Displaying similar documents to “Effective computation of the first Lyapunov quantities for a planar differential equation”

Preparation theorems for matrix valued functions

Nils Dencker (1993)

Annales de l'institut Fourier

Similarity:

We generalize the Malgrange preparation theorem to matrix valued functions F ( t , x ) ∈ C ∞ ( R × R n ) satisfying the condition that t ↦ det F ( t , 0 ) vanishes to finite order at t = 0 . Then we can factor F ( t , x ) = C ( t , x ) P ( t , x ) near (0,0), where C ( t , x ) ∈ C ∞ is inversible and P ( t , x ) is polynomial function of t depending C ∞ on x . The preparation is (essentially) unique, up to functions vanishing to infinite order at x = 0 , if we impose some additional conditions on P ( t , x ) . We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass...

A necessary condition of local solvability for pseudo-differential equations with double characteristics

Fernando Cardoso, François Trèves (1974)

Annales de l'institut Fourier

Similarity:

Pseudodifferential operators P ( x , D ) ∼ ∑ j = 0 + ∞ P m - j ( x , D ) are studied, from the viewpoint of local solvability and under the assumption that, micro-locally, the principal symbol factorizes as P m = Q L 2 with Q elliptic, homogeneous of degree m - 2 , and L homogeneous of degree one, satisfying the following condition : there is a point ( x 0 , ξ 0 ) in the characteristic variety L = 0 and a complex number z such that d ξ Re ( z L ) ≠ 0 at ( x 0 , ξ 0 ) and such that the restriction of Im ( z L ) to the bicharacteristic strip of Re ( z L ) vanishes of order k < + ∞ at ( x 0 , ξ 0 ) , changing sign there from...

On the closure of spaces of sums of ridge functions and the range of the X -ray transform

Jan Boman (1984)

Annales de l'institut Fourier

Similarity:

For a ∈ R n ∖ { 0 } and Ω an open bounded subset of R n definie L p ( Ω , a ) as the closed subset of L p ( Ω ) consisting of all functions that are constant almost everywhere on almost all lines parallel to a . For a given set of directions a ν ∈ R n ∖ { 0 } , ν = 1 , ... , m , we study for which Ω it is true that the vector space ( * ) L p ( Ω , a 1 ) + ⋯ + L p ( Ω , a m ) is a closed subspace of L p ( Ω ) . This problem arizes naturally in the study of image reconstruction from projections (tomography). An essentially equivalent problem is to decide whether a certain matrix-valued differential operator...

Partial differential operators depending analytically on a parameter

Frank Mantlik (1991)

Annales de l'institut Fourier

Similarity:

Let P ( λ , D ) = ∑ | α | ≤ m a α ( λ ) D α be a differential operator with constant coefficients a α depending analytically on a parameter λ . Assume that the family { P( λ ,D) } is of constant strength. We investigate the equation P ( λ , D ) 𝔣 λ ≡ g λ where 𝔤 λ is a given analytic function of λ with values in some space of distributions and the solution 𝔣 λ is required to depend analytically on λ , too. As a special case we obtain a regular fundamental solution of P( λ ,D) which depends analytically on λ . This result answers a question of L. Hörmander. ...