Facts,methods, programmes and paradigms

The concept of OR has widened at the Swedish Defence Research Institute (FOA) as at many other places during the last ten years. It has grown from academically faithful frameworks of science and mathematics into many other subjects and also into practical, purposeful activities with a weaker academic affiliation. Holism has been promoted “the hardway” and not only by inspiration from the Ackoff alarms. The awareness of work processes and working paradigms has likewise grown by the simulataneous stimulation from modern OR-literature and as uneasy bureaucratic context.The following taxonomy has a solid foundation in how OR is actually performed at the Institute:

     

2.

Construction of BIB designs with various support sizes—with special emphasis for v = 8 and k = 4     

A Hedayat  H.L Hwang 《Journal of Combinatorial Theory, Series A》1984,36(2):163-173

This paper deals with methods of constructing new BIB designs from known designs. First, we provide a method of finding trades on a given BIB design so that new designs with the same parameters ν, b, k can be constructed. Second, we point out that the method of combining designs mentioned in [9] is particularly useful for constructing designs based on ν = 2k, k even, when designs based on ν = 2k ? 1 are available. By utilizing the above methods with the aid of Table I in [4], we have constructed a table of 53 BIB designs based on V = 8 and k = 4 with support sizes $\text{14 ? b 1 (}\text{8}\text{4}\text{)=70}$ except for $\text{b}{}^1\text{= 15, 16, 17, 19}$. BIB designs with ν = 8, k = 4 and $\text{b}{}^1\text{< 14}$ are shown to be nonexistent.     

3.

Influence of a singular perturbation on the infimum of some functionals     

A Haraux  F Murat 《Journal of Differential Equations》1985,58(1):43-75

     

4.

On an infinite class of steiner systems with t = 3 and k = 6     

《Journal of Combinatorial Theory, Series A》1986,42(1):55-60

     

5.

Cardinal interpolation and spline fucntions V. The B-splines for cardinal Hermite interpolation     

I.J. Schoenberg  A. Sharma 《Linear algebra and its applications》1973,7(1):1-42

In the third paper of this series on cardinal spline interpolation [4] Lipow and Schoenberg study the problem of Hermite interpolation

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′,…,S}{}^{\mathrm{(r?1)}}\text{(v) = Y}{}_{\mathrm{v}}{}^{\mathrm{(r?1)}}\text{for all}\text{v}$

. The B-splines are there conspicuous by their absence, although they were found very useful for the case γ = 1 of ordinary (or Lagrange) interpolation (see [5–10]). The purpose of the present paper is to investigate the B-splines for the case of Hermite interpolation (γ > 1). In this sense the present paper is a supplement to [4] and is based on its results. This is done in Part I. Part II is devoted to the special case when we want to solve the problem

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′}\text{for all}\text{v}$

by quintic spline functions of the class C?(– ∞, ∞). This is the simplest nontrivial example for the general theory. In Part II we derive an explicit solution for the problem (1), where v = 0, 1,…, n.     

6.

Diffusion with interactions between two types of particles and Pressureless gas equations     

Azzouz Dermoune  Siham Filali 《Comptes Rendus Mathematique》2003,337(11):731-735

We construct two d-dimensional independent diffusions $\text{X}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{=a+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{a}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{,}\text{X}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{=b+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{b}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{b}}$, with the same viscosity ν≠0 and the same drift u(x,t)=(pρ_t^a(x)v₁+(1?p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1?p)ρ_t^b(x)), where ρ_t^a,ρ_t^b are respectively the density of X_t^a and X_t^b. Here $\text{a,b,v}{}_1\text{,v}{}_2\text{∈}\text{R}{}^{\mathrm{d}}$ and p∈(0,1) are given. We show that $\text{(ρ}{}_{\mathrm{t}}\text{(x)=pρ}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{(x)+(1?p)ρ}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{(x),u(x,t):}\text{t?0,}\text{x∈}\text{R}{}^{\mathrm{d}}\text{)}$ is the unique weak solution of the following pressureless gas system

such that $\text{ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pδ}{}_{\mathrm{a}}\text{+(1?p)δ}{}_{\mathrm{b}}\text{,}\text{u(x,t)ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pv}{}_1\text{δ}{}_{\mathrm{a}}\text{+(1?p)v}{}_2\text{δ}{}_{\mathrm{b}}$ as t→0+. To cite this article: A. Dermoune, S. Filali, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

7.

Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datumExistence et unicité de solutions renormalisées d'équations elliptiques non linéaires avec des termes d'ordre inférieur et données mesures     

M.Francesca Betta  Anna Mercaldo  François Murat  M.Michaela Porzio 《Comptes Rendus Mathematique》2002,334(9):757-762

In this Note we consider a class of noncoercive nonlinear problems whose prototype is

$\text{?△}{}_{\mathrm{p}}\text{u+b(x)|?u|}{}^{\mathrm{\lambda }}\text{=μ}\text{in}\text{Ω,}\text{u=0}\text{on}\text{?Ω,}$

where $\text{Ω}$ is a bounded open subset of $\text{R}{}^{\mathrm{N}}$ (N?2), △_p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on $\text{Ω}$ or a function in $\text{L}{}^1\text{(Ω)}$, λ?0 and b belongs to the Lorentz space $\text{L}{}^{\mathrm{N,1}}\text{(Ω)}$ or to the Lebesgue space $\text{L}{}^{\mathrm{\infty }}\text{(Ω)}$. We prove existence and uniqueness of renormalized solutions. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 757–762.     

8.

Kolmogorov-Landau inequalities for monotone functions     

A.M Fink 《Journal of Mathematical Analysis and Applications》1982,90(1):251-258

Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetN_n (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g⁽ⁿ⁾(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofN_nis

$\text{g}{}_1\text{(x) = 0}\text{if}\text{?∞ < x < ?1, g}{}_1\text{(x) = (1 + x)}{}^{\mathrm{n}}\text{if}\text{?1 ? x ? 0}$

. Ifg(x) ∈ N_nis such that

$\text{g(0) ? g}{}_1\text{(0) = 1, g}{}^{\mathrm{(n)}}\text{(0) ? g}{}_1{}^{\mathrm{(n)}}\text{(0) = n!}$

, then

$\text{g}{}^{\mathrm{(v)}}\text{(0) ? g}{}_1{}^{\mathrm{(v)}}\text{(0)}$

for

1$\text{v = 1,…, n ? 1}$

. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if $\text{g(x) = g}{}_1\text{(x)}$ in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4).     

9.

Singular solutions of some nonlinear elliptic equations     

Laurent Veron 《Nonlinear Analysis: Theory, Methods & Applications》1981,5(3):225-242

     

10.

On unique solvability of nonlocal drift–diffusion-type problems     

《Nonlinear Analysis: Theory, Methods & Applications》2004,56(6):803-830

     

11.

On convolution tails     

Paul Embrechts  Charles M. Goldie 《Stochastic Processes and their Applications》1982,13(3):263-278

     

12.

Multiple solutions for an elliptic system on bounded and unbounded domains     

《Nonlinear Analysis: Theory, Methods & Applications》2004,56(4):555-568

     

13.

Convergence properties of the strongly damped nonlinear Klein-Gordon equation     

《Journal of Differential Equations》1987,67(2):243-255

     

14.

On the number of zeros of diagonal cubic forms     

S Chowla  J Cowles  M Cowles 《Journal of Number Theory》1977,9(4):502-506

Let M_s, be the number of solutions of the equation

$\text{X}{}_1{}^3\text{+ X}{}_2{}^3\text{+ … + X}{}_{\mathrm{s}}{}^3\text{=0}$

in the finite field GF(p). For a prime p ≡ 1(mod 3),

$\text{∑}\text{s=1}\text{∞}\text{M}{}_{\mathrm{s}}\text{X}{}^{\mathrm{s}}\text{=}\text{x}\text{1 ? px}\text{+}\text{x}{}^2\text{(p ? 1)(2 + dx)}\text{1 ? 3px}{}^2\text{? pdx}{}^3$

,

$\text{M}{}_3\text{= p}{}^2\text{+ d(p ? 1)}$

, and

$\text{M}{}_4\text{= p}{}^2\text{+ 6(p}{}^2\text{? p)}$

. Here d is uniquely determined by

$\text{4}{}_{\mathrm{p}}\text{= d}{}^2\text{+ 27b}{}^2\text{and}\text{d ≡ 1(}\text{mod}\text{3)}$

.     

15.

Sublinear perturbations of the differential equation y(n) = 0 and of the analogous difference equation     

Attila Máté  Paul Nevai 《Journal of Differential Equations》1984,53(2):234-257

According to a result of A. Ghizzetti, for any solution y(t) of the differential equation where $\text{y}{}^{\mathrm{(n)}}\text{(t)+}\text{∑}\text{i=0}\text{n?1}\text{g}{}_{\mathrm{i}}\text{(t) y}{}^{\mathrm{i}}\text{(t)=0}\text{(t ? 1)}$, $\text{∝}{}_1{}^{\mathrm{\infty }}\text{¦g}{}_{\mathrm{i}}\text{(x)¦x}{}^{\mathrm{n?I?1}}\text{dx < ∞}$ (0 ?i ? n ?1, either y(t) = 0 for t ? 1 or there is an integer r with 0 ? r ? n ? 1 such that $\text{lim}\text{t → ∞}\text{y(t)}\text{t}{}^{\mathrm{r}}$ exists and ≠0. Related results are obtained for difference and differential inequalities. A special case of the former has interesting applications in the study of orthogonal polynomials.     

16.

Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form ut=Lu+Vu−γup in Rn     

《Journal of Differential Equations》2003,192(2):396-428

     

17.

Taylor expansion of best lp-approximations about p = 1     

《Applied Mathematics Letters》2003,16(7):993-998

     

18.

Realizing a weak solution on a probability space     

V.E. Beneŝ 《Stochastic Processes and their Applications》1978,7(2):205-225

     

19.

Approximations to generalized renewal measures     

《Stochastic Processes and their Applications》2004,113(1):127-142

     

20.

Critical exponents for the Pucci's extremal operatorsLes exposants critiques pour l'opérateur extrémal de Pucci     

Patricio L. Felmer  Alexander Quaas 《Comptes Rendus Mathematique》2002,335(11):909-914

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

(1)$\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}\text{(D}{}^2\text{u)+u}{}^{\mathrm{p}}\text{=0,u?0}\text{in}\text{R}{}^{\mathrm{N}}\text{.}$

Here N?3, p>1 and $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}$ denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents $\text{1<p}{}^{\mathrm{s}}{}_{\mathrm{+}}\text{<p}{}^1{}_{\mathrm{+}}\text{<p}{}^{\mathrm{p}}{}_{\mathrm{+}}$, that satisfy: (i) If $\text{1<p<p}{}^1{}_{\mathrm{+}}$ then there is no nontrivial solution of ($\text{1}$). (ii) If $\text{p=p}{}^1{}_{\mathrm{+}}$ then there is a unique fast decaying solution of ($\text{1}$). (iii) If $\text{p}{}^1\text{<p?p}{}^{\mathrm{p}}{}_{\mathrm{+}}$ then there is a unique pseudo-slow decaying solution to ($\text{1}$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($\text{1}$). Similar results are obtained for the operator $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{?}}$. To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.     

System/facts:	subject	object
method:	creativity	analysis
purpose/process:	interactive	advisory
paradigm:	intuitive	rational

Construction of BIB designs with various support sizes—with special emphasis for v = 8 and k = 4

This paper deals with methods of constructing new BIB designs from known designs. First, we provide a method of finding trades on a given BIB design so that new designs with the same parameters ν, b, k can be constructed. Second, we point out that the method of combining designs mentioned in [9] is particularly useful for constructing designs based on ν = 2k, k even, when designs based on ν = 2k ? 1 are available. By utilizing the above methods with the aid of Table I in [4], we have constructed a table of 53 BIB designs based on V = 8 and k = 4 with support sizes $\text{14 ? b 1 (}\text{8}\text{4}\text{)=70}$ except for $\text{b}{}^1\text{= 15, 16, 17, 19}$. BIB designs with ν = 8, k = 4 and $\text{b}{}^1\text{< 14}$ are shown to be nonexistent.     

Cardinal interpolation and spline fucntions V. The B-splines for cardinal Hermite interpolation

In the third paper of this series on cardinal spline interpolation [4] Lipow and Schoenberg study the problem of Hermite interpolation

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′,…,S}{}^{\mathrm{(r?1)}}\text{(v) = Y}{}_{\mathrm{v}}{}^{\mathrm{(r?1)}}\text{for all}\text{v}$

. The B-splines are there conspicuous by their absence, although they were found very useful for the case γ = 1 of ordinary (or Lagrange) interpolation (see [5–10]). The purpose of the present paper is to investigate the B-splines for the case of Hermite interpolation (γ > 1). In this sense the present paper is a supplement to [4] and is based on its results. This is done in Part I. Part II is devoted to the special case when we want to solve the problem

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′}\text{for all}\text{v}$

by quintic spline functions of the class C?(– ∞, ∞). This is the simplest nontrivial example for the general theory. In Part II we derive an explicit solution for the problem (1), where v = 0, 1,…, n.     

Diffusion with interactions between two types of particles and Pressureless gas equations

We construct two d-dimensional independent diffusions $\text{X}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{=a+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{a}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{,}\text{X}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{=b+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{b}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{b}}$, with the same viscosity ν≠0 and the same drift u(x,t)=(pρ_t^a(x)v₁+(1?p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1?p)ρ_t^b(x)), where ρ_t^a,ρ_t^b are respectively the density of X_t^a and X_t^b. Here $\text{a,b,v}{}_1\text{,v}{}_2\text{∈}\text{R}{}^{\mathrm{d}}$ and p∈(0,1) are given. We show that $\text{(ρ}{}_{\mathrm{t}}\text{(x)=pρ}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{(x)+(1?p)ρ}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{(x),u(x,t):}\text{t?0,}\text{x∈}\text{R}{}^{\mathrm{d}}\text{)}$ is the unique weak solution of the following pressureless gas system

such that $\text{ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pδ}{}_{\mathrm{a}}\text{+(1?p)δ}{}_{\mathrm{b}}\text{,}\text{u(x,t)ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pv}{}_1\text{δ}{}_{\mathrm{a}}\text{+(1?p)v}{}_2\text{δ}{}_{\mathrm{b}}$ as t→0+. To cite this article: A. Dermoune, S. Filali, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datumExistence et unicité de solutions renormalisées d'équations elliptiques non linéaires avec des termes d'ordre inférieur et données mesures

In this Note we consider a class of noncoercive nonlinear problems whose prototype is

$\text{?△}{}_{\mathrm{p}}\text{u+b(x)|?u|}{}^{\mathrm{\lambda }}\text{=μ}\text{in}\text{Ω,}\text{u=0}\text{on}\text{?Ω,}$

where $\text{Ω}$ is a bounded open subset of $\text{R}{}^{\mathrm{N}}$ (N?2), △_p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on $\text{Ω}$ or a function in $\text{L}{}^1\text{(Ω)}$, λ?0 and b belongs to the Lorentz space $\text{L}{}^{\mathrm{N,1}}\text{(Ω)}$ or to the Lebesgue space $\text{L}{}^{\mathrm{\infty }}\text{(Ω)}$. We prove existence and uniqueness of renormalized solutions. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 757–762.     

Kolmogorov-Landau inequalities for monotone functions

Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetN_n (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g⁽ⁿ⁾(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofN_nis

$\text{g}{}_1\text{(x) = 0}\text{if}\text{?∞ < x < ?1, g}{}_1\text{(x) = (1 + x)}{}^{\mathrm{n}}\text{if}\text{?1 ? x ? 0}$

. Ifg(x) ∈ N_nis such that

$\text{g(0) ? g}{}_1\text{(0) = 1, g}{}^{\mathrm{(n)}}\text{(0) ? g}{}_1{}^{\mathrm{(n)}}\text{(0) = n!}$

, then

$\text{g}{}^{\mathrm{(v)}}\text{(0) ? g}{}_1{}^{\mathrm{(v)}}\text{(0)}$

for

1$\text{v = 1,…, n ? 1}$

. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if $\text{g(x) = g}{}_1\text{(x)}$ in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4).     

On the number of zeros of diagonal cubic forms

Let M_s, be the number of solutions of the equation

$\text{X}{}_1{}^3\text{+ X}{}_2{}^3\text{+ … + X}{}_{\mathrm{s}}{}^3\text{=0}$

in the finite field GF(p). For a prime p ≡ 1(mod 3),

$\text{∑}\text{s=1}\text{∞}\text{M}{}_{\mathrm{s}}\text{X}{}^{\mathrm{s}}\text{=}\text{x}\text{1 ? px}\text{+}\text{x}{}^2\text{(p ? 1)(2 + dx)}\text{1 ? 3px}{}^2\text{? pdx}{}^3$

,

$\text{M}{}_3\text{= p}{}^2\text{+ d(p ? 1)}$

, and

$\text{M}{}_4\text{= p}{}^2\text{+ 6(p}{}^2\text{? p)}$

. Here d is uniquely determined by

$\text{4}{}_{\mathrm{p}}\text{= d}{}^2\text{+ 27b}{}^2\text{and}\text{d ≡ 1(}\text{mod}\text{3)}$

.     

Sublinear perturbations of the differential equation y(n) = 0 and of the analogous difference equation

According to a result of A. Ghizzetti, for any solution y(t) of the differential equation where $\text{y}{}^{\mathrm{(n)}}\text{(t)+}\text{∑}\text{i=0}\text{n?1}\text{g}{}_{\mathrm{i}}\text{(t) y}{}^{\mathrm{i}}\text{(t)=0}\text{(t ? 1)}$, $\text{∝}{}_1{}^{\mathrm{\infty }}\text{¦g}{}_{\mathrm{i}}\text{(x)¦x}{}^{\mathrm{n?I?1}}\text{dx < ∞}$ (0 ?i ? n ?1, either y(t) = 0 for t ? 1 or there is an integer r with 0 ? r ? n ? 1 such that $\text{lim}\text{t → ∞}\text{y(t)}\text{t}{}^{\mathrm{r}}$ exists and ≠0. Related results are obtained for difference and differential inequalities. A special case of the former has interesting applications in the study of orthogonal polynomials.     

Critical exponents for the Pucci's extremal operatorsLes exposants critiques pour l'opérateur extrémal de Pucci

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

(1)$\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}\text{(D}{}^2\text{u)+u}{}^{\mathrm{p}}\text{=0,u?0}\text{in}\text{R}{}^{\mathrm{N}}\text{.}$

Here N?3, p>1 and $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}$ denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents $\text{1<p}{}^{\mathrm{s}}{}_{\mathrm{+}}\text{<p}{}^1{}_{\mathrm{+}}\text{<p}{}^{\mathrm{p}}{}_{\mathrm{+}}$, that satisfy: (i) If $\text{1<p<p}{}^1{}_{\mathrm{+}}$ then there is no nontrivial solution of ($\text{1}$). (ii) If $\text{p=p}{}^1{}_{\mathrm{+}}$ then there is a unique fast decaying solution of ($\text{1}$). (iii) If $\text{p}{}^1\text{<p?p}{}^{\mathrm{p}}{}_{\mathrm{+}}$ then there is a unique pseudo-slow decaying solution to ($\text{1}$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($\text{1}$). Similar results are obtained for the operator $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{?}}$. To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.