For p a prime number, let X0(p)Q be the modular curve, over Q, parametrizing isogenies of degree p between elliptic curves, and let J0(p)Q be its jacobian variety. Let f : X0(p)Q → J0(p)Q be the morphism of varieties over Q that sends a point P to the class of the divisor P − ∞, where ∞ is the Q-valued point of X0(p)Q that corresponds to PQ with 0 and ∞ identified, equipped with the subgroup sc...

We study connected branches of non-constant 2π-periodic solutions of the Hamilton equation ẋ(t) = λJ∇H(x(t)), where λ ∈ (0,+∞), H ∈ C(R ×Rn,R) and ∇H(x0) = [ A 0 0 B ] for x0 ∈ ∇H−1(0). The Hessian ∇H(x0) can be singular. We formulate sufficient conditions for the existence of such branches bifurcating from given (x0, λ0). As a consequence we prove theorems concerning the existence of connected...

The papers [15], [16], [4], [12], [2], [14], [5], [1], [3], [7], [6], [10], [11], [8], [9], [17], and [13] provide the notation and terminology for this paper. The following proposition is true (1) For every real number x and for every natural number n holds |x| = |x|. Let f be a partial function from R to R, let Z be a subset of R, and let a be a real number. The functor Maclaurin(f, Z, a) yie...

We provide necessary and sucient conditions for psi-conditional as-ymptotic stability of the solution of a linear matrix Lyapunov system and sucientconditions for psi -conditional asymptotic stability of the solution of a rst ordernon-linear matrix Lyapunov system X0 = A(t)X + XB(t) + F(t;X).

Antoine and Zouaki ([1] and [16]) have given the existence of a continuous selection of (1.1) in the particular case where M − x0 and N − y0 are closed convex cones . In this paper, we extend this result when M (resp. N) approximates continuously its Clarke tangent cone at x0 (resp. y0) (see Definition 2.2). This property is verified at x0 for an important class of sets. We get among others: se...

Let m2 < m1 be two given nonnegative integers with n = m1+m2+1. For suitably differentiable f , we let P,Q ∈ πn be the Hermite polynomial interpolants to f which satisfy P (a) = f (a), j = 0, 1, ..., m1 and P (b) = f (b), j = 0, 1, ..., m2, Q (a) = f (a), j = 0, 1, ..., m2 and Q(b) = f (b), j = 0, 1, ..., m1. Suppose that f ∈ C (I) with f (x) 6= 0 for x ∈ (a, b). If m1 − m2 is even, then there ...

and Applied Analysis 3 Let us consider the functional I x defined on H1 0 0, 2γτ by I x ∫2γτ 0 [ x′ t x′ t − τ − F t, x t , x t − τ dt. 2.4 For all x, y ∈ H1 0 0, 2γτ and ε > 0, we know that I ( x εy ) I x ε (∫2γτ 0 [ x′ t y′ t − τ x′ t − τ y′ t −Ft, x t εy t , x t − τ εy t − τ ) − F t, x t , x t − τ dt ) ε2 ∫2γτ 0 y′ t y′ t − τ dt. 2.5 It is then easy to see that 〈 I ′ x , y 〉 ∫2γτ 0 [ x′ t y′...

For a local po-space X and a base point x0 ∈ X, we define the universal dicovering space Π : X̃x0 → X. The image of Π is the future ↑ x0 of x0 in X and X̃x0 is a local po-space such that | π 1 (X̃, [x0], x1)| = 1 for the constant dipath [x0] ∈ Π−1(x0) and x1 ∈ X̃x0 . Moreover, dipaths and dihomotopies of dipaths (with a fixed starting point) in ↑ x0 lift uniquely to X̃x0 . The fibers Π −1(x) are dis...