1 Metrical multi-time Lagrange spaces

for metrical Multi-Time Lagrange Spaces

Mircea Neagu and Constantin Udri¸ste

Dedicated to the memory of Radu Rosca (1908-2005)

Abstract. In this paper we construct a geometrization on the 1-jet fiber bundleJ¹(T, M) for the multi-time quadratic Lagrangian function

L=h^αβ(t)gij(t, x)xⁱ_αx^j_β+U_(i)^(α)(t, x)xⁱ_α+F(t, x).

Our geometrization includes a nonlinear connection Γ, a generalized Cartan canonical Γ-linear connection CΓ together with its d-torsions and d-curvatures, naturally provided by the given multi-time quadratic Lagrangian functionL.

Mathematics Subject Classification:53B40, 53C60, 53C80.

Key words: metrical multi-time Lagrange spaces, nonlinear connections, Cartan canonical connection,d-torsions,d-curvatures.

1 Metrical multi-time Lagrange spaces

It is important to note that quadratic multi-time Lagrangians dominates most sci- entific domains. We can remind only the theory of elasticity [19], the dynamics of ideal fluids, the magnetohydrodynamics [6], [7], the theory of bosonic strings [5] or the multi-time evolution (p-flow) of some physical or economical phenomena [21, 22, 23, 24, 25, 26]. This fact emphasizes the importance of the geometrization of quadratic multi-time Lagrangians. In conclusion, a Riemann-Lagrange geometry on 1-jet spaces was required. Such a geometry was initially developed by Saunders [20]

and Asanov [2], and continued, in a Miron’s approach [10], by Udri¸ste ([21]-[26]) and Neagu ([13], [16]).

In the sequel, let us fixh= (hαβ(t^γ)) a semi-Riemannian metric on the temporal manifold T and let g = (gij(t^γ, x^k, x^k_γ)) be a symmetric d-tensor onE =J¹(T, M), of ranknand having a constant signature.

Generally, a smooth multi-time Lagrangian function

L:E→R, EI 3(t^α, xⁱ, xⁱ_α)→L(t^α, xⁱ, xⁱ_α)∈R,I (1.1.1)

Balkan Journal of Geometry and Its Applications, Vol.11, No.1, 2006, pp. 87-98.∗

°c Balkan Society of Geometers, Geometry Balkan Press 2006.

produces afundamental vertical metrical d-tensor G^(α)(β)_(i)(j) = 1

2

∂²L

∂xⁱ_α∂x^j_β, (1.1.2)

wherei, j= 1, ..., nandα, β= 1, ..., p.

Definition 1.1. A multi-time Lagrangian functionL:E→R, having the fundamen-I tal vertical metrical d-tensor of the form

G^(α)(β)_(i)(j) (t^γ, x^k, x^k_γ) =h^αβ(t^γ)gij(t^γ, x^k, x^k_γ), (1.1.3)

is called aKronecker h-regular multi-time Lagrangian function.

In this context, we can introduce the following important concept:

Definition 1.2. A pair M Lⁿ_p = (J¹(T, M), L), p = dimT, n = dimM, consisting of the 1-jet fibre bundle and a Kronecker h-regular multi-time Lagrangian function L:J¹(T, M)→R, is called aI multi-time Lagrange space.

Remark 1.3. i) In the particular case(T, h) = (IR, δ), a multi-time Lagrange space is called arelativistic rheonomic Lagrange space and is denoted by

RLⁿ= (J¹(IR, M), L).

For more details about the relativistic rheonomic Lagrangian geometry, the reader may consult [15].

ii) If the temporal manifold T is 1-dimensional one, then, via a temporal repara- metrization, we have

J¹(T, M)≡J¹(IR, M).

In other words, a multi-time Lagrange space, having dimT = 1, is a reparametrized relativistic rheonomic Lagrange space.

Example 1.4. Let us suppose that the spatial manifold M is also endowed with a semi-Riemannian metricg= (gij(x)). Then, the multi-time Lagrangian function

L1:E→R,I L1=h^αβ(t)gij(x)xⁱ_αx^j_β (1.1.4)

is a Kroneckerh-regular one. It follows that the pair BSM Lⁿ_p = (J¹(T, M), L1)

is a multi-time Lagrange space. It is important to note that the multi-time Lagrangian L1=L1

p|h|is exactly the ”energy” Lagrangian, whose extremals are ultra-harmonic maps between the semi-Riemannian manifolds(T, h)and(M, g)[4]. At the same time, the multi-time Lagrangian that governs the physical theory of bosonic strings is of type L1 [6].

Example 1.5. (geometric dynamics [21, 22, 23, 24, 25, 26]) Let us start with a p-flow described by the completely integrable PDES system

∂xⁱ

∂t^α =X_αⁱ(t, x(t)), i= 1, ..., n; α= 1, ..., p.

This system and the semi-Riemannian metrics h^αβ(t) and gij(t, x) determine the quadratic Lagrangian function

L2:E→R,I L2=h^αβ(t)gij(t, x)(xⁱ_α−X_αⁱ(t, x(t))(x^j_β−X_β^j(t, x(t)), (1.1.5)

which is Kronecker h-regular. If the metricshandgare positive definite, then this is the least squares Lagrangian. Also, we remark that any PDEs system can be replaced with a p-flow (multi-time evolution), and consequently it produces a Lgrange-Hamilton problem via any quadratic Lagrange function of preceding form.

Example 1.6. In the above notations, taking U_(i)^(α)(t, x) a d-tensor field on E and F :T×M →RI a smooth function, the quadratic multi-time Lagrangian function

L2:E→R,I L2=h^αβ(t)gij(x)xⁱ_αx^j_β+U_(i)^(α)(t, x)xⁱ_α+F(t, x), (1.1.6)

is also a Kroneckerh-regular one. The multi-time Lagrange space EDM Lⁿ_p = (J¹(T, M), L2)

is called theautonomous multi-time Lagrange space of electrodynamics. This is because, in the particular case(T, h) = (IR, δ), the spaceEDM Lⁿ₁ naturally genera- lizes the classical Lagrange space of electrodynamics, that governs the movement law of a particle placed concomitently into a gravitational field and an electromagnetic one. From physical point of view, the semi-Riemannian metric hαβ(t) (resp. gij(x)) represents the gravitational potentials of the manifoldT (resp. M), the d-tensor U_(i)^(α)(t, x) plays the role of the electromagnetic potentials which produce a gyro- scopic force, and F is a potential function. The non-dynamical character of the spatial gravitational potentials gij(x)motivates us to use the term”autonomous”.

Example 1.7. More general, if we consider the symmetrical d-tensorgij(t, x)onE, of ranknand having a constant signature onE, we can define the Kroneckerh-regular multi-time Lagrangian function

L3:E →R,I L3=h^αβ(t)gij(t, x)xⁱ_αx^j_β+U_(i)^(α)(t, x)xⁱ_α+F(t, x).

(1.1.7)

The multi-time Lagrange space

N EDM Lⁿ_p = (J¹(T, M), L₃)

is called thenon-autonomous multi-time Lagrange space of electrodynamics.

We use the term ”non-autonomous”, in order to emphasize the dynamical character of spatial gravitational potentialsgij(t, x), i.e., their dependence of the temporal coor- dinates t^γ.

An important role and, at the same time, an obstruction in the subsequent devel- opment of the theory of the multi-time Lagrange spaces, is played by the following theorem, proved in [12]:

Theorem 1.8. (of characterization of multi-time Lagrange spaces) If p= dimT ≥2, then the following statements are equivalent:

i)L is a Kroneckerh-regular Lagrangian function on J¹(T, M).

ii) The multi-time Lagrangian functionLreduces to a multi-time Lagrangian func- tion of non-autonomous electrodynamic type, that is

L=h^αβ(t)gij(t, x)xⁱ_αx^j_β+U_(i)^(α)(t, x)xⁱ_α+F(t, x).

Corollary 1.9. The fundamental vertical metrical d-tensor of an arbitrary Kronecker h-regular multi-time Lagrangian functionLis of the form

G^(α)(β)_(i)(j) =1 2

∂²L

∂xⁱ_α∂x^j_β =

( h¹¹(t)gij(t, x^k, y^k), p= dimT = 1 h^αβ(t^γ)gij(t^γ, x^k), p= dimT ≥2.

(1.1.8)

Remark 1.10. i) It is obvious that the preceding theorem is an obstruction in the development of a fertile geometrical theory for the multi-time Lagrange spaces. This obstruction was surpassed in the paper [13], by introducing the more general notion of ageneralized multi-time Lagrange space. The generalized multi-time Riemann- Lagrange geometry onJ¹(T, M)will be constructed using only a Kronecker h-regular vertical metrical d-tensorG^(α)(β)_(i)(j) and a nonlinear connectionΓ,”a priori” given on the 1-jet space J¹(T, M).

ii) In the case p= dimT ≥2, the preceding theorem obliges us to continue our geometrical study of the multi-time Lagrange spaces, directing our attention upon the non-autonomous multi-time Lagrange spaces of electrodynamics.

Let M Lⁿ_p = (J¹(T, M), L), where dimT =p, dimM = n, be a multi-time La- grange space whose fundamental vertical metrical d-tensor metric is

G^(α)(β)_(i)(j) = 1 2

∂²L

∂xⁱ_α∂x^j_β =

( h¹¹(t)gij(t, x^k, y^k), p= 1 h^αβ(t^γ)gij(t^γ, x^k), p≥2.

Supposing that the semi-Riemannian temporal manifold (T, h) is compact and orientable, by integration on the manifold T, we can define the energy functional associated to the multi-time Lagrange functionL, taking

EL:C^∞(T, M)→R,I EL(f) = Z

T

L(t^α, xⁱ, xⁱ_α)p

|h|dt¹∧dt²∧. . .∧dt^p,

where the smooth mapf is locally expressed by (t^α)→(xⁱ(t^α)) andxⁱ_α= ∂xⁱ

∂t^α. The extremals of the energy functional ELverify the Euler-Lagrange PDEs

2G^(α)(β)_(i)(j) x^j_αβ+ ∂²L

∂x^j∂xⁱ_αx^j_α− ∂L

∂xⁱ + ∂²L

∂t^α∂xⁱ_α + ∂L

∂xⁱ_αH_αγ^γ = 0, (1.1.9)

where x^j_αβ= ∂²x^j

∂t^α∂t^β and H_αβ^γ are the Christoffel symbols of the semi-Riemannian temporal metrichαβ.

Taking into account the Kroneckerh-regularity of the Lagrangian functionL, it is possible to rearrange the Euler-Lagrange equations of the LagrangianL=Lp

|h|in the followinggeneralized Poisson form (ultra-hyperbolic partial differential equations):

Œhx^k+ 2G^k(t^µ, x^m, x^m_µ) = 0, (1.1.10)

where

Œhx^k=h^αβ{x^k_αβ−H_αβ^γ x^k_γ},

2G^k= g^ki 2

½ ∂²L

∂x^j∂xⁱ_αx^j_α− ∂L

∂xⁱ + ∂²L

∂t^α∂xⁱ_α + ∂L

∂xⁱ_αH_αγ^γ + 2gijh^αβH_αβ^γ x^j_γ

¾ . Proposition 1.11. i) The geometrical object G = (G^r) is a multi-time dependent spatialh-spray.

ii) Moreover, the spatialh-sprayG= (G^l)is theh-trace of a multi-time dependent spatial spray G= (G⁽ⁱ⁾_(α)β), that isG^l=h^αβG^(l)_(α)β.

Proof. The proof of this proposition is given in [12].

Following previous reasonings and the preceding result, we can regard the equa- tions (1.1.10) as being the equations of the ultra-harmonic maps of a multi-time dependent spray.

Theorem 1.12. The extremals of the energy functionalELattached to the Kronecker h-regular Lagrangian function Lare ultra-harmonic maps on J¹(T, M)of the multi- time dependent spray(H, G) defined by the temporal components

H_(α)β⁽ⁱ⁾ =









−1

2H₁₁¹ (t)yⁱ, p= 1

−1

2H_αβ^γ xⁱ_γ, p≥2 and the local spatial componentsG⁽ⁱ⁾_(α)β=

=







 h11g^ik

4

· ∂²L

∂x^j∂y^ky^j− ∂L

∂x^k + ∂²L

∂t∂y^k + ∂L

∂x^kH₁₁¹ + 2h¹¹H₁₁¹gkly^l

¸

, p= 1 1

2Γⁱ_jkx^j_αx^k_β+T_(α)β⁽ⁱ⁾ , p≥2,

wherep= dimT.

Definition 1.13. The multi-time dependent spray(H, G)constructed in the preceding Theorem is called thecanonical multi-time spray attached to the multi-time Lagrange spaceM Lⁿ_p.

In the sequel, by local computations, the canonical multi-time spray (H, G) of the multi-time Lagrange spaceM Lⁿ_p induces naturally a nonlinear connection Γ on J¹(T, M).

Theorem 1.14. The canonical nonlinear connection Γ = (M_(α)β⁽ⁱ⁾ , N_(α)j⁽ⁱ⁾ )

of the multi-time Lagrange space M Lⁿ_p is defined by the temporal components

M_(α)β⁽ⁱ⁾ = 2H_(α)β⁽ⁱ⁾ =





−H₁₁¹ yⁱ, p= 1

−H_αβ^γ xⁱ_γ, p≥2, (1.1.11)

and the spatial components

N_(α)j⁽ⁱ⁾ = ∂Gⁱ

∂x^jγ

hαγ=











h11∂Gⁱ

∂y^j, p= 1

Γⁱ_jkx^k_α+g^ik 2

∂gjk

∂t^α +g^ik

4 hαγU_(k)j^(γ), p≥2, (1.1.12)

whereGⁱ=h^αβG⁽ⁱ⁾_(α)β.

Remark 1.15. In the particular case(T, h) = (IR, δ), the canonical nonlinear connec- tionΓ = (0, N_(1)j⁽ⁱ⁾ )of the relativistic rheonomic Lagrange spaceRLⁿ = (J¹(IR, M), L) generalizes naturally the canonical nonlinear connection of the classical rheonomic Lagrange space Lⁿ= (IR×TM, L) [10].

2 Generalized Cartan canonical connection CΓ of a metrical multi-time Lagrange space

Now, let us consider thatM Lⁿ_p = (J¹(T, M), L) is a multi-time Lagrange space, whose fundamental vertical metrical d-tensor is

G^(α)(β)_(i)(j) = 1 2

∂²L

∂xⁱ_α∂x^j_β =

( h¹¹(t)gij(t, x^k, y^k), p= 1 h^αβ(t^γ)gij(t^γ, x^k), p≥2.

Let Γ = (M_(α)β⁽ⁱ⁾ , N_(α)j⁽ⁱ⁾ ) be the canonical nonlinear connection of the multi-time La- grange spaceM Lⁿ_p.

The main result of this Section is the Theorem of existence and uniqueness of thegeneralized Cartan canonical connectionCΓ, which allowed us to develop in the paper [14] the multi-time Riemann-Lagrange geometry of physical fields, theory that represents a natural generalization of the classical field theories (theFinslerian theory [1], [2] and theordinary Lagrangian theory[10]).

Theorem 2.1. (the generalized Cartan canonical connection)

On the multi-time Lagrange spaceM Lⁿ_p = (J¹(T, M), L), endowed with the canonical nonlinear connectionΓ, there is a unique h-normal Γ-linear connection

CΓ = (H_αβ^γ , G^k_jγ, Lⁱ_jk, C_j(k)^i(γ)), having the metrical properties:

i)gij|k= 0, gij|^(γ)_(k)= 0, ii) G^k_jγ =g^ki

2 δgij

δt^γ , L^k_ij=L^k_ji, C_j(k)^i(γ)=C_k(j)^i(γ),

where”_|α”,”_|i”and”|^(α)_(i)” are the local covariant derivatives of theh-normalΓ-linear connectionCΓ.

Proof. Let CΓ = ( ¯G^γ_αβ, G^k_jγ, Lⁱ_jk, C_j(k)^i(γ)) be an h-normal Γ-linear connection, whose local coefficients are defined by the relations ¯G^γ_αβ=H_αβ^γ ,G^k_jγ =g^ki

2 δgij

δt^γ and Lⁱ_jk= g^im

2

µδgjm

δx^k +δgkm

δx^j −δgjk

δx^m

¶ , C_j(k)^i(γ)= g^im

2

̶gjm

∂x^k_γ +∂gkm

∂x^jγ

−∂gjk

∂x^m_γ

! . (2.2.1)

Taking into account the local expressions of the local covariant derivatives induced by the connection Γ, by a local calculation, we deduce that CΓ satisfies the conditions i) and ii).

Conversely, let us consider an h-normal Γ-linear connection CΓ = ( ˜¯˜ G^γ_αβ,G˜^k_jγ,L˜ⁱ_jk,C˜_j(k)^i(γ))

which satisfies the metrical conditionsi)andii). In this context, we have G˜¯^γ_αβ=H_αβ^γ , G˜^k_jγ= g^ki

2 δgij

δt^γ . Moreover, the condition gij|k = 0 is equivalent to

δg_ij

δx^k =gmjL˜^m_ik+gimL˜^m_jk.

Applying now a Christoffel process to the indices{i, j, k}, we find L˜ⁱ_jk= g^im

2

µδg_jm

δx^k +δg_km δx^j −δg_jk

δx^m

¶ .

By analogy, using the relations C_j(k)^i(γ) = C_k(j)^i(γ) and gij|^(γ)_(k) = 0 and using also a Christoffel process applied to the indices{i, j, k}, we obtain

C˜_j(k)^i(γ)= g^im 2

Ã

∂gjm

∂x^k_γ +∂gkm

∂x^jγ

−∂gjk

∂x^m_γ

! .

In conclusion, the uniqueness of the generalized Cartan canonical connection CΓ is clear.

Remark 2.2. i) Replacing the canonical nonlinear connection Γ with an arbitrary nonlinear connection, the preceding Theorem holds good.

ii) In the particular case(T, h) = (IR, δ), the generalizedδ-normal Γ-linear Cartan connection associated to the relativistic rheonomic Lagrange space

RLⁿ= (J¹(IR, M), L)

generalizes naturally the canonical Cartan connection of a classical rheonomic La- grange space Lⁿ= (IR×TM, L), constructed in [10].

iii) The generalized Cartan canonical connection of the multi-time Lagrange space M Lⁿ_p verifies also the metrical properties

h_αβ/γ =h_αβ|k=h_αβ|^(γ)_(k)= 0, g_ij/γ = 0.

iv) In the case p= dimT ≥2, the coefficients of the generalized Cartan canonical connection of the multi-time Lagrange space M Lⁿ_p reduce to

G¯^γ_αβ=H_αβ^γ , G^k_jγ= g^ki 2

∂gij

∂t^γ , Lⁱ_jk= Γⁱ_jk, C_j(k)^i(γ)= 0.

3 Local d-torsions and d-curvatures of CΓ

Applying the formulas that determine the local d-torsions and d-curvatures of an h-normal Γ-linear connection∇Γ (see [16]) to the generalized Cartan canonical con- nectionCΓ, we obtain the following results:

Theorem 3.1. The torsion d-tensor T of the generalized Cartan canonical connec- tionCΓof the multi-time Lagrange spaceM Lⁿ_p is determined by the local components

hT hM v

p= 1 p≥2 p= 1 p≥2 p= 1 p≥2

hThT 0 0 0 0 0 R_(µ)αβ^(m)

hMhT 0 0 T_1j^m T_αj^m R^(m)_(1)1j R^(m)_(µ)αj

hMhM 0 0 0 0 R^(m)_(1)ij R^(m)_(µ)ij

vhT 0 0 0 0 P_(1)1(j)^{(m) (1)} P_(µ)α(j)^{(m) (β)}

vhM 0 0 P_i(j)^m(1) 0 P_(1)i(j)^{(m) (1)} 0

vv 0 0 0 0 0 0

(3.3.1)

where,

i) forp= dimT = 1 we have

T_1j^m=−G^m_j1 , P_i(j)^m(1)=C_i(j)^m(1) , P_(1)1(j)^{(m) (1)}=−G^m_j1 , P_(1)i(j)^{(m) (1)}= ∂N_(1)i^(m)

∂y^j −L^m_ji , R^(m)_(1)ij = δN_(1)i^(m)

δx^j −δN_(1)j^(m) δxⁱ ,

R^(m)_(1)1j =−∂N_(1)j^(m)

∂t +H₁₁¹



N_(1)j^(m)−∂N_(1)j^(m)

∂y^k y^k



;

ii) for p= dimT ≥2, denoting F_i(µ)^m =g^mp

2

·∂gpi

∂t^µ +1

2hµβU_(p)i^(β)

¸ , H_µαβ^γ = ∂H_µα^γ

∂t^β −∂H_µβ^γ

∂t^α +H_µα^η H_ηβ^γ −H_µβ^η H_ηα^γ , r_pij^m = ∂Γ^m_pi

∂x^j −∂Γ^m_pj

∂xⁱ + Γ^k_piΓ^m_kj−Γ^k_pjΓ^m_ki, we have

T_αj^m=−G^m_jα, P_(µ)α(j)^{m (β)} =−δ_γ^βG^m_jα, R^(m)_(µ)α(j)=−H_µαβ^γ x^m_γ, R_(µ)αj^(m) =−∂N_(µ)j^(m)

∂t^α +g^mk 2 H_µα^β

·∂gjk

∂t^β +hβγ

2 U_(k)j^(γ)

¸ , R_(µ)ij^(m) =r_ijk^mx^k_µ+

h

F_i(µ)|j^m −F_j(µ)|i^m i

;

Theorem 3.2. The curvature d-tensorRof the generalized Cartan canonical connec- tionCΓof the multi-time Lagrange spaceM Lⁿ_p is determined by the local components

hT hM v

p= 1 p≥2 p= 1 p≥2 p= 1 p≥2

hThT 0 H_ηβγ^α 0 R^l_iβγ 0 R^(l)(α)_(η)(i)βγ

h_Mh_T 0 0 R^l_i1k R^l_iβk R^(l)(1)_(1)(i)1k =R^l_i1k R_(η)(i)βk^(l)(α) hMhM 0 0 R_ijk^l R^l_ijk R^(l)(1)_(1)(i)jk=R^l_ijk R^(l)(α)_(η)(i)jk

vhT 0 0 P_i1(k)^{(l) (1)} 0 P_(1)(i)1(k)^{(l)(1) (1)} =P_i1(k)^{(l) (1)} 0 vhM 0 0 P_ij(k)^l(1) 0 P_(1)(i)j(k)^{(l)(1) (1)} =P_ij(k)^l(1) 0

vv 0 0 S_i(j)(k)^l(1)(1) 0 S(l)(1)(1)(1)

(1)(i)(j)(k)=S_i(j)(k)^l(1)(1) 0 whereR^(l)(α)_(η)(i)βγ=δ^α_ηR^l_iβγ+δ_i^lH_ηβγ^α ,R^(l)(α)_(η)(i)βk=δ_η^αR^l_iβk ,R_(η)(i)jk^(l)(α) =δ_η^αR^l_ijk and

i) forp= dimT = 1 we have R^l_i1k= δG^l_i1

δx^k −δL^l_ik

δt +G^m_i1L^l_mk−L^m_ikG^l_m1+C_i(m)^l(1)R^(m)_(1)1k, R^l_ijk= δL^l_ij

δx^k −δL^l_ik

δx^j +L^m_ijL^l_mk−L^m_ikL^l_mj+C_i(m)^l(1)R^(m)_(1)jk, P_i1(k)^l(1) = ∂G^l_i1

∂y^k −C_i(k)/1^l(1) +C_i(m)^l(1)P_(1)1(k)^{(m) (1)}, P_ij(k)^l(1) =∂L^l_ij

∂y^k −C_i(k)|j^l(1) +C_i(m)^l(1)P_(1)j(k)^{(m) (1)} ,

S^l(1)(1)_i(j)(k)=∂C_i(j)^l(1)

∂y^k −∂C_i(k)^l(1)

∂y^j +C_i(j)^m(1)C_m(k)^l(1) −C_i(k)^m(1)C_m(j)^l(1) ; ii) for p= dimT ≥2we have

H_ηβγ^α =∂H_ηβ^α

∂t^γ −∂H_ηγ^α

∂t^β +H_ηβ^µ H_µγ^α −H_ηγ^µ H_µβ^α , R^l_iβγ= δG^l_iβ

δt^γ −δG^l_iγ

δt^β +G^m_iβG^l_mγ−G^m_iγG^l_mβ , R^l_iβk=δG^l_iβ

δx^k −δΓ^l_ik

δt^β +G^m_iβΓ^l_mk−Γ^m_ikG^l_mβ , R^l_ijk=r^l_ijk=∂Γ^l_ij

∂x^k −∂Γ^l_ik

∂x^j + Γ^m_ijΓ^l_mk−Γ^m_ikΓ^l_mj .

Acknowledgments.The authors would like to thank to Professor Vladimir Balan and to the reviewers of the Acta Applicandae Matematicae for their valuable com- ments upon a previous version of this paper. It seems that only now our insights are accepted as new and interesting, without mathematical grimaces.

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Authors’ addresses:

Mircea Neagu,

Str. L˘amˆait¸ei, Nr. 66, Bl. 93, Sc. G, Ap. 10, Bra¸sov, BV 500371, Romˆania email: [email protected]

Constantin Udri¸ste

Department of Mathematics I, University ”Politehnica” of Bucharest, Splaiul Independent¸ei 313, RO-060042, Bucharest, Romania.

email: [email protected]