Mihai Mihȃilescu

Vicent ¸iu Rȃdulescu
email: vicentiu.radulescu@math.cnrs.fr

A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces *

Keywords: 2000 Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02 nonhomogeneous differential operator, nonlinear eigenvalue problem, continuous spectrum, Orlicz-Sobolev space

We study the nonlinear eigenvalue problem -div(a(|∇u|)∇u) = λ|u| q(x)-2 u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R N with smooth boundary, q is a continuous function, and a is a nonhomogeneous potential. We establish sufficient conditions on a and q such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments.

Introduction and preliminary results

Let Ω be a bounded domain in R N (N ≥ 3) with smooth boundary ∂Ω. In this paper we are concerned with the following eigenvalue problem:

    
-div(a(|∇u|)∇u) = λ|u| q(x)-2 u, for x ∈ Ω u = 0, for x ∈ ∂Ω .

(

) 1
We assume that the function a : (0, ∞) → R is such that the mapping ϕ : R → R defined by

ϕ(t) =     
a(|t|)t, for t = 0 0, for t = 0 , 1 is an odd, increasing homeomorphism from R onto R. We also suppose throughout this paper that λ > 0 and q : Ω → (0, ∞) is a continuous function.

Since the operator in the divergence form is nonhomogeneous we introduce an Orlicz-Sobolev space setting for problems of this type. On the other hand, the term arising in the right hand side of equation ( 1) is also nonhomogeneous and its particular form appeals to a suitable variable exponent Lebesgue space setting.

We point out that eigenvalue problems involving quasilinear nonhomogeneous problems in Orlicz-Sobolev spaces were studied in [START_REF] Garciá-Huidobro | On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting[END_REF] but in a different framework. In what concerns the case when a(|∇u|) = |∇u| q(x)-2 , problem (1) was studied by Fan et al. in [START_REF] Fan | A class of De Giorgi type and Hölder continuity[END_REF][START_REF] Fan | Eigenvalues of p(x)-Laplacian Dirichlet problem[END_REF] who established the existence of a sequence of eigenvalues, by means of the Ljusternik-Schnirelmann critical point theory. Denoting by Λ the set of all nonnegative eigenvalues, Fan, Zhang and Zhao showed that sup Λ = +∞ and they pointed out that only under additional assumptions we have inf Λ > 0. We remark that in the homogeneous case corresponding the p-Laplace operator ( p(x) ≡ p) we always have inf Λ > 0. A different approach of the eigenvalue problem (1) corresponding to a(|∇u|) = |∇u| p(x)-2 and p(x) = q(x) is given in Mihȃilescu and Rȃdulescu [START_REF] Mihȃilescu | On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent[END_REF].

We first recall some basic facts about Orlicz spaces. Define

Φ(t) = t 0 ϕ(s) ds, Φ ⋆ (t) = t 0 ϕ -1 (s) ds, for all t ∈ R.
We observe that Φ is a Young function, that is, Φ(0) = 0, Φ is convex, and lim x→∞ Φ(x) = +∞. Furthermore, since Φ(x) = 0 if and only if x = 0, lim x→0 Φ(x)/x = 0, and lim x→∞ Φ(x)/x = +∞, then Φ is called an N -function. The function Φ ⋆ is called the complementary function of Φ and it satisfies Φ ⋆ (t) = sup{st -Φ(s); s ≥ 0}, for all t ≥ 0 .

We observe that Φ ⋆ is also an N -function and the following Young's inequality holds true:

st ≤ Φ(s) + Φ ⋆ (t), for all s, t ≥ 0 .

The Orlicz space L Φ (Ω) defined by the N -function Φ (see [START_REF] Adams | Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Adams | Sobolev Spaces[END_REF][START_REF] Ph | Mountain pass type solutions for quasilinear elliptic equations[END_REF]) is the space of measurable functions u : Ω → R such that

u L Φ := sup Ω uv dx; Ω Φ ⋆ (|v|) dx ≤ 1 < ∞ . Then (L Φ (Ω), • L Φ ) is a Banach space whose norm is equivalent to the Luxemburg norm u Φ := inf k > 0; Ω Φ u(x) k dx ≤ 1 .
For Orlicz spaces the Hölder's inequality reads as follows (see [21, Inequality 4, p. 79]):

Ω uvdx ≤ 2 u L Φ v L Φ ⋆ for all u ∈ L Φ (Ω) and v ∈ L Φ ⋆ (Ω) .
We denote by W 1 0 L Φ (Ω) the corresponding Orlicz-Sobolev space for problem (1), equipped with the norm u = |∇u| Φ (see [START_REF] Adams | Sobolev Spaces[END_REF][START_REF] Ph | Mountain pass type solutions for quasilinear elliptic equations[END_REF][START_REF] Gossez | Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients[END_REF]). The space W 1 0 L Φ (Ω) is also a Banach space. In this paper we assume that Φ and Φ ⋆ satisfy the ∆ 2 -condition (at infinity), namely

1 < lim inf t→∞ tϕ(t) Φ(t) ≤ lim sup t>0 tϕ(t) Φ(t) < ∞.
Then L Φ (Ω) and W 1 0 L Φ (Ω) are reflexive Banach spaces. Now we introduce the Orlicz-Sobolev conjugate Φ ⋆ of Φ, defined as

Φ -1 ⋆ (t) = t 0 Φ -1 (s) s (N +1)/N ds.
We assume that lim

t→0 1 t Φ -1 (s) s (N +1)/N ds < ∞, and lim t→∞ t 1 Φ -1 (s) s (N +1)/N ds = ∞. (2)
Finally, we define

p 0 := inf t>0 tϕ(t) Φ(t) and p 0 := sup t>0 tϕ(t) Φ(t) .
Next, we recall some background facts concerning the variable exponent Lebesgue spaces. For more details we refer to the book by Musielak [START_REF] Musielak | Orlicz Spaces and Modular Spaces[END_REF] and the papers by Acerbi at al., Edmunds et al. [START_REF] Edmunds | On L p(x) norms[END_REF][START_REF] Edmunds | Density of smooth functions in W k,p(x) (Ω)[END_REF][START_REF] Edmunds | Sobolev embedding with variable exponent[END_REF], Kovacik and Rákosník [START_REF] Kováčik | On spaces L p(x) and W 1,p(x)[END_REF], Mihȃilescu and Rȃdulescu [START_REF] Mihȃilescu | A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids[END_REF], Samko and Vakulov [START_REF] Samko | Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators[END_REF], Zhikov [START_REF] Zhikov | On some variational problems[END_REF].

Set C + (Ω) = {h; h ∈ C(Ω), h(x) > 1 for all x ∈ Ω}.

For any h ∈ C + (Ω) we define

h + = sup x∈Ω h(x) and h -= inf x∈Ω h(x).
For any q(x) ∈ C + (Ω) we define the variable exponent Lebesgue space L q(x) (Ω) (see [START_REF] Kováčik | On spaces L p(x) and W 1,p(x)[END_REF]). On L q(x) (Ω) we define the Luxemburg norm by the formula

|u| q(x) = inf µ > 0; Ω u(x) µ q(x) dx ≤ 1 .
We remember that the variable exponent Lebesgue spaces are separable and reflexive Banach spaces. If 0 < |Ω| < ∞ and q 1 , q 2 are variable exponents so that q 1 (x) ≤ q 2 (x) almost everywhere in Ω then there exists the continuous embedding L q 2 (x) (Ω) ֒→ L q 1 (x) (Ω). If (u n ), u ∈ L q(x) (Ω) then the following relations hold true

|u| q(x) > 1 ⇒ |u| q - q(x) ≤ Ω |u| q(x) dx ≤ |u| q + q(x) (3)
|u| q(x) < 1 ⇒ |u| q + q(x) ≤ Ω |u| q(x) dx ≤ |u| q - q(x) (4)
|u n -u| q(x) → 0 ⇔ Ω |u n -u| q(x) dx → 0. (5)

The main results

We say that λ ∈ R is an eigenvalue of problem (1) if there exists

u ∈ W 1 0 L Φ (Ω) \ {0} such that Ω a(|∇u|)∇u∇v dx -λ Ω |u| q(x)-2 uv dx = 0, for all v ∈ W 1 0 L Φ (Ω). We point out that if λ is an eigenvalue of problem (1) then the corresponding u ∈ W 1 0 L Φ (Ω) \ {0} is a weak solution of (1)
, called an eigenvector of equation ( 1) corresponding to the eigenvalue λ.

Our first main result shows that, in certain circumstances, any positive and sufficiently small λ is an eigenvalue of (1).

Theorem 1. Assume that relation ( 2) is fulfilled and furthermore

1 < inf x∈Ω q(x) < p 0 , (6)
and

lim t→∞ |t| q + Φ ⋆ (kt) = 0, for all k > 0. (7)
Then there exists λ ⋆ > 0 such that any λ ∈ (0, λ ⋆ ) is an eigenvalue for problem [START_REF] Acerbi | Regularity results for a class of functionals with non-standard growth[END_REF].

The above result implies inf

u∈W 1 0 L Φ (Ω)\{0} Ω Φ(|∇u|) dx Ω |u| q(x) dx = 0.
The second main result of this paper asserts that in certain cases the set of eigenvalues may coincide with the whole positive semiaxis.

Theorem 2. Assume that relations (2) and ( 7) are fulfilled and furthermore

sup x∈Ω q(x) < p 0 . ( 8
)
Then every λ > 0 is an eigenvalue for problem [START_REF] Acerbi | Regularity results for a class of functionals with non-standard growth[END_REF]. Moreover, for any λ > 0 there exists a sequence of eigenvectors

{u n } ⊂ E such that lim n→∞ u n = 0 in W 1 0 L Φ (Ω).
Remark 1. Relations ( 2) and ( 7) enable us to apply Theorem 2.2 in [START_REF] Garciá-Huidobro | On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting[END_REF] (see also Theorem 8.33 in [START_REF] Adams | Sobolev Spaces[END_REF]) in order to obtain that W 1 0 L Φ (Ω) is compactly embedded in L q + (Ω). This fact combined with the continuous embedding of

L q + (Ω) in L q(x) (Ω) ensures that W 1 0 L Φ (Ω) is compactly embedded in L q(x) (Ω).
Remark 2. The conclusion of Theorems 1 and 2 still remains valid if we replace the hypothesis [START_REF] Edmunds | Density of smooth functions in W k,p(x) (Ω)[END_REF] in Theorems 1 and 2 by the following relation

N < p 0 < lim inf t→∞ log(Φ(t)) log(t) . (9)
Indeed, using Lemma D.2 in [START_REF] Ph | Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces[END_REF], it follows that

W 1 0 L Φ (Ω) is continuously embedded in W 1,p 0 0
(Ω). On the other hand, since we assume p 0 > N , we deduce that W 1,p 0 0 (Ω) is compactly embedded in C(Ω). Thus, we obtain that

W 1 0 L Φ (Ω) is compactly embedded in C(Ω). Since Ω is bounded it follows that W 1 0 L Φ (Ω) is continuously embedded in L q(x) (Ω).
3 Proof of Theorem 1

Let E denote the Orlicz-Sobolev space W 1 0 L Φ (Ω). For any λ > 0 the energy functional J λ : E → R corresponding to problem (1) is defined by

J λ (u) = Ω Φ(|∇u|) dx -λ Ω 1 q(x) |u| q(x) dx.
Standard arguments imply that J λ ∈ C 1 (E, R) and

J ′ λ (u), v = Ω a(|∇u|)∇u∇v dx -λ Ω |u| q(x)-2 uv dx,
for all u, v ∈ E. Thus the weak solutions of (1) coincide with the critical points of J λ . If such a weak solution exists and is nontrivial then the corresponding λ is an eigenvalue of problem (1).

Lemma 1. There is some λ ⋆ > 0 such that for any λ ∈ (0, λ ⋆ ) there exist ρ, α > 0 such that J λ (u) ≥ α > 0 for any u ∈ E with u = ρ.

Proof. By the definition of p 0 and since d dτ (τ p 0 Φ(t/τ )) ≥ 0 we obtain Φ(t) ≥ τ p 0 Φ(t/τ ), ∀ t > 0 and τ ∈ (0, 1] ,

(see page 44 in [START_REF] Ph | Mountain pass type solutions for quasilinear elliptic equations[END_REF]). Combining this fact with Proposition 6 in [21, page 77] we find that

Ω Φ(|∇u(x)|) dx ≥ u p 0 , ∀ u ∈ E with u < 1. ( 10
)
On the other hand, since E is continuously embedded in L q(x) (Ω), there exists a positive constant c 1 such that

|u| q(x) ≤ c 1 u , ∀ u ∈ E. (11)
We fix ρ ∈ (0, 1) such that ρ < 1/c 1 . Then relation [START_REF] Fan | Eigenvalues of p(x)-Laplacian Dirichlet problem[END_REF] implies

|u| q(x) < 1, ∀ u ∈ E, with u = ρ. (12)
Furthermore, relation (4) yields

Ω |u| q(x) dx ≤ |u| q - q(x) , ∀ u ∈ E, with u = ρ. (13)
Relations [START_REF] Fan | Eigenvalues of p(x)-Laplacian Dirichlet problem[END_REF] and [START_REF] Gossez | Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients[END_REF] imply

Ω |u| q(x) dx ≤ c q - 1 u q -, ∀ u ∈ E, with u = ρ. (14)
Taking into account relations [START_REF] Fan | A class of De Giorgi type and Hölder continuity[END_REF], ( 4) and ( 14) we deduce that for any u ∈ E with u = ρ the following inequalities hold true

J λ (u) ≥ u p 0 - λ q -Ω |u| q(x) dx = ρ q -ρ p 0 -q -- λ q -c q - 1 .
We point out that by relation ( 6) and the definition of p 0 we have q -< l ≤ p 0 . By the above inequality we remark that if we define

λ ⋆ = ρ p 0 -q - 2 • q - c q - 1 (15)
then for any λ ∈ (0, λ ⋆ ) and any u ∈ E with u = ρ there exists α = ρ p 0 2 > 0 such that

J λ (u) ≥ α > 0.
The proof of Lemma 1 is complete.

Lemma 2. There exists ϕ ∈ E such that ϕ ≥ 0, ϕ = 0 and J λ (tϕ) < 0, for t > 0 small enough.

Proof. Assumption (6) implies that q -< p 0 . Let ǫ 0 > 0 be such that q -+ ǫ 0 < p 0 . On the other hand, since q ∈ C(Ω) it follows that there exists an open set Ω 0 ⊂ Ω such that |q(x)q -| < ǫ 0 for all x ∈ Ω 0 . Thus, we conclude that q(x) ≤ q -+ ǫ 0 < p 0 for all x ∈ Ω 0 .

Let ϕ ∈ C ∞ 0 (Ω) be such that supp(ϕ) ⊃ Ω 0 , ϕ(x) = 1 for all x ∈ Ω 0 and 0 ≤ ϕ ≤ 1 in Ω. We also point out that there exists t 0 ∈ (0, 1) such that for any t ∈ (0, t 0 ) we have

t|∇ϕ| Φ = t ϕ < 1.
Taking into account all the above information and using Lemma C.9 in [START_REF] Ph | Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces[END_REF] we have

J λ (tϕ) = Ω Φ(t|∇ϕ(x)|) dx -λ Ω t q(x) q(x) |ϕ| q(x) dx ≤ Ω Φ(t|∇ϕ(x)|) dx - λ q + Ω t q(x) |ϕ| q(x) dx ≤ Ω Φ(t|∇ϕ(x)|) dx - λ q + Ω 0 t q(x) |ϕ| q(x) dx ≤ t p 0 ϕ p 0 - λ • t q -+ǫ 0 q + |Ω 0 |,
for any t ∈ (0, 1), where |Ω 0 | denotes the Lebesgue measure of Ω 0 . Therefore J λ (tϕ) < 0 for t < δ 1/(p 0 -q --ǫ 0 ) , where 0 < δ < min t 0 ,

λ q + |Ω 0 | ϕ p 0 .
The proof of Lemma 2 is complete.

Proof of Theorem 1. Let λ ⋆ > 0 be defined as in [START_REF] Kováčik | On spaces L p(x) and W 1,p(x)[END_REF] and λ ∈ (0, λ ⋆ ). By Lemma 1 it follows that on the boundary of the ball centered at the origin and of radius ρ in E, denoted by B ρ (0), we have inf ∂Bρ(0)

J λ > 0. ( 16
)
On the other hand, by Lemma 2, there exists ϕ ∈ E such that J λ (tϕ) < 0 for all t > 0 small enough. Moreover, relations ( 10), ( 14) and ( 4) imply that for any u ∈ B ρ (0) we have

J λ (u) ≥ u p 0 - λ q -c q - 1 u q -. It follows that -∞ < c := inf Bρ(0) J λ < 0.
We let now 0 < ǫ < inf ∂Bρ(0) J λinf Bρ(0) J λ . Applying Ekeland's variational principle [START_REF] Ekeland | On the variational principle[END_REF] to the functional J λ : B ρ (0) → R, we find u ǫ ∈ B ρ (0) such that

J λ (u ǫ ) < inf Bρ(0) J λ + ǫ J λ (u ǫ ) < J λ (u) + ǫ • u -u ǫ , u = u ǫ . Since J λ (u ǫ ) ≤ inf Bρ(0) J λ + ǫ ≤ inf Bρ(0) J λ + ǫ < inf ∂Bρ(0) J λ ,
we deduce that u ǫ ∈ B ρ (0). Now, we define

I λ : B ρ (0) → R by I λ (u) = J λ (u) + ǫ • u -u ǫ .
It is clear that u ǫ is a minimum point of I λ and thus

I λ (u ǫ + t • v) -I λ (u ǫ ) t ≥ 0
for small t > 0 and any v ∈ B 1 (0). The above relation yields

J λ (u ǫ + t • v) -J λ (u ǫ ) t + ǫ • v ≥ 0. Letting t → 0 it follows that J ′ λ (u ǫ ), v + ǫ • v > 0 and we infer that J ′ λ (u ǫ ) ≤ ǫ.
We deduce that there exists a sequence {w n } ⊂ B ρ (0) such that

J λ (w n ) → c and J ′ λ (w n ) → 0. ( 17
)
It is clear that {w n } is bounded in E. Thus, there exists w ∈ E such that, up to a subsequence, {w n } converges weakly to w in E. By Remark 2 we deduce that E is compactly embeddded in L q(x) (Ω), hence {w n } converges strongly to w in L q(x) (Ω). So, by relations ( 5) and Hölder's inequality for variable exponent spaces (see e.g. [START_REF] Kováčik | On spaces L p(x) and W 1,p(x)[END_REF]),

lim n→∞ Ω |w n | q(x) dx = Ω |w| q(x) dx and lim n→∞ Ω |w n | q(x)-2 w n v dx = Ω |w| q(x)-2 wv dx for any v ∈ E.
We conclude that w is a nontrivial weak solution for problem [START_REF] Acerbi | Regularity results for a class of functionals with non-standard growth[END_REF] and thus any λ ∈ (0, λ ⋆ ) is an eigenvalue of problem [START_REF] Acerbi | Regularity results for a class of functionals with non-standard growth[END_REF]. Similar arguments as those used on page 50 in [START_REF] Ph | Mountain pass type solutions for quasilinear elliptic equations[END_REF] imply that {w n } converges strongly to w in E. So, by [START_REF] Mihȃilescu | Nonhomogeneous boundary value problems in Orlicz-Sobolev spaces[END_REF],

J λ (w) = c < 0 and J ′ λ (w) = 0. ( 18
)
The proof of Theorem 1 is complete.

Proof of Theorem 2

We still denote by E the Orlicz-Sobolev space W 1 0 L Φ (Ω). For any λ > 0 let J λ be defined as in the above section of the paper.

In order to prove Theorem 2 we apply to the functional J λ a symmetric version of the mountain pass lemma, recently developed by Kajikia in [START_REF] Kajikia | A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations[END_REF]. Before presenting the result in [START_REF] Kajikia | A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations[END_REF] we remember the following definition. Definition 1. Let X be a real Banach space. We say that a subset A of X is symmetric if u ∈ A implies -u ∈ A. For a closed symmetric set A which does not contain the origin, we define the genus γ(A) of A as the smallest integer k such that there exists an odd continuous mapping from A to R k \ {0}. If there does not exist such an integer k, we define γ(A) = +∞. Moreover, we set γ(∅) = 0. Finally, we denote by Γ k the family Γ k = {A ⊂ X; 0 ∈ A and γ(A) ≥ k}.

We state now the symmetric mountain pass lemma of Kajikia (see Theorem 1 in [START_REF] Kajikia | A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations[END_REF]).

Theorem 3. Assume X is an infinite dimensional Banach space and Λ ∈ C 1 (X, R) satisfies conditions (A1) and (A2) below.

(A1) Λ(u) is even, bounded from below, Λ(0) = 0 and Λ(u) satisfies the Palais-Smale condition (i.e., any sequence {u n } in X such that {Λ(u n )} is bounded and Λ

′ (u n ) → 0 in X ⋆ as n → ∞ has a convergent subsequence); (A2) For each k ∈ N, there exists an A k ∈ Γ k such that sup u∈A k Λ(u) < 0.
Under the above assumptions, either (i) or (ii) below holds true.

(i) There exists a sequence {u n } such that Λ ′ (u n ) = 0, Λ(u n ) < 0 and {u n } converges to zero;

(ii) There exist two sequences {u n } and {v n } such that Λ ′ (u n ) = 0, Λ(u n ) = 0, u n = 0, lim n→∞ u n = 0, Λ ′ (v n ) = 0, Λ(v n ) = 0, and v n converges to a non-zero limit.

In order to apply Theorem 3 to the functional J λ we prove two auxiliary results.

Lemma 3. The functional J λ satisfies condition (A1) from Theorem 3.

Proof. Clearly, J λ (u) = J λ (-u) for any u ∈ E, i.e. J λ is even, and J λ (0) = 0. On the other hand, since by relation [START_REF] Fan | A class of De Giorgi type and Hölder continuity[END_REF] we have

Ω Φ(|∇u(x)|) dx ≥ u p 0 , ∀ u ∈ E with u < 1,
while by Lemma C.9 in [START_REF] Ph | Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces[END_REF] we have

Ω Φ(|∇u(x)|) dx ≥ u p 0 , ∀ u ∈ E with u > 1, we deduce that Ω Φ(|∇u(x)|) dx ≥ α( u ), ∀ u ∈ E, (19)
where

α : [0, ∞) → R, α(t) = t p 0 if t < 0 and α(t) = t p 0 if t > 1.
By Remark 1, the space E is continuously embedded in L q ± (Ω). Thus, there exist two positive constants d 1 and d 2 such that

Ω |u| q + dx ≤ d 1 u q + , Ω |u| q -dx ≤ d 2 u q -, ∀ u ∈ E. ( 20
)
Combining relations [START_REF] Mihȃilescu | On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent[END_REF] and [START_REF] Musielak | Orlicz Spaces and Modular Spaces[END_REF] we get

J λ (u) ≥ α( u ) - d 1 λ q -u q + - d 2 λ q -u q -, ∀ u%inE.
Since by relation [START_REF] Edmunds | Sobolev embedding with variable exponent[END_REF] we have q + < p 0 the above relation shows that J λ is bounded from below. Next, we show that J λ satisfies the Palais-Smale condition. Let {u n } be a sequence in E such that {J λ (u n )} is bounded and J ′ (u n ) → 0 in E ′ , as n → ∞. We show that {u n } is bounded in E. Assume by contradiction the contrary. Then, passing eventually to a subsequence, still denoted by {u n }, we may assume that u n → ∞ as n → ∞. Thus we may consider that u n > 1 for any integer n.

For any t ∈ (0, 1) we have sup

u∈A k (t) J λ (u) ≤ sup θ∈S 1 ∩F J λ (tθ) = sup θ∈S 1 ∩F Ω Φ(t|∇θ|) dx -λ Ω 1 q(x) t q(x) |θ| q(x) dx ≤ sup θ∈S 1 ∩F t p 0 Ω Φ(|∇θ|) dx - λt q + q + Ω |θ| q(x) dx = sup θ∈S 1 ∩F t p 0 1 - λ q + • 1 t p 0 -q + • Ω |θ| q(x) dx Since S 1 ∩ F is compact we have m = min θ∈S 1 ∩F Ω |θ| q(x) dx > 0.
Combining that fact with the information given by relation [START_REF] Edmunds | Sobolev embedding with variable exponent[END_REF], that is p 0 > q + , we deduce that we can choose t k ∈ (0, 1) small enough such that

1 - λ q + • 1 t p 0 -q + • m < 0. The above relations yield sup u∈A k (t k ) J λ (u) < 0.
The proof of Lemma 4 is complete.

Proof of Theorem 2. Using Lemmas 3 and 4 we deduce that we can apply Theorem 3 to the functional J λ . So, there exists a sequence {u n } ⊂ E such that J ′ (u n ) = 0, for each n, J λ (u n ) ≤ 0 and

{u n } converges to zero in E.
The proof of Theorem 2 is complete.

Examples

In this section we point out two concrete examples of problems to which we can apply the main results of this paper. Example 1. We consider the problem

     -div(log(1 + |∇u| r )|∇u| p-2 ∇u) = λ|u| q(x)-2 u, for x ∈ Ω u = 0, for x ∈ ∂Ω, (21)
where p and r are real numbers such that 1 < p, r, N > p + r and q(x) is a continuous function on Ω such that 1 < q(x) for all x ∈ Ω and furthermore inf Ω q(x) < p and sup Ω q(x) < N p Np .

In this case we have ϕ(t) = log(1 + |t| r ) • |t| p-2 t, for all t ∈ R and Φ(t) = t 0 ϕ(s), for all t ∈ R.

Clearly, ϕ is an odd, increasing homeomorphism of R into R, while Φ is convex and even on R and increasing from R + to R + . By Example 2 on p. 243 in [START_REF] Ph | Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces[END_REF] we know that p 0 = p and p 0 = p + r and thus relation [START_REF] Edmunds | On L p(x) norms[END_REF] in Theorem 1 is satisfied. On the other hand, by Proposition 1 in [START_REF] Mihȃilescu | Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting[END_REF] (see also [START_REF] Mihȃilescu | Nonhomogeneous boundary value problems in Orlicz-Sobolev spaces[END_REF]) we deduce that relations (2) and ( 7) are fulfilled. Thus, we verified that we can apply Theorem 1 in order to find out that there exists λ ⋆ > 0 such that any λ ∈ (0, λ ⋆ ) is an eigenvalue for problem [START_REF] Rao | Theory of Orlicz Spaces[END_REF].

Example 2. We consider the problem

    
-div |∇u| p-2 ∇u log(1 + |∇u|) = λ|u| q(x)-2 u, for x ∈ Ω u = 0, for x ∈ ∂Ω ,

where p is a real number such that p > N + 1 and q ∈ C(Ω) satisfies 1 < q(x) < p -1 for any x ∈ Ω.

In this case we have Thus, conditions (2), ( 8) and ( 9) from Theorem 2 and Remark 2 are verified. We deduce that every λ > 0 is an eigenvalue for problem [START_REF] Samko | Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators[END_REF]. Moreover, for each λ > 0 there exists a sequence of eigenvectors {u n } such that lim n→∞ u n = 0 in W 1 0 L Φ (Ω).

  continuous function from R + to R + , with Φ(0) = 0 and such that the function Φ( √ t) is convex. By Example 3 on p. 243 in[START_REF] Ph | Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces[END_REF] we havep 0 = p -1 < p 0 = p = lim inf t→∞ log(Φ(t))log(t) .

Acknowledgements. The authors have been supported by Grant CNCSIS PNII-79/2007 "Procese Neliniare Degenerate şi Singulare".

By our assumptions, there is a positive constant M such that for all n large enough we have

Since p 0 > 1, letting n → ∞ we obtain a contradiction. It follows that {u n } is bounded in E. Similar arguments as those used in the end of the proof of Theorem 1 imply that, up to a subsequence, {u n } converges strongly in E.

The proof of Lemma 3 is complete.

Lemma 4. The functional J λ satisfies condition (A2) from Theorem 3.

Proof. We construct a sequence of subsets

Define Ω 1 = Ω \ B r 1 (x 1 ). Next, let x 2 ∈ Ω and r 2 > 0 be such that B r 2 (x 2 ) ⊂ Ω 1 and |B r 2 (x 2 )| < |Ω 1 |/2. Consider θ 2 ∈ C ∞ 0 (Ω) be a function with supp(θ 2 ) = B r 2 (x 2 ). Continuing the process described above we can construct by recurrence a sequence of functions θ 1 , θ 2 ,..., θ k ∈ C ∞ 0 (Ω) such that supp(θ i ) = supp(θ j ) if i = j and |supp(θ i )| > 0 for any i, j ∈ {1, ..., k}. We define the finite dimensional subspace of E, F = span{θ 1 , θ 2 , ..., θ k }.

Clearly, dimF = k and Ω |θ| q(x) dx > 0, for any θ ∈ F \ {0}. We denote by S 1 the unit sphere in E, i.e. S 1 = {u ∈ E; u = 1}. For any number t ∈ (0, 1) we define the set

Since for any bounded symmetric neighborhood ω of the origin in R k there holds γ(∂ω) = k (see Proposition 5.2 in [START_REF] Struwe | Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems[END_REF]) we deduce that γ(A k (t)) = k for any t ∈ (0, 1).

Finally, we show that for each integer k there exists t k ∈ (0, 1) such that sup u∈A k (t k )

J λ (u) < 0.


Mehdi Hage-Hassan

Hurwitz transformations and Pauli, Dirac matrices and the generators of SO(4),SO(5)

may come

Hurwitz transformations and Pauli, Dirac matrices and the generators of SO(4),SO(5)

Introduction

The real division algebras, or Octonion algebra, are: real numbers, complex numbers, quaternion, and Octonion. These algebras and the classical groups SO(n) and SU(n) are very important in mathematics and physics [START_REF] Wybourne | Classical Groups for Physicists[END_REF][START_REF] Messiah | Mécanique Tomes I et II E[END_REF][START_REF] Schiff | Quantum Mechanics[END_REF][START_REF] Merzbacher | Quantum Mechanics[END_REF][START_REF] Sakurai | Modern Quantum Mechanics[END_REF]. This real division algebra is originated of the famous problem solving "the sums of squares" [START_REF] Dickson | On Quaternions and their Generalization and the History of Eight Square Theorem[END_REF][START_REF] Okubo | Introduction to Octonion and Other Non-Associative Algebras in Physics[END_REF][START_REF] Dixon | Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics[END_REF][START_REF] Conrad | The Hurwitz theorem on sums of squares[END_REF]. So we try to determine the values of n so that there is a solution of the identity:

( )( ) 2

=

This problem was solved by Euler and Fermat for n = 2, by Hamilton for n = 4 and by Grave and Cayley for n = 8. Hurwitz [START_REF] Conrad | The Hurwitz theorem on sums of squares[END_REF] showed that the equation (1.1) has solutions only for n = 1, 2, 4, and 8. Note also that every solution corresponds to an algebra which is isomorphic to the anti-symmetric and orthogonal matrices H n (u),(u)= (u 1 ... u n ), and representing the division algebras [START_REF] Conrad | The Hurwitz theorem on sums of squares[END_REF][START_REF] Tian | matrix Representations of Octonion and their applications[END_REF][START_REF] Hage-Hassan | The matrices representation of division algebras, new properties, the Cartan-Weyl basis of SO(n) and the 2 n -dimensional harmonic oscillator basis[END_REF].

It is important to emphasize that in the work of the Indian mathematician Brahma-gupta (598-668) figures the famous identity [START_REF] Lambert | Le probleme de Hurwitz,une histoires de mathematiciens[END_REF]:

( )( ) ( ) 2 1 2 2 1 2 2 2 1 1 2 2 2 1 2 2 2 1 ) ( v u v u D v u v u D v v D u u + + - = + + (1.
2) The nonbijective canonical transformations [START_REF] Hage-Hassan | An elementary introduction to Quantum mechanic[END_REF][START_REF]Octonion generalization of Pauli and Dirac matrices Intern[END_REF][START_REF] Levi | Civita opera Mathematiche[END_REF][16][17][18] or Hurwitz transformations [START_REF] Hage-Hassan | An elementary introduction to Quantum mechanic[END_REF] are simply deduced from Hurwitz matrics H n .

In recent years, these transformations, generalizing the so called Kustaanheimo-Stiefel transformation, 3 4 R R → , have been the object of numerous studies in particular: the algebra of quaternion's and, more generally, the Cayley-Dickson algebras and the Connection between the Hydrogen atom and the Harmonic oscillator [START_REF] Levi | Civita opera Mathematiche[END_REF][16][17][18][START_REF] Hage-Hassan | Inertia tensor and cross product in n-dimensions space[END_REF].

In this work we determine the representation matrices of division algebras taking as starting point the trigonometric formulas and using a recurrence method [START_REF] Hage-Hassan | Some applications of Wigner's D-matrix to the addition formulas for the spherical harmonics and to Wigner D-matrix for the SO(4) groups (hal-01791820)[END_REF]. We derive the matrices

R R → .

By expressing these transformations using complex variables, we find that the matrices associated with quadratic forms By expressing the matrices of transformations using reel numbers, we find from the commutators of the matrices associated with the quadratic forms, or departure matrices, three and five: the two commuting spins which generate the SO (4) groups [START_REF] Hage-Hassan | Some applications of Wigner's D-matrix to the addition formulas for the spherical harmonics and to Wigner D-matrix for the SO(4) groups (hal-01791820)[END_REF][START_REF] Beveren | Cours:Some notes on group theory[END_REF] and the generators of SO(4) SO ( 5) groups [START_REF] Demler | SO(5) Theory of Antiferromagnetism and Superconductivity[END_REF][START_REF] Hecht | [END_REF][START_REF] Sharp And | O(5) Polynomial Bases[END_REF]. The matrices associated with the quadratic forms x i =0, (i=5-->8) have the commutation relations of angular momentum.

We derive also a simple graphical method for the calculation of Gauss Sums.

In this paper we present the derivation of the representation matrices of division algebras in section two. Quadratic transformations Multiplying these expressions by r≥0 we find:

2 1 2 2 2 2 1 1 2 , u u x u u x = - = (2.2) With ) 2 / sin( ), 2 / cos( 2 1 θ θ r u r u = = (2.
3) We write (2.2) and (2.3) in matrix form:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 2 1 2 2 1 1 2 2 1 2 1 )) ( ( u u u H u u u u u u x x ( 2

2.2The problem of sums of squares

It is simple to make the derivation of the problem of sums of squares:

( )( ) [ ]( ) ( )( ) ( ) ( ) 2 1 2 2 1 2 2 2 1 1 2 2 2 1 2 2 2 1 2 1 2 2 2 1 v u v u v u v u v v u u v v H H v v t + + - = + + = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ (2.6)

Quadratic transformations

3 2 2 1 1 , iu u z iu u z + = + =
and its complexes conjugate we find that these transformations are expressed in terms of Pauli matrices.

The representation matrix of quadratic transformation

3 4 R R →
The generalization of the transformation (2.2) and (2.4) is obtained by setting:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 4 3 1 2 2 1 2 1 u u 2 x x u u u u (3.1) Add ) ( ) ( 2 4 2 3 2 2 2 1 3 u u u u x + - + =
and so let us write us:

, ) ( 0 2 1 4 3 4 3 2 1 ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ u u u u u H x x x t (3.2)
So we derive the representation matrix of Quaternion, ) ( 4 u H [START_REF] Tian | matrix Representations of Octonion and their applications[END_REF][START_REF] Hage-Hassan | The matrices representation of division algebras, new properties, the Cartan-Weyl basis of SO(n) and the 2 n -dimensional harmonic oscillator basis[END_REF][START_REF] Hage-Hassan | An elementary introduction to Quantum mechanic[END_REF],

⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - - - - - = 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 4 ) ( u u u u u u u u u u u u u u u u u H (3.3)

Quaternion's notations

In quaternion's notations we write:

L e K e J e I e = = = = 4 3 2 1 , , , .
And Q= 1 , ) (

The Octonion quadratic transformation

3 4 R R → From expression (3.2) we deduce the Hurwitz transformations 3 4 R R → 0 , , 2 2 , 2 2 4 2 4 2 3 2 2 2 1 3 3 2 4 1 2 4 2 3 1 1 = - - + = - = + = x u u u u x u u u u x u u u u x (3.5)

Quadratic transformations and Pauli matrices

If we put t (u) = (u 1 +iu 2 , u 3 +iu 4 ), we find that the Hurwitz transformations expressed with the help of Pauli matrix in the form:

( ) ( ) ( ) ( ) ( ) ( ) u u x u u x u u x z t y t x t σ σ σ = = = 3 2 1 , , (3.6)
The Pauli matrices are

z y x k j i σ σ σ σ r r r r + + =
and they are given by:

, 1 0 0 1 , 0 0 , 0 1 1 0 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = z y x i i σ σ σ (3.7)
3.5 Complex Quadratic transformations and Pauli matrices

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u z iy x iy x z u u u z u u y u u x t z t y t x t ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + - = + + σ σ σ (3.8)

Quadratic transformations R 4 R 3 and SO(4) matrices

The number of generators of SO (n) is n (n -1) / 2 and consequently the number of generators of SO ( 4) is six. Keeping the quadratic transformations, 3 4 R R → , in terms of real variables (u). We find from the commutators of the matrices associated with the quadratic forms R 4 R 3, or departures matrices, the first three generators of SO [START_REF] Merzbacher | Quantum Mechanics[END_REF]. From the development of ) ( 4 u H t we find the other three generators of SO [START_REF] Merzbacher | Quantum Mechanics[END_REF]. It is interesting to note that, we find without any calculation the two commuting spins which generate the SO (4) groups [START_REF] Hage-Hassan | Some applications of Wigner's D-matrix to the addition formulas for the spherical harmonics and to Wigner D-matrix for the SO(4) groups (hal-01791820)[END_REF][START_REF] Beveren | Cours:Some notes on group theory[END_REF]

The first spin of SO (4)

We write the quadratic transformation R 4 R 3 in the form:

( )( )( ) 2 4 2 3 2 2 2 1 1 1 u u u u u a u x t - - + = = ( )( )( ) ( )( )( ) 4 2 3 1 3 3 3 2 4 1 2 2 2 2 2 2 u u u u u a u x u u u u u a u x t t + = = - = = (4.1)
With ( )

) ( 4 3 2 1 u u u u u t =
. The matrices associated with the quadratic forms, or the departure matrices, are:

( ) ( ) ( ) ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 , 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 , 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 3 2 1 a a a (4.2)
By analogy with the kinetic momentum and make the commutations of matrices ( ) ( ) ( )

3 2 1 , , a a a
we write:

( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = = = 4 / , 4 / , 4 / , 3 2 3 3 1 2 2 1 1 a a i N a a i N a a i N (4.3)
We find that N i , i=1, 2, 3, are the first three generators of SO (4) [START_REF] Hage-Hassan | Inertia tensor and cross product in n-dimensions space[END_REF][START_REF] Hage-Hassan | Some applications of Wigner's D-matrix to the addition formulas for the spherical harmonics and to Wigner D-matrix for the SO(4) groups (hal-01791820)[END_REF]:

⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = = 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 2 , 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 2 , 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 2 3 2 1 i N i N i N N r (4.4)

The second spins of SO (4)

It is important to note that we can write ) ( 4 u H t in the form:

( )

1 4 2 3 3 2 4 2 ) ( M u M u M u u H i t - + - =
By analogy with the kinetic momentum we write

k M j M i M M r r r r 3 2 1 + + = : ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 2 , 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 2 , 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 2 3 2 1 i M i M i M (4.5)
The commutation relations for the generators M r and N r can easily be obtained,

[ ] [ ] [ ] , 0 , , , , , = = = j i k ijk j i k ijk j i N M N i N N M i M M ε ε (4.6)
Where ijk ε is the Levi-Civita-symbol.

The commutation relations (4.6) are equivalent either to SO(3) or to SU(2). And by expressing the transformations with the help of , ,

Quadratic transformations

4 3 2 2 1 1 iu u z iu u z + = + = 8 7 4 6 5 3 , iu u z iu u z + = + =
and its conjugate complexes we find that these transformations are expressed in terms of the Dirac matrices [START_REF] Wybourne | Classical Groups for Physicists[END_REF][START_REF] Messiah | Mécanique Tomes I et II E[END_REF][START_REF] Schiff | Quantum Mechanics[END_REF][START_REF] Merzbacher | Quantum Mechanics[END_REF][START_REF] Sakurai | Modern Quantum Mechanics[END_REF].

The representation matrix of Octonion

We derive the orthogonal matrix H 8 (u) by generalization of the transformation (3.1) We find the Octonion quadratic transformation

⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ u u u u u H x x x x t (5.1) By setting ) ( ) ( 2 8 2 7 2 6 2 5 2 4 2 3 2 2 2 1 5 u u u u u u u u x + + + - + + + = so that 2 2 2 ) (u r r r = , We find: ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - - - - - - - - - - - - - - - - - - - - - - - - - - - = 1 2 3 4
) ( 0 1 5 5 0 5 v v x j v v i x v v x t i t j j t γ γ γ = = - = = - - γ i are the Dirac matrices: ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 , 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 2 1 0 i i i i γ γ γ (5.5) ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - = 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 , 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 5 3 γ γ

Complex Quadratic transformations and Dirac matrices

It is easy to verify that:

⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - + - - - + - + - + - + - + = + + + + 1 5 4 3 2 1 3 2 5 4 5 4 3 2 1 3 2 5 4 1 5 5 3 4 2 3 1 2 0 1 0 0 0 0 ix ix x ix x ix ix x ix x ix x ix x ix ix x ix x ix x x x x ix γ γ γ γ γ (5.6)
It is obvious that this matrix is unitary.

Quadratic transformations and SO(4) SO (5)

In this case, the quadratic transformations are expressed with the help of real variables. Therefore we derive from the commutators of the matrices associated with the quadratic forms 5 8 R R → (or departures matrices) the two commuting spins which generate the SO ( 4) groups [START_REF] Hage-Hassan | Some applications of Wigner's D-matrix to the addition formulas for the spherical harmonics and to Wigner D-matrix for the SO(4) groups (hal-01791820)[END_REF][START_REF] Beveren | Cours:Some notes on group theory[END_REF] and the generators of SO(4) SO ( 5) groups [START_REF] Demler | SO(5) Theory of Antiferromagnetism and Superconductivity[END_REF][START_REF] Hecht | [END_REF][START_REF] Sharp And | O(5) Polynomial Bases[END_REF]. We find also that the representations matrix of x 6 =x 7 =x 8 =0 are the generators of .

The quadratic transformation

5 8

R R → in real field

We have:

( )( )( ) . 5 , 4 , 3 , 2 , 1 ,
= = i u a u x i t i (6.1)
With ( ) ( )

8 7 6 5 4 3 2 1 u u u u u u u u u t = And ( ) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 a .... ( ) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - - - = 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 5 a (6.2)
The departure matrices are ( ) ( ) [ ] 5 1 ,..., a a .

The generators of SO(5)

We put: 5

,..., 1 , 4 ,..., 1 , 2 / )] ( ), [( + = = = m n m a a i L n m mn (6.3)
And the matrix form of ( ) mn L is : 5). Moreover we can choose in several ways the generators of SO (4) SO (5). But we will follow Hecht and others [START_REF] Beveren | Cours:Some notes on group theory[END_REF][START_REF] Demler | SO(5) Theory of Antiferromagnetism and Superconductivity[END_REF][START_REF] Hecht | [END_REF][START_REF] Sharp And | O(5) Polynomial Bases[END_REF]:

⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ 45
( ), 2
1 ), ( 2
1 iL L V L L U L L T L L T L L T m = ± = - = - = - = ± ± (6.5) S r 2
and T r are two commuting spins which generate SO(4).

± U and ± V are generators that do not commute with S r and T r

The matrices associated with the quadratic forms { x i =0,i > 5}

To determine the matrices associated with the quadratic forms {x i = 0} we put: ( ) ( ) ( )

( )( )( ) . 8 ,..., 1 , ' ' = = i u a u x i t i
8 3 7 2 6 1 2 , 2 , 2 a i J a i J a i J = - = = (6.8)
Note that J 1 , J 2 , J 3 verify the relationships of the angular momentum (4.6).

Appendix: Gauss sum for Students

We deduce from Fig.

(1) a simple method for calculating Gauss sums. And I found great interest by students (10 years) by this method and is the reason I mention it.

1-Calculation of Sum S = 1 + 2 + 3.... + n.

In the square, Fig.

(1), we replace L ij by points. So the sum S represents the number of the points which have the form of a right triangle and which represents the half of the square which has n 2 points. The other zone represents a right triangle but it misses the points of the diagonal (n). So 2S-n = n 2 and S = n (n + 1) / 2.

  themselves in terms of Dirac matrices (γ).

3 4 R

 34 R → and Pauli matrices are given in section three. The Quadratic transformations R 4 R 3 and the generators of SO(4) given in sections four and five we present in section five and six the Quadratic transformations θ = θ(2.1)

4

 4

  Fig.(1)

a-From the orthogonality of H n , n = 1, 2, 4 it is easy to verify the identity (1.1). b-We derive by the same method the matrix H n (u) and then the relation (1.1) is no longer verified for n = 16.

The Octonion quadratic transformation

(5.4) 0 ,

R R → and γ-matrices

If we put: t (v) = (u 1 +iu 2 ,u 3 +iu 4 ,u 5 +iu 6 ,u 7 +u 8 ) , it's simple to shows the relations:

are the matrices with x i # 0 and ( ) 6,7,8 = i a i are the matrices with x i = 0. We Write:

2-Calculation of Sum with a jump of 2

The equilateral triangle is formed of the collage of two right triangles, 2S, minus the points of height, n: S 1 = 2S-n = n 2 = 1 + 3 + 5 + 7 + 9 ٠٠٠ we can therefore make the collages of several rectangles to obtain other sums of Gauss.