Burnett coefficients and laminates

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Burnett coefficients and laminatesC. Conca a , J. San Martín a , L. Smaranda b & M. Vanninathan ca Facultad de Ciencias Físicas y Matemáticas, Departamentode Ingeniería Matemática, Universidad de Chile and Centro deModelamiento Matemático, UMR 2071 CNRS-UChile, Casilla 170/3– Correo 3, Santiago, Chileb Faculty of Mathematics and Computer Science, Department ofMathematics and Computer Science, University of Piteşti, 110040Piteşti, Str. Târgu din Vale nr.1, Argeş, Romaniac TIFR-CAM, Post Bag 6503, GKVK Post, Bangalore 560065,Karnataka, India

Available online: 01 Nov 2011

To cite this article: C. Conca, J. San Martín, L. Smaranda & M. Vanninathan (2011): Burnettcoefficients and laminates, Applicable Analysis, DOI:10.1080/00036811.2011.625017

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Applicable Analysis2011, 1–22, iFirst

Burnett coefficients and laminates

C. Concaa, J. San Martına, L. Smarandab* and M. Vanninathanc

aFacultad de Ciencias Fısicas y Matematicas, Departamento de Ingenierıa Matematica,Universidad de Chile and Centro de Modelamiento Matematico,

UMR 2071 CNRS-UChile, Casilla 170/3 – Correo 3, Santiago, Chile; bFaculty ofMathematics and Computer Science, Department of Mathematics and Computer Science,

University of Pitesti, 110040 Pitesti, Str. Targu din Vale nr.1, Arges, Romania;cTIFR-CAM, Post Bag 6503, GKVK Post, Bangalore 560065, Karnataka, India

Communicated by M. Ptashnyk

(Received 15 July 2011; final version received 9 September 2011)

The object of discussion of this article is the fourth-order tensor dintroduced as a set of macro coefficients associated with fine periodicstructures. Focus of attention is its variation on laminated microstructures.Complete bounds are obtained on its quartic form along with thecorresponding optimal structures. Differences with corresponding resultsfor the homogenized matrix are pointed out. Using Blossoming Principle, itis shown that d is not negative in the sense of Legendre–Hadamard, eventhough its quartic form is negative.

Keywords: homogenization; Burnett coefficients; laminates

AMS Subject Classifications: 35B27; 74Q20; 78M40

1. Introduction

Following our previous publications [1–3], we continue to analyse in this work fineperiodic structures and their macroscopic behaviour. To fix ideas, let us consider theproblem of steady state heat conduction in such structures. This model wasconsidered in [4,5] and a first set of macro coefficients (called the homogenizedmatrix and denoted by q) describing macro behaviour was obtained. Next, usingBloch waves, the articles [6,7] gave a spectral characterization of the homogenizedmatrix. This result may be interpreted as follows: homogenized medium describes thebehaviour of the periodic structure in the lowest energy and in the largest scale.Thanks to this interpretation, we can proceed in two directions: either we can pursuehomogenization at any increased energy level [8] or we can decrease the scalemaintaining the lowest energy level.

Our work in this article is in the second direction. It was noted in [1,2,9] thatother macro coefficients apart from the homogenized matrix appear naturally whenwe decrease the scale. One such set is a fourth-order tensor d called Dispersion tensor

*Corresponding author. Email: [email protected]

ISSN 0003–6811 print/ISSN 1563–504X online

� 2011 Taylor & Francis

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or Burnett coefficients in our previous publications. In general, these macrocoefficients depend heavily on the microstructure of the medium. Motivated byquestions coming from optimal design, metamaterials etc., one of the goals of thetheory is to bring out this dependence. The books by Allaire [10], Cherkaev [11],Milton [12] and Tartar [13], accomplish this task quite admirably in the case of thehomogenized matrix. One therefore seeks similar results in the case of d.

Complete one-dimensional results obtained in [3] say that d is negative and itsminimum value is attained at a simple microstructure when we vary microstructures.Needless to say that such results are attractive to provide possible explanationto metamaterialistic behaviours, especially in degenerate situations where thehomogenized coefficient is negligible. In this article, we begin to investigate multi-dimensional properties of d, by restricting our attention only to laminates with two-phases, as the microstructure varies preserving the volume proportion denoted by �.(However detailed study of d on general structures is rather difficult [1]). Wheneverpossible, we make comparison with the corresponding result for q so that the readerscan readily note new phenomena that appear at smaller scale. Following presen-tation of highlights of this article is made with this in mind. After introducing Blochwaves in Section 2, the matrix q and the tensor d are defined on the spectral side.Next, in the same section, we recall spatial integral representations for the quadraticform of q and the quartic form of d valid for arbitrary microstructures.

The main reason for considering the quadratic form for q and the quartic formfor d is that they appear directly in the Taylor expansion of the first Bloch eigenvalue(representing lowest energy level) around zero (quasi) wave number (representing theinverse of the scale). While the quadratic form for q is positive, the quartic form for dis non-positive. The main feature of the representation is the presence of two cell testfunctions X

ð1ÞðT Þ and X

ð2ÞðT Þ interacting among themselves in the quartic form for d.

No such feature is seen for the quadratic form for q.In the Section 3, these representations are taken up for detailed analysis in the

case of special microstructures, namely laminates with two-phases, their volumeproportion being fixed. Let us recall that such structures are important in that theydisplay optimal properties for q (see [13,14]). While the quadratic form for q remainsfixed, the quartic form for d varies over an interval whose end points are computedexplicitly. One can notice the presence of an interaction term between longitudinaland transverse modes at the minimum value. There is no such interaction in thequadratic form for q.

Negativity of the quartic form for d poses naturally the following question on d:is the tensor d negative in the sense of Legendre–Hadamard? In the second part ofthis work, we answer this question negatively in Section 4. To this end, let us considerthe linear action of the tensor d on all matrices. Because of its symmetry, itannihilates all anti-symmetric matrices. It is thus enough to consider its action on thespace SymN of all symmetric N�N matrices. The quartic form of d corresponds tothe action of d on the subset RSymN of all rank-one symmetric matrices. Thanks tothe Blossoming Principle (recalled in Section 4) and to the results of Section 3, we getan explicit expression for the action of d on SymN when the microstructure is alaminate. Spectral decomposition of this linear operator is obtained. Apart fromdegeneracies of eigenvalues, it is important to note that there is a unique positiveeigenvalue. In particular, there are vectors � and � such that d(�� ) � (�� ) ispositive showing that d does not possess Legendre–Hadamard negativity property.

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Further consequences of these computations will appear in a future publication.

Usual summation convention is followed throughout the article.

2. Preliminaries

Let us introduce the problem to be studied in this work. We denote by Y the

reference cell ]0, 2�[N and for any real number � 2 [0, 1], we consider measurable

subsets T of Y such that

jT j

jYj¼ �:

For any positive real numbers c1, c2, we denote byM(c1, c2,Y ) the set of coercive and

periodic bounded measurable matrices defined on Y, i.e.

Mðc1, c2,Y Þ ¼n�2L1# ðY Þ

N�N : �ðyÞ� � � � c1j�j2, j�ðyÞ�j � c2j�j a.e. in Y, 8�2R

No:

We consider the operator

A ¼def�

@

@yk�k‘ð yÞ

@

@y‘

� �, y2R

N,

where the coefficient a¼ (�kl)2M(�1,�0,Y ) and in the reference cell is given by

að yÞ ¼ �ð yÞI, with �ð yÞ ¼ �0vTC ð yÞ þ �1vTð yÞ, y2Y, ð1Þ

with 05 �15�0, I represents the identity matrix in RN�N and sT(y) denotes the

characteristic function on T. Here, we consider the mixture of two homogeneous

materials characterized by the scalar conductivities �0 and �1.For each "4 0, we also consider the "Y-periodic operator A" defined by

A" ¼def�

@

@xk�"k‘ðxÞ

@

@x‘

� �with �"k‘ðxÞ ¼

def�k‘

x

"

� �, x2R

N:

Let us define the following spaces:

L2#ðY Þ ¼

nf2L2ðR

NÞ : f is Y-periodic

o,

H1#ðY Þ ¼

nf2L2

#ðY Þ : rf2L2#ðY Þ

No,

H1#,0ðY Þ ¼

nf2H1

#ðY Þ : mð f Þ ¼ 0o,

where mð f Þ denotes the average of f over the reference cell Y, that is,

mð f Þ ¼ 1

jYj

ZY

f ð yÞdy:

We are interested in studying properties of the homogenized and dispersion

coefficients denoted by q, respectively d. These macro quantities (q, d) are defined in

terms of Bloch waves associated with the operator A which we introduce now.

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Let us consider the following spectral problem parameterized by �2RN: find

�¼ �(�)2R and ¼ (y ; �) (not identically zero) such that

A ð�; �Þ ¼ �ð�Þ ð�; �Þ in RN, ð�; �Þ is ð�;Y Þ-periodic, i:e:

ð yþ 2�m; �Þ ¼ e2�im�� ð y; �Þ 8m2ZN, y2R

N:

(ð2Þ

Next, by the Floquet theory, we define �(y;�)¼ e�iy� � (y;�) and (2) can be rewritten

in terms of � as follows:

Að�Þ� ¼ �� in RN, � is Y-periodic: ð3Þ

Here, the operator A(�) is called the translated operator and is defined by

Að�Þ ¼ e�iy��Aeiy��:

It is well known [4,15] that for each �2Y0 ¼��1

2 ,12

�N, the above spectral problem

(3) admits a discrete sequence of eigenvalues �m(�). Their associated eigenfunctions

�m(y; �) (referred to as Bloch waves) enable us to describe the spectral resolution of A(as an unbounded self-adjoint operator in L2(RN)) in the orthogonal basis

{eiy��m(y;�) :m� 1, �2Y0}.Let us introduce Bloch waves at the "-scale:

�"mð�Þ ¼ "�2�mð�Þ, �"mðx; �Þ ¼ �mð y; �Þ, "mðx; �Þ ¼ mð y; �Þ,

where the variables (x, �) and (y, �) are related by y ¼ x" and �¼ " �.

We consider a sequence {u"} bounded in H1(RN) and f2L2(RN) satisfying

A"u" ¼ f in RN: ð4Þ

We assume that u" * u weakly in H1(RN). The homogenization problem consists of

passing to the limit, as "! 0, in the previous equation and obtain the equationsatisfied by u, namely,

Qu ¼def�

@

@xkqk‘

@u

@x‘

� �¼ f in R

N,

where q¼ (qkl) is a constant matrix known as the homogenized matrix [4].

Simple relation linking q with Bloch waves is the following: qk‘ ¼12@2�1@�k@�‘ð0Þ (see

[7]). At this point, it is appropriate to recall that derivatives of the first eigenvalue

and eigenfunction at �¼ 0 exist, thanks to the regularity property established in [6].

In fact, we know that there exists 04 0 such that the first eigenvalue �1(�) is ananalytic function on B0 ¼ f� : j�j5 0g, and there is a choice of the first eigenvector

�1(y; �) satisfying

��1ð�; �Þ 2H1#ðY Þ is analytic on B0 , �1ð y; 0Þ ¼ jYj�1=2 ¼

1

ð2�ÞN=2:

To see how d arises, let us consider wave propagation problem in the periodic

structure governed by the operator @ttþA" with appropriate initial conditions. If weconsider short waves of low energy with wave number satisfying "2 j�j4¼O (1) and

"4j�j6¼ o(1) then a simplified description is obtained with the operator @ttþQþ "2D,

where D is the fourth-order operator whose symbol is 14!

@4�1@�k@�‘@�m@�n

ð0Þ�k�‘�m�n:

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This was noted in [1]. The tensor (dk‘mn) with dk‘mn ¼14!

@4�1@�k@�‘@�m@�n

ð0Þ, which captures

dispersive effects on such waves, represents a corrector to the periodic medium.

It was studied in [1] and in particular, a physical space representation for it was

obtained. We recall it in the below result.

PROPOSITION 2.1 For any �2RN, the quantities q(�) and d(�) defined by

qð�Þ ¼def

qk‘�k�‘ ¼1

2

@2�1@�k@�‘

ð0Þ�k�‘,

d ð�Þ ¼def

dk‘mn�k�‘�m�n ¼1

4!

@4�1@�k@�‘@�m@�n

ð0Þ�k�‘�m�n

admit the following representations:

qð�Þ ¼m �ð yÞðrXð1ÞðT Þ þ �Þ � �

� �, ð5Þ

d ð�Þ ¼ �m �ð yÞr�Xð2ÞðT Þ �

1

2ðXð1ÞðT ÞÞ

2�� r�Xð2ÞðT Þ �

1

2ðXð1ÞðT ÞÞ

2��

, ð6Þ

with the test functions Xð1ÞðT Þ and X

ð2ÞðT Þ defined by the following cell problems:

�div ð�ð yÞrXð1ÞðT ÞÞ ¼ div ð�ð yÞ�Þ in R

N,

Xð1ÞðT Þ 2H

1#,0ðY Þ,

8<: ð7Þ

�div ð�ð yÞrXð2ÞðT ÞÞ ¼ �ð yÞ rX

ð1ÞðT Þ þ �Þ � �� qð�Þ þ divð�ð yÞ�Xð1Þ

ðT Þ

� �in R

N,

Xð2ÞðT Þ 2H

1#,0ðY Þ:

8<: ð8Þ

Remark 2.2 Let us first note that since Xð1ÞðT Þ satisfies Equation (7), we get that the

homogenized tensor admits the following representation:

qð�Þ ¼m �ð yÞðrXð1ÞðT Þ þ �Þ � ðrX

ð1ÞðT Þ þ �Þ

� �: ð9Þ

A celebrated theorem [13,16] describes the variation of the eigenvalues of the

homogenized matrix q as the microstructure varies. More precisely, the eigenvalues

{j} of q are characterized by the following bounds (see [13] for a proof):

�ð�Þ � j � þð�Þ,

XNj¼1

1

j � �1�

1

�ð�Þ � �1þ

N� 1

þð�Þ � �1,

XNj¼1

1

�0 � j�

1

�0 � �ð�Þþ

N� 1

�0 � þð�Þ:


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Here, we have used the following notations for the arithmetic and harmonic means:

þð�Þ ¼mð�Þ ¼ ð1� �Þ�0 þ ��1, ð10Þ

1

�ð�Þ¼mð1=�Þ ¼ 1� �

�0þ�

�1: ð11Þ

In the above description, laminates play a crucial role. Indeed, they are optimal

micro-structures for q in the sense that the eigenvalues of q on laminates provide

upper and lower bounds for the eigenvalues of q on arbitrary microstructures.In this article, our attention turns to multi-dimensional aspects of macro

coefficients by studying their variation on laminates. One of the first phenomena

which comes to our mind is the direction-dependence of physical properties, i.e. lack

of isotropy at the macro scale even when the materials at microstructure level are

isotropic. In particular, we are aware of different longitudinal and transverse

behaviour in laminates. Further, there is no interaction between longitudinal and

transverse modes in these structures. Somewhat surprisingly, this is no more true

when we decrease the scale, as seen below.

3. First main result: bounds on dispersion tensor in periodic laminates

Periodic laminates, by definition, correspond to the following choice of the

conductivity matrix: �kl(y)¼�(y) kl with �(y)¼ �(y1) which is 2�-periodic functionof a single variable. It is implicitly assumed that T as a subset of the basic reference

cell Y has one-dimensional structure: T¼T1� ]0, 2�[N�1 where T1 is an arbitrary

measurable subset of ]0, 2�[. Examples include vertical strips and their unions.

Further, since we are considering two-phase media, we have

�ð y1Þ ¼ �0vYnT1ð y1Þ þ �1vT1

ð y1Þ, with � ¼jT j

jYj¼jT1j

2�:

Note that there are two types of directions in this microstructure: parallel

(longitudinal) and perpendicular (transverse) directions to lamination.

Accordingly, we split the (momentum) variable � ¼ ð�1, ~�Þ with �1 2R, ~� ¼ð�2, �3 . . . �NÞ 2R

N�1:The aim is to study the variation of d(�) as the laminated microstructure T varies

inside the basic cell Y, assuming that the volume proportion � ¼ jT jjYj remains fixed.

We observe that if � 2 {0, 1} or �¼ 0, the dispersion coefficient d(�) is equal to 0. For

this reason, we just consider � 2 ]0, 1[ and � 6¼ 0 in the sequel.Let us recall that the homogenized conductivities are the harmonic mean

(denoted by �(�)) on the longitudinal direction and the arithmetic mean (denoted

by þ(�)) on the transverse direction (for more details, see [12,13]). Therefore, the

value q(�)¼ qkl�k�l remains fixed at

qð�Þ ¼ �ð�Þ�21 þ þð�Þj ~�j

2 8�2RN, ð12Þ

as microstructure varies. For the variation of d(�), we obtain the following theorem.

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THEOREM 3.1 Let us consider that the coefficient � is given by

�ð yÞ ¼ �ð y1Þ 8y2Y:

Then, for any � 2 ]0, 1[ and for any �2Y0 n {0}, we have:

(i) The following equalities hold:

infd ð�Þ : jT j ¼ �jYj

¼

�1

12jYj2�2ð1� �Þ2

��ð�Þ

��1 ��ð�Þ

�2�21

1

�1�

1

�0

� �� j ~�j2ð�0 � �1Þ

� �2

,

supd ð�Þ : jT j ¼ �jYj

¼ 0:

(ii) Moreover, there exists a minimizing microstructure which is unique up to

translations and is characterised by the classical laminate consisting of a single

vertical strip inside Y, that is for Tmin¼ ]0, 2��[� ]0, 2�[N�1�Y we have

jTminj ¼ �jYj and

infd ð�Þ : jT j ¼ �jYj

¼ min

d ð�Þ : jT j ¼ �jYj

¼ dTmin

ð�Þ:

(iii) In contrast, there are no classical maximizing microstructures; the only

possibility to achieve the upper bound (d(�)¼ 0) is to introduce the relaxed

laminated microstructure characterized by the local density �(y1)¼ � for all

y12 ]0, 2�[.(iv) As microstructure varies among classical laminates, the dispersion coefficient

fills the entire interval [inf d(�), sup d(�)[, that is,d ð�Þ : jT j ¼ �jYj

¼

�1

12jYj2�2ð1� �Þ2

��ð�Þ

��1 ��ð�Þ

�2�21

1

�1�

1

�0

� �� j ~�j2ð�0 � �1Þ

� �2

, 0

:

Remark 3.2 A few remarks about the previous theorem are in order. It is a natural

generalization of our one-dimensional result in [3]; indeed, if ~� ¼ 0, the above result

is reduced to the result in one dimension. On the other hand, the presence of the cross

term in statement (i) of Theorem 3.1 shows there is interaction between the

longitudinal and the transverse modes in the case of higher order macro coefficient

d(�), which does not appear in q(�) given in (12). This is a new phenomenon

occurring because of the decrease of scale. The origin of this macro phenomenon lies

in the presence of interaction terms�Xð1ÞðT Þ

�4,Xð2ÞðT Þ

�Xð1ÞðT Þ

�2, etc., in the spatial

representation formula for d(�) (see (6)).

Finally, taking successively one after the other ~� ¼ 0 or �1¼ 0 in the statement (i),

and comparing it with the identity (12), we realize that the quantities

�1

12jYj2�2ð1� �Þ2 �ð�Þð Þ

3 1

�1�

1

�0

� �2

,

�1

12jYj2�2ð1� �Þ2 �ð�Þð Þ

�1ð�1 � �0Þ

2,


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represent the next order corrections to the harmonic and the arithmetic means,

respectively, as the scale is decreased.

Before starting the proof of Theorem 3.1, let us state the following technical

lemma.

LEMMA 3.3 Let us consider the function aT: ]0, 2�[!R defined by

aTð y1Þ ¼

Z y1

0

ðvTðsÞ � �Þds: ð13Þ

For any m2N and k2Z, we have

mðamT�kÞ ¼mðamT Þmð�kÞ: ð14Þ

Proof We remark that for any k2Z,

�k ¼ �k0 þ ð�k1 � �

k0ÞvT and mð�kÞ ¼ �k0 þ ð�k1 � �k0Þ�,

then

�k �mð�kÞ ¼ ð�k1 � �k0ÞðvT � �Þ ¼ ð�k1 � �k0ÞdaTdy1

:

Therefore, for any m2N we haveZ 2�

0

amT ðsÞ �k �mð�kÞ

� �ds ¼ ð�k1 � �

k0Þ

Z 2�

0

amT ðsÞdaTdy1ðsÞds

¼ ð�k1 � �k0Þ

amþ1T ðsÞ

mþ 1

��2�

0

¼ 0:

The property (14) is obtained from the above identity, dividing by 2�. g

Proof of Theorem 3.1 Since we have considered

�ð yÞ ¼ �ð y1Þ,

then we look for solutions of type

Xð1ÞðT Þð yÞ ¼ X

ð1ÞðT Þð y1Þ and X

ð2ÞðT Þð yÞ ¼ X

ð2ÞðT Þð y1Þ:

With this, the problems (7) and (8) could be written as follows:

�@

@y1�ð y1Þ

@Xð1ÞðT Þ

@y1

!¼

@

@y1

��ð y1Þ�1

�in R,

Xð1ÞðT Þ 2H

1#,0ð0, 2�Þ

8>>><>>>: ð15Þ

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and

�@

@y1�ð y1Þ

@Xð2ÞðT Þ

@y1

!¼ �ð y1Þ

@Xð1ÞðT Þ

@y1þ �1

!�1 þ �ð y1Þje�j2 � qð�Þ

þ@

@y1�ð y1Þ�1X

ð1ÞðT Þ

� �in R,

Xð2ÞðT Þ 2H

1#,0ð0, 2�Þ,

8>>>>>>><>>>>>>>:ð16Þ

where �¼ (�1, . . . , �N)T.

Let us integrate the first equation in (15) and we get

��ð y1Þ@Xð1ÞðT Þ

@y1þ �1

!¼ C1:

In order to compute the constant C1, we first divide by �, then we take the average on

(0, 2�) and since Xð1ÞðT Þ 2H

1#,0ð0, 2�Þ, we obtain

C1 ¼ ��1

mð1=�Þ :

Therefore, Xð1ÞðT Þ satisfies

@Xð1ÞðT Þ

@y1þ �1 ¼

�1mð1=�Þ�ð y1Þ

in R: ð17Þ

With these computations, we get that the homogenized matrix given in (9) could be

written as follows:

qð�Þ ¼m �ð y1Þ�� @Xð1ÞðT Þ@y1þ �1

��2 þ �ð y1Þj ~�j2 !

¼�21

mð1=�Þ þmð�Þj ~�j2: ð18Þ

Integrating Equation (17), we get

Xð1ÞðT Þð y1Þ ¼ C2 þ

�1mð1=�Þ

Z y1

0

1

�ðsÞ�mð1=�Þ

� �ds: ð19Þ

In order to compute the integral from the right-hand side, we observe that due to

definition (1) we have

��1ðsÞ ¼ ��10 vTC ðsÞ þ ��11 vTðsÞ ¼ ��10 þ ð�

�11 � �

�10 ÞvTðsÞ

and

mð��1Þ ¼ ��10 ð1� �Þ þ ��11 � ¼ �

�10 þ ð�

�11 � �

�10 Þ�,

then

��1ðsÞ �mð��1Þ ¼ ð��11 � ��10 ÞðvTðsÞ � �Þ:


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Then, by integrating between 0 and y1, we getZ y1

0

ð��1ðsÞ �mð��1ÞÞds ¼ ð��11 � ��10 ÞaTð y1Þ,

where aT is defined in (13).Therefore, Equation (19) becomes

Xð1ÞðT Þð y1Þ ¼ C2 þ

�1mð1=�Þ ð�

�11 � �

�10 ÞaTð y1Þ;

since Xð1ÞðT Þ 2H

1#,0ð0, 2�Þ, we obtain

C2 ¼ ��1

mð1=�Þ ð��11 � �

�10 ÞmðaTÞ:

It follows that

Xð1ÞðT Þð y1Þ ¼

�1mð1=�Þ ð�

�11 � �

�10 Þ aTð y1Þ �mðaTÞð Þ: ð20Þ

Let us now observe that using the identity (17), the right-hand side of

Equation (16) becomes

�21mð1=�Þ þ �ð y1Þje�j2 � qð�Þ þ

@

@y1�ð y1Þ�1X

ð1ÞðT Þ

� �: ð21Þ

Therefore, using (18) and (21), the first equation in (16) yields

�@

@y1�ð y1Þ

@Xð2ÞðT Þ

@y1

!¼ j ~�j2 �ð y1Þ �mð�Þð Þ þ

@

@y1�ð y1Þ�1X

ð1ÞðT Þ

� �,

then by integrating we get


@y1¼ C3 þ j ~�j

2

Z y1

0

ð�ðsÞ �mð�ÞÞdsþ �ð y1Þ�1Xð1ÞðT Þð y1Þ: ð22Þ

In order to compute the integral from the right-hand side of (22), we observe that

due to definition (1), we have

�ðsÞ ¼ �0 þ ð�1 � �0ÞvTðsÞ and mð�Þ ¼ �0 þ ð�1 � �0Þ�,

then Z y1

0

ð�ðsÞ �mð�ÞÞds ¼ �ð�0 � �1ÞaTð y1Þ:

Therefore, Equation (22) becomes


@y1¼ C3 � j ~�j

2ð�0 � �1ÞaTð y1Þ þ �ð y1Þ�1Xð1ÞðT Þð y1Þ:

ð23Þ

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In order to compute the constant C3, we first divide by � and then taking the average

on ]0, 2�[, we obtain

C3mð1=�Þ ¼ j ~�j2ð�0 � �1ÞmðaT=�Þ,

and using identity (14) from Lemma 3.3, we get

C3 ¼ j ~�j2ð�0 � �1ÞmðaTÞ:

Thus, Equation (23) yields

@Xð2ÞðT Þ

@y1¼ j ~�j2ð�0 � �1Þ

aTð y1Þ �mðaT Þ�ð y1Þ

� �1Xð1ÞðT Þð y1Þ:

ð24Þ

Then, using (17), (20) and (24), we deduce that

@

@y1Xð2ÞðT Þ �

1

2ðXð1ÞðT ÞÞ

2

� �¼

"j ~�j2ð�0 � �1Þ �

�21m2ð1=�Þ

1

�1�

1

�0

� �#aTð y1Þ �mðaT Þ

�ð y1Þ:

With these computations and using the representation of the dispersion tensor given

in (6), we get

�d ð�Þ ¼

"j ~�j2ð�0 � �1Þ �

�21m2ð1=�Þ

1

�1�

1

�0

� �#2

m ðaTð y1Þ �mðaTÞÞ2�ð y1Þ

� �:

Using again Lemma 3.3, �d(�) yields

�d ð�Þ ¼

"j ~�j2ð�0 � �1Þ �

�21m2ð1=�Þ

1

�1�

1

�0

� �#2

mð1=�Þ mða2TÞ �m2ðaTÞ� �

: ð25Þ

Let us note that Equation (25) has the following structure: the first multiplicative

term depends only on the dimension N through the parameter �, while the last

multiplicative term is only dependent on the microstructure T through the function

aT(y1) defined in (13). Thus, to find the set in which the dispersion tensor lies as the

microstructure varies preserving the volume proportion �, we need to concentrate

our attention on the last multiplicative term from (25). We first remark that the

function aT is independent of the dimension N, that is

aTð y1Þ ¼ aT1ð y1Þ 8T ¼ T1�0, 2�½

N�1, 8y1 2T1, 8T1 �0, 2�½: ð26Þ

For this purpose, let us write the formula (25) in the particular case of dimension one

(N¼ 1):

�d1ðvT1Þ¼

1

m2ð1=�Þ

1

�1�

1

�0

� � �2mð1=�Þ mða2T1

Þ�m2ðaT1Þ

� �8T10,2�½, jT1j¼2��:

ð27Þ


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Combining relations (25)–(27), we get

d ð�Þ ¼j ~�j2ð�0 � �1Þ �

�21

m2ð1=�Þ

1�1� 1

�0

� �h i21

m2ð1=�Þ

1�1� 1

�0

� �h i2 d1ðvT1Þ 8T ¼ T1� 0, 2�½

N�1, jT j ¼ �jYj:

ð28Þ

Using the results from [3, Theorem 3.1], we know that

inffd1ðvT1Þ : jT1j ¼ 2��g ¼ �

1

12�2ð1� �Þ2j2�j2mð1=�Þ 1

m2ð1=�Þ

1

�1�

1

�0

� � �2,

supfd1ðvT1Þ : jT1j ¼ 2��g ¼ 0,

thus, due to notation (11), we conclude the item (i) of the theorem.Moreover, items (ii) and (iii) are direct consequences of (28) and the results of

Theorem 3.1 in [3].Finally, we remark that

nd ð�Þ : jT j ¼ �jYj

o¼j ~�j2ð�0 � �1Þ �

�21

m2ð1=�Þ

1�1� 1

�0

� �h i21

m2ð1=�Þ

1�1� 1

�0

� �h i2 nd1ðvT1

Þ : jT1j ¼ 2��o,

then using Theorem 3.2 in [3], we conclude the item (iv). g

4. Second main result: spectral decomposition of dispersion tensor in laminates

In order to study the spectral decomposition of the dispersion tensor d starting fromits quartic form, we are going to use the so-called Blossoming principle. Thisprinciple was independently developed by de Casteljau [17] and Ramshaw [18,19] inthe context of Computer-Aided Geometric Design, as a new way to looking at Beziercurves. For the sake of completeness, we recall this principle here, in the case ofvectorial polynomial functions:

THEOREM 4.1 (Blossoming Principle) For any multivariate polynomial pn: Rm!R

of total degree n, there exists a unique symmetric n-affine function Pn: Rm� � � � �

Rm!R satisfying

Pnðx, . . . , xÞ ¼ pnðxÞ 8x2Rm:

After Ramshaw [18], Pn is called the blossoming of pn and pn is called the polar ofPn. For the proof of this version of the Blossoming principle, we refer the readerto [20].

Using this principle, we can prove our second main result, which is stated in thefollowing theorem.

THEOREM 4.2 Let us denote by SymN the space of symmetric matrices of orderN�N, with N� 2. If we consider the tensor d¼ (dklmn) as a self-adjoint linear

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map on SymN:

d : SymN�!SymN

ðMmnÞ 7�!ðdklmnMmnÞ,

then the eigenvalues of d admit the following formulas:

�1 ¼ �A

2þ Nþ1

3 B2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA

2� Nþ1

3 B2

2

!2

þN� 1

9A

2B2

vuut 5 0 of multiplicity 1,

ð29Þ

�2 ¼ �A

2þ Nþ1

3 B2

2�


2� Nþ1

3 B2

2

!2

þN� 1

9A

2B2

vuut 5 0 of multiplicity 1,

ð30Þ

�3 ¼2

3AB4 0 of multiplicity N� 1, ð31Þ

�4 ¼ �2

3B2 5 0 of multiplicity

ðNþ 1ÞðN� 2Þ

2, ð32Þ

where

A ¼1

m2ð1=�Þ

1

�1�

1

�0

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimð1=�Þ mða2TÞ �m2ðaTÞ

� �q4 0, ð33Þ

B ¼ ð�0 � �1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimð1=�Þ mða2TÞ �m2ðaTÞ

� �q4 0: ð34Þ

Moreover, the corresponding spaces of eigenmatrices are given by:

V1 ¼

� 0 � � � 0

0 �ffiffiffiffiffiffiffiN�1p � � � 0

..

. ... . .

. ...

0 0 � � � �ffiffiffiffiffiffiffiN�1p

266666664

377777775: � 2R, � ¼ � tan 1,

8>>>>>>><>>>>>>>:

tan 1 ¼

A2�Nþ1

3 B2

2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA

2�Nþ1

3 B2

2

� �2þ N�1

9 A2B2

r13

ffiffiffiffiffiffiffiffiffiffiffiffiN� 1p

AB

9>>>>>>>=>>>>>>>;, ð35Þ


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V2 ¼

� 0 � � � 0

0 �ffiffiffiffiffiffiffiN�1p � � � 0

..

. ... . .

. ...

0 0 � � � �ffiffiffiffiffiffiffiN�1p

2666664

3777775 : � 2R, � ¼ � tan 2,

8>>>>><>>>>>:

tan 2 ¼

A2�Nþ1

3 B2

2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA

2�Nþ1

3 B2

2

� �2þ N�1

9 A2B2

r13


AB

9>>>>>=>>>>>;, ð36Þ

V3 ¼

0 M12 � � � M1N

M12 0 � � � 0

..

. ... . .

. ...

M1N 0 � � � 0

266664377775 : M12, . . . ,M1N 2R

8>>>><>>>>:

9>>>>=>>>>;, ð37Þ

V4 ¼

0 0 � � � 0

0 M22 � � � M2N

..

. ... . .

. ...

0 M2N � � � MNN

266664377775 : Mrs 2R, 8r, s2 f2, . . . ,Ng,

XNr¼2

Mrr ¼ 0

8>>>><>>>>:

9>>>>=>>>>;: ð38ÞRemark 4.3 There exists � 2 [0, 1] such that A¼B. In fact, for � ¼

ffiffi�p

1ffiffi�p

0þffiffi�p

1we have

m2ð1=�Þ ¼ 1�0�1

. In this case, we get �1 ¼ � Nþ23 B

2 and �2 ¼ �4 ¼ � 23B

2 and

�3 ¼ þ 23B

2. Thus, there are only three different eigenvalues ð�iÞ3i¼1 of multiplicities

1, NðN�1Þ2 and N� 1, respectively.

Proof

Step 1. Application of Blossoming principle. Using the notation (33)–(34), relation

(25) can be written as follows:

�d ð�Þ ¼ �dijkl�i�j�k�l ¼ ðA�21 � Bje�j2Þ2 ¼ �ðA þ BÞ�21 � Bj�j2�2: ð39Þ

Let us take �1, �2, �3, �42RN four arbitrary vectors. Then, using Blossoming

principle given in Theorem 4.1 (see also [21]), we get that Dð�1, �2, �3, �4Þ ¼dijkl�

1i �

2j �

3k�

4l corresponds to a unique symmetric 4-affine function D(�, �, �, �) called the

blossom of d, such that d(�)¼D(�, �, �, �) corresponds to the polar representation

of d. In our particular case (39), it is clear that the function D(�, �, �, �) is given by

�Dð�1, �2, �3, �4Þ ¼1

3

(ðA þ BÞ�11�

21 � B�

1 � �2� �

ðA þ BÞ�31�41 � B�

3 � �4� �

þ ðA þ BÞ�11�31 � B�

1 � �3� �

ðA þ BÞ�21�41 � B�

2 � �4� �

þ ðA þ BÞ�11�41 � B�

1 � �4� �

ðA þ BÞ�21�31 � B�

2 � �3� �)

: ð40Þ

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For given p, q, r, s2 {1, . . . ,N}, to compute dpqrs is equivalent to evaluate the

blossom function D(�, �, �, �) at the canonical vectors ep, eq, er, es, that is,

�dpqrs ¼ �Dðep, eq, er, esÞ ¼

1

3

(ðA þ BÞp1q1 � Bpq� ��

ðA þ BÞr1s1 � BrsÞ

þ ðA þ BÞp1r1 � Bpr� �

ðA þ BÞq1s1 � Bqs� �

þ ðA þ BÞp1s1 � Bps� �

ðA þ BÞq1r1 � Bqr� �)

, ð41Þ

where ij denotes the classical Kronecker’s symbol given by

ij ¼1 if i ¼ j,

0 elsewhere.

�Let us now take the symmetric matrix M¼ (Mrs) of order N�N and compute

�dM¼ (�dpqrsMrs). We get

�dpqrsMrs ¼1

3

(ðAþBÞp1q1�Bpq� ��

ðAþBÞM11�BMrrÞ

þ 2 ðAþBÞ2p1q1M11� ðAþBÞBq1Mp1� ðAþBÞBp1M1qþB2Mpq

� �):

Step 2. A natural orthogonal decomposition of SymN. Let us decompose the space of

symmetric matrices of order N�N as the direct sum

SymN ¼M1 �M2,

where

M1 ¼

nðMrsÞ : Mrr ¼ 0 for all r2 f1, . . . ,Ng

o,

M2 ¼

nðMrsÞ : Mrs ¼ 0 for all r, s2 f1, . . . ,Ng, r 6¼ s

o:

In the following two steps, we are going to look for the spectral decomposition of

fourth-order tensor d on spacesM1, respectivelyM2.

Step 3. Spectral decomposition of d on M1. Considering matrices (Mrs)2M1, we

obtain that

�dpqrsMrs ¼2

3

(�ðA þ BÞBq1Mp1 � ðA þ BÞBp1M1q þ B

2Mpq

)

¼

0 if p ¼ q,

�2

3ABM1q if p ¼ 1, q4 1,

�2

3ABMp1 if p4 1, q ¼ 1,

2

3B2Mpq if p4 1, q4 1:

8>>>>>>>><>>>>>>>>:


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Here we observe that (�dpqrsMrs)2M1 and the real numbers � 23AB and 2

3B2 are

eigenvalues of tensor �d whose corresponding eigenmatrices belong to the followingspaces:

M1,1 ¼ fðMrsÞ : Mrs ¼ 0 if r, s4 1 or if r ¼ s ¼ 1g, ð42Þ

M1,2 ¼ fðMrsÞ : Mrs ¼ 0 if p ¼ 1 or if q ¼ 1 or if p ¼ qg: ð43Þ

The multiplicities of these eigenvalues inM1 are N� 1 and ðN�2ÞðN�1Þ2 , respectively.

Step 4. Spectral decomposition of d on M2. Let us now consider matrices whichbelong toM2. In this case, we get that

�dpqrsMrs ¼1

3

(ðAþBÞp1q1�Bpq� �

ðAþBÞM11�BMrrð Þ þ 2 ðAþBÞ2p1q1M11

��ðAþBÞBq1Mp1� ðAþBÞBp1M1qþB

2Mpq

�): ð44Þ

For p 6¼ q, the identity (44) becomes

�dpqrsMrs ¼1

3ðAþBÞp1q1|fflffl{zfflffl}

¼0

�B pq|{z}¼0

0@ 1A ðAþBÞM11�BMrrð Þ

8<:þ2 ðAþBÞ2 p1q1|fflffl{zfflffl}

¼0

M11�ðAþBÞB q1Mp1|fflfflffl{zfflfflffl}¼0

�ðAþBÞB p1M1q|fflfflffl{zfflfflffl}¼0

þB2Mpq|{z}¼0

0@ 1A9=;¼ 0, ð45Þ

which implies that (�dpqrsMrs)2M2.For p¼ q¼ 1, the identity (44) yields

�dpqrsMrs ¼1

3

(A ðA þ BÞM11 � BMrrð Þ þ 2A2M11

)¼ A

2M11 �1

3AB

XNr¼2

Mrr:

ð46Þ

Moreover, for p¼ q4 1, the identity (44) has the following form:

�dpqrsMrs ¼ �1

3ABM11 þ

1

3B2XNr¼2r6¼p

Mrr þ B2Mpp: ð47Þ

Therefore, the formulas (46)–(47) can be written as follows:

�

ðdMÞ11

ðdMÞ22

..

.

ðdMÞNN

0BBBB@1CCCCA ¼

A2

� 13AB � 1

3AB � � � � 13AB

� 13AB B

2 13B

2� � � 1

3B2

� 13AB

13B

2B2

� � � 13B

2

..

. ... ..

. . .. ..

.

� 13AB

13B

2 13B

2� � � B

2

2666666664

3777777775

M11

M22

..

.

MNN

0BBBB@1CCCCA: ð48Þ

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In order to find the eigenvalues and eigenvectors associated to the above matrix,

we look for �2R such that the following homogeneous system admits a nontrivial

solution:

A2� � � 1

3AB � 13AB � � � � 1

3AB � 13AB

� 13AB B

2� � 1

3B2� � � 1

3B2 1

3B2

� 13AB

13B

2B2� � � � � 1

3B2 1

3B2

..

. ... ..

. . .. ..

. ...

� 13AB

13B

2 13B

2� � � B

2� � 1

3B2

� 13AB

13B

2 13B

2� � � 1

3B2B2� �

2666666666664

3777777777775

M11

M22

..

.

MNN

0BBBB@1CCCCA ¼ 0,

which is equivalent to

A2� � � 1

3AB � 13AB � � � � 1

3AB � 13AB

0 23B

2� � 0 � � � 0 �ð23B

2� �Þ

0 0 23B

2� � � � � 0 �ð23B

2� �Þ

..

. ... ..

. . .. ..

. ...

0 0 0 � � � 23B

2� � �ð23B

2� �Þ

� 13AB

13B

2 13B

2� � � 1

3B2

B2� �

2666666666664

3777777777775

M11

M22

..

.

MNN

0BBBB@1CCCCA ¼ 0: ð49Þ

From the above relation, we deduce that � ¼ 23B

2 is an eigenvalue of �d and its

corresponding eigenmatrix belongs to

M2,1 ¼ ðMrsÞ : Mrs ¼ 0 if r 6¼ s or if r ¼ s ¼ 1,XNr¼2

Mrr ¼ 0

( ): ð50Þ

The multiplicity of this eigenvalue inM2 is at least N� 2.If � 6¼ 2

3B2, the system (49) will be reduced to: M11¼ � and M22 ¼M33 ¼ � � � ¼

MNN ¼�ffiffiffiffiffiffiffiN�1p , with �,� solutions of the following system

A2� � �


3AB

�


3AB

Nþ 1

3B2� �

26643775 �

�

� �¼ 0: ð51Þ

In order to get a nontrivial solution of the above system, � has to be a root of the

characteristic polynomial

pð�Þ ¼ �2 � A2þNþ 1

3B2

� ��þ

Nþ 1

3A

2B2�N� 1

9A

2B2,

that is,

�1,2 ¼A

2þ Nþ1

3 B2

2�


2� Nþ1

3 B2

2

!2

þN� 1

9A

2B2

vuut :


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The associated eigenvectors are of the form ��

� �with �, � 2R such that

� ¼ � tan 1,2,

where

tan 1,2 ¼

Nþ13 B

2�A2

2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNþ13 B

2�A2

2

� �2þ N�1

9 A2B2

r� 1

3


AB:

With this, we get that (29)–(30) and (35)–(36) hold. Moreover, we deduce that the

multiplicity of the eigenvalue 23B

2 is equal to N� 2 inM2 andðN�2ÞðN�1Þ

2 inM1, then

we conclude (32) and (38). Additionally, the identities (31), (37) are direct

consequences of (42). g

COROLLARY 4.4 Let us assume N¼ 2. The eigenvalues of d are simple and admit the

following formulas:

�1,2 ¼ �A

2þ B

2

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA

2� B

2

2

!2

þ1

9A

2B2

vuut ,

�3 ¼2

3AB:

Moreover, the corresponding spaces of eigenmatrices are given by

V1,2 ¼1 0

0 tan 1,2

�M11 : M11 2R, tan 1,2 ¼

A2�B2

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA

2�B2

2

� �2þ 1

9A2B2

r13AB

8>><>>:9>>=>>;,

V3 ¼0 1

1 0

�M12 : M12 2R

� �:

COROLLARY 4.5 In the particular case �¼ e1 and �¼ ep with p4 1, we have

d(�� ) � (�� )4 0.

Proof By symmetry of tensor d, we have

d ð�� Þ � ð�� Þ ¼1

4d ð�� þ � � �Þ � ð�� þ � � �Þ

¼1

4d

0 0 � � � 1 � � � 0

0 0 � � � 0 � � � 0

..

. ... . .

. ... . .

. ...

1 0 � � � 0 � � � 0

..

. ... . .

. ... . .

. ...

0 0 � � � 0 � � � 0

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA�

0 0 � � � 1 � � � 0

0 0 � � � 0 � � � 0

..

. ... . .

. ... . .

. ...

1 0 � � � 0 � � � 0

..

. ... . .

. ... . .

. ...

0 0 � � � 0 � � � 0

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

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¼�34

0 0 � � � 1 � � � 0

0 0 � � � 0 � � � 0

..

. ... . .

. ... . .

. ...

1 0 � � � 0 � � � 0

..

. ... . .

. ... . .

. ...

0 0 � � � 0 � � � 0

0BBBBBBBBB@

1CCCCCCCCCA�

0 0 � � � 1 � � � 0

0 0 � � � 0 � � � 0

..

. ... . .

. ... . .

. ...

1 0 � � � 0 � � � 0

..

. ... . .

. ... . .

. ...

0 0 � � � 0 � � � 0

0BBBBBBBBB@

1CCCCCCCCCA¼�32

4 0:

Using Theorem 4.2, we can deduce the following spectral decomposition of

dispersion tensor d(�). g

THEOREM 4.6 The spectral decomposition of d in terms of the eigenvectors of each

matrix which belongs to the spanning sets of Vi, i¼ 1, . . . , 4, is the following:

d¼ �1 cos 1e1� e1þ

sin 1ffiffiffiffiffiffiffiffiffiffiffiffiN� 1p

XNk¼2

ek� ek

!� cos 1e

1� e1þsin 1ffiffiffiffiffiffiffiffiffiffiffiffiN� 1p

XNk¼2

ek� ek

!

þ �2 cos 2e1� e1þ


XNk¼2

ek� ek

!� cos 2e


XNk¼2

ek� ek

!

þ1

2�3XNp¼2

e1þ epffiffiffi2p �


e1� epffiffiffi2p �

e1� epffiffiffi2p

� �

�e1þ epffiffiffi

2p �


e1� epffiffiffi2p �

e1� epffiffiffi2p

� �þ1

2�4

XNp¼2

XNq¼pþ1

epþ eqffiffiffi2p �


ep� eqffiffiffi2p �

ep� eqffiffiffi2p

� �

�epþ eqffiffiffi

2p �


ep� eqffiffiffi2p �

ep� eqffiffiffi2p

� �þXNp¼3

e2� e2� ep� ep� �

� e2� e2� ep� ep� �!

: ð52Þ

Proof For the first eigenvalue �1 of d, we have V1¼ span{M1} where

M1 ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ tan2 1p

1 0 � � � 0

0tan 1ffiffiffiffiffiffiffiffiffiffiffiffiN� 1p � � � 0

..

. ... . .

. ...

0 0 � � �tan 1ffiffiffiffiffiffiffiffiffiffiffiffiN� 1p

26666666664

37777777775¼

cos 1 0 � � � 0

0sin 1ffiffiffiffiffiffiffiffiffiffiffiffiN� 1p � � � 0

..

. ... . .

. ...

0 0 � � �sin 1ffiffiffiffiffiffiffiffiffiffiffiffiN� 1p

26666666664

37777777775:

The eigenvalues of matrix M1 are cos 1 (multiplicity 1) and sin 1ffiffiffiffiffiffiffiN�1p (multiplicity

N� 1). Their associated orthonormalized eigenvectors are e1 and ek,


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k2 {2, . . . ,N}, respectively. Then, the spectral decomposition of matrix M1 is asfollows:

M1 ¼ cos 1e1 � e1 þ


XNk¼2

ek � ek: ð53Þ

Analogously, V2¼ span{M2}, where the spectral decomposition of matrix M2 is

M2 ¼ cos 2e1 � e1 þ


XNk¼2

ek � ek: ð54Þ

In order to get the spectral decomposition of eigenmatrices in the spanning setsof V3 and V4, it is convenient to introduce the following notation: for anyp, q2 {1, . . . ,N}, p 6¼ q, we define the matrix Ep,q

¼ ((Ep,q)ij) by

ðEp,qÞij ¼1ffiffiffi2p

1 if ði, j Þ 2 fð p, qÞ, ðq, pÞg,

0 elsewhere.

�It is clear that the eigenvalues of matrix Ep,q are 1ffiffi

2p (multiplicity 1), �1ffiffi

2p (multiplicity 1)

and 0 (multiplicity N� 2). Their associated orthonormalized eigenvectors are epþeqffiffi2p ,

ep�eqffiffi2p and ek, k2 {1, . . . ,N}n{p, q}, respectively. Then, the spectral decomposition ofmatrix Ep,q is

Ep,q ¼1ffiffiffi2p

ep þ eqffiffiffi2p �

ep þ eqffiffiffi2p �

1ffiffiffi2p

ep � eqffiffiffi2p �

ep � eqffiffiffi2p ¼

1ffiffiffi2p ep � eq þ eq � epð Þ: ð55Þ

With the above remarks, for the third eigenvalue �3 of d, we getV3¼ span{E1,2, . . . ,E1,N}. Similarly, for �4 we have

V4 ¼ spannEp,q : p, q2 f2, . . . ,Ng, p5 q

o� span

nM4,p : p2 f3, . . . ,Ng

o,

where

ðM4,pÞij ¼1ffiffiffi2p

1 if ði, j Þ ¼ ð2, 2Þ,

�1 if ði, j Þ ¼ ð p, pÞ,

0 elsewhere.

8><>:The eigenvalues of matrixM4,p are 1ffiffi

2p (multiplicity 1), �1ffiffi

2p (multiplicity 1) and 0 (multi-

plicity N� 2). The associated orthonormalized eigenvectors are e2, ep and ek, k2

{1, . . . ,N}n{2, p}, respectively. Then, the spectral decomposition of matrix M4,p is

M4,p ¼1ffiffiffi2p ðe2 � e2Þ �

1ffiffiffi2p ðep � epÞ: ð56Þ

Since the following spectral decomposition of tensor d holds

d ¼ �1M1 �M1 þ �2M2 �M2 þ �3XNp¼2

E1,p � E1,p

þ �4XNp¼2

XNq¼pþ1

Ep,q � Ep,q þXNp¼3

M4,p �M4,p

!,

then using (53)–(56) we obtain the decomposition (52). g

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Remark 4.7 The spectral decomposition (52) can be rewritten as follows:

d¼ �1 cos 1e1� e1þ


XNk¼2

ek� ek

!� cos 1e


XNk¼2

ek� ek

!

þ �2 cos 2e1� e1þ


XNk¼2

ek� ek

!� cos 2e


XNk¼2

ek� ek

!

þ1

2�3XNp¼2

e1� epþ ep� e1� �

� e1� epþ ep� e1� �

þ1

2�4

XNp¼2

XNq¼pþ1

ep� eqþ eq� epð Þ � ep� eqþ eq� epð Þ

þXNp¼3

e2� e2� ep� ep� �

� e2� e2� ep� ep� �!

:

However, in the above decomposition the vectors involved in the tensor products are

not all eigenvectors of the corresponding matrix in the spanning sets of V3, V4.

Acknowledgements

The authors thank the anonymous referee for his/her comments on our work. Their suggestionsare noted and will be addressed in a future work. C. Conca thanks the MICDB for partialsupport through Grant ICM P05-001-F, Fondap-Basal-Conicyt. J. San Martın was partiallysupported by Grant Fondecyt 1090239 and BASAL-CMM Project. L. Smaranda was par-tially supported by Grant PN-II-RU-TE-2011-3-0059 of CNCS-UEFISCDI. M. Vanninathanthanks French CNRS and Institut de Mathematiques de Toulouse for hospitality and supportfor his stay in France during which this work was carried out.

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Burnett coefficients and laminates

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Transcript of Burnett coefficients and laminates