Implicit Social Cues Influence the Interpretation of Intonation

James Sneed German Aix Marseille Univ, CNRS, LPL, Aix-en-Provence, France Individuals may have significant experience with more than one variety of their native language. If the non-native varieties are saliently linked to specific social identities, then an individual's production or perception can be biased towards a particular variety through contextual cues that exemplify the associated identity (e.g., [START_REF] Niedzielski | The effect of social information on the perception of sociolinguistic variables[END_REF], [START_REF] Hay | Factors influencing speech perception in the context of a merger-in-progress[END_REF]). In this study, we explore whether the interpretation of intonational patterns by Singapore English (SgE) listeners is influenced by implicit social cues. Specifically, we ask whether the relationship between accentuation and pronoun reference, which is robust in American (and other) varieties, but weak in SgE, varies with exposure to either a "Singaporean" or "American" contextual cue.

It is well-documented that the reference of personal pronouns in many English varieties depends on intonation, and specifically, whether a pronoun bears a pitch accent or not. Consider the following sentence: (1) Tim laughed at Shawn, then Stacy laughed at him. American English (AmE) listeners generally interpret unaccented him as referring to Shawn, and accented him to Tim. While these effects have been shown to depend on focus presuppositions associated with clause-level intonation patterns (e.g., [START_REF] Mcclay | Accented Pronouns and Contrast[END_REF]), what is of interest to our study is the role of prominence. The realization of this contrast depends on the fact that AmE freely places pitch accents in service of information structure. By comparison, the intonation system of SgE is edge-based: it does not have pitch accents and prominence is determined primarily by prosodic phrasing. Specifically, each Accentual Phrase (AP) begins with a L tone at its left edge, a H tone at its right edge, and has significant lengthening on the final syllable [START_REF] Chong | The accentual phrase in Singapore English[END_REF]. As a consequence, the prominence of a pronoun depends on its position in a phrase and does not correlate with information structure. Object pronouns tend to fall at the right edge of an AP and generally receive high prominence (which is likely responsible for the impression that SgE speakers stress pronouns inappropriately). Together, these facts predict that SgE listeners are insensitive to prominence in computing pronoun reference as compared with AmE listeners, which we confirmed in a pilot study. Nevertheless, most SgE individuals have substantial contact with AmE, suggesting that their system may adapt based on the regional identity of the speaker and other cues.

In our study, 40 SgE listeners were presented with spoken sentences, which varied in the accentual status of the object pronoun (as in ( 1)). Participants responded by choosing from two paraphrases, which reflected reference to either the subject or the object of the previous clause. In addition, participants were exposed before and during the experiment to either a "Singaporean" cue or an "American" cue, which took the form of an image of either a popular Singaporean or American television series. The image was construed as being present by accident and was therefore not explicitly linked to the identity of the speaker. The speaker was a native Singaporean with training which allowed him to produce AmE intonational contrasts while maintaining SgE segmental features. Our hypothesis was that if the contextual cues bias the listeners' toward specific systems, then they should show more sensitivity to accentual status in the American condition than in the Singaporean condition.

The results confirmed our hypothesis. Specifically, there was a significant interaction between cue type and accentual status in the expected direction. Contrary to pilot results, however, sensitivity to accentual status was not correlated with listeners' self-reported exposure to AmE, though surprisingly, it was negatively correlated with listeners' attitudes towards American culture. Our findings point towards an exemplar basis for the representation of intonation, in line with findings from explicit training [START_REF] Kurumada | Is it or isn't it: Listeners make rapid use of prosody to infer speaker meanings[END_REF] and collocational frequency [START_REF] Calhoun | Can intonation contours be lexicalised? Implications for discourse meanings[END_REF], though this is the first study that shows socio-indexical effects at the intonation-meaning interface. Future studies will explore at which level of representation the socio-indexical cues enter into the overall mapping from phonetic inputs to discourse-level meaning.


J Moreau

FREE BOUNDARIES AND SOME FIELD EQUATIONS : THROUGH THE NON-SMOOTH SOLUTIONS TO VARIATIONAL CHARACTERIZATION TRANSPORT METHOD

I. INTRODUCTION

Some methods primarily devised for the optimization of domains are currently used also in the numerical treatment of problems, arising from Physics or Engineering, which involve unknown boundaries.

This requires, of course, that the location of the said boundaries could be characterized variationally.

To the author's knowledge, it is in the dynamics Of inviscid fluids that variational statements of such a sort have been first

proposed. In that field of applications, the unknown surfaces may represent the free boundary through which a liquid confines an ~tmosphere with negligible inertia and given pressure ; they may also describe a jet boundary, in a possibly compressible flow, i.e. a discontinuity locus of the hydrodynamic field, separating two parts of the fluid with preserved material identity (shock waves do not fall into the scope of this lecture). Results in that llne have been known since the fifties [I] [2] In recent papers [START_REF] Moreau | Fluid dynamics and the calculus of horizontal variations, n~t[END_REF] ~] ,

the author has shown that, when considered from the viewpoint of the dynamics of the whole material in presence, the determination of such surfaces is a problem whose nonlinearity has the same intimate structure as the nonlinearity of the conventional equations of fluid dynamics holding in the regions of smooth flow. This is made clear by expressing dynamics in terms of Schwartz's distributions ;

in doing it, one puts forward the vector distribution divergence of a second order tensor measure associated with the time-space distribution of mass and velocity. For instance, the free boundary of a liquid appears as a surface, interior to the investigated region of time-space, across which the material density abruptly drops to the zero value corresponding to the assumedly mass-less atmosphere.

Expressing the dynamics of the whole in terms of distributions encompasses the usual equations, verified in the liquid domain, as well as the conditions to be satisfied on th~free surface.

Some details on this aspect of dynamics are given in Sect. 2 below ; for brevity only steady flows are considered.

The next step consists in giving a variational significance to the divergence operator acting on second order tensor measures. When the traditional calculus of variation is applied, with a view to characterize the possible solutions to some field equations, an alteration of the investigated field is performed by adding to it a term, arbitrary in a certain class of functions. This additive variation has to be smooth enough for the familiar trick of integration by part to work ; such a procedure leaves invariant the location of possible singularities and so is unable to characterize it. For this reason, we have instead proposed the transport method (called in [START_REF] Moreau | Fluid dynamics and the calculus of horizontal variations, n~t[END_REF] and [4] the method of horizontal variations ; it seems preferable to abandon this denomination which could generate confusion with some other uses of the word "horizontal", in Differential Geometry).

In this method every alteration of the investigated object is effeeted by transporting it along an arbitrarily chosen smooth vector field, say ~, with compact support in the considered region of ~n . This vector field may be viewed as the velocity field of some imagined continuous medium A , called a carrier. When the technique is used in a problem of continuum mechanics, one should keep in mind that the carrier has nothing to d O with the material in presence ; in particular, the real variable indexing the evolution of A in ~n is denoted by E , not to be confused with the time t of Dynamics, when the latter figures among the problem variables.

For every position of the carrier, a certain real functional, involving the transported object, is calculated. The result is a real function of • ; expressing that its derivative vanishes at T = 0 , whatever is the test vector field ~ , yields the expected characterization. Such a derivation, applied to a geometric object transported along a vector field is known in Differential Geometry as a Lie derivation. For the applications we have in view, it seems more efficient to describe the transport process in terms of the classical kinematics of continua, using but a little of the formalism of Differential Geometry.

Section 3 provides the necessary information about the concepts of vector and tensor distributions on a differential manifold, with emphasis on the special case where distributions are actually measures.

The transport of such objects by what we have called a carrier is described in Section 4.

This yields in Section 5 the very simple formula by which the divergence operation, acting on a doubly contravariant tensor measure, is interpreted variationally in terms of transport. The real functional to be extremized is the integral of the Euclidean trace of the investigated tensor measure. The reader specially interested in Mechanics could additionally refer to [START_REF] Moreau | Fluid dynamics and the calculus of horizontal variations, n~t[END_REF] , where Hamilton's principle of the least action is connected with this formula.

Section 6 gives some examples of applications of the preceding to hydrodynamical situations.

Then comes in Section 7 the study of the second derivative of the considered real functional, in the course of any twice differentiable transport, if the investigated tensor measure makes the first derivative vanish. This provides a necessary condition for the functional to achieve a local minimum. The calculation results in some positivity property, concerning the investigated tensor measure, which tends to explain the preeminence of measures, in that context, over distributions of higher order.

The final Section 8 summarizes the logical pattern of the transport method and sketches its application to more general situations.

STEADY FLOW OF AN INVISCID FLUID

In some region of a tridimensional reference frame, with orthonormal Cartesian coordinates x I , x 2 , x 3 , the steady motion of an inviscid fluid is considered. Let u] , u 2 , u 3 denote the components of the velocity field ; let p and p be the pressure and density scalar fields. The components of the gravity field are given, equal to the partial derivatives U . of some real function U.

,i

Then, under the usual smoothness assumption for the investigated fields, the Euler time-independent equations of fluid dynamics write down as 0 uj ui, j = -P,i + p U,i ' to be joined with the equation of mass conservation

(2.1)

(puj),j = 0 (2.2)
By combination, this yields (6ij is the Kronecker symbol) (Puiu j + p 6ij),j = PU, i .

(2.

3)

The left-hand side may be seen as the i-component of the vector field divergence of the symmetric tensor field with components in brackets.

If on the contrary the involved functions are not smooth enough for the partial derivatives to exist in the elementary sense, it is generally admitted that such partial differential equations as above have to be understood with reference to the partial derivatives of distributions. This is an abuse of language ; in Schwartz's theory of distributions, a real function f of the x-variables cannot constitute a distribution. But, as soon as f is locally integrable relative to the Lebesgue measure £ , there is defined the real measure f£ i.e. the measure admitting f as density relative to £ .

A real measure on some region of x-space is a special sort of Schwartz distribution ; then (2.3) will precisely be replaced by (p uiuj£ + p 6ijg) j = P U , i £ (2.4)

In order that the tensor measure with components in brackets make P u i u j n j + p n i = p u i u j n j + p n i • + Since O = 0 and o-u?n. = 0 , + ] J tion p = p this is just the pressure condi-

VECTOR AND TENSOR MEASURES ON A MANIFOLD

A possible way of constructing measure theory consists in taking the Riesz representation theorem as a definition. From that standpoint, by a real (signed) measure on a locally compact topological space X (one may precise : a Radon measure) is meant a real linear functional, meeting some continuity requirements we shall recall later, defined on the space D°(X) of the continuous real functions with compact support in X . The treatise of Bourbaki E5] is developed in that line, up to include the more sophisticated matter of measures with values in topological linear spaces.

In particular, the considered locally compact space may be an Real distributions of order £ on M make a topological linear space denoted by D'~M) ; real measures on M are the elements of D'°(M)

In the author's view , the above duality construction serves the purpose of Mechanics very well. Generally, dual linear spaces have been a basic ingredient of Classical Mechanics much before the concept was mathematically formalized. This duality is in fact the essence of the method of "virtual power" or "virtual work" which has played a central role in Mechanics since the 17 th century at least (some authors trace it back to Aristotle).

Another definite advantage of this approach Of measures is that it readily adapts to the introduction of vector measures on the mani- x + T'(x) 6 M ' ~M ' by its components T '13 , which are elements

X X
of L oc(X,~;]R); one may even choose ~ in order that they belong to L~oc(X,u;~) .

CARRIERS AND LIE DERIVATIVES

For all the sequel, X denotes a fixed n-dimensional C k manifold, k ~ 2 ; in usual applications, X simply reduces to an open subset of ~n .

Let ~ 6 VI(X,X ') , i.e. ~ is a C ! vector field of X , with compact support. We are to look at it as the Eulerian velocity field of some continuum A in motion over X . This precisely means that every element, or particle, of A is a moving point in X , Let us express as T + p(T,%) the motion of a particle

v z • m , v x • A : ~( z , x )
=¢(pCv,X)) . ' ÷ X' the tangent linear mapping to with ~ = p-l(x) and p% A~ x p at point A . This image belongs to C°(X,X ') iff ~ 6 C°(A,A').

For a covector field on A , the image is analogously defined, only Symmetrically, l e t us consider now a z-constant object, say a , related to X • Its image under p-| is a T-dependent object of T the same nature, related to A, say a T . Recall that the various classes of objects we agreed to consider constitutes each a linear spaces ; there is usually no difficulty in endowing those respective linear spaces with topologies and, due to the assumptions made at the start about the vector field ~ , to show that the derivative d~T/dT , at T = 0 , exists, an element of the same linear space.

In turn, this derivative admits an image under the placement Po ; k this is an object, related to the C -structure of X , of the same nature as a . Traditionnally this object is denoted by L~a and called the Lie derivative of a along ~ .

We finish this section by recalling an example of Lie derivative which plays an essential rule in the classical kinematics of continua.

As a T-constant object of the geometry of X , let us take a second order doubly covariant symmetric tensor field g assumed to be C I at least. Practically, this will be the tensor field involved in the definition of a Riemannian metric on X (or a pseudo-Riemannian metric, in Relativity theory) through the writing ds 2 = gijdxidxJ .

THE METRIC TRACE INTEGRAL OF A TENSOR MEASURE

C k

As before, X denotes a manifold, k ~ 2 . A doubly covariant symmetric C I tensor field g is supposed given in X ;

it is intended to define a Riemannian metric on X but, at the present stage, we do not need it to satisfy any positivity condition.

One considers a doubly contravariant symmetric tensor measure

T in X such that the integral J =~g'T>>= f gij dTIJ makes sense, i.e. the real measure number J ve to g .

(5.1)

gijdT 13 is bounded. The r e a l .

T 13>>

T I] .>> .

(5.3)

1,3 ' = -<~i ' , 3
Therefore the linear functional ~ -~.

. , TI]>> is a vector disl,] tribution of order one, whose Cartesian component of rank i equals T l] . . This vector distribution will naturally be called the diver-,3

gence of the tensor measure T , by analogy with the divergence of a tensor field, commonly used in the Mechanics of Continua. In the latter case, s iJ " denote the components of some C | (symmetric)

tensor field, relative to some orthonormal Cartesian frame ; partial derivation is understood in the traditional sense, so that the diverij is a C ° vector field. Here is the connection between g e n c e S ,j the two concepts : let £ denote the Lehesgue measure in our (locally) Euclidean space X ; one easily checks that slJg are the components of a tensor measure whose divergence, in distribution sense, equals the vector measure with components s zj .£ .

,J

As a result of the above calculations, we may formulate the following: More generally, (5.2) with (5.3) gives a variational meaning to the divergence operator, acting on symmetric tensor measures. This may be applied in characterizing variationally the solutions to some field equations which assert that such a divergence equals some vector distribution of adequate form. An example of this sort will be presented in the next Section.

Of course, it is not necessary to restrict oneself to locally Euclidean manifolds ; if X is pseudo-Riemannian everythings works the same way, provided we define the operator div : D'°(X'X' ~s X') ÷ D'I(x,x '~)

as the negative transpose of def : D](X,X ') + D°(X,X '~ ~ X '~) ; s recall that def denotes the operator ~ + (L g)/2 .

HYDRODYNAMICAL EXAMPLE

This Section is to show the ability of the transport method to provide a variational treatment of equations in which the divergence of a tensor measure appears jointly with other terms. The setting is that of Section 2 ; we write again equation (2.4) under the form (PUiUj~),j + (P£),i -PU, i£ = 0 . (6.1)

The tensor measure with components pu.u.£ has a peculiar 1 3 structure ; relatively to the nonnegative real measure m = p£ , it admits as density the tensor field with components u.u. , name-1 3 ly the tensor product of the vector u by itself. In order that such a tensor measure he transported by the carrier A in the way described in the preceding Section, i.e. that it equal for every T the image under pm of some T-constant doubly contravariant tensor measure on A , it is enough : firstly to have the measure m con- where ~ • pI(x,x') denotes the velocity field of the carrier A .

The transport method also gives a variational meaning to the T last term in (6.1) . Assuming as above that the measure m = pT£ is convected by the carrier, one considers the integral Let us finish by stressing that the condition (puj£),j of mass conservation is not involved in the above variational characterization. In that respect, one has to recall the observation made in Section 3 : the concept of a divergence-free vector measure belongs to the geometry of C l manifolds ; hence it is preserved under any C l diffeomorphism such as the placement p of the T carrier A into X . Consequently, for a vector measure convected by a carrier, the vanishing of divergence is a ~-invariant property. Now the law of transport defined in the preceding implies that the vector measure with components pu.~ is convected. It will 3 be pointed out in the sequel that the essence of the transport method consists in characterizing some investigated object as a critical point for some real functional in an infinite dimensional manifold. The elements of this manifold result from each other through the transport by carriers. In the present case all of them are divergence-free vector measures ; hence the vanishing of divergence does not constitute a constraint regarding the variational procedures.

An analogous remark applies to the fact that, since the investigated flow is supposed steady, the variable K must assume a constant value along each streamline. This property is evidently T T preserved when u and K are transported by any carrier in the imposed way.

THE SECOND VARIATION RATE

Let us come back to the setting of Proposition 5.1 . Some symmetric doubly contravariant tensor measure T on the locally Euclidean manifold X is supposed to have zero divergence ; equivalently, in view of the proposition, the trace integral J(T) of T T has zero time-derivative at t = 0 , whatever is the carrier A , with velocity field ~ 6 ~I(X,X') .

Incidentally, since the distribution T ij is by assumption a measure, the vanishing of the left-hand member of (5.3) for every 6 DI(x,X ') is secured, in view of a density argument, as soon as the same holds for ~ ranging over the subset D=(X,X ') , the C ~ vector fields with compact support in X .

We are now to investigate the second derivative of the function

T + J(T)
This of course requires some additional assumption concerning the carrier motion ; we shall suppose that ~ is twice differentiahle in T and x , i. Recall that the components of e , as well as the components of

I n
, essentially are functions of T,x ,...,x ; the notation ,t refers to the partial derivatives of such functions with respect to T , evaluated at point x = pT(k) . Then, using (7. The following Proposition allows one to discuss this condition.

PROPOSITION 7.2 As before, X denotes a locally Euclidean manifold. The expression (7.6) is nonnegative for every ~ 6 ~|(X,X')

(equivalently for every ~ e D~(X,X'))

if and only if one these equivalent conditions is satisfied :

(a) For every ~ e D°(X,X ') (equivalently for every ~6D (X,X))

one has (see e.g. [START_REF] Ros | Formes quadratiques positives sur des espaces de gradients[END_REF]) this may be proved to imply that the quadratic f functional is nonnegative for every ~ 6 ~ (X,X) and therefore, by density, for every ~ E 0°(X,X') ; this is property (a)

Conversely, the special form that expression In the applications of the transport method, the nonnegativity of J(0) appears as a natural "stability" requirement. For this reason the presentation of the method may practically be restricted to the case of tensor measures.

CONCLUSION

The logical pattern of the foregoing may be summarized as follows.

In a reference manifold X (this equals in practical instances an open subset of Euclidean ]R n) some investigated object U is required to satisfy a certain system of partial differential equations. Nonsmooth solutions are expected, i.e. the unknown object U is an element of some space J of distributions with prescribed tensorial type and distributional order.

Here are the essentials of the transport method.

A reference field r is specified in X . The preceding sections have been restricted to the case where r equals the doubly covariant tensor field g defining in X some Riemannian metric (and for simplicity most calculations have been performed with a Euclidean metric). Generally r may be a field whose tensoriel type and order of differentiability match the tensorial type and distributional order of U ; in other words, for every U in J , the real valued linear functional U ~ ~r,U>> makes sense.

A carrier A is defined in X by its velocity field, say E Dk(x,x ') , with k large enough for the transport of elements of J to be defined. Of course, the existence of solutions can only be expected in a weak sense, involving placements which are no more C I mappings of A onto X but only elements of some Soholev spaces. The numerical treatment of this system of partial differential equations requires a mesh in the manifold A : this is the same as the moving mesh considered in the preceding.

This throws some light on the structure of the set of the solutions to the equations of Hydrodynamics, as investigated in Section 6. These equations do no actually reduce to the simple form T 13 . = 0 , but the analogy with a problem of hyperelastic equili-,J brium is not destroyed by the additional terms ; the gravity term PU,i£ in (6.1) only plays the part of a loading (a "dead loading" in the usual case of constant gravity) while the pressure term modifies the density of elastic energy by adding a function of the Jacobian determinant of the placement. The essential nonlinearity of the equations of Hydrodynamics lies in the algebraic structure of the tensor measure T lj : its density, relative to the scalar measure p£ , equals the tensor product of the vector field u by ~tself.

Provided that the approximation process is initiated with a tentative solution T which meets such a requirement, the transport O method handles this condition automatically ; the nonnegativity property of Proposition 7.2 is also secured by itself. Observe that, if the hyperelastic analogy is brought about in numerical procedures, the constraint imposed to the fictitious elastic medium A may be relaxed : instead of assuming each boundary particle fixed, oue may permit the medium to slide along some part S of the boundary of the region X . If the approximation process is initiated with a fluid velocity field u tangential to S -a usual circumstance O in hydrodynamical problems-the transport by A will preserve this condition.

  sense, one supposes p 6 [~loc ' ui E i21oc ' p E [|loe (the components U • of the gravity field are essentially smooth functions). ,l Similarly (2.2) expresses that the vector measure, admitting as Cartesian components the three real measures pui£ , has zero divergence in the sense of distributions. Before showing how this formulation of hydrodynamics may encompass some free boundary situations, let us recall an elementary calculation rule for distributions. Assume that the considered domain ~ of the x-space is divided by a surface S determining two subdomains ~+ and £-At every point of S , denote by n. the components of the normal I unit, directed toward £+ and assumed continuous on S . If a function f is C l in £+ and £-and possesses respective onesided limits f+ and f-at every point of S , the measure f£ is a distribution on ~ whose partial derivative relative to x. l is easily found equal to the sum of the two following measures : area measure of S , a nonnegative real measure on ~ , with S as support. Coming back to hydrodynamics, suppose that ~-contains an + inviscid fluid, while ~ corresponds to an atmosphere with p = 0 . Mass conservation is expressed by (pui£)~ ,j = 0 ; through the above calculation rule, this is equivalent to (2.2) being satisfied in ~+ and ~-and to the vanishing of the measure concentrated on S whose density relative to the area measures equals [(puj) + -(pu.)-Jn.. Since p = 0 in ~+ the latter simply

  to define measures as distributions of a special sort. Let £ he an integer, 0 ~ £ Z k ; denote by Q£(M) the linear space of the C £ real functions on M with compact support. For every compact subset K of ~I , denote by ~( M ) the subspace of D£(M) consisting of the functions f with support contained in K . A Banach norm llflIK,£ is defined on ~(M)_ as the sup of the absolute values of f and of its partial derivatives up to order £ at all points of ~{ ; of course, this is conditioned by the choice of coordinates in the C k manifold M , but in view of £ ~ k , any admissible change of coordinates (more precisely the change of a covering of ~ with local charts for another one) replaces the said norm by an equivalent one. By definition, a real distribution of order £ (strictly speaking, one should say "of order ~ £") is is a linear functional on D£(M) whose restriction to every D K continuous.

  fold M . With every point x of M is associated the dual pair of n-dimensional linear spac~ M' x ' the tangent space, and M'x~ the cotangent space. By definition a field of covectors on M (or section of the cotangent fiber bundle) is an assignement associating with every x 6 M an element, say v(x) of M '~ . Basically, the x choice of a (local) coordinate system (xl,x 2,...,x n) in M induces respective bases in the linear spaces M' and A|'~ . Then the x x covector field v may be described by n functions vi(xl,x 2 expressing the components of v(x) in M '~ . Since, by axiom, any x ck change of (local) coordinate system in M is , k ~ l , the concept of the continuity of v is coordinate-free, as well as that of the support of such a field. We shall denote by D°(M,~ '~) the linear space of the continuous covector fields on M with compact support and by DK(M,~ '~) the subspace consisting of those fields whose support is contained in some compact subset K of M . The of the absolute values of the components of v E D~(M,M'~) sup constitutes a Banach norm llvl[ K on this space ; changing the coordinate system (more precisely changing the covering of ~ by local charts) replaces this norm by an equivalent one. By definition, a vector measure on M is a real linear functional on D°(M,M '~) whose restriction to each ~K(M,M'~) is continuous. More generally, such a duality procedure has been used by G. de Rham when constructing his theory of currents on C m manifolds ~] . Vector measures on M make a topological linear space denoted by D'O(M,M ') . Observe that, in contrast with the special case where M is an open subset of ~n , a vector measure on an arbitrary C k manifold M can by no means be seen as an additive vec-tor function of sets since it does not make sense to add vectors localized at different points of the manifold. When applied to a covector field v e D°(M,~ '~) , a vector measure m e D'°(M,M') yields, by definition, a real number denoted by ~v,m>> or, more expressively by f <v,dm> . A soon as m is fixed the meaning of this symbol may be extended to a larger class of covector fields than D°(M,M '~) , said integrable relative to m. From the Radon-Nikodym theorem one easily deduces that every m e D'°(~,~ ') can be (non uniquely) represented under the form m = m' B where ~ is a nonnegative real measure on M and m' is a locally ~-integrable vector field. This means that for every v e D % M , M '~) , one has f<v,dm> = f<v(x),m~(x)> d~ ; h e r e < . , . > d e n o t e s , a t e v e r y p o i n t x o f M , t h e r e a l -v a l u e d duality pairing of the cotangent and tangent spaces. Incidentally, a vector measure m 6 D'°(M,M ') is said divergence-free if, for every ~ 6 DI(~o , one has f<grad ~ , dm> = 0 ; we denote by grad ~ the gradient field of (or differential of ~) , naturally an element of D°(M,M ) .In[START_REF] Moreau | Le transport d'une mesure vectorielle par un fluide et le th~or~me de Kelvin-Helmholtz[END_REF] this concept has been applied to classical hydrodynamics, yielding a generalization of the Kelvin-Helmholtz theorem on vorticity which encompasses more recent results as the conservation of the he2icit9 of a flow. More generally, the divergence of a vector measure on M may be defined as a scalar distribution of order IThe point to be stressed is that this operation makes sense in the simple framework of the differential manifold M , without reference to any metric or connection ; this contrasts with the divergence of vector fields.A similar duality device may be generally used in defining tensor distributions, in particular tensor measures, on the C k manifold ~ . For instance, there exist four sorts of second order tensor fields on ~ : they are assignements associating respectively with every x 6 M an element of M i a M i , M I ~M ~ ~ , M ~ Mi or M~ ~ M~ ~ . This allows for the definition of spaces ~£ , D K£ consisting of such fields and, through duality, the definition of spaces of tensor distributions of order £ with specified tensorial type. In particular, a doubly contravariant tensor measure on ~ is an element of D ' °( M , M ' ~ M ') , the dual space of D °( M , M ' ~ '$) ; the latter consists of continuous doubly covariant tensor fields on ~ , with compact supports. As in the case of vector measures, the Radon-Nikodym theorem may be used in order to prove that, for every T 6 D ' °( ~, ~' ~ M ') , there exist (non uniquely) a nonnegative real measure ~ and a doubly contravariant locally ~-integrable tensor field T' such that T = T' ~ . Through the use of local coordinates (x ~,...,x n) in X , one defines the tensor field • o

  of (local) coordinates in X , the study of this differential equation in the manifold may be reduced to the similar problem in ~n , for which standard theory is available. The assumptions made about ~ secure that, for every ~o 6 X , there exists a unique solution • ÷ ~(T) to (4,]) , defined for T ranging over the whole real line, such that ~(0) -~o " Furthermore, standard results concerning the dependence on initial conditions of the solutions to differential equations, entail that, for every fixed T , the mapping ~ + ~(T) commonly denoted by O * exp T~ , is C | of X to itself. Since this mapping admits exp(-T)~ as inverse, it constitutes a C | diffeomorphism of X , leaving invarlant every point of the subset X\support ~ .

  a c a r r i e r such moving differential manifold as A , elaborated from a given C | vector field ~ in X . For simplicity, we started with ~ independent of T , i.e. the motion of A over X is a steady flow. It is sometimes useful to consider more generally a vector field x ÷ ~(~,x) in X , depending on T, at least for T ranging over some open real interval I containing zero ; then ~ will be supposed C I in T and x jointly (equivalently (T,x) + (l,~(T,x)) is a C I vector field of the product manifold I × X) ; in addition the support of x + ~(~,x) , for every T 6 I , will be assumed contained in a T-constant compact subset of X . Again this allows one to equip A with the structure of a C I manifold, in such a way that every placement + p(T,l) , T 6 1 , is a C I diffeomorphism of A onto X • With a C l manifold, as are A or X above, one may associate various linear spaces, respectively consisting of real functions, vector or tensor fields, distributions of order ~ l , etc. For every element of any of these spaces, there is a natural definition of its imaye under any C l diffeomorphism of the considered manifold onto another one ; this image is an object of the same nature attached to the target manifold. The spaces po , D] , p,o , P'| introduced in the preceding section generate examples of this. As another instance, consider the linear space cl(h,~) of the continuously differentiable real functions on the C l manifold A and some C l diffeomorphism p : A + X . For every real function K : h + ~, the image under p is naturally defined as the function k = K o p | : X ÷ ~ , which belongs to CI(x,~) iff K 6 C I ( A , ~) . Similarly may be considered a vector field on A , say A + a(%) 6 A~ . Its image under p is defined as x + p~(a(A)), ' :

I

  using instead of Pl its inverse transpose. Let us come back to the kinematical setting ; then p , T 6 1 T is the T-depending C 1 diffeomorphism of A onto X , generated by the given velocity field ~ . Let a be any of the objects we associated above with the C l differential structure of A . Assume a independent of T ; then its image under PT ' say a T , is a T-dependent object of the same nature related to X . Generally a T-varying object of X derived ~n that way from a T-constant object of A is said convected by the moving continuum, or carrier, A • This belongs to the vocabulary of usual physics : a function a which assigns to every particle of the moving continuum a timeindependent real value is commonly called a convected quantity. When observed from the "reference manifold" X , such a function ~ : A ÷ • is reflected as aT : X + ~ ; in the familiar case of C ] real functions, one elementarily characterizes convection by the transport equation 8aT/ST + <~,grad aT> = 0 .

  For every placement PT of the moving continuum A , the image of -I g under PT is a doubly eovariant symmetric tensor field on A ~jd~d~] that we shall denote by ~ Through the writing do 2 = y , there is defined a metric on A , actually the metric induced on A by its placement at time T in the Riemann space X . Saying that d~ 2 depends on ~ means that the continuum A is not expected to move "rigidly". For every particle I e A , y~(%) is a ~dependent element of the symmetrized tensor product A'~ ~s A~ ~ Under the smoothness assumptions previously made, one easily finds that the z-derivative ~T exists at ~ = 0 ; thereby is defined Y °e C°(A'A'~ ~s A'~) The image of ~o under the placement Po : C ° A + X constitutes the Lie derivative L~ g , a doubly covariant symmetric tensor field on X . The calculation of L~ g is performed, under diverse notations, in any textbook on the Mechanics of Continua ; we shall come back to this in Section 7 . Most books are restricted to the usual case where X equals an open subset of a Euclidean space ; then some l orthonormal Cartesian coordinates are used as x . If the (possibly T-dependent) velocity field ~ is described by its components ~i(xl,...,x n) relative to this Cartesian frame, the components of the tensor __ L g are found equal to ~i / ~x j + ~j / ~x i . In this setting of orthonormal Cartesian coordinates, it is usual to define the differential operator def , acting on differentiable vector fields, by writing e = def ~ for eij = (~i,j + ~j,i )/2 being orthonormal, in our Euclidean space, no distinction has to be made here between covariance and contravariance. In the elementary kinematics of continua, e is usually called the spatial strain rate tensor associated with the Eulerian velocity field ~ . In fact, knowing 2e = L~ g enables one to calculate what we have denoted above by yY ; thereby may be computed the time-rate of change of the length of any "infinitesimal material curve element", as soon as is known the image of this element in X under the placement p~ Actually an expression of L~ g formally as simple as (4.3) may be written in the general case where g defines a regular pseudo-Riemannian metric on X ; one finds (ef. [7]) t--~,~g, ij = ~ilj + ~jli 9i(xl,...,x n) are the covariant components of the possibly where z-dependent velocity field ~ and where lj refers to the covariant derivative in the j direction, relative to the pseudo-Riemannian connection.

  will be called the metric trace integral of T , relati-This name is suggested by the special case where X equals an open subset of some Euclidean space and g the tensor l n associated with the Euclidean metric ; then, if x ,...,x are orthonormal Cartesian coordinates, one has gij = I for i = j and zero otherwise, therefore J = f gij dTij = f dTii = i trace aT . In this special situation, no distinction has to be made between upper and lower indices ; T ii is a scalar measure, easily proved invariant under any orthogonal change of coordinates. The integr~ J is meaningful provided this measure is bounded. Let us define a carrier A by its velocity field ~ E DI(X,X'), possibly depending on T E I , an open real interval containing zero, in a C l way. Incidentally, the possible lack of boundedness of the real measure gijT zj comes from g not having a compact support. The essential of the calculations we shall perform below may be adapted to such a situation by restricting the integral to a compact subset of X containing, for every T E I , the support of ~ . Denote by T ~ the z-dependent doubly contravariant symmetric tensor measure on X , equal to T for T = 0 and convected by the carrier. According to the system of definitions developed under PT " We are to calculate the z-derivative, images of fields and measures or distributions are precisely devised in order to preserve the various duality pairings ; in particular, for every T 6 1 , one has ~g,TT>> = ~yT,8>> We have seen that, under the smoothness assumptions made, concerning the carrier motion, the x-derivative ~T exists, a continuous doubly covariant symmetric tensor field in A with compact support. Then a standard argument of derivation under the integral symbol (see e.g. ~3) yields -~-d ~g , T ~ = 2~def ~,In the case where X equals some open subset of an Euclidean space, with orthonormal Cartesian coordinates, no distinction is made between upper and lower indices and, in view of the symmetry of of partial derivatives in the theory of distributions, yields ~.

  X has zero divergence if and only if, for every carrier, as defined in Section 4 , the real function T ÷ f trace dT T has zero derivative at T = 0 ; here T T denotes the doublg contravariant symmetric tensor measure convected by the carrier, equal to T for T = 0 .

  vected by A , secondly to have the vector field u convected. The former means that a r-dependent real measure m r on the space X of the x-coordinates, reducing to m for T = 0 , equals the image under PT of some T-constant real measure ~ on A . This is the same as the familiar situation of the Mechanics of Continua : then ~ denotes the mass measure, defined independently of time on T a material continuum ; the image m of this real measure under the placement mapping pm : A + X admits a density, say pm , relatively to the Lebesgue measure ~ . Elementarily, the law of transport for pT consists in the following : the value of p corresponding to any determined particle X of A verifies p~J~ = const, where jr denotes the Jacobian determinant of pT at point (relative to an arbitrarily chosen coordinate system in A) . The metric trace integral of the above tensor measure equals = uiuiP d£ ; that is, formally, twice the kinetic energy of the investigated portion of fluid. As a result of Section 5 , by calculating the T-derivative at m = 0 of this expression

  C(T) = f UpTd£ • In order to calculate its T-derivative, one uses the fact that m T equals the image of a T-constant measure on A • When U is evaluated by following up the motion of a particle A • A in the reference manifold X { = ~ U ~( p T ( ~) ) d ~ = ~ U , i ~Z P d £ . (6.3)There finally remains to treat the middle term in (6.1) .Mechanically, the dynamical equation of inviscid flows have to be exploited in conjunctions with the knowledge of some compressibility law for the investigated fluid. We shall make here the traditional assumption of a barotropic flow, i.e. for every fluid particle, a certain relation between the density p and the pressure p is asserted, without explicit recourse to temperature. This holds in particular in the very usual situation where the fluid may be admitted to evolve isentropically. Let us write down this denoting the volume of unit mass. Here K is a variable of arbitrary mathematical nature, assumed to be a constant for every fluid particle ; this may refer as suggested above to the entropy of the particle, but, since our main motivation lies in the treatment of sharp inhomogeneities, one has also to be prepared to make K account for the chemical nature, possibly different in various parts of the flow. Traditionally, there is introduced a primitive of o ÷ p(K,o) , say P(<,o)We shall apply the transport method to the integral f P(K,o)~d£ , supposing as before that the measure m = p£ is convected, i.e. m T = pr£ with pTjT = const, for every ~ 6 A .On the other hand we agree to effect the transport with K kept independent of • for every % and, naturally, with o related to p by r = (OT)-I Under these assumptions one calculates the r-derivative of f P(K,~r)0rd£ . By applying to the carrier A the dilatation formula, classical in the Kinematics of Continua, one has, if o r is evaluated for a x-constant element of A , do r = oT ~i . . dr ,I Then, keeping in mind that ptd£ equals the image under PT the T-constant measure d~ on A , one finds, for T = 0 , d d--? ~P(K~'~T)Prd£ = f 8P da T dv ^ 3o dT = ~X P(K'O)~t'iOTDTd£ since ~ has compact support• Adding up the various terms, one obtains (cf. for every carrier with velocity field in ~I(X,X') ) = f (~ uiu i + p(KT,Ù "c) + U)O~d£ ; are transported by the carrier in the way defined in the preceding.

  e. the vector field (|,~) of the product manifold I × X is C 2 . T As before, y denotes the image of the tensor field g under -I Pr For every ~ 6 A , some element ~T(~) of A ~s A~ ~ constitutes the derivative of T + yT(I) if and only if, for every ! pair e,B of z-constant elements of A l , one has a(T) , b(T) of ~ and 8 under p~(X) (the tangent linear mapping to PT at point %) are moving vectors in X , associated with the moving point T ÷ pT(A) . Since X is locally Euclidean, we shall make use in it of Cartesian coordii nares x , nonneeessarily orthonormal. By the definition of y , one has, for every T E I , In the terminology of Section 4 , the moving vectors a(T) b(T) are convected by the carrier ; this is known to be and expressed by d ak(T) = %0k,i(T,x)ai(T ) dT (7°3) with x = pz(l) , and the similar law for b(z)(such a law is nothing hut the classical formula concerning the derivation, relative to initial data, for the solutions to the differential equation (4.1)) . Since x i are Cartesian coordinates in X , g is a constant, hence the right-hand side of (7'"" ~n denote the covariant components of the velocity .y field ~.This proves the existence of y , namely the image, -I under the tangent mapping to p , of the element 2e of Xx~ ~s X'~x with components ~£,k + @k,£ " What we have done here is only establishing that the tensor field 2e equals the Lie derivative L g , the classical fact from which Proposition 5.1 was deduced. We shall now apply the same technique in calculating the T-derivative of the expression (

  3) , one calculates the derivative of T ÷ ek£(T,pr(X))ak(T)b£(holds whatever is the couple s,fi of elements of A~ one starts with, this proves the existence of the second derivative "'T y of the function T + yT(A) ; it equals the image of the tensor -! 2h (~,pz(l)) under the tangent mapping to PT We shall use this to establish : PROPOSITION 7.1 Let T , a bounded s g m m e t r i c tensor m e a s u r e in the locally Euclidean manifold X, have zero divergence. Let T T denote the doubly contravariant tensor m e a s u r e convected by the carrier A , equal to T for T = 0 Then the real function T + J(~) = / trace dT T , whose first derivative vanishes at T = 0 in view of Proposition 5.1 , admits, as soon as the carrier velocitg is C 2, a second derivative at T = 0 equal to J(0) = 2 fgij ~i,k~J, £dT k£ • (7.61 °°T ° Proof. In view of the preceding• y Is a continuous doubly covariant tensor field of A , with compact support ; T T equals the image in X of a z-constant doubly contravariant tensor measure 0 in A.Therefore the standard procedure of derivation under the as follows the use of Proposition 7.1 .Suppose that a doubly eontravariant symmetric tensor measure, with bounded trace, is given in X . The totality of the tensor measures which can be obtained from this one through the transport by carriers with D l velocity fields constitutes, roughly speaking, an infinite-dimensional manifold, say M. Proposition 5.] states that, in this manifold, the elements with zero divergence are the critical points of the trace integral, a real function. Exploring the vicinity of such a critical point through the transport by carriers with D 2 velocity fields is the object of Proposition 7.1 . In particular, the nonnegativity of J is a necessary condition for the functional to achieve a local minimum at the considered point.

  every x 6 X the quadratic form ~ -> ~i~jTl lj (x) is nonnegative in ~n . Proof. It is understood that ~. denote the covariant components 1 • °of the vector field @ , while the components T IJ of the tensor measure are contravariant. Thus, the properties (a) and (b) are invariant under any change of coordinates. Let us take profit of this by assuming that orthonormal Cartesian coordinates are used t i i k£ in X ; then the expression (7.6) reduces to 2 J~ ,k ~ ,£dT Suppose this expression is nonnegative for every ~ E ~ (X,X'); in particular, one may fix i 6 {l,...,n} and assume i • Then the quadratic functional defined on D~(X,X ') as ~ + f $k$£ dTk£ is nonnegative for every vector field $ which equals the gradient of some 8 6 D (X,~) . Through the use of standard test functions

  implies (a) . Conversely, we have observed in Section 3 that a representation of the form T ij = T'iJ~ , with T 'ij 6 Lloo(X,p ) exists for every tensor measure. Suppose that (a) holds ; take ~i = ~i ~-, with 8 6 D°(X,~ +) and = (~|,...,~n) E ~n ; then S O ~i~j T 'ij~ d~ ~ 0 . This shows that ~i~j T~iJ(x) ~ 0 for every x 6 X ble exception of a ~-negligible subset N(~) . If a countable dense subset E of ~n , the union of ~-negligible. Hence, for ~-almost every x form ~ + ~i~; T~XJ(x) in nonnegative on negative on the whole of ~n. [] J REMARK]. We have defined in the foregoing some infinite-dimensional manifold M as the totality of the doubly contravariant tensor measures in X which result from one of them through the transport by carriers with D ! velocity fields. Clearly properties (a) and (b) are possessed by every element of M as soon as this is true for one of them. Such is the ease for the tensor measure considered restricted ourselves in the preceding to the transport of tensor measures. Actually, supposing again for simplicity that the manifold X is locally Euclidean, one may take as T 13 the components of a symmetric tensor distribution of order m . if the velocity field of a carrier A belongs to Dm+|(X,X ') , the transport of the tensor distribution T makes sense, generating wlth the possidensity it is non-a moving tensor distribution T ~ in X . Suppose, on the other hand, the real expression J = ~gij,T13>> meaningful ; this is true in particular if T has compact support. Thereby a real function + J(T) is defined for T in a neighborhood of zero. The derivative J of this function may be calculated as in Section 5 ; again it turns out that J(O) vanishes iff Tij,j = 0 . When this holds, one may come to the calculation of the second derivative ; like in Proposition 7.1 one finds i " J(0) = 2 ~g i j ~, k ~] , A 'Tk£>> a meaningful expression since ~ e Dm+I(x,x ') Similarly to Proposition 7.2 , this expression is found nonnegative iff, for every e D~(X,X ') , one has ~i#j,TIJ>> ~ 0 . Now this property implies that the tensor distribution T is actually a measure. The proof ~] is easily based on the classical fact that any nonnegative real distribution equals a measure•

  For instance, k = l allows for the transport of tensor measures ; generally the transport of tensorial objects with distributional order k-l is meaningful. As indicated in Section 4, ~ may also depend on the formal time T 6 1 (I is an open real interval containing zero) with continuous differentiability up to order k in I x X ; in that case it is assumed that the sup-port of x ÷ ~(T,x) is contained in a z-constant compact subset of X .Since ~ belongs to D k , the carrier A may be endowed with the structure of a Ck-manifold, in such a way that, for every 6 I , the placement mapping PT : A ~ X is a C k diffeomorphism.Assume that the tensor field r is C kcertain tensorial type, possibly astrained to some conditions of symmetry or skew-symmetry. Therefore, ,.÷ L~r is a continuous linear mapping, s a y D , of ~(X,X') to ~-I ( x , 7) . In the preceding sections, restricted to the case r = g , D equalled twice the operator def ; the divergence operator, acting on symmetric tensor measures, emerged as the negative transpose of def. Generally, the transpose of D is a linear opera-tor acting on distributions, say D ~ : D'k-I(x,T ~) ÷ D'k(x,x '~) ; here T ~ refers to the tensorial type dual of T . The transport method provides a variational characterization of the solutions U to D~U = 0 , or more generally to D~U = f , provided the righthand member f has an adequate form. For more comments on the prospect of this method, let us again restrict ourselves to the special case r = g , with X equal to an open subset of a Euclidean space. Using orthonormal Cartesian coordinates in this space, one looks for a symmetric tensor measure T ij with zero divergence, i.e. T ij . = 0 in X . We have inter-,] preted Proposition 5.1 by saying that T verifies this equation iff it constitutes a critical point of the real functional J on some infinite-dimensional manifold M . This manifold consists of the tensor measures obtained by transporting T along carriers with velocity fields in VI(x,x ') . The computational use of this remark may be contemplated by discretizing the carrier as a moving finite element mesh. The approximate representation of a tensor measure relatively to such a mesh, and of its transport when the mesh deforms, are easily imagined. Starting with a chosen tensor measure T , the mesh will be deformed stepwise. If T and o o therefore all the tensor measures obtained by transporting it possess the nonnegativity property involved in Proposition 7.2 , each step of mesh displacement will be devised so as to generate a walk toward a minimal point of J . The convergence of the process is naturally related to the boundary conditions one intends to satisfy. All the preceding has been developed with a carrier A whose velocity field has compact support in X . The mechanical analog is a continuous medium whose boundary particles are fixed. Determining a placement p : A + X which minimizes the real functional J under such a boundary constraint may then be seen as a special problem of hyperelastic equilibrium. With the notations of Sections 4 and 5, the corresponding elastic energy is expressed as & Yij deij ; here recall that 8 is a doubly contravariant tensor measure, constant on the manifold A • The doubly covariant tensor field y , i.e. the image of g -! under p , depends quadratically on the tangent mapping ~p/~% ; in fact, if pk(Al,...,~n) denote the components of p(X) relative to some orthonormal Cartesian frame of X , one has YiJ (A) = Bpk ~pk a~ i ~J This makes the determination of the equilibrium placement p a boundary value problem concerning a linear system of partial differential equations in the Ai variables, with a priori nonsmooth coefficients.


Laurent Praly

Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases

Convergence of the gradient algorithm for linear regression models in the continuous and discrete time cases

Laurent Praly

Continuous time case

Given a continuous vector function 1 t ∈ R → φ(t) ∈ R n , we let T(t, s) be the transition matrix associated with the non autonomous differential equation 2 ˙θ(t) = -φ(t)φ(t) θ(t) [START_REF] Aeyels | On the convergence of a time-variant linear differential equation arising in identification[END_REF] i.e. satisfying ∂ 1 T(t, s) = -φ(t)φ(t) T(t, s) , T(s, s) = I

where ∂ 1 denotes the partial derivative with respect to the first argument, t here. Let x be an arbitrary (constant) vector. We have

∂ 1 |T(t, s)x| 2 = -2|φ(t) T(t, s)x| 2
By defining the function fs as fs (t) = φ(t) T(t, s)x , fs (s) = φ(s) x

and integrating, we get

|T(t, s)x| 2 = |x| 2 -2 t s fs (r) 2 dr (4)
On another hand, by integrating (2), we get T(t, s) = I -t s φ(r)φ(r) T(r, s)dr [START_REF] Ren | Stochastic adaptive prediction and model reference control[END_REF] Incorporating this in the definition of fs yields fs (t) = φ(t) T(t, s)x = φ(t) I -

2 dr ≤ |x| 2 - 1 1 + t s |φ(u)| 2 du 2 t s f (u) 2 du ≤ |x| 2 - 1 1 + t s |φ(u)| 2 du 2 t s [φ(u) x] 2 du ≤ x    I - 1 1 + t s |φ(u)| 2 du 2 t s φ(u)φ(u) du    x
In other words we have simply

T(t, s) T(t, s) ≤ I - t s φ(u)φ(u) du 1 + t s |φ(u)| 2 du 2 (9)
Remark 1

1. Up to (7), we have identities. The conservativeness we may have in this last inequality is only in the majoration obtained in (8). See footnote 3.

2. The steps used up to (??) for the differential equation (1) can also be used for :

˙θ(t) = -ψ(t)φ(t) θ(t)
as we have for example in the least square algorithm.

3. With adapting the arguments used in the proof of the claim p.369 of Appendix B.2 of [START_REF] Marino | Nonlinear control design, geometric, adaptive and robust[END_REF], it should be possible to extend the above result to the system

η(t) = Aη(t) + Bφ(t) θ(t) , ˙θ(t) = -φ(t)Cη(t)
where the triple (B, A, C) is strictly positive real.

Other relations

We have also

∂ 2 T(t, s) = T(t, s)φ(s)φ(s)
By direct integration, we obtain the following other expression for T(t, s)

T(t, s) = I + s t T(t, u)φ(u)φ(u) du = I - t s T(t, u)φ(u)φ(u) du (10)
We "merge" the two expression ( 5) and (10) by substituting one inside the integral of the other. This gives where the kernel k is defined as

T(t, s) = I - t s φ(r)φ(r) I -
k(r, u) = φ(r) T(r, u)φ(u)
By inserting the expression (5) of T in this definition of k we get

k(t, s) = φ(t) T(t, s)φ(s) = φ(t) I - t s φ(r)φ(r) T(r, s)dr φ(s) = φ(t) φ(s) - t s φ(t) φ(r)φ(r) T(r, s)φ(s)dr = φ(t) φ(s) - t s φ(t) φ(r)k(r, s)dr (12)
With using (10), we obtain :

k(t, s) = φ(t) φ(s) - t s k(t, r)φ(r) φ(s)dr
A way to view the identity (12) is that, letting ϕ(t, s) = φ(t) φ(s), the kernel δ + ϕ is the inverse of the kernel δ -k where δ is the Dirac distribution. Precisely, for any C 1 test function f we obtain formally (a rigorous computation is given below)

t s (δ(t -u) + ϕ(t, u)) u s (δ(u -r) -k(u, r)) f (r)dr du = t s (δ(t -u) + ϕ(t, u)) f (u) - u s k(u, r)f (r)dr du = f (t) - t s k(r, s)f (r)dr + t s ϕ(t, u) f (u) - u s k(u, r)f (r)dr du = f (t) - t s k(r, s)f (r)dr + t s ϕ(t, u)f (u)du - t s t r ϕ(t, u)k(u, r)du f (r)dr = f (t) - t s k(r, s)f (r)dr + t s ϕ(t, u)f (u)du - t s (ϕ(t, r) -k(t, r)) f (r)dr = f (t) (13)
We exploit this remark to reestablish the relation (6) between fs defined in (3) and f defined in (7). With (11) and (12), we obtain :

fs (t) = φ(t) x - t s φ(t) φ(r)φ(r) xdr + t s φ(t) φ(r) r s k(r, u)φ(u) xdu dr = [φ(t) x] - t s φ(t) φ(u)[φ(u) x]du + t s t u φ(t) φ(r)k(r, u)dr [φ(u) x]du = [φ(t) x] - t s φ(t) φ(u) - t u φ(t) φ(r)k(r, u)dr [φ(u) x]du = [φ(t) x] - t s k(t, u)[φ(u) x]du
With the definition of f this yields :

fs (r) = f (r) - r s k(r, u)f (u)du or equivalently fs (u) = u s (δ(u -r) -k(u, r)) f (r)dr
So, in view of (13), we suspect that we have

f (t) = t s (δ(t -u) + ϕ(t, u)) fs (u)du .
This is indeed the case since, with (12), we obtain

fs (t) + t s φ(t) φ(r) fs (r)dr = f (t) - t s k(t, r)f (r)dr + t s φ(t) φ(r) f (r) - r s k(r, u)f (u)du dr = f (t) + t s (φ(t) φ(r) -k(t, r))f (r)dr - t s φ(t) φ(r) r s k(r, u)f (u)du dr = f (t) + t s t r φ(t) φ(u)k(u, r)du f (r)dr - t s t u φ(t) φ(r)k(r, u)dr f (u)du = f (t)
This proves that the transformation f → fs is invertible and resestablishes (6). Similarly, to any arbitrary C 1 test function f , its transform ft defined as

ft (s) = f (s) - t s f (r)k(r, s)dr It satisfies ft (s) + t s φ(u) φ(s) ft (u)du = f (s)
And, when f is given by (10), we have

ft (s) = x φ(s) - t s
x φ(r)φ(r) T(r, s)φ(s)dr ,

= x I - t s φ(r)φ(r) T(r, s)dr φ(s) , = x T(t, s)φ(s) ,
where we have used [START_REF] Ren | Stochastic adaptive prediction and model reference control[END_REF] to obtain the last identity.

From this, we can for instance obtain an upperbound for T(t, s)T(t, s) .

Discrete time case

Given a sequence of vectors φ(t) ∈ R n bounded4 in norm by φ ≤ √ 2, we let T(t, s) be the transition matrix associated with the non autonomous discrete time system

θ(t) = I -φ(t)φ(t) θ(t -1) i.e. satisfying T(t, s) = I -φ(t)φ(t) T(t -1, s) , T(s, s) = I Our problem is to find sufficient conditions implying lim t→+∞ |T(t, s)| = lim t→+∞ sup x |T(t, s)x| |x| = 0 .
Let x be an arbitrary unit vector. We have

T(t, s)x = T(t -1, s)x -φ(t)φ(t) T(t -1, s)x
With denoting, for t ≥ s + 1,

fs (t) = φ(t) T(t -1, s)x , fs (s + 1) = φ(s + 1) x we get |T(t, s)x| 2 = T(t -1, s)x -φ(t) fs (t) T(t -1, s)x -φ(t) fs (t) = |T(t -1, s)x| 2 -2x T(t -1, s) φ(t) fs (t)φ(t) φ(t) fs (t) 2 = |T(t -1, s)x| 2 -2 -φ(t) φ(t) fs (t) 2 . . . |T(s + 1, s)x| 2 = |x| 2 -2 -φ(s + 1) φ(s + 1) fs (s + 1) 2
So summation gives

|T(t, s)x| 2 = |x| 2 - t r=s+1 2 -φ(r) φ(r) fs (r) 2 (14)
On another hand, we have

T(t, s) = T(t -1, s) -φ(t)φ(t) T(t -1, s) T(t -1, s) = T(t -2, s) -φ(t -1)φ(t -1) T(t -2, s)
. . .

T(s + 1, s) = I -φ(s + 1)φ(s + 1) T(s, s)
So again summation gives

T(t, s) = I - t r=s+1 φ(r)φ(r) T(r -1, s) (15)
Incorporating this in the expression of fs yields, for t ≥ s + 2,

fs (t) = φ(t) T(t -1, s)x = φ(t) I - t-1 r=s+1 φ(r)φ(r) T(r -1, s) x = φ(t) x - t-1 r=s+1 φ(t) φ(r) φ(r) T(r -1, s)x = f (t) - t-1 r=s+1 φ(t) φ(r) fs (r) ∀t ≥ s + 2 ,
where we have let

f (t) = φ(t) x (16)
We have also fs (s + 1) = f (s + 1)

With the Cauchy-Schwarz inequality we obtain

f (u) 2 ≤ 1 + u-1 r=s+1 [φ(u) φ(r)] 2 u r=s+1 fs (r) 2 ≤ 1 + |φ(u)| 2 u-1 v=s+1 |φ(v)| 2 u r=s+1 fs (r) 2 ∀u ≥ s + 2 f (s + 1) 2 = fs (s + 1) 2
and therefore

t u=s+1 f (u) 2 ≤ fs (s + 1) 2 + t u=s+2 1 + |φ(u)| 2 u-1 v=s+1 |φ(v)| 2 u r=s+1 fs (r) 2 ≤ 1 + t u=s+2 1 + |φ(u)| 2 u-1 v=s+1 |φ(v)| 2 fs (s + 1) 2 + t r=s+2 t u=r 1 + |φ(u)| 2 u-1 v=s+1 |φ(v)| 2 fs (r) 2 ≤ (t -s) + t u=s+2 |φ(u)| 2 t-1 v=s+1 |φ(v)| 2 fs (s + 1) 2 + t r=s+2 (t -r + 1) + t u=r |φ(u)| 2 t-1 v=s+1 |φ(v)| 2 fs (r) 2 ≤ (t -s) + t u=s+2 |φ(u)| 2 t-1 v=s+1 |φ(v)| 2 t r=s+1 fs (r) 2 ≤   (t -s) + t u=s+1 |φ(u)| 2 2   t r=s+1 fs (r) 2 (17)
With the definition (16) of f (u) this inequality is

x t u=s+1 φ(u)φ(u) x ≤   (t -s) + t u=s+1 |φ(u)|
2 -φ(r) φ(r)   (t -s) + t u=s+1 |φ(u)| 2 2   x t u=s+1 φ(u)φ(u) x
In other words we have

T(t, s) T(t, s) ≤ I - min r∈{s+1,...,t} 2 -|φ(r)| 2   (t -s) + t u=s+1 |φ(u)| 2 2   t u=s+1 φ(u)φ(u) . ( 18
)
Remark 2 The same final remarks as for the continuous time case can be done here. In particular about the majoration (17), a less conservative bound is obtained in the proof of [2, Theorem 4.5] (or of [5, Theorem 2.2, 1st column, p. 2052], for the case where the preprocessing mentioned at the beginning of this section is

φ(t) = Φ(t) r(t) , r(t) = r(t -1) + |Φ(t)| 2
and the analysis in carried out exploiting the assumption that the sequence r goes to infinity in some specific way.

Other relations

We have also

T(t, s) = T(t, s + 1) I -φ(s + 1)φ(s + 1)
and therefore

T(t, s) = T(t, s + 1) -T(t, s + 1)φ(s + 1)φ(s + 1) T(t, s + 1) = T(t, s + 2) -T(t, s + 2)φ(s + 2)φ(s + 2)
. . .

T(t, t -1) = I -T(t, t)φ(t)φ(t)
So again by summation, we get an indentity similar to ( 15)

T(t, s) = I - t u=s+1 T(t, u)φ(u)φ(u) (19)
Let us mix the two expressions (15) and ( 19) of T we have found :

T(t, s) = I - t r=s+1 φ(r)φ(r) T(r -1, s) = I -φ(s + 1)φ(s + 1) - t r=s+2 φ(r)φ(r) I - r-1 u=s+1 T(r -1, u)φ(u)φ(u) = I - t r=s+1 φ(r)φ(r) + t r=s+2 φ(r) r-1 u=s+1 k(r, u)φ(u) = I - t u=s+1 φ(u)φ(u) + t-1 u=s+1 t r=u+1 φ(r)k(r, u) φ(u) = I + t u=s+1 t r=u φ(r)k(r, u) φ(u)
with denoting k(r, u) = φ(r) T(r -1, u)φ(u) ∀r ≥ u + 1 .

Note that we have k(r, r -1) = φ(r) φ(r -1) .

(20) Also, with the expression (15) of T, we see that k satisfies, for t ≥ s + 1,

k(t, s) = φ(t) T(t -1, s)φ(s) = φ(t) I - t-1 r=s+1 φ(r)φ(r) T(r -1, s) φ(s) = φ(t) φ(s) - t-1 r=s+1 φ(t) φ(r)φ(r) T(r -1, s)φ(s) = φ(t) φ(s) - t-1 r=s+1 φ(t) φ(r)k(r, s) (21)
Note that we have also

k(t, s) = φ(t) φ(s) - t-1 r=s+1 k(t, u)φ(u) φ(s) (22)
The identities ( 21) and ( 22) are is very meaningful. A closer look shows that by denoting :

-K(t, s) be the lower triangular matrix with 1 on the diagonal and k(u, v) as element of raw u and column v, -L(t, s) be the lower triangular matrix with 1 on the diagonal and φ(u) φ(v) as element of raw u and column v, we have L(t, s)K(t, s) = K(t, s)L(t, s) = I Precisely, let f (t) be arbitrary. We obtain, with (20) and (21),

f (t) - t-1 u=s+1 k(t, u) f (u) + t-1 r=s+1 φ(t) φ(r) f (r) - r-1 u=s+1 k(r, u) f (u) = f (t) - t-1 u=s+2 k(t, u)f (u) -k(t, s + 1)f (s + 1) + t-1 r=s+2 φ(t) φ(r) f (r) - r-1 u=s+1 k(r, u)f (u) + φ(t) φ(s + 1)f (s + 1) = f (t) + t-1 u=s+1 φ(t) φ(u) -k(t, u) f (u) - t-2 u=s+1 t-1 r=u+1 φ(t) φ(r)k(r, u) f (u) = f (t) + t-1 u=s+1 φ(t) φ(u) -k(t, u) f (u) - t-2 u=s+1 k(t, u) -φ(t) φ(u) f (u) = f (t) + φ(t) φ(t -1) -k(t, t -1) f (t -1) = f (t) (23)
A similar result can be obtained from (22).

Convergence for the discrete and continuous time case

Let t i be strictly positive real numbers going to +∞ with t 0 = 0. For any t, there exists τ (t) such that t is between t τ (t) and t 1+τ (t) . We have seen that both in the discrete and continuous time case, there exist a real number π i in [0, 1] satisfying

|T(t i , t i-1 )| ≤ 1 -π i Specifically, since √ 1 -a ≤ 1 -a 2
in the continuous time case, (9) gives

π i ≥ 1 2 λ min t i t i-1 φ(u)φ(u) du 1 + t i t i-1 |φ(u)| 2 du 2 (24)
With denoting φi = esssup t∈[t i-1 ,t i ] |φ(t)|, a more conservative lowerbound for π i is

π i ≥ 1 2 λ min t i t i-1 ≤ φ(u)φ(u) du 1 + (t i -t i-1 ) 2 φ4 i (25)
Also, in [START_REF] Aeyels | On the convergence of a time-variant linear differential equation arising in identification[END_REF], the following assumption is introduced: There exist α and β such that there exists a function T such that we have, for all t ≥ 0 α I ≤ T (t) t φ(r)φ(r) dr ≤ β I .

Since we have

T (t) t |φ(r)| 2 dr = trace T (t) t φ(r)φ(r) dr ≤ nλ max T (t) t φ(r)φ(r) du
The above assumption implies that,by letting

t i = T i (0) ,
we have, for all i α ≤ λ min

t i t i-1
φ(r)φ(r) dr ,

t i t i-1 |φ(r)| 2 dr ≤ n β
and therefore

π i = α 1 + n 2 β 2
in the discrete time case, with φ2 smaller than 2, (18) gives

π i ≥ min r∈{1+t i-1 ,...,t i } 2 -|φ(r)| 2 2   (t i -t i-1 -1) +   t i u=1+t i-1 |φ(u)| 2   2   λ min   t i 1+t i-1 φ(u)φ(u)   (26)
or the more conservative lower bound

π i = 2 -φ2 2(t i -t i-1 -1)(1 + (t i -t i-1 -1) φ4 ) λ min   t i 1+t i-1 φ(u)φ(u)   So we have |T(t, 0)| = T(t, t τ (t) ) τ (t) i=1 T(t i , t i-1 ) ≤ T(t, t τ (t) ) τ (t) i=1 |T(t i , t i-1 )| ≤ τ (t) i=1 (1 -π i ) ≤ exp   - τ (t) i=1 π i  
where to obtain the last inequality we have used the property

(1 -x) ≤ exp(-x) ∀x ∈ R .
We conclude that |T(t, 0)| tends to 0 if we can find T and the t i 's such that we get

∞ i=1 π i = +∞ .
Discussion : To show the interest of this result, we compare it with the persistent excitation (spanning) condition. We do this here for the continuous time case only, but the same holds for the discrete time case. The vector function φ is said persistently exciting or spanning if there exist two strictly positive real numbers ε and T such that, for any t, the Gram matrix It is established inn [START_REF] Morgan | On the uniform asymptotic stability of certain linear nonautonomous differential equations[END_REF] that this condition is necessary and sufficient to have the uniform asymptotic stability of the origin for (1)

The condition of non summability of π i above implies attractiveness but not uniform attractiveness. It can be seen weaker than the persistent excitation or spanning condition in two ways: the level ε may decrease with t, the width T of the time window may increase with t.

Specifically, let T be fixed and let ε i be the level reached by the Gram matrix on the ith time window [(i -1)T, iT ], i.e.

ε i = λ min iT (i-1)T
φ(s)φ(s) ds then, with (25), π i is not summable if ε i is not, i.e. Now, let ε be fixed and, with t 0 = 0, let t i be the smallest time such that the Gram matrix on the time window [t i-1 , t] is larger than the level ε, i.e. Addition of the reference [START_REF] Aeyels | On the convergence of a time-variant linear differential equation arising in identification[END_REF] suggested by Romeo Ortega and of the comment on how π i can be chosen constant when the assumption proposed in that paper holds.

Modifications on January 15, 2017 Addition of the discussion on the relation between non summability of π i and the persistent excitation (spanning) condition.

Modifications on January 13, 2017

The lower bounds (24) and ( 26) have been changed to follow a suggestion of Romeo Ortega of giving less conservative lower bound for π i . Before they were, with t i+1 -t i ≤ T , -for the continuous time case,

π i = 1 2(1 + T 2 φ4 ) λ min t i t i-1
φ(u)φ(u) du -for the discrete time case

π i = 2 -φ2 2T (1 + T φ4 ) λ min   t i 1+t i-1 φ(u)φ(u)  

Modifications on may 18, 2018

The paragraphs "Other relations" have been added. They are actually extracted from a previous note.

  )φ(r) T(r, s)dr x = φ(t) x -t s φ(t) φ(r) φ(r) T(r, s)x dr = f (t) -t s φ(t) φ(r) fs (r)dr(6)where we have letf (t) = φ(t) x .(7)It follows that we have 3 |f (u) -fs (u)| 2 = φ(u)

  , u)φ(u) du dr (11)

  φ(s) ds on a time window with width T is above the level ε, i.e. λ min t+T t φ(s)φ(s) ds ≥ ε ∀t ≥ 0 .

  )φ(s) ds = +∞ .

1 1 +

 1 t i = min t : λ min t t i-1 φ(s)φ(s) ds ≥ ε t then, with (25), π i is not summable if ∞ i=1 (t i -t i-1 ) 2 = +∞ .4 History of the versionsModifications on January 20, 2017

See footnote 2