S/N,Col. Lomas de Chamilpa,62210 Cuernavaca, Morelos, [email protected]

Prof. Yuri V. Muranov,Vitebsk State University,Moskovskii pr.33,210026 Vitebsk,[email protected];[email protected]

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[17] Muranov Yu. V.; Repovs D., Groups of obstructions to surgery andsplitting for a manifold pair, Sb. Math. 188 No. 3 (1997), 449–463.

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[20] Ranicki A. A., Exact Sequences in the Algebraic Theory of Surgery,Math. Notes 26, Princeton Univ. Press, Princeton, N. J., 1981.

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[23] Wall C. T. C., Surgery on Compact Manifolds, Academic Press,London–New York, 1970. (Second Edition, Mathematical Surveysand Monographs 69, A. A. Ranicki Editor, Amer. Math. Soc., Prov-idence, R. I., 1999.)


On information measures and prior distributions:a synthesis

Francisco Venegas-Martınez

Abstract

This paper suggests a new approach to reconciling, in a sys-tematic way, all inferential methods that maximize a specific crite-rion functional to produce non-informative and informative priors.In particular, Good’s (1968) Minimax Evidence Priors (MEP),Zellner’s (1971) Maximal Data Information Priors (MDIP) andBernardo’s (1979) Reference Priors (RP) are seen as special casesof maximizing a more general criterion functional. In a unifyingapproach Good-Bernardo-Zellner’s priors are introduced and ap-plied to a number of Bayesian inference problems, including theKalman filter and Normal linear model. Moreover, the paper fo-cuses, under plausible conditions, on the existence and uniquenessof the solutions of the derived optimization problems.

2000 Mathematics Subject Classification: 62F15, 49K20.Keywords and phrases: information measures, Bayesian inference.

1 Introduction

The distinctive task in Bayesian inference of deriving priors, in such away that the inferential content of the data is minimally affected in theposterior, has been of great interest for more than two centuries sincethe early work of Bayes (1763). More current approaches to this prob-lem, based on the maximization of a specific criterion functional, havebeen suggested by Good (1968), Zellner (1971) and Bernardo (1979),among others. It is also important to mention that recent literaturehas included inference procedures to provide a posterior without havinga prior, like the Bayesian method of moments (BMOM) introduced byZellner (1996) and (1998).

27

28 Francisco Venegas-Martınez

In Good’s (1968) principle of maximum invariantized negative cross-entropy, the minimax evidence method of deriving priors was presentedfor the first time. In this approach the initial density is taken as thesquare root of Fisher’s information. Zellner’s (1971) book introduceda method to obtain priors through the maximization of the total in-formation about the parameters provided by independent replicationsof an experiment (prior average information in the data minus the in-formation in the prior). In Bernardo (1979) a procedure was proposedto produce reference priors by maximizing the expected informationabout the parameters provided by independent replications of an ex-periment (average information in the posterior minus the informationin the prior). All of the above methods have comparative and absoluteadvantages in several respects and have been applied to a large numberof inference problems:

(i) While Zellner’s method is based on an exact finite sample criterionfunctional, Good’s approach uses a limiting criterion functional,and Bernardo’s procedure lies in asymptotic results. In Bernardo’sproposal a reference prior (posterior) is defined as the limit of asequence of priors (posteriors) that maximize finite-sample crite-ria. In a pragmatic approach in which results are most important,many reference prior algorithms have been developed. For in-stance, Berger, Bernardo and Mendoza (1989), and Berger andBernardo (1989), (1992a), (1992b), Bernardo and Smith (1994 ,ch. 5), and Bernardo and Ramon (1997).

(ii) The criterion functional used by Bernardo is a cross-entropy, whichsatisfies a number of remarkable properties, in particular, invari-ance with respect to one-to-one transformations of the parameters(Lindley 1956). In contrast, the total information functional em-ployed by Zellner is invariant only for the location-scale familyand under linear transformations of the parameters. To generateinvariance under other relevant transformations, not necessarilyone-to-one, side conditions could be needed, as suggested by Zell-ner (1971).

(iii) These methods have been tested by seeing how well they per-form in particular examples. The evaluation is often based oncontrasting the derived priors with Jeffreys’ (1961) priors, usu-ally improper. Even though improper priors can be associatedwith unbounded measures consistent with Renyi’s (1970) axioms

Information measures and prior distributions 29

on probability measures, some technical difficulties remain, see:Box and Tiao (1973), p. 314; Akaike (1978), p. 58; and Bergerand Bernardo (1992a), p. 37. It is also important to mention thatJeffreys’ priors can lead to singularities producing inadequate re-sults at certain values of the parameters; see Jeffreys (1967, p.359). Of course, if MEP, MDIP, and RP priors were to be used tocontrast the performance of other priors, the former priors couldalso produce unsatisfactory results under certain circumstances.

In this paper, we attempt to reconcile all inferential methods thatmaximize a criterion functional to produce non-informative and infor-mative priors. In our general approach, Good’s Minimax Evidence Pri-ors (1968 and 1969), Zellner’s Maximal Data Information Priors (1971,1977, 1991, 1993, 1995, 1996a and 1996b) and Bernardo’s Reference Pri-ors (1979 and 1997) are seen as special cases of maximizing a more gen-eral indexed criterion functional. Thus, properties of the derived priorswill depend on the choice of indexes from a wide range of possibilities,instead of on a few personal points of view with ad hoc modifications.In the spirit of Akaike (1978) and Smith (1979), we can say that thiswill look more like Mathematics than Psychology–without underesti-mating the importance of the latter in the Bayesian framework. Thisunified approach will enable us to explore a vast range of possibilities forconstructing priors. It is worthwhile to note that our general methodextends in a natural way Soofi’s (1994) pyramid by adding more ver-tices and including their convex hull. In any event, a good choice willdepend on the specific characteristics of the problem we are concernedwith. Needless to say, the chosen method should also provide goodpredictions.

This work is organized as follows. In section 2, we will introduce anindexed family of information functionals. In section 3, on the basis ofasymptotic normality, we will state a relationship between Bernardo’s(1979) criterion functional and some members of the indexed family. Insection 4, we will study a Bayesian inference problem associated withconvex combinations of relevant members of the proposed indexed fam-ily. Here, we will introduce Good-Bernardo-Zellner’s priors as well astheir controlled versions as solutions of maximizing discounted entropy.We will pay special attention to the existence and uniqueness of the so-lution of the corresponding optimization problems. In section 5, we willstudy Good-Bernardo-Zellner’s priors as Kalman Filtering priors. Insection 6, we examine the relationship between Good-Bernardo-Zellner’s


priors and the Normal linear model. Finally, in section 7, we will drawconclusions, acknowledge limitations, and make suggestions for furtherresearch.

2 An indexed family of information functionals

In this section, we define an indexed family of information functionalsand study some distinguished members. For the sake of simplicity, wewill remain in the single parameter case. The extension to the multi-dimensional parameter case will lead to conceptual complications. Thisis not surprising when dealing with information measures and priors;see Jeffreys (1961), Zellner (1971), Box and Tiao (1973), and Bergerand Bernardo (1992a).

Suppose that we wish to make inferences about an unknown param-eter θ ∈ Θ ⊆ R of a distribution Pθ, from which there is available anobservation, say, X. Assume that Pθ has density f(x|θ) (Radon-Nikodymderivative) with respect to some fixed dominating σ-finite measure λ onR for all θ ∈ Θ ⊆ R, that is, dPθ/dλ = f(x|θ) for all θ ∈ Θ ⊆ R, thusPθ(A) =

!A f(x|θ)dλ(x) for all Borel sets A ⊂ R.

The Bayesian approach is to assume that there is a prior density,π(θ), describing initial knowledge about the likelihood of the values ofthe parameter, θ. We will assume that π(θ) is a density with respectto some σ-finite measure µ on R. Once a prior distribution, π(θ), hasbeen prescribed, then the information provided by the data, x, aboutthe parameter is used to modify the initial knowledge, as expressed inπ(θ), via Bayes’ theorem to obtain a posterior distribution of θ, namely,f(θ|x) ∝ f(x|θ)π(θ) for every x ∈ R (using f generically to representdensities). The normalized posterior distribution is then used to makeinferences about θ. Let us define an infinite system of nested functionals:

Vγ,α,δ(π) =1

1− γ

"π(θ)G(I(θ),F(θ), γ,α, δ)dµ(θ)(1)

where

G(I(θ),F(θ), γ,α, δ)

= log

#exp[F(θ)/I(θ)]1−δ[I(θ)]

1−γ1+α − δ[I(θ)]1−α

π(θ)1−γ

$,


0 ≤ γ < 1, α ∈ 0, 1, δ ∈ 0, 1, and

I(θ) =! "

∂

∂θlog f(x|θ)

#2

f(x|θ)dλ(x)(2)

is Fisher’s information about θ provided by an observation X with den-sity f(x|θ), and

F(θ) =

!f(x|θ) log f(x|θ)dλ(x)(3)

is the negative Shannon’s information of f(x|θ), provided I(θ) and F(θ)exist. In the case that n independent observations of X are drawn fromPθ, say, (X1, X2, ..., Xn), then I(θ) and F(θ) will still stand for theaverage Fisher’s information and the average negative Shannon’s infor-mation of f(x|θ) respectively. It is not unsual to deal with indexed func-tionals in inference problems about a distribution, as in Good (1968). Itis worthwhile pointing out that for each triad (γ,α, δ) taking values in0 ≤ γ < 1, α ∈ 0, 1, δ ∈ 0, 1, then Vγ,α,δ(π) is a criterion functionalthat can be used to derive a prior π(θ), θ ∈ Θ, belonging to a feasibleset C. Usually, C is defined by constraints in terms of potential valuesof θ.

Note now that for the location parameter family f(x|θ) = f(x −θ), θ ∈ R, with the properties

![f ′(x)]2/f(x) dλ(x) < ∞

and !f(x) log f(x) dλ(x) < ∞,

where λ = µ stands for the Lebesgue measure, we have that both I(θ)and F(θ) are constant. Observe also that the scale parameter familyf(x|θ) = (1/θ)f(x/θ), θ > 0, with the above properties, satisfies thefollowing relationship:

F(θ) = 12 log I(θ) + constant.(4)

The indexed family in which we will be concerned with is given by:

A = conv[ Vγ,α,δ(π) ]

=convex hull of the closure of the familyVγ,α,δ(π).

We readily identify a number of distinguished members of A:


(i) Criterion for Maximum Entropy Priors (MAXENTP):

V0,0,1(π) = −!

π(θ) log π(θ)dµ(θ),

which is just Shannon’s information measure of a density π(θ),or Jaynes’ (1957) criterion functional to derive maximum entropypriors. Notice also that (3) can be rewritten in a simpler way asF(θ) = −V0,0,1(f(x|θ)).

(ii) Criterion for Minimax Evidence Priors (MEP):

V1,1,1(π)def= lim

γ→1Vγ,1,1(π) = −

!π(θ) log

π(θ)

p(θ)dµ(θ)− logC,(5)

which is Good’s invariantized negative cross-entropy, taking asinitial density p(θ) = C[I(θ)]

12 with C =

"[I(θ)]

12dµ(θ)−1, pro-

vided that"[I(θ)]

12dµ(θ) < ∞. We can also write (5) as

V1,1,1(π)− V0,0,1(π) =

!π(θ) log[I(θ)]

12dµ(θ).(6)

(iii) Criterion for Maximal Data Information Priors (MDIP):

V0,0,0(π) =

! !f(x)f(θ|x) log ℓ(θ|x)

π(θ)dµ(θ)dλ(x),(7)

which is Zellner’s criterion functional in his MDIP approach. Here,as usual,

f(θ|x) = f(x|θ)π(θ)f(x)

, f(x) =

!f(x|θ)π(θ)dµ(θ),

and ℓ(θ|x) = f(x|θ) is the likelihood function. An alternativeformulation of (7), which is often useful, is given by

V0,0,0(π)− V0,0,1(π) =

!π(θ)F(θ)dµ(θ).(8)

Some members of A define new criterion functionals in which theinformation provided by the sampling model, I(θ), plays a role:


(iv) Criterion for Maximal Modified Data Information Priors(MMDIP):

V0,1,0(π) =

! !f(x)f(θ|x) log [ℓ(θ|x)][I(θ)]

12

π(θ)dµ(θ)dλ(x),(9)

which is the prior average information in the data modified byFisher’s information minus the information in the prior. Note thatwhen I(θ) is constant, (9) reduces to Zellner’s criterion functional(up to a constant factor).

(v) Criterion for Maximal Fisher Information Priors (MFIP):

V0,1,1(π) = −!

π(θ) logπ(θ)

exp[I(θ)]12

dµ(θ)− 1,(10)

which is the prior average Fisher’s information minus the infor-mation in the prior.

3 Revisiting Bernardo’s reference priors

The maximization of Bernardo’s (1979) criterion is usually a difficultproblem to deal with. In order to get a simpler alternative procedureunder specific conditions, we will derive a useful asymptotic approxima-tion between Bernardo’s criterion functional (or Lindley’s informationmeasure, 1956) and some members of the class A. As stated in Bernardo(1979), the concept of reference prior is very general. However, in orderto keep the analysis tractable, we will restrict ourselves to the continu-ous one-dimensional parameter case.

Suppose that there are available n independent observations, say,(X1, X2, . . . , Xn), of a distribution Pθ, θ ∈ Θ ⊆ R. Accordingly, therandom vector (X1, X2, . . . , Xn) has density

dPθ/dν = f(ξ|θ) =n"

k=1

f(xk|θ),

for all ξ = (x1, x2, ..., xn) and all θ ∈ Θ ⊆ R, where

Pθ = Pθ ⊗ Pθ ⊗ · · ·⊗ Pθ# $% &n

and ν = λ⊗ λ⊗ · · ·⊗ λ# $% &n

.


Following Lindley (1956), a measure of the expected information aboutθ of a sampling model f(x|θ) provided by a random sample of size nwhen the prior distribution of θ is π(θ), is defined to be

L(n)(π) =

!f(ξ)

!f(θ|ξ) log f(θ|ξ)

π(θ)dµ(θ)dν(ξ).(11)

In order to obtain an asymptotic approximation of (11) in terms ofV1,1,1 and V0,0,1, we state a limit theorem which justifies the passage ofthe limit under the integral signs in (11). The theorem rules out thepossibility that the essentials of the statistical model, f(ξ|θ), changewhen samples grow in size. Let us rewrite (11) as:

L(n)(π) =Vγ,0,1(π) + log√n

−! !

log

"!Tn(ω)Wn(ω)dµ(ω)

#f(ξ|θ)π(θ)dν(ξ)dµ(θ),(12)

where

Tn(ω) =f(X1, X2, ..., Xn|θ + ω√

n)

f(X1, X2, ..., Xn|θ)(13)

and

Wn(ω) =π(θ + ω√

n)

π(θ).(14)

Throughout the paper, both λ and µ will stand for the Lebesgue measureon R. Also, we will assume that all densities involved are Lebesguemeasurable in both arguments, x and θ.

Theorem 3.1 Assume that the following conditions hold:

(I) Θ is an open interval in R;

(II) The function$

f(x|θ) is absolutely continuous on θ, and

x|f(x|θ) > 0

is independent of θ;

(III) If θ, θ′ ∈ Θ, then θ = θ′ implies λx|f(x|θ) = f(x|θ′) > 0;


(IV) ∂∂θ log f(x|θ) exists for all θ ∈ Θ and every x;

(V) I(θ) is a continuous and bounded function in Θ;

(VI) For all δ > 0, and all θ ∈ Θ

!

Bδ(ω√n)

"#f(x|θ + ω√

n)−

$f(x|θ)

%2dλ(x) = o( 1n),

where

Bδ(ω√n) = x : |

#f(x|θ + ω√

n)−

$f(x|θ) | > δ

$f(x|θ) ;

(VII) There exist c > 0 and τ > 0 such that

! &&π(θ + u)− π(θ)&&dµ(θ) ≤ c|u|τ ;

(VIII) For all ρ > 0

!

|ω|>nρ

"Tn(ω)Wn(ω)− Tn(ω)

%dµ(ω)

P−→0;

(IX) The sequence of random variables logUn∞n=1 where

Un =

!Tn(ω)Wn(ω)dµ(ω)

satisfies

limε→∞

supn≥1

!

| logUn|≥ε| logUn|dP = 0,

where

Pξ ∈ A, θ ∈ B =

!

Bπ(θ)

!

Af(ξ|θ)dν(ξ)dµ(θ),

for all A ∈ Rn and B ∈ Θ.

Then, as n → ∞,

L(n)(π)− V1,1,1(π) = −V0,0,1(ϕ) + logC√n+ o(1),(15)

where ϕ(z) is the density of Z ∼ N (0, 1), and C is taken as in (4).


Some comments are in order: (I)-(IV) are standard regularity con-ditions, (V) states desirable properties for I(θ), (VI) is a bounded vari-ance condition, (VII) is a smoothness condition, (VIII) is a convergencecondition, and (IX) says that the sequence logUn∞n=1 is uniformlyintegrable with respect to P . It can be shown that (I)-(VI) lead to

Tn(ω)L−→ exp

!ω"

I(θ)#Z − 1

2ω"

I(θ)$%

,(16)

where Z ∼ N (0, 1), and (16) along with (VII)-(IX) imply

logUn = log

&Tn(ω)Wn(ω)dµ(ω)

L−→ log"

2π/I(θ) + 12Z

2,

from where the conclusion of the theorem follows. Notice that the right-hand side of (3.5) is independent of π. Thus, if conditions (I)-(IX) arefulfilled, instead of maximizing L(∞)(π), which is usually a difficult prob-lem, we have as an alternative procedure maximizing V1,1,1(π), which isindependent of n. Notice that for maximization purposes the right-handside of (15) becomes a constant.

Finally, it is worthwhile to note that the location parameter fam-ily f(x|θ) = f(x − θ), with

"f(x) absolutely continuous on R, and'

[f ′(x)]2/f(x) dλ(x) < ∞, fully satisfies the conditions of Theorem3.1.

4 Good-Bernardo-Zellner priors

In this section we introduce Good-Bernardo-Zellner’s priors as solutionsof convex combination of relevant members of the class A. Very often,there exist priors for which entropy becomes infinite, specially whendealing with the non-informative case. In order to overcome this diffi-culty, we suggest the concept of discounted entropy. We also introduceGood-Bernardo-Zellner’s controlled priors as solutions of maximizingdiscounted entropy. We emphasize the existence and uniqueness of thesolutions of the corresponding variational and optimal control problems.

Throughout this section, we will be studying a number of Bayesianinferential problems related to convex combinations of distinctive ele-ments of A. Let

Mφ(π)def=φV1,1,1(π) + (1− φ)V0,0,0(π),

0 ≤ φ ≤ 1. Plainly, Mφ(π) ∈ A. To see that Mφ(π) is concave w.r.t.π, it is enough to observe, as in Zellner (1991), that

V0,0,0(π(θ)) = L(1)(π(θ)) + V0,0,1(π(θ))− V0,0,1(f(x)),


is a sum of concave functions w.r.t. π (up to the constant V0,0,1(f(x))).Since V1,1,1(π) is concave w.r.t. π, Mφ(π) is also concave w.r.t. π.Zellner (1996b) provides a criterion functional that agrees with Mφ(π)given by

Gφ[π(θ)]

=

! "φF(θ) + (1− φ) log[I(θ)]

12

#π(θ)dµ(θ)−

!π(θ) log π(θ)dµ(θ).

Indeed, from (5), (6) and (8), we get

Gφ[π(θ)] =φ (V0,0,0 − V0,0,1) + (1− φ) (V1,1,1 − V0,0,1) + V0,0,1

=φV0,0,0 + (1− φ)V1,1,1 − V0,0,1 + V0,0,1

=Mφ(π).

Usually, in the absence of data supplementary information, in termsof expectations about the parameter, comes from additional knowledgeof the experiment, or from the experience of the experimenter, namely,

!ak(θ)π(θ)dµ(θ) = ak, k = 1, 2, ..., s,(17)

where both the functions ak and the constants ak, k = 1, 2, ..., s, areknown. Hereafter, we will assume that (17) does not lead to any con-tradiction about π(θ). We will now concern with maximizing Mφ(π)subject to supplementary information.

Proposition 4.1 Consider the Good-Bernardo-Zellner problem:

Maximize Mφ(π) (with respect to π)

subject to C :

!ak(θ)π(θ)dµ(θ) = ak, k = 0, 1, 2, ..., s, a0 ≡ 1 = a0.

Then a necessary condition for a maximum is

π∗φ(θ) ∝ [I(θ)]

φ2 exp(1− φ)F(θ) +

s$

k=0

λkak(θ),(18)

where λk, k = 0, 1, ..., s, are the Lagrange multipliers associated withthe constraints C (cf. Zellner 1995).


Notice that when no supplementary information is available, π∗φ(θ)

is appropiate for an unprejudiced experimenter, otherwise it will besuitable for an informed experimenter who is in favor of C. Observealso that π∗

1(θ) is Good-Bernardo’s prior, and π∗0(θ) is Zellner’s prior.

Consider the binomial distribution for a single observation, f(x|θ) =

θx(1 − θ)1−x, 0 ≤ θ ≤ 1. In such a case, π∗1(θ) ∝ θ−

12 (1 − θ)−

12 and

π∗0(θ) ∝ θθ(1−θ)1−θ for θ ∈ [0, 1], which are quite different. Notice that

π∗1(θ) becomes infinite at θ = 0 and θ = 1. On the other hand π∗

0(θ)rises monotonically to 1.6186 at θ = 0 and θ = 1. Yet, another viewin this regard (Geisser, 1993) states that when the sample size is fairlylarge it does not matter which prior is employed, and the uniform priormay as well be used for θ.

Corolary 4.1 Consider the location and scale parameter families,

f(x|θ) = f(x− θ), θ ∈ R,

and

f(x|θ) = (1/θ)f(x/θ), θ > 0,

respectively, both satisfying

![f ′(x)]2/f(x) dλ(x) < ∞

and"f(x) log f(x) dλ(x) < ∞. Then, Good-Bernardo’s and Zellner’s

priors agree regardless of the value of φ ∈ (0, 1).

The proof of the above corollary for the scale parameter case followsfrom (4). It is important to point out that when there is no supplemen-tary information, we require µ(Θ) < ∞. Of course, the parameter spaceΘ can have bounds as large as needed to consider where the likelihoodfor θ is appreciable.

Notice that Proposition 4.1 can be used recursively when there ismore supplementary information to be added, say,

!ak(θ)π(θ)dµ(θ) = ak, k = s+ 1, s+ 2, ..., t.(19)

In such a case, in a cross-entropy formulation (Kullback 1959), we take(18) as the initial density, and (19) as the additional information. Hence,


π∗φ(θ) ∝ [I(θ)]

φ2 exp(1− φ)F(θ) +

s!

k=0

λkak(θ) expt!

k=s+1

λkak(θ)

= [I(θ)]φ2 exp(1− φ)F(θ) +

t!

k=0

λkak(θ) .

To deal with the (local) uniqueness of the solution of the problemstated in Proposition 4.1, we rewrite the constraints, C, as a function ofthe multipliers in the form A(Λ) = [

"ak(θ)π∗

φ(θ)dµ(θ)]sk=0 = A, where

AT = (a0, a1, ..., as), and ΛT = (λ0,λ1, ...,λs) (the superindex T denotesthe usual vector or matrix transposing operation).

Proposition 4.2 Let π∗φ(θ) be as in (4.2), and suppose that ak, k =

0, 1, ..., s, are linearly independent continuous functions in L2[Θ,π∗φdµ]

(the space of all π∗φdµ-measurable functions a(θ) defined on Θ such that

|a(θ)|2 is π∗φdµ-integrable). Suppose that A(Λ) is defined on an open set

∆ ⊂ Rs+1, and let Λo be a solution of A(Λ) = A for a fixed value ofA = Ao. Then, there exists a neighborhood of Λo, N(Λo), in which Λo

is the unique solution of A(Λ) = Ao in N(Λo).

The proof follows from the fact that A(Λ) is continuously differen-tiable on ∆, with nonsingular derivative

A′(Λ) = [

#aȷ(θ)aℓ(θ)π

∗φ(θ)dµ(θ)]0≤ȷ,ℓ≤s,

and from a straightforward application of inverse function theorem.From (4.1) we may derive the following necessary condition, which

is useful in practical situations.

Proposition 4.3 The multipliers ΛT = (λ0, λ1, ..., λs) appearing in(18) satisfy the following non-linear system of s+ 1 equations:

1 = λ0 + log

$#[I(θ)]

φ2 e(1−φ)F(θ)

s%

k=1

eλkak(θ)dµ(θ)

&,

1 = λ0 − log ak + log

$#ak(θ)[I(θ)]

φ2 e(1−φ)F(θ)

s%

u=1

eλuau(θ)dµ(θ)

&,

k = 1, 2, ..., s.

Moreover,


(i) if the integral in the first equality has a closed-form solution, thenthe rest of the multipliers can be found from the relations:

∂λ0

∂λk= ak, k = 1, 2, ..., s,

(ii) the formula

φV1,1,1(π∗φ) + (1− φ)[V0,0,0(π

∗φ)− 2V0,0,1(π

∗φ)] = 1−

s∑

k=0

λkak,

holds for all 0 ≤ φ ≤ 1.

Very often, experimenters are concerned with assigning weights ak,k = 1, 2, ..., s, to regions Ak, k = 1, 2, ..., s, to express, according toexperience, how likely it is that θ belongs to each region. The followingresult, based on Proposition 4.3, characterizes Good-Bernardo-Zellner’spriors when such a supplementary information comes in the form ofquantiles, and both I(θ) and F(θ) are constant. Under such assump-tions, the non-linear system of s+ 1 equations given in Proposition 4.3is transformed into a homogeneous linear system of the same dimensionas shown below:

Proposition 4.4 Suppose that the sets Ak = (bk, bk+1], k = 1, 2, ..., s−1 and As = (bs, bs+1) with b1 < b2 < · · · < bs+1, s ≥ 2, constitute apartition of Θ, 0 < µ(Θ) < ∞. Suppose also that both I(θ) and F(θ)are constant. Let a1, a2, ..., as > 0 be such that

∑sk=1 ak = 1, and∫

IAk(θ)π(θ)dµ(θ) = ak, k = 1, 2, ..., s. If we define new multipliers:

ω0 = e1−λ0/D where D = [I(θ)]φ2 e(1−φ)F(θ), and ωk = eλk , k =

1, 2, ..., s. Then, Ω = (ω0,ω1, ...,ωs) can be found from the followinghomogeneous linear system:

(20)

⎛

⎜⎜⎜⎜⎜⎝

−1 u1 u2 . . . us−1 v1 0 . . . 0−1 0 v2 . . . 0...

......

. . ....

−1 0 0 . . . vs

⎞

⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎝

ω0

ω1

ω2...ωs

⎞

⎟⎟⎟⎟⎟⎠=

⎛

⎜⎜⎜⎜⎜⎝

000...0

⎞

⎟⎟⎟⎟⎟⎠,

where uk = µ(Ak), and vk = a−1k uk, k = 1, 2, ..., s.


Observe that the determinant, ∆, of the matrix in (20) is given by

∆ =

!"sk=1 ak − 1#s

k=1 ak

$#sk=1 uk,

which guarantees that there exists a unique nontrivial solution since"sk=1 ak = 1. In this case, the solution is Ω∗T = (1, v−1

1 , v−12 , ..., v−1

s ),and π∗φ =

"sk=1 v

−1k IAk .

The following proposition extends Good-Bernardo-Zellner’s priorsto a richer family by using the MMDIP and MFIP criteria:

Proposition 4.5 Let

Nφ,ψ(π)def=φV1,1,1(π)+(1−φ)(1−ψ)V0,0,0(π)+(ψ(1−φ)/2)[V0,1,1+V0,1,0],

0 ≤ φ,ψ ≤ 1. Then

(i) Nφ,ψ(π) ∈ A and is concave w.r.t. π.

(ii) A necessary condition for π to be a maximum of the problem

Maximize Nφ,ψ(π)

subject to C :

%ak(θ)π(θ)dµ(θ) = ak, k = 0, 1, 2, ..., s,

where a0 ≡ 1 = a0, is given by

π∗φ,ψ(θ) ∝[I(θ)]φ2 exp

&(1− φ)(1− ψ)F(θ)

+ψ(1− φ)

2

'[I(θ)]

12 +

F(θ)

[I(θ)]12

(+

s)

k=0

λkak(θ)

*,(21)

where λk, k = 0, 1, ..., s, are the Lagrange multipliers associatedwith the constraints C.

The second term inside the exponential of (21) is the average be-tween Fisher’s information and the negative relative Shannon-Fisher’sinformation. Notice that π∗φ,0(θ) is just Good-Bernardo-Zellner’s prior.

In the following proposition, Good-Bernardo-Zellner type priors arederived as MAXENTP solutions by treating (5) and (8) as constraints(for the rationale of MAXENTP methods see Jaynes’ 1982 seminal pa-per).


Proposition 4.6 Consider the Jaynes–Good–Bernardo–Zellner prob-lem:

Maximize V0,0,1(π)

subject to:

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

V1,1,1(π)− V0,0,1(π) = b1,

V0,0,0(π)− V0,0,1(π) = b2,∫

ak(θ)π(θ)dµ(θ) = ak, k = 0, 1, 2, ..., s, a0 ≡ 1 = a0.

Then a necessary condition for a maximum is

π∗(θ) ∝ [I(θ)]ρ12 expρ2F(θ) +

s∑

k=0

λkak(θ),(22)

where ρj , j = 1, 2, and λk, k = 0, 1, ..., s, are the Lagrange multipliersassociated with the constraints.

Unlike the coefficients φ and 1−φ appearing in (4.6), the multipliersρj , j = 1, 2, do not necessarily add up to 1.

There typically exist priors for which Shannon-Jaynes entropy be-comes infinite. One way to overcome this problem consists of discount-ing entropy at a constant rate ν > 0. The following proposition intro-duces Good-Bernardo-Zellner’s controlled priors as solutions of maxi-mizing discounted entropy:

Proposition 4.7 Consider the discounted version of the problem statedin the preceding proposition:

Maximize −∫

e−νθπ(θ) log π(θ)dµ(θ),

subject to:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1π(θ)

dh1(θ)dµ(θ) = log[I(θ)]

12 , h1(−∞) = 0,

h1(∞) = V1,1,1(π)− V0,0,1(π) < ∞,

1π(θ)

dh2(θ)dµ(θ) = F(θ), h2(−∞) = 0,

h2(∞) = V0,0,0(π)− V0,0,1(π) < ∞,

1π(θ)

dgk(θ)dµ(θ) = ak(θ), gk(−∞) = 0, gk(∞) < ∞, k = 0, 1, 2, ..., s


where a0 ≡ 1 = a0. Then, a necessary condition for π∗(θ) to be anoptimal control is given by

π∗(θ) ∝ [I(θ)]ρ1(θ)

2 expρ2(θ)F(θ) +s!

k=0

λk(θ)ak(θ),(23)

where ρj(θ) = ρj0eνθ, j = 1, 2, and λk(θ) = λk0eνθ, k = 0, 1, ..., s,are the costate variables associated with the state variables hj(θ), j =1, 2, and gk(θ), k = 0, 1, ..., s, respectively. Furthermore, the constantsρj0, j = 1, 2, and λk0, k = 0, 1, ..., s, can be computed from the followingnon-linear system of s+ 3 equations:

1 + log h1(∞) = log

"#log[I(θ)]

12m(ρ10, ρ20,λ00,λ10, ...,λs0; θ)dµ(θ)

$,

1 + log h2(∞) = log

"#F(θ)m(ρ10, ρ20,λ00,λ10, ...,λs0; θ)dµ(θ)

$,

1 + log gk(∞) = log

"#ak(θ)m(ρ10, ρ20,λ00,λ10, ...,λs0; θ)dµ(θ)

$,

k = 0, 1, 2, ..., s

wherem(ρ10, ρ20,λ00,λ10, ...,λs0; θ)

=

%[I(θ)]

ρ102 eρ20F(θ) eλ00

&su=1 e

λu0au(θ)

'eνθ

.

5 Kalman filtering priors

In this section, we will study Good-Bernardo-Zellner’s priors as KalmanFiltering priors (Kalman 1960, and Kalman and Bucy 1961). We willcontinue to work with the single parameter case, and focus our attentionon both the location and scale parameter families.

Let Y1, Y2, ..., Yt be a set of indirect measurements, from a pollingsystem or a sample survey, of an unobserved state variable βt. Theobjective is to make inferences about βt. The relationship between Ytand βt is specified by the measurement equation, sometimes also calledthe observation equation:

(24) Yt = Atβt + εt,

where At = 0 is known, and εt is the observation error distributed asN (0,σ2

εt) with σ2εt known. Notice that the main difference between the


measurement equation and the linear model is that, in the former, thecoefficient βt changes with time. Furthermore, we suppose that βt isdriven by a first order autoregressive process, that is,

(25) βt = Ztβt−1 + ηt−1,

where Zt = 0 is known, and ηt ∼ N (0,σ2ηt) with σ2

ηt known. In whatfollows, we will assume that β0, εt, and ηt are independent randomvariables. We might state nonlinear versions of (24) and (25), but thiswould not make any essential differences in the subsequent analysis.

Suppose now, that at time t = 0, supplementary information is givenby β0 and σ2

0, the mean and variance of β0 respectively. That is,

(26) C :

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫ ∞

−∞π(β0)dβ0 = 1,

∫ ∞

−∞β0π(β0)dβ0 = β0,

∫ ∞

−∞(β0 − β0)

2π(β0)dβ0 = σ20.

In this case, Good-Bernardo-Zellner’s prior is given by

(27) π∗φ(β0) ∝ [I(β0)]

φ2 exp(1− φ)F(β0) + λ0 + λ1β0 + λ2(β0 − β0)

2,

where λj , j = 0, 1, 2, are Lagrange multipliers.Suppose that, at time t, we wish to make inferences about the condi-

tional state variable θt = βt|It, where It = Y1, Y2, ..., Yt−1. To obtaina posterior distribution of θt, the information provided by the measure-ment Yt, with density f(Yt|θt), is used to modify the initial knowledgein π∗

φ(θt) according to Bayes’ theorem:

(28) f(θt|Yt) ∝ f(Yt|θt)π∗φ(θt).

We are now in a position to state the Bayesian recursive updat-ing procedure of the Kalman Filter (KF) for both the location andscale parameter families f(Yt|θ) = f(Yt − θ), θ ∈ R, and f(Yt|θ) =(1/θ)f(Yt/θ), θ > 0, respectively. To start off the KF procedure, wesubstitute (27) in (26), obtaining that Good-Bernardo-Zellner’s priorat time t = 0, is given by N (β0, σ2

0), which is describing the initial

knowledge of the system. Proceeding inductively, at time t, βt−1 and


σ2t−1 become supplementary information, and therefore Good-Bernardo-

Zellner’s prior at time t is represented by

(29) θt = βt|It ∼ N (Ztβt−1,Mt),

where

(30) Mt = Z2t σ

2t−1 + σ2

ηt−1.

The sampling model (or likelihood function) is determined by

(31) Yt|θt ∼ N (Atβt,σ2εt).

The posterior distribution, at time t, is then obtained by substitutingboth (29) and (30) in (28), so

f(θt|Yt) ∝ exp−1

2 [(Atβt − Yt)2σ−2

εt + (βt − Ztβt−1)2M−1

t ].

Noting that π∗φ(θt) is a natural conjugate prior, we may complete the

squares to get

θt|Yt ∼ N[Ztβt−1 +Kt(Yt −AtZtβt−1),Mt −KtAtMt

],

where

(32) Kt = MtAt(σ2εt +A2

tMt)−1.

This, of course, means that

(33)

βt = Ztβt−1 +Kt(Yt −AtZtβt−1),σ2t = Mt −KtAtMt.

We then proceed with the next iteration. Equations (33), (30), and (32)are known in the literature as the KF.

The above analysis can be summarized in the following proposition:

Proposition 5.1 Consider the state-space representation:⎧⎨

⎩

Yt = Atβt + εt,

βt = Ztβt−1 + ηt−1,

defined as in (24) and (25). Suppose that supplementary informationon the mean and variance of β0 is available. Let θt = βt|It, where


It = Y1, Y2, ..., Yt−1, and consider the location and scale parameterfamilies f(Yt|θ) = f(Yt− θ), θ ∈ R, and f(Yt|θ) = (1/θ)f(Yt/θ), θ > 0,respectively, along with the properties stated in Corollary 4.1. Then,under Good-Bernardo-Zellner’s prior, π∗

φ(θt), the posterior estimate of

βt, !βt, is given by

!βt = ωtZt!βt−1 + (1− ωt)(Yt/At),

where ωt = σ2εt(σ

2εt +A2

tMt)−1.

6 Revisiting the normal linear model

The results on Good-Bernardo-Zellner priors given so far can be eas-ily extended to the multi-dimensional parameter case, namely, θ =(θ1, θ2, ..., θm) ∈ Θ ⊆ Rm, m > 1. Consider a vector of independent andidentically distributed normal random variables (X1, X2, ..., Xn) withcommon and known variance σ2 satisfying

(34) E(Xk) = ak1θ1 + ak2θ2 + · · ·+ akmθm, k = 1, 2, ..., n,

where A = (aij) is a matrix of known coefficients for which (ATA)−1

exists.Let X and θ stand for the column vectors of variables Xk and pa-

rameters θj , respectively. Then (34) can be written in matrix notationas, E(X) = Aθ. In this case, we have

(35) f(ξ|θ) = ( 12πσ2 )

n2 exp− 1

2σ2 ∥ξ −Aθ∥2,

where ξ = (x1, x2, ..., xn). Since σ2 has been assumed known, only thelocation parameter is unknown. The analogue of (2) is now given bythe matrix:

In(θ) ≡"# $

∂∂θȷ

log f(x|θ)%$

∂∂θℓ

log f(x|θ)%f(x|θ)dλ(x)

&

1≤ȷ,ℓ≤m

=1

σ2ATA,

and so det[In(θ)] is constant, which implies that the Good-Bernardo-Zellner prior distribution π∗

φ(θ), describing a situation of vague infor-mation on θ, must be a locally uniform prior distribution.


Let !θ be the least squares estimate for θ, then it is known thatATA!θ = ATX, E(!θ ) = θ, and Var(!θ ) = σ2(ATA)−1. Noting fromequation (35) that

f(ξ|θ) = ( 12πσ2 )

n2 exp− 1

2σ2 (∥ξ −A!θ ∥2 + ⟨ATA(θ − !θ ), θ − !θ ⟩),

and applying Bayes’ theorem, we get as the posterior distribution of θ

f(θ|ξ) = (2π)−m2 (det[ 1

σ2ATA])

12 exp

"−1

2

#1σ2A

TA(θ − !θ ), θ − !θ$%

.

If supplementary information in mean, c, and variance-covariancematrix, D, is now incorporated, then the (informative) Good-Bernardo-Zellner prior is given by

π∗φ(θ) = (2π)−

m2 (det[D])−

12 exp

"−1

2⟨D−1(θ − c), θ − c⟩

%.

The posterior distribution is now

f(θ|ξ) = (2π)−m2 (det[B])

12

× exp&−1

2⟨B[θ − ((DB)−1c+ 1σ2B

−1ATA!θ )],

θ − ((DB)−1c+ 1σ2B

−1ATA!θ )⟩',

where B = D−1 + 1σ2ATA.

7 Summary and conclusions

We have presented, in a unified framework, a number of well-knownmethods that maximize a criterion functional to obtain non-informativeand informative priors. Our general procedure is, by itself, capable ofdealing with a range of interesting issues in Bayesian analysis. However,in this paper, we have limited our attention to Good-Bernardo-Zellner’spriors as well as their application to some Bayesian inference problems,including the Kalman filter and the Normal linear model.

There exist priors for which Shannon-Jaynes entropy becomes infi-nite. In order to overcome this difficulty we proposed discounted en-tropy. We introduced Good-Bernardo-Zellner’s controlled priors whichmaximize discounted entropy at a constant rate. Throughout the paper,we have emphasized the existence and uniqueness of the solutions of thecorresponding variational and optimal control problems. There are, of


course, many other members of the class A that deserve much more at-tention than that we have attempted here. Needless to say, more workwill be required in this direction. Results will be reported elsewhere.

AcknowledgementThe author wishes to thank an anonymous referee for valuable guid-

ance and numerous suggestions. The author is indebted to Arnold Zell-ner, Jose M. Bernardo, Jim Berger, and George C. Tiao for helpfulcomments on earlier drafts of this paper. As usual, the author bearssole responsability for opinions and errors.

Francisco Venegas-MartınezDepartment of Finance,Tecnologico de Monterrey,Calle del Puente 222,14380 Mexico D. [email protected]

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On a problem of Steinhausconcerning binary sequences

Shalom Eliahou Delphine Hachez1

Abstract

A finite ±1 sequence X yields a binary triangle ∆X whose firstrow is X, and whose (k + 1)st row is the sequence of pairwiseproducts of consecutive entries of its kth row, for all k ≥ 1. Wesay that X is balanced if its derived triangle ∆X contains as many+1’s as −1’s. In 1963, Steinhaus asked whether there exist bal-anced binary sequences of every length n ≡ 0 or 3 mod 4. Whilethis problem has been solved in the affirmative by Harborth in1972, we present here a different solution. We do so by construct-ing strongly balanced binary sequences, i.e. binary sequences oflength n all of whose initial segments of length n − 4t are bal-anced, for 0 ≤ t ≤ n/4. Our strongly balanced sequences do occurin every length n ≡ 0 or 3 mod 4. Moreover, we provide a com-plete classification of sufficiently long strongly balanced binarysequences.

2000 Mathematics Subject Classification: 05A05, 05A15.Keywords and phrases: Steinhaus, balanced binary sequence, derivedsequence, derived triangle.

1 Introduction

Let X = (x1, x2, . . . , xn) be a binary sequence of length n, i.e. a se-quence with xi = ±1 for all i. We define the derived sequence ∂X of Xby ∂X = (y1, y2, . . . , yn−1) where yi = xixi+1 for all i. By convention,we agree that ∂X = ∅ whenever n = 0 or 1, where ∅ stands for the

1The present work is part of the PhD thesis work of the second author at the.uohailE.SrosseforPfonoitceridehtrednuelapO’detoClarottiLudetisrevinU

51

52 Eliahou and Hachez

empty binary sequence of length 0. More generally, for k ≥ 0, we shalldenote by ∂kX the kth derived sequence of X, defined recursively asusual by ∂0X = X and ∂kX = ∂(∂k−1X) for k ≥ 1.

We shall denote by ∆X the collection of the derived sequences X,∂X,. . . , ∂n−1X ofX. This collection may be pictured as a triangle, as inthe following example: ifX = (+1,+1,−1,+1,−1,+1,+1), abbreviatedas + +−+−++, then ∆X =

++−+−+++−−−−+−+++−−++−−+−−−+

We shall henceforth refer to ∆X as the derived triangle of X. If Y =(y1, . . . , ym) is any finite collection of numbers, we denote the sum ofits entries by S(Y ) =

!mi=1 yi. For instance, if X = (x1, x2, . . . , xn) is a

binary sequence, then S(∆X) represents the sum of the entries in thederived triangle ∆X of X, i.e. S(∆X) =

!n−1k=0 S(∂

kX).

Definition 1.1 A binary sequence X = (x1, x2, . . . , xn) is balanced ifS(∆X) = 0. In other words, X is balanced if its derived triangle ∆Xcontains as many +1’s as −1’s.

For example, the above binary sequence X = ++−+−++ is balanced,as its derived triangle contains 14 positive signs and 14 negative signs intotal. This sequence, as well as other balanced sequences of length 11,12, 19 and 20, appear in [3], where the author proposed the followingproblem.

Problem Is there a balanced binary sequence of length n for everyn ≡ 0 or 3 mod 4?

(The term “balanced” is not used by Steinhaus.) Note that the con-dition n ≡ 0 or 3 mod 4 is necessary for the existence of a balancedbinary sequence X of length n. Indeed, the derived triangle of X con-tains n(n + 1)/2 entries; if n ≡ 1 or 2 mod 4, this number of entries isodd, and therefore S(∆X) cannot vanish.

The above problem has been solved in the affirmative in [Harborth1972]. In this paper, we shall present a new solution to the problem of

Steinhaus’ binary sequences 53

Steinhaus, by constructing binary sequences satisfying a much strongercondition.

Definition 1.2 A binary sequence X = (x1, . . . , xn) is strongly bal-anced if the initial segment (x1, . . . , xn−4t) of X is balanced, for every0 ≤ t ≤ n/4.

Alternatively, strongly balanced sequences may be defined recursively,as follows. As initial conditions, balanced sequences of length 0 or 3 areconsidered as strongly balanced. For n ≥ 4, the sequence (x1, . . . , xn)is defined as strongly balanced if and only if it is balanced and (x1, . . . ,xn−4) is strongly balanced.

For instance, the above binary sequence X = + + − + − + + isstrongly balanced of length 7, as X and its initial segment of length3, namely + + −, are both balanced. Another example of a stronglybalanced binary sequence is given by P = + − + + − + + + + − −−,of length 12. Indeed, the initial segments of length 4, 8 and 12 of P ,namely + − ++, + − + + − + ++ and P itself, are all balanced aseasily seen. On the other hand, the sequences Y7 = + + + − + + −and Y8 = + + + + − + −− are both balanced, but not strongly so.Indeed, the initial segments of length 3 of Y7 and length 4 of Y8 areboth constant +1 sequences, and therefore cannot be balanced.

We shall denote by sb(n) the number of strongly balanced binarysequences of length n. There is no a priori reason to expect that stronglybalanced sequences should exist at all for n large. But fortunately, thetask of searching for all such sequences lends itself very well to computerexperimentation (see below).

The outcome of our experiments is quite surprising. Initially, thenumber sb(n) for n ≡ 0 mod 4 strictly increases, from n = 4 up ton = 36. Then it starts to decrease (non-strictly) up to length n = 92,where it finally stabilizes to the constant sb(n) = 4 for all n = 4m ≥ 92.For n ≡ 3 mod 4, the situation is similar, though more complicated:provided n ≥ 127, we find that sb(n) = 14 if n ≡ 3, 7 mod 12, andsb(n) = 12 if n ≡ 11 mod 12.

A convenient way to summarize the behavior of the numbers sb(n) isto exhibit properties of their generating function g(t) =

!∞n=0 sb(n)t

n.For example, the eventual periodicity of sb(n) for n large is reflected bythe property of the generating function g(t) of being a rational function.Our main result in this paper is the following


Theorem 1.3 The generating function g(t) =!∞

n=0 sb(n)tn of the

number sb(n) of strongly balanced binary sequences of length n is givenby the following rational function:

g(t) = 4t92/(1− t4) + f0(t) + (14 + 12t4 + 14t8)t127/(1− t12) + f3(t),

where f0(t) and f3(t)are the following polynomials:

f0(t) = 1 + 6t4 + 18t8 + 30t12 + 52t16 + 80t20 + 88t24 + 106t28

+ 116t32 + 124t36 + 106t40 + 92t44 + 92t48 + 90t52 + 64t56

+ 44t60 + 38t64 + 32t68 + 20t72 + 20t76 + 8t80 + 8t84 + 6t88,

f3(t) = 4t3 + 8t7 + 16t11 + 26t15 + 36t19 + 48t23 + 48t27 + 66t31

+ 88t35 + 108t39 + 114t43 + 90t47 + 88t51 + 104t55 + 92t59

+ 60t63 + 48t67 + 28t71 + 26t75 + 26t79 + 20t83 + 16t87

+ 18t91 + 14t95 + 14t99 + 14t103 + 14t107 + 16t111 + 14t115

+ 14t119 + 16t123.

In the above formula for g(t), the terms tn are separated according asn ≡ 0 or 3 mod 4, for better readability and because their behavior isdifferent.

Corollary 1.4 For every natural number n ≡ 0 or 3 mod 4, there existsa strongly balanced binary sequence of length n.

Proof: Consider first the case n ≡ 0 mod 4. By expanding the sum-mand 4t92/(1−t4) as 4t92+4t96+4t100+. . . in the formula for g(t), we seethat sb(n) = 4 for every n = 4m ≥ 92, as stated earlier. And the sum-mand f0(t) in g(t) gives the exact value of sb(n) for 0 ≤ n = 4m ≤ 88,which is nowhere zero. Similarly, for the case n ≡ 3 mod 4, we see thatsb(n) = 14 for every n ≡ 3 or 7 mod 12 with n ≥ 127, and sb(n) = 12for every n ≡ 11 mod 12 with n ≥ 131. This follows from expanding thesummand (14 + 12t4 + 14t8)t127/(1− t12) as an infinite series. Smallervalues of n are taken care of by the polynomial f3(t). For example,sb(51) = 88, sb(55) = 104 and sb(59) = 92. Alternatively, one may notethat, if there exists a strongly balanced binary sequence X of length n,then the initial segment of length n− 4 of X is also a strongly balancedbinary sequence. This follows directly from the definition. !

The set of all strongly balanced binary sequences of small length n(n ≤ 127, say) may be constructed by the method described in Section 3.


The eventual periodicity of sb(n) is a consequence of Theorems 2.1and 2.2 below.

2 A classification of long strongly balancedsequences

In this section, we shall describe the set of all strongly balanced binarysequences of length n ≥ 92 for n ≡ 0 mod 4, and n ≥ 127 for n ≡ 3mod 4. These two sets admit periodic structures. In order to presentthe results, we introduce the following notation.

Notation If P , Q are finite binary sequences, we shall denote by PQ∞

the infinite eventually periodic sequence which starts with P and con-tinues with Q repeated periodically thereafter. If R is yet another finitebinary sequence, and if k ∈ N, we shall denote by PQkR the sequencestarting with P , continuing with Q repeated k times, and ending withR. Finally, if T = (t1, . . . , tm, . . .) is any finite or infinite sequence oflength ≥ m, we shall denote by T [m] = (t1, . . . , tm) the initial segmentof length m of T .

2.1 The case n ≡ 0 mod 4

Let Q1, . . . , Q4 denote the following infinite eventually periodic binarysequences. We will show that every initial segmentQi[n] with n ≡ 0 mod4 is strongly balanced, and that there are no other strongly balancedbinary sequences of length n, provided n = 4m ≥ 92. These statementsare formalized in the next theorem.

Q1 = +−++ (+ +−++−+−−−++)∞,

Q2 = (+−++−++++−−−)∞,

Q3 = +−+− (+−−++++−−+++)∞,

Q4 = +−+− (−+−+++−+−+++)∞.

Theorem 2.1 For every n ≡ 0 mod 4, the initial segment of length nof each of Q1, Q2, Q3 and Q4 is a strongly balanced binary sequence.Conversely, every strongly balanced binary sequence of length n withn ≡ 0 mod 4 and n ≥ 92 is an initial segment of either of Q1, Q2, Q3

or Q4.

Parts of the proof of this result can be found in Section 6.


2.2 The case n ≡ 3 mod 4

This case is more complicated. Let R1, . . . , R12 denote the followinginfinite eventually periodic binary sequences. Their initial segments oflength n ≡ 3 mod 4 are all strongly balanced. Moreover, they accountfor all sufficiently long strongly balanced binary sequences, except fortwo more exotic ones in length n ≡ 3 mod 12 and n ≡ 7 mod 12. Forinstance, one of these extra sequences for n ≡ 3 mod 12 is R5[n− 4] +−+−, that is, the initial segment of length n− 4 of R5 appended withthe sequence +−+−.

R1 = ++−(+−++++−++++−)∞,

R2 = ++−−−−+(+ +−−+−+−+−−+)∞,

R3 = +−+(+ ++−+−++++−+)∞,

R4 = +−++++−(+−++−+−−−−++++−+−+−−−−−−)∞,

R5 = +−++++−(−++−++++−++−)∞,

R6 = +−+−+−−(+−+−+−−+++−−)∞,

R7 = +−+−+−−(+−+−−−−−−−+−+−−+−++−−−+−)∞,

R8 = +−+(−+−−−+−−++−+)∞,

R9 = −++(+ +−++++−+−++)∞,

R10 = −++++−+(−−+++−−+−+−+)∞,

R11 = −−−−−+−(+−−+++−−+−+−)∞,

R12 = −−−(−−+−−++++−−−)∞.

Theorem 2.2 Let n ≡ 3 mod 4. Then, the initial segment of length nof each of R1, . . . , R12 is a strongly balanced binary sequence. Moreover,if n ≥ 127, then every strongly balanced binary sequence of length n isan initial segment of one of R1, . . . , R12, with the following exceptions:

• If n ≡ 3 mod 12, there are two more strongly balanced binary se-quences of length n, namely R5[n−4]+−+− and R8[n−4]+−++.

• If n ≡ 7 mod 12, there are also two more strongly balanced binarysequences of length n, namely R8[n−8]+−++−+++, and eitherR5[n−8]+−+−−−−− if n ≡ 7 mod 24, or R5[n−8]+−+−−+−+if n ≡ 19 mod 24.


The proof is similar to that of Theorem 2.1. See the last comment inSection 6.

Even though Theorems 2.1 and 2.2 achieve the complete descriptionof all sufficiently long strongly balanced binary sequences, we shouldpoint out that there are other infinite families of (simply) balancedbinary sequences. For example, for all n ≡ 3 mod 4, the sequenceQ1[n]+ happens to be balanced. Similarly, for all n ≡ 8 mod 12, thesequence R1[n]+−−+ is balanced as well. And of course, there are thesequences in [2] which originally solved the problem of Steinhaus. Noneof the presently discussed sequences are strongly balanced, though.

3 The method

We shall explain now the method by which we have obtained the resultsabove, and shall also supply our specific Mathematica implementationof it.

The idea is quite simple. Assume X is a strongly balanced binarysequence of length n. An extension of X is any binary sequence Ycontaining X as an initial segment. Let Y be any one of the 16 possibleextensions of X of length n + 4. Then, Y is strongly balanced if andonly if Y is balanced. This holds because X itself is strongly balanced.

Consequently, if we know the set SB(n) of all strongly balancedbinary sequences of length n, and if card(SB(n)) = t, then in order toconstruct the set SB(n+ 4), it is enough to consider the 16t extensionsof length n+ 4 of all the elements in SB(n), and select those which aresimply (hence strongly) balanced. This is a computational task of lowcomplexity.

In summary, our method is a greedy algorithm, which aims to con-struct all strongly balanced sequences at increasing lengths. For lengthsdivisible by 4, the algorithm may start with the set ∅ of (strongly)balanced sequences of length 0. In length 3 mod 4, it will start with theset ++−,+−+,−++,−−− of all (strongly) balanced sequencesof length 3.

Here are the very concise Mathematica functions which we havewritten to implement the method. The first four functions (derive,triangle, weight and ext4) take as argument an arbitrary finite binary


sequence s, e.g. s = 1, 1,−1, 1 in Mathematica syntax.

1. The function derive[s] outputs the derived sequence ∂s of s,that is, the sequence of pairwise products of consecutive terms ins.

derive[s_] := Table[s[[i]]s[[i+1]], i, 1, Length[s] - 1]

2. Then, the function triangle[s] outputs the derived triangle ∆sof s, i.e. the list of all higher order derived sequences of s.

triangle[s_] := Block[s1, tri, s1 = s; tri = s1;

While[Length[s1] > 1, s1 = derive[s1];

AppendTo[tri, s1]]; tri]

3. The function weight[s] outputs the sum of the entries in thederived triangle ∆s of s.

weight[s_] := Apply[Plus, Flatten[triangle[s]]]

4. The function ext4[s] outputs the list of all balanced binary se-quences containing s as an initial segment and 4 units longer. Notethat, if s is strongly balanced, then ext4[s] outputs the list ofall strongly balanced sequences containing s as an initial segmentand 4 units longer.

ext4[s_] := Block[l, sext, l = ;

Do[sext = Join[s, x1, x2, x3, x4];

If[weight[sext] == 0, AppendTo[l, sext]],

x1, -1, 1, 2, x2, -1, 1, 2,

x3, -1, 1, 2, x4, -1, 1, 2];

l]

5. Finally, given a non-negative integer n ≡ 0 or 3 mod 4, the func-tion strong[n] successively builds all strongly balanced binarysequences of length m with m ≤ n and m ≡ n mod 4.

strong[n_] := strong[n] = (If[n == 0, Return[]];

If[n == 3,

Return[1, 1, -1, 1, -1, 1,

-1, 1, 1, -1, -1, -1]];

Flatten[Map[ext4, strong[n - 4]], 1])


For instance, the command Sum[Length[strong[n]]*t^n, n, 0,

88, 4] will output the polynomial f0(t) of Theorem 1.3, wheref0(t) =

!22i=0 sb(4i)t

4i displays the numbers sb(n) for each length n =4i ≤ 88. This computation takes about 90 seconds on a standard PCwith a Pentium 4m processor clocked at 1.6 GHZ.

4 Other possible strengthenings

We describe here two other attempts of strengthening the notion of bal-anced sequences. However, in contrast to strongly balanced sequences,these other strengthenings turn out to admit only finitely many com-plying binary sequences.

4.1 M-sequences

In our first attempt, we shall be seeking binary sequences X = (x1, . . . ,xn) having the property M defined recursively as follows: X is balanced,and its middle segment (x3, . . . , xn−2) of length n − 4 is also balancedand satisfies property M. By convention, balanced binary sequences oflength 0 or 3 satisfy property M. (Compare with the similar-lookingrecursive definition of strongly balanced sequences.) For brevity, se-quences satisfying property M will be called M-sequences.

We shall restrict our attention to lengths n ≡ 0 mod 4. As it turnsout, there are binary M-sequences of length n for every n = 4, 8, . . . , 96.In length 96, there remain exactly two binary M-sequences. Quite sur-prisingly, none of these two sequences can be extended to a sequenceof length 100 still satisfying property M. Consequently, there are nobinary M-sequences X of length n ≡ 0 mod 4 with n ≥ 100. Thus, thegenerating function gM (t) =

!X tl(X), where X runs over the set of all

balanced binary M-sequences of even length, and where l(X) denotesthe length of X, is a polynomial of degree 96, given by the followingexpression:

gM (t) = 2t96 + 8t92 + 10t88 + 14t84 + 22t80 + 22t76 + 30t72

+48t68 + 76t64 + 88t60 + 108t56 + 130t52 + 174t48

+226t44 + 222t40 + 198t36 + 172t32 + 144t28

+138t24 + 94t20 + 60t16 + 40t12 + 20t8 + 6t4 + 1.


For definiteness, here are the two binary M-sequences of length 96:

++++−−−−−+++−−−−+−++−+−−+−++−+−−++−−−−+−−−−−−+++−−+++++−++++−++−+−−++++−−−−−−−−−++−−−++−+−−+++++,

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

4.2 Universal balanced binary sequences

In our second attempt, we seek universal balanced binary sequences,i.e. balanced binary sequences X = (x1, . . . , xn) with the propertythat every initial segment (x1, . . . , xk), with k ≡ 0 or 3 mod 4, is alsobalanced. There are exactly 6 universal balanced binary sequences oflength 11, namely + − + + + + − + − + +, + − + + + + − − + + −,+−++++−−+−+, +−+−+−−+−+−, +−+−+−−−+++and +−+−+−−−+−−. As easily checked, by adding one more ±sign at the end of each of these 6 sequences, we find that there are nouniversal balanced binary sequences in length 12 or higher.

5 Related open problems

We propose here a few open problems in the same spirit as that ofSteinhaus.

Problem 1 Are there infinitely many symmetric balanced binary se-quences, such as X = ++−+−++ ? More generally, what is the set oflengths of all such sequences? For instance, it may be shown that thereexists no symmetric balanced binary sequences of length n ≡ 4 mod 8.

Problem 2 The balanced sequences X of length 12 and 20 given in[3] have the property that S(X) = 0, where S(X) is the sum of theentries in X. As a consequence, their derived sequences, of length 11and 19, respectively, are also balanced. It would be of great interest toknow, more generally, whether for every n divisible by 4, there exists abalanced binary sequence X of length n satisfying S(X) = 0. We did


find such sequences in every length n = 4k with n ≤ 36. However, wedo not know whether they exist in higher length. This problem wassuggested by Michel Kervaire during a phone conversation with one ofthe authors.

Problem 3 For every binary sequence X of length n ≡ 1 or 2 mod 4,the sum S(∆X) of the entries of the derived triangle ∆X of X is an oddnumber. It is natural to ask whether the value S(∆X) = 1 (respectivelyS(∆X) = −1) is attained for every n ≡ 1 or 2 mod 4. More generally,given any integer v, are there infinitely many finite binary sequences Xsuch that S(∆X) = v ? We know at least that the answer is positivefor v = −3,−2, 1, 2, 4 and 5, by taking suitable initial segments of someof the Qi and the Ri. The answer is also positive for v = −1, with thesequence Q1[n] +− for every n ≡ 11 mod 12. Still more generally, whatcan be said about the generating function Gn(t) =

!X tS(∆X), where

X runs over the set of all binary sequences of length n ?

Problem 4 The notion of balanced sequence makes sense not only withentries ±1, but more generally with entries taken in any (commutative)ring R. Indeed, let X = (x1, . . . , xn) be a sequence with entries xi ∈ Rfor all i. The derived sequence ∂X = (y1, . . . , yn−1) of X can still bedefined by yi = xixi+1 for all 1 ≤ i ≤ n − 1, and this gives rise againto the derived triangle ∆X of X, namely the collection of the ∂kX. Ofcourse, the sequence X is said to be balanced if the sum of the entries in∆X is 0 ∈ R.Are there interesting infinite families of balanced sequencesin this more general setting?

For instance, let p be a prime number, let ζ be a primitive pth rootof unity, and let R = Z[ζ]. In a forthcoming note, we shall show that, forp = 3, the ring R contains infinitely many balanced sequences of powersof ζ. We do not know whether this remains true for larger primes p.

The referee has suggested the following related problem. Let Gbe a finite group, even a non-abelian one. Are there infinitely manysequences X with entries in G whose derived triangle ∆X contains thesame number of occurrences of each group element?

Problem 5 This is really a family of problems. We may consider higher-dimensional analogues of balanced sequences, such as balanced binarymatrices, balanced binary 3-dimensional tensors, or balanced binarysimplices for example. In general, the concept of a balanced object X


will make sense whenever there is a suitable notion of derived objectX !→ ∂X, with strictly decreasing sizes. The derived object should beconstructed by taking the product of the neighbours for each suitableposition in X, as is the case for sequences. A given object X will thenbe said to be balanced whenever the sum of the entries in the collectionof its iterated derived objects ∂kX is zero.

Consider, for example, the following notion of a balanced binarysquare matrix. If A = (ai,j)1≤i,j≤n is a binary matrix of order n, define∂A as the binary matrix (bi,j)1≤i,j≤n−1 of order n − 1, where bi,j =ai,jai,j+1ai+1,jai+1,j+1. The derived pyramid ∆A is then defined as thecollection of ∂kA for 0 ≤ k ≤ n− 1. Note that again, the total numberof binary entries in ∆A is even if and only if n is congruent to 0 or 3mod 4. Are there infinitely many balanced binary matrices?

Problem 6 Let X be an arbitrary binary sequence of length n. Doesthere exist a balanced binary sequence Y havingX as an initial segment?(This problem is due to Pierre Duchet.)

For instance, let Jn be the constant +1 sequence of length n. Whatis the length j(n) of a shortest possible balanced binary extension of Jn,if one exists at all? We know by construction that j(100) ≤ 236.

6 Highlights of the proof of Theorem 2.1

We shall give here parts of the proof of Theorem 2.1. There are twothings to prove. First, that the initial segments Qi[n] are balanced,for every n ≡ 0 mod 4. And second, that there are no other stronglybalanced binary sequences of length n ≡ 0 mod 4, provided n ≥ 92.

We shall restrict our attention to Q1. (The phenomena are similarforQ2, Q3, Q4.) The fact that S(∆Q1[n]) = 0 for n ≡ 0 mod 4 will followfrom a certain periodic structure of the derived triangle ∆Q1[n]. Thisstructure then allows us to control which extensions of Q1[n] remainstrongly balanced, leading to the classification statement.

This is already quite tedious. Consequently, we shall not discussTheorem 2.2 concerning sequences of length n ≡ 3 mod 4. However,the phenomena are similar again, and it should become clear that acomplete proof can be written in this case as well.


Figure 1: Structure of the derived triangle of Q1[52].

Recall from subsection 2.1 that Q1 = + − + + (+ + − + + − + −−−++)∞. Let n ≡ 0 mod 4 be a given positive integer. We claim that∆Q1[n] has a periodic structure, as illustrated in Figure 1. More specif-ically, we will prove that, if n ≥ 16, there are nine types of NE/SWdiagonal strips of width 4, denoted A1, A2, A3, B1, B2, B3, C1, C2,C3, such that the derived triangle ∆Q1[n] is the periodic assembly ofT16 = ∆Q1[16] and of the components Ai, Bi, Ci, as depicted in Figure 1.Note that the components A1, B1, C1 appear on the top of the derivedtriangle, the components A3, B3, C3 on its SW side, and A2, B2, C2 oc-cupy the rest of the triangle (except T16). The sum of each componentis as indicated (e.g. A1 has sum S(A1) = 0, B1 has sum S(B1) = −4,and so on).

According to this structure of ∆Q1[n], we see that each full NE/SWdiagonal strip of width 4 on the right of T16 has sum zero, and therefore

T16

S = 0

A1 A1 A1

0 0 0

A3

A3

A3

0

0

0

A2 A2

A2

0 0

0

B1 B1 B1

-4 -4 -4

B3

B3

B3

4

4

4

B2 B2

B2

0 0

0

C1 C1 C1

-4 -4 -4

C3

C3

C3

4

4

4

C2 C2

C2

0 0

0

1


S(∆Q1[n]) = 0, as claimed.

In order to establish this structure, we need to introduce a few no-tations selecting certain specific parts of these NE/SW diagonal strips.

Notation.

• xqp denotes the pth digit in the qth row of∆Q1[n], for all 1 ≤ p ≤ nand 1 ≤ q ≤ n− p+ 1. In particular, the first row of ∆Q1[n], i.e.Q1[n] itself, is constituted by the elements x11, x

12, . . . , x

1n, and the

left side of the triangle∆Q1[n] consists of x11, x21, . . . , x

n1 . The basic

defining property of the triangle∆Q1[n] thus reads xq+1p = xqpx

qp+1.

• di denotes the ith NE/SW diagonal of ∆Q1[n], i.e. di is the rightside of the triangle ∆Q1[i], for all 1 ≤ i ≤ n;

• For i ≡ 1 mod 4 and j ≡ 1 mod 4, 1 ≤ j ≤ i, T ji denotes the

trapezoid of figure 3.

• For i ≡ 1 mod 4, we set Si = T ii . This special trapezoid Si

corresponds to the last four NE/SW diagonals of ∆Q1[i+ 3], andwill be called a strip.

• For i ≡ 1 mod 4 and j ≡ 2 mod 4, 2 ≤ j ≤ i, P ji denotes the

parallelogram of figure 2, of width 4 and length 12.

A few remarks are in order. First observe that, because of the basicproperty xq+1

p = xqpxqp+1, the trapezoid T j

i is completely determined by

its top row and its left side, namely by x1i , x1i+1, x

1i+2, x

1i+3 and x1i ,

x2i−1, . . . , xji+1−j . Now, this left side of T j

i is itself determined by x1iand by the right side of the adjacent trapezoid T j−4

i−4 . We record theseobservations as follows.

Fact 1 The trapezoid T ji is completely determined by its top row and

by the right side of T j−4i−4 .

Similar remarks can be made about the parallelogram P ji , and we

have:

Fact 2 The parallelogram P ji is completely determined by the bottom

of the quadrilateral just above it and by the right side of P j−4i−4 .

Finally, given i ≡ 1 mod 4, let j be the unique element in the set1, 5, 9 which is congruent to i mod 12. Clearly, with these notations,


xji

xj+1i−1 xj+1

i

xj+2i−2 xj+2

i−1 xj+2i

xj+3i−3 xj+3

i−2 xj+3i−1 xj+3

i

xj+4i−4 xj+4

i−3 xj+4i−2 xj+4

i−1. . . .. . . .. . . .

xj+11i−11 xj+11

i−10 xj+11i−9 xj+11

i−8

xj+12i−11 xj+12

i−10 xj+12i−9

xj+13i−11 xj+13

i−10

xj+14i−11

Figure 2: Parallelogram P ji .

x1i x1i+1 x1i+2 x1i+3

x2i−1 x2i x2i+1 x2i+2

x3i−2 x3i−1 x3i x3i+1. . . .. . . .. . . .

xji+1−j xji+2−j xji+3−j xji+4−j

xj+1i+1−j xj+1

i+2−j xj+1i+3−j

xj+2i+1−j xj+2

i+2−j

xj+3i+1−j

Figure 3: Trapezoid T ji .


the strip Si is the concatenation, in the NE/SW direction, of the trape-zoid T j

i and of the (i− j)/12 parallelograms P j+1i , P j+13

i , . . ., P i−11i .

We will denote the NE/SW concatenation by the symbol +. Withthis notation, we have Si = T j

i + P j+1i + P j+13

i + . . .+ P i−11i .

We now define the 9 special components Ai, Bi, Ci, where A1, B1, C1

are trapezoids, whereas A2, B2, C2, A3, B3, C3 are parallelograms:

∗ A1 := T 517 = + + − +

+ + − −+ + − +

− + − −− − − ++ + −+ −−

∗ A2 := P 629 = +

− ++ − +

+ − − −+ − + +

− − − ++ + + −

− + + −+ − + −

+ − − −+ − + +

− − − ++ + −+ −−

∗ A3 := P 617 = +

− ++ − +

+ − − −+ − + +

− − − ++ + + −

− + + −+ − + −

+ − − −+ − + +

+ − − +− + −− −+

∗ B1 := T 921 = + − + −

+ − − −− − + +

− + − ++ − − −

+ − + +− − − +

+ + + −− + + −− + −− −+


∗ B2 := P 1033 = +

+ −+ − +

− − − ++ + + −

− + + −+ − + −

+ − − −+ − + +

− − − ++ + + −

− + + −− + −− −+

∗ B3 := P 1021 = +

+ −+ − +

− − − ++ + + −

− + + −+ − + −

+ − − −+ − + +

− − − ++ + + −

+ + + −+ + −+ −−

∗ C1 := T 1325 = − − + +

+ + − +− + − −

− − − +− + + −

+ − + −+ − − −

+ − + +− − − +

+ + + −− + + −

+ − + −+ − − −− + +− +−

∗ C2 := P 1437 = +

− −+ + +

− + + −+ − + −

+ − − −+ − + +

− − − ++ + + −

− + + −+ − + −

+ − − −− + +− +−


∗ C3 := P 1425 = +

− −+ + +

− + + −+ − + −

+ − − −+ − + +

− − − ++ + + −

− + + −+ − + −

− − − −+ + ++ ++

We shall need to observe some resemblances between some of thesecomponents, to be used with Facts 1 and 2.

• The SW edge of A1 (respectively B1, C1) is equal to the SW edgeof A2 (respectively B2, C2).

• The 12-tuple composed by the last 12 digits of the right side ofC1 is equal to the 12-tuple containing the digits of the right sideof C2.

We claim that the strips Si come in 3 different types, depending on theclass i ≡ 1, 5 or 9 mod 12. Here is the general key formula we want toprove:

Claim 1

∀k ∈ N, k ≥ 1, S12k+5 = A1 + (k − 1)A2 +A3,S12k+9 = B1 + (k − 1)B2 +B3,

S12(k+1)+1 = C1 + (k − 1)C2 + C3.

As we will see, this results from the structure of the 9 componentsAi, Bi, Ci and Facts 1 and 2, and may be proved by induction on k.

To start the induction, one verifies the claim in ∆Q1[40] by directobservation.


Assume now that the claim is true for k = 1, 2. In particular, weknow that S37 = C1+C2+C3. We will show that S41 = A1+2A2+A3.By periodicity of the sequence Q1, we know that the top of S41 is equalto the top of A1. Thus, using Fact 1, we derive that the trapezoid T 17

41 isequal to A1 +A2. Indeed, it is completely determined by the top of A1

and the right side of C1 (by the hypothesis for S37), and the same is truefor A1 + A2 in S29, by the hypothesis for S29. Thus, the parallelogramjust under A1 + A2 in S41 is completely determined by the bottom ofA2 and the right side of C2, which is equal to the last 12 digits of theright side of C1. According to the verifications we have just made forthe previous trapezoid, the same holds for A2, whence Fact 2 implies:T 2941 = A1 +A2 +A2.

Finally, similar arguments enable us to show that the last parallelo-gram of S41 is equal to the last parallelogram of S29, namely A3. Hencewe get S41 = A1 + 2A2 +A3, and we are done.

The case k ≥ 3 can be treated in the same way, by induction.

Claim 2 ∀n ≡ 1 mod 4, S(Sn) = 0.

Using Claim 1, it suffices to compute the sum of each of the 9 com-ponents Ai, Bi, Ci and of the first irregular strips.

For every n ≤ 37, we check the equality by direct computations ofsums in the triangle ∆Q1[40].

For n ≥ 41, we have to consider three possibilities, according toClaim 1:

• If n = 12k + 1, k ≥ 2, then

S(Sn) = S(C1) + (k − 2)S(C2) + S(C3)

= −4 + (k − 2)× 0 + 4

= 0 ;

• if n = 12k + 5, k ≥ 2, then

S(Sn) = S(A1) + (k − 1)S(A2) + S(A3)

= 0 + (k − 1)× 0 + 0

= 0 ;

• if n = 12k + 9, k ≥ 2, then

S(Sn) = S(B1) + (k − 1)S(B2) + S(B3)


= −4 + (k − 1)× 0 + 4

= 0 .

This proves Claim 2. It follows that S(∆Q1[n]) = 0, i.e. that Q1[n] isbalanced, for every n ≡ 0 mod 4.

We now turn to the proof of the second part of Theorem 2.1, namelythat every strongly balanced binary sequence of length n with n = 4m ≥92 is equal to Qi[n] for some 1 ≤ i ≤ 4.

We do this by induction on n, starting at n = 92. In order toconstruct all strongly balanced binary sequences of length 92, we usethe method of Section 3, implemented in the given Mathematica func-tions. For example, issuing the command strong[92] to Mathematicawill output exactly four sequences, namely Q1[92], Q2[92], Q3[92] andQ4[92]. This computation uses exact integer arithmetic only. This es-tablishes the case n = 92.

Let n ≥ 92 with n ≡ 0 mod 4. It remains to show that, if X = Qi[n]for some i ∈ 1, 2, 3, 4, then there is a unique extension X ′ of X, oflength n + 4, such that X ′ is (simply, hence strongly) balanced, andX ′ = Qi[n+4]. (In fact, this statement already holds true for n ≥ 52 ifi = 1 or 3, and for n ≥ 64 if i = 2 or 4.)

Once again, we restrict our attention to Q1, so X = Q1[n]. Wedenote an arbitrary extension of length n+4 of X as the concatenationY = Y (x1, x2, x3, x4) = Xx1x2x3x4, where x1, x2, x3, x4 are unknownbinary digits satisfying x2i = 1. Our task is to determine those values ofxi ∈ ±1 for which S(∆Y ) = 0.

In order to do this, we need to determine the structure of the derivedtriangle ∆Y (x1, x2, x3, x4) in terms of the unknown x1, x2, x3, x4.

Claim 3 For every n ∈ N, n ≡ 0 mod 4, the last strip Sn+1(x1, x2, x3,x4) of the triangle ∆(Q1[n]x1x2x3x4) has the following structure:

Sn+1(x1, x2, x3, x4) =

⎧⎨

⎩

C ′1 + (k − 2)C ′

2 + C ′3 if n = 12k

A′1 + (k − 1)A′

2 +A′3 if n = 12k + 4

B′1 + (k − 1)B′

2 +B′3 if n = 12k + 8,

where A′1, B

′1, C

′1 are trapezoids and A′

2, B′2, C

′2, A

′3, B

′3, C

′3 parallel-

ograms. These components have the same size as the correspondingcomponents Ai, Bi, Ci, and are depicted below.


Not surprisingly, A′i, B

′i, C

′i share similar properties as Ai, Bi, Ci, i.e.

the bottom of A′1 (respectively B′

1, C′1) is equal to the bottom of A′

2

(respectively B′2, C

′2), and the 12-tuple composed by the last 12 digits

of the right side of C ′1 is equal to the 12-tuple containing the digits of

the right side of C ′2. Thus, the proof of Claim 3 is similar to that of

Claim 1. Here are A′i, B

′i, C

′i, explicitly.

∗ A′1 = x1 x2 x3 x4

x1 x1x2 x2x3 x3x4x1 x2 x1x3 x2x4

−x1 x1x2 x1x2x3 x1x2x3x4−x1 − x2 x3 x4x1 x1x2 −x2x3 x3x4

−x1 x2 −x1x3 −x2x4x1 −x1x2 −x1x2x3 x1x2x3x4

x1 − x2 x3 − x4x1 −x1x2 −x2x3 −x3x4

−x1 − x2 x1x3 x2x4x1 x1x2 −x1x2x3 x1x2x3x4

−x1 x2 − x3 − x4x1 −x1x2 −x2x3 x3x4

x1 − x2 x1x3 −x2x4x1 −x1x2 −x1x2x3 −x1x2x3x4

−x1 − x2 x3 x4x1x2 −x2x3 x3x4

−x1x3 −x2x4x1x2x3x4


∗ A′2 = x1

−x1 x2x1 −x1x2 −x1x2x3

x1 − x2 x3 − x4x1 −x1x2 −x2x3 −x3x4


−x1 x2 − x3 − x4x1 −x1x2 −x2x3 x3x4


−x1 − x2 x3 x4x1x2 −x2x3 x3x4


∗ A′3 = x1

−x1 x2x1 −x1x2 −x1x2x3

x1 − x2 x3 − x4x1 −x1x2 −x2x3 −x3x4


−x1 x2 − x3 − x4x1 −x1x2 −x2x3 x3x4


x1 − x2 x3 x4−x1x2 −x2x3 x3x4

x1x3 −x2x4−x1x2x3x4


∗ B′1 = x1 x2 x3 x4

x1 x1x2 x2x3 x3x4−x1 x2 x1x3 x2x4

−x1 −x1x2 x1x2x3 x1x2x3x4x1 x2 − x3 x4

x1 x1x2 −x2x3 −x3x4−x1 x2 −x1x3 x2x4x1 −x1x2 −x1x2x3 −x1x2x3x4

−x1 − x2 x3 x4x1x2 −x2x3 x3x4

−x1x3 − x2x4x1x2x3x4

∗ B′2 = x1

x1 x2x1 x1x2 −x1x2x3

−x1 x2 −x3 −x4x1 −x1x2 −x2x3 x3x4

−x1 − x2 x1x3 −x2x4x1 x1x2 −x1x2x3 −x1x2x3x4

x1 x2 − x3 x4x1 x1x2 −x2x3 −x3x4

−x1 x2 −x1x3 x2x4x1 −x1x2 −x1x2x3 −x1x2x3x4

−x1 − x2 x3 x4x1x2 −x2x3 x3x4



∗ B′3 = x1

x1 x2x1 x1x2 −x1x2x3

−x1 x2 −x3 −x4x1 −x1x2 −x2x3 x3x4

−x1 − x2 x1x3 −x2x4x1 x1x2 −x1x2x3 −x1x2x3x4

x1 x2 − x3 x4x1 x1x2 −x2x3 −x3x4

−x1 x2 −x1x3 x2x4x1 −x1x2 −x1x2x3 −x1x2x3x4

x1 − x2 x3 x4−x1x2 −x2x3 x3x4


∗ C ′1 = x1 x2 x3 x4

−x1 x1x2 x2x3 x3x4x1 − x2 x1x3 x2x4

x1 −x1x2 −x1x2x3 x1x2x3x4x1 − x2 x3 − x4

−x1 −x1x2 −x2x3 −x3x4−x1 x2 x1x3 x2x4

−x1 −x1x2 x1x2x3 x1x2x3x4x1 x2 − x3 x4

−x1 x1x2 −x2x3 −x3x4x1 −x2 −x1x3 x2x4

−x1 −x1x2 x1x2x3 −x1x2x3x4−x1 x2 − x3 − x4−x1x2 −x2x3 x3x4



∗ C ′2 = −x1

x1 x2−x1 x1x2 x1x2x3x1 − x2 x3 − x4

−x1 −x1x2 −x2x3 −x3x4−x1 x2 x1x3 x2x4

−x1 −x1x2 x1x2x3 x1x2x3x4x1 x2 − x3 x4

−x1 x1x2 −x2x3 −x3x4x1 −x2 −x1x3 x2x4

−x1 −x1x2 x1x2x3 −x1x2x3x4−x1 x2 − x3 − x4−x1x2 −x2x3 x3x4


∗ C ′3 = −x1

x1 x2−x1 x1x2 x1x2x3x1 − x2 x3 − x4

−x1 −x1x2 −x2x3 −x3x4−x1 x2 x1x3 x2x4

−x1 −x1x2 x1x2x3 x1x2x3x4x1 x2 − x3 x4

−x1 x1x2 −x2x3 −x3x4x1 −x2 −x1x3 x2x4

−x1 −x1x2 x1x2x3 −x1x2x3x4x1 x2 − x3 − x4

x1x2 −x2x3 x3x4−x1x3 −x2x4

x1x2x3x4

We are now in a position to determine the sum S(∆Q1[n]x1x2x3x4)in terms of the xi. Since we already know that S(∆Q1[n]) = 0, it follows


that S(∆Q1[n]x1x2x3x4) = S(Sn+1(x1, x2, x3, x4)). From this remarkand Claim 3, we have, for all n ≥ 36:

S(∆Q1[n]x1x2x3x4) =⎧⎪⎨

⎪⎩

S(C ′1) + (k − 2)S(C ′

2) + S(C ′3) if n = 12k

S(A′1) + (k − 1)S(A′

2) + S(A′3) if n = 12k + 4

S(B′1) + (k − 1)S(B′

2) + S(B′3) if n = 12k + 8 .

Writing n = 12k + r with r ∈ 0, 4, 8, we shall use the notation

wk,r(x1, x2, x3, x4) = S(∆Q1[n]x1x2x3x4).

Computing explicitly S(A′i), S(B

′i), S(C

′i) from the above figures, we get:

wk,0(x1, x2, x3, x4) = 5x1 + x2(−1 + x1)+x3(1− x1 + x2 − 2x1x2)+x4(1 + x2 + x3 + 3x1x2x3)+k[−4x1 + 2x2(1− x1)

+x3(−1 + x1 − 3x2 + 3x1x2)+x4(−1 + x2 − x3 − x1x2x3)] ,

wk,4(x1, x2, x3, x4) = 3x1 + x2(1 + x1) + x3(2 + 2x1 + x1x2)+2x4(1 + x3)+k[4x1 − 2x2(1 + x1)

+x3(1 + x1 − 3x2 − 3x1x2)+x4(−1− x2 + x3 + x1x2x3)] ,

wk,8(x1, x2, x3, x4) = 3x1 + x2(3− x1) + x3(1 + x1 − x2)+x4(3 + x2 + x3 − x1x2x3)+k[4x1 + 2x2(1 + x1)

+x3(−1− x1 − 3x2 − 3x1x2)+x4(1− x2 + x3 − x1x2x3)] .

Successively replacing (x1, x2, x3, x4) by each of the 16 binary sequencesof length 4, we obtain 48 polynomial functions of degree 1 in k. Wemust then determine the zeroes of these polynomials.

Case 1 : r = 0, i.e. we consider sequences of the type Q1[12k]x1x2x3x4.We obtain the following values of:

wk,0(x1, x2, x3, x4) = S(∆Q1[12k]x1x2x3x4) :


wk,0(1, 1, 1, 1) = 10− 6k

wk,0(1, 1, 1,−1) = −2− 2k

wk,0(1, 1,−1, 1) = 4− 2k

wk,0(1, 1,−1,−1) = 8− 6k

wk,0(1,−1, 1, 1) = 4− 6k

wk,0(1,−1, 1,−1) = 8− 2k

wk,0(1,−1,−1, 1) = 6− 6k

wk,0(1,−1,−1,−1) = 2− 2k

wk,0(−1, 1, 1, 1) = −2

wk,0(−1, 1, 1,−1) = −2

wk,0(−1, 1,−1, 1) = −8 + 16k

wk,0(−1, 1,−1,−1) = −16 + 16k

wk,0(−1,−1, 1, 1) = 0

wk,0(−1,−1, 1,−1) = −8 + 8k

wk,0(−1,−1,−1, 1) = −6− 4k

wk,0(−1,−1,−1,−1) = 2− 4k.

Given that wk,0(−1,−1, 1, 1) = 0, independently of k, we see that thesequence Q1[12k]−−++ is (simply, hence strongly) balanced. But, aseasily checked, Q1[12k]−−++ = Q1[12k+4]. The 15 other polynomialsmay vanish for small values of k, yielding “exotic” short strongly bal-anced sequences. However, direct inspection reveals that none of theseother functions vanishes for k ≥ 5.

Consequently, Q1[12k+4] is the unique balanced extension of length12k + 4 of Q1[12k], provided k ≥ 5. Note that, in the context of thisproof, we have k ≥ 7 in fact, since we are assuming n ≥ 92.

Case 2 : r = 4, i.e. we consider the binary sequences Q1[12k +4]x1x2x3x4. We obtain the following values of wk,4(x1, x2, x3, x4) :

wk,4(1, 1, 1, 1) = 14− 4k

wk,4(1, 1, 1,−1) = 6− 4k

wk,4(1, 1,−1, 1) = 0

wk,4(1, 1,−1,−1) = 8k

wk,4(1,−1, 1, 1) = 8 + 16k

wk,4(1,−1, 1,−1) = 16k


wk,4(1,−1,−1, 1) = −2

wk,4(1,−1,−1,−1) = −2

wk,4(−1, 1, 1, 1) = −6k

wk,4(−1, 1, 1,−1) = −8− 2k

wk,4(−1, 1,−1, 1) = −2− 6k

wk,4(−1, 1,−1,−1) = −2− 2k

wk,4(−1,−1, 1, 1) = 2− 2k

wk,4(−1,−1, 1,−1) = −6− 6k

wk,4(−1,−1,−1, 1) = −4− 6k

wk,4(−1,−1,−1,−1) = −4− 2k.

Here we have wk,4(1, 1,−1, 1) = 0, independently of k. Thus, the se-quence Q1[12k+4]++−+ is (simply, hence strongly) balanced. Again,one easily checks that Q1[12k + 4] + + − + = Q1[12k + 8]. The other15 functions do not vanish for k ≥ 2.

Therefore, Q1[12k + 8] is the unique balanced extension of length12k + 8 of Q1[12k + 4], provided k ≥ 2.

Case 3 : r = 8, i.e. we consider the binary sequences Q1[12k +8]x1x2x3x4. Here are the values of wk,8(x1, x2, x3, x4) :

wk,8(1, 1, 1, 1) = 10

wk,8(1, 1, 1,−1) = 2

wk,8(1, 1,−1, 1) = 8 + 16k

wk,8(1, 1,−1,−1) = 16k

wk,8(1,−1, 1, 1) = 8 + 8k

wk,8(1,−1, 1,−1) = 0

wk,8(1,−1,−1, 1) = −2− 4k

wk,8(1,−1,−1,−1) = −2− 4k

wk,8(−1, 1, 1, 1) = 6− 2k

wk,8(−1, 1, 1,−1) = −6− 6k

wk,8(−1, 1,−1, 1) = 4− 6k

wk,8(−1, 1,−1,−1) = −2k

wk,8(−1,−1, 1, 1) = −4− 2k

wk,8(−1,−1, 1,−1) = −8− 6k


wk,8(−1,−1,−1, 1) = −6− 2k

wk,8(−1,−1,−1,−1) = −10− 6k.

Again, wk,8(1,−1, 1,−1) = 0, but none of the other 15 functions van-ishes for k ≥ 4. Moreover, Q1[12k+ 8] +−+− = Q1[12k+ 12]. There-fore, Q1[12k+12] is the unique balanced extension of length 12k+12 ofQ1[12k + 8] for k ≥ 4.

With the above three cases, we have verified that, for every n ≡ 0mod 4 with n ≥ 52, the sequence Q1[n] admits a unique balanced binaryextension of length n+ 4, namely Q1[n+ 4].

Similar phenomena as those described here for Q1 occur for the othersequences Q2, Q3, Q4, R1, . . . , R12, and for the supplementary stronglybalanced sequences described in Theorem 2.2. This explains why, aftera somewhat chaotic initial behavior, the set SB(n) of strongly balancedbinary sequences of length n ultimately becomes periodic.

Acknowledgement

We are grateful to Pierre Duchet for attracting our attention tothe problem of Steinhaus. It is worthwile to note that Duchet usesthis problem with school pupils, by challenging them to find as largebalanced binary sequences as possible. Interestingly, some pupils wereable to discover such sequences in length as high as 123, or even 240 [1].

We also thank Michel Kervaire, Pierre de la Harpe and the anony-mous referee for useful comments about this paper.

Shalom EliahouLaboratoire de MathematiquesPures et Appliquees JosephLiouvilleUniversite du Littoral Coted’OpaleB.P. 699, 62228 Calais cedex,[email protected]

Delphine HachezLaboratoire de MathematiquesPures et Appliquees JosephLiouvilleUniversite du Littoral Coted’OpaleB.P. 699, 62228 Calais cedex,[email protected]


References

[1] Duchet P., La regle des signes, MATh.en.JEANS (1995), 139–140.

[2] Harborth H., Solution of Steinhaus’s problem with plus and minussigns, J. Comb. Th. (A) 12 (1972), 253–259.

[3] Steinhaus H., One Hundred Problems in Elementary Mathematics,Pergamon, Elinsford, N.Y., 1963.

Morfismos, Vol. 8, No. 2, 2004

Errata

En el Vol. 8, No. 1 (junio 2004),nebed4y3saicnerefersal,55.gap

ser:

In Vol. 8, No. 1 (June 2004), page55, references 3 and 4 should be:

[3] de Mier A.; Noy M., On graphs determined by their Tutte polyno-mials, Graphs Combin. 20 (2004), 105–119.

,.P.MatleuveR;.MyoN;.AreiMed;.AzeuqraM]4[ Locally gridgraphs: Classification and Tutte uniqueness, Discr. Math. 266(2003), 327–352.

81

Morfismos, Comunicaciones Estudiantiles del Departamento de Matematicas delCINVESTAV, se termino de imprimir en el mes de junio de 2005 en el taller de re-produccion del mismo departamento localizado en Av. IPN 2508, Col. San PedroZacatenco, Mexico, D.F. 07300. El tiraje en papel opalina importada de 36 kilo-gramos de 34 × 25.5 cm consta de 500 ejemplares en pasta tintoreto color verde.

Apoyo tecnico: Omar Hernandez Orozco.

Contenido

Homotopy triangulations of a manifold triple

Rolando Jimenez and Yuri V. Muranov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

On information measures and prior distributions: a synthesis

Francisco Venegas-Martınez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

On a problem of Steinhaus concerning binary sequences

Shalom Eliahou and Delphine Hachez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Errata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

morfismos, vol 8, no 2, 2004

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