We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological... more We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. In the new version references added: Most of the results are known. So this paper will not be submitted to
Journal of Fourier Analysis and Applications, 2014
We consider the operator R, which sends a function on R 2n to its integrals over all affine Lagra... more We consider the operator R, which sends a function on R 2n to its integrals over all affine Lagrangian subspaces in R 2n . We discuss properties of the operator R and of the representation of the affine symplectic group in the space of functions on R 2n .
Pulling back sets of functions in involution by Poisson mappings and adding Casimir functions dur... more Pulling back sets of functions in involution by Poisson mappings and adding Casimir functions during the process allows to construct completely integrable systems. Some examples are investigated in detail. 1991 Mathematics Subject Classification. 58F07.
This paper studies a specific metric on plane curves that has the property of being isometric to ... more This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc. . . ) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided Date: April 24, 2012. 1991 Mathematics Subject Classification. Primary 58B20, 58D15, 58E40.
Let ρ : G → GL(V ) be a rational representation of a reductive linear algebraic group G defined o... more Let ρ : G → GL(V ) be a rational representation of a reductive linear algebraic group G defined over C on a finite dimensional complex vector space V . We show that, for any generic smooth (resp. C M ) curve c : R → V / /G in the categorical quotient V / /G (viewed as affine variety in some C n ) and for any t 0 ∈ R, there exists a positive integer N such that t → c(t 0 ± (t − t 0 ) N ) allows a smooth (resp. C M ) lift to the representation space near t 0 . (C M denotes the Denjoy-Carleman class associated with M = (M k ), which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in V / /G admits locally absolutely continuous (not better!) lifts. Assume that G is finite. We characterize curves admitting differentiable lifts. We show that any germ of a C ∞ curve which represents a lift of a germ of a quasianalytic C M curve in V / /G is actually C M . There are applications to polar representations.
If u → A(u) is a C 0,α -mapping, for 0 < α ≤ 1, having as values unbounded self-adjoint operators... more If u → A(u) is a C 0,α -mapping, for 0 < α ≤ 1, having as values unbounded self-adjoint operators with compact resolvents and common domain of definition, parametrized by u in an (even infinite dimensional) space, then any continuous (in u) arrangement of the eigenvalues of A(u) is indeed C 0,α in u.
We prove the exponential law A(E × F, G) ∼ = A(E, A(F, G)) (bornological isomorphism) for the fol... more We prove the exponential law A(E × F, G) ∼ = A(E, A(F, G)) (bornological isomorphism) for the following classes A of test functions: B (globally bounded derivatives), W ∞,p (globally p-integrable derivatives), S (Schwartz space), B [M ] (globally Denjoy-Carleman), W [M ],p (Sobolev-Denjoy-Carleman), and S [M ] [L] (Gelfand-Shilov). Here E, F, G are convenient vector spaces (finite dimensional in the cases of W ∞,p and W [M ],p ) and M = (M k ) is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms DiffB, DiffW ∞,p , and DiffS are C ∞ Lie groups, and DiffB {M } , DiffW {M },p , and DiffS {M } {L} , for non-quasianalytic M , are C {M } Lie groups, where DiffA
For Denjoy-Carleman differentiable function classes C M where the weight sequence M = (M k ) is l... more For Denjoy-Carleman differentiable function classes C M where the weight sequence M = (M k ) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is C M if it maps C M -curves to C M -curves. The category of C M -mappings is cartesian closed in the sense that C M (E, C M (F, G)) ∼ = C M (E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C M -diffeomorphisms is a C M -Lie group but not better.
We characterize those regular, holomorphic or formal maps into the orbit space $V/G$ of a complex... more We characterize those regular, holomorphic or formal maps into the orbit space $V/G$ of a complex representation of a finite group $G$ which admit a regular, holomorphic or formal lift to the representation space $V$. In particular, the case of complex reflection groups is investigated.
Let C [M ] be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight seque... more Let C [M ] be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence M = (M k ) is log-convex and has moderate growth. We prove that the groups DiffB [M ] (R n ), DiffW [M ],p (R n ), DiffS [M ] [L] (R n ), and DiffD [M ] (R n ) of C [M ] -diffeomorphisms on R n which differ from the identity by a mapping in B [M ] (global Denjoy-Carleman), W [M ],p (Sobolev-Denjoy-Carleman), S [M ] [L] (Gelfand-Shilov), or D [M ] (Denjoy-Carleman with compact support) are C [M ] -regular Lie groups. As an application we use the R-transform to show that the Hunter-Saxton PDE on the real line is well-posed in any of the classes W [M ],1 , S [M ]
We improve the main results in the paper from the title using a recent refinement of Bronshtein&#... more We improve the main results in the paper from the title using a recent refinement of Bronshtein's theorem due to Colombini, Orr\'u, and Pernazza. They are then in general best possible both in the hypothesis and in the outcome. As a consequence we obtain a result on lifting smooth mappings in several variables.
Let P (x)(z) = z n + P n j=1 (−1) j a j (x)z n−j be a family of polynomials of fixed degree n who... more Let P (x)(z) = z n + P n j=1 (−1) j a j (x)z n−j be a family of polynomials of fixed degree n whose coefficients a j are germs at 0 of smooth (C ∞ ) complex valued functions defined near 0 ∈ R q . We show: If P is generic there exists a finite collection T of transformations Ψ : R q , 0 → R q , 0 such that S {im(Ψ) : Ψ ∈ T } is a neighborhood of 0 and, for each Ψ ∈ T , the family P • Ψ allows smooth parameterizations of its roots near 0. Any Ψ ∈ T is a finite composition of linear coordinate changes and transforma-
For quasianalytic Denjoy-Carleman differentiable function classes C Q where the weight sequence Q... more For quasianalytic Denjoy-Carleman differentiable function classes C Q where the weight sequence Q = (Q k ) is log-convex, stable under derivations, of moderate growth and also an L-intersection (see (1.6)), we prove the following: The category of C Q -mappings is cartesian closed in the sense that C Q (E, C Q (F, G)) ∼ = C Q (E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C Q -diffeomorphisms is a regular C Q -Lie group but not better.
If u 7→ A(u) is a C1,�-mapping having as values unbounded self- adjoint operators with compact re... more If u 7→ A(u) is a C1,�-mapping having as values unbounded self- adjoint operators with compact resolvents and common domain of definition, parametrized by u in an (even infinite dimensional) space then any continuous arrangement of the eigenvalues u 7→ �i(u) is C0,1 in u. If u 7→ A(u) is C0,1, then the eigenvalues may be chosen C0,1/N (even C0,1 if N = 2), locally in u, where N is locally the maximal multiplicity of the eigenvalues.
We prove in a uniform way that all Denjoy-Carleman differentiable function classes of Beurling ty... more We prove in a uniform way that all Denjoy-Carleman differentiable function classes of Beurling type C (M ) and of Roumieu type C {M } , admit a convenient setting if the weight sequence M = (M k
Let C [M ] be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight seque... more Let C [M ] be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence M = (M k ) is log-convex and has moderate growth. We prove that the groups DiffB [M ] (R n ), DiffW [M ],p (R n ), DiffS [M ] [L] (R n ), and DiffD [M ] (R n ) of C [M ] -diffeomorphisms on R n which differ from the identity by a mapping in B [M ] (global Denjoy-Carleman), W [M ],p (Sobolev-Denjoy-Carleman), S [M ] [L] (Gelfand-Shilov), or D [M ] (Denjoy-Carleman with compact support) are C [M ] -regular Lie groups. As an application we use the R-transform to show that the Hunter-Saxton PDE on the real line is well-posed in any of the classes W [M ],1 , S [M ]
We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological... more We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. In the new version references added: Most of the results are known. So this paper will not be submitted to
Journal of Fourier Analysis and Applications, 2014
We consider the operator R, which sends a function on R 2n to its integrals over all affine Lagra... more We consider the operator R, which sends a function on R 2n to its integrals over all affine Lagrangian subspaces in R 2n . We discuss properties of the operator R and of the representation of the affine symplectic group in the space of functions on R 2n .
Pulling back sets of functions in involution by Poisson mappings and adding Casimir functions dur... more Pulling back sets of functions in involution by Poisson mappings and adding Casimir functions during the process allows to construct completely integrable systems. Some examples are investigated in detail. 1991 Mathematics Subject Classification. 58F07.
This paper studies a specific metric on plane curves that has the property of being isometric to ... more This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc. . . ) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided Date: April 24, 2012. 1991 Mathematics Subject Classification. Primary 58B20, 58D15, 58E40.
Let ρ : G → GL(V ) be a rational representation of a reductive linear algebraic group G defined o... more Let ρ : G → GL(V ) be a rational representation of a reductive linear algebraic group G defined over C on a finite dimensional complex vector space V . We show that, for any generic smooth (resp. C M ) curve c : R → V / /G in the categorical quotient V / /G (viewed as affine variety in some C n ) and for any t 0 ∈ R, there exists a positive integer N such that t → c(t 0 ± (t − t 0 ) N ) allows a smooth (resp. C M ) lift to the representation space near t 0 . (C M denotes the Denjoy-Carleman class associated with M = (M k ), which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in V / /G admits locally absolutely continuous (not better!) lifts. Assume that G is finite. We characterize curves admitting differentiable lifts. We show that any germ of a C ∞ curve which represents a lift of a germ of a quasianalytic C M curve in V / /G is actually C M . There are applications to polar representations.
If u → A(u) is a C 0,α -mapping, for 0 < α ≤ 1, having as values unbounded self-adjoint operators... more If u → A(u) is a C 0,α -mapping, for 0 < α ≤ 1, having as values unbounded self-adjoint operators with compact resolvents and common domain of definition, parametrized by u in an (even infinite dimensional) space, then any continuous (in u) arrangement of the eigenvalues of A(u) is indeed C 0,α in u.
We prove the exponential law A(E × F, G) ∼ = A(E, A(F, G)) (bornological isomorphism) for the fol... more We prove the exponential law A(E × F, G) ∼ = A(E, A(F, G)) (bornological isomorphism) for the following classes A of test functions: B (globally bounded derivatives), W ∞,p (globally p-integrable derivatives), S (Schwartz space), B [M ] (globally Denjoy-Carleman), W [M ],p (Sobolev-Denjoy-Carleman), and S [M ] [L] (Gelfand-Shilov). Here E, F, G are convenient vector spaces (finite dimensional in the cases of W ∞,p and W [M ],p ) and M = (M k ) is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms DiffB, DiffW ∞,p , and DiffS are C ∞ Lie groups, and DiffB {M } , DiffW {M },p , and DiffS {M } {L} , for non-quasianalytic M , are C {M } Lie groups, where DiffA
For Denjoy-Carleman differentiable function classes C M where the weight sequence M = (M k ) is l... more For Denjoy-Carleman differentiable function classes C M where the weight sequence M = (M k ) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is C M if it maps C M -curves to C M -curves. The category of C M -mappings is cartesian closed in the sense that C M (E, C M (F, G)) ∼ = C M (E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C M -diffeomorphisms is a C M -Lie group but not better.
We characterize those regular, holomorphic or formal maps into the orbit space $V/G$ of a complex... more We characterize those regular, holomorphic or formal maps into the orbit space $V/G$ of a complex representation of a finite group $G$ which admit a regular, holomorphic or formal lift to the representation space $V$. In particular, the case of complex reflection groups is investigated.
Let C [M ] be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight seque... more Let C [M ] be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence M = (M k ) is log-convex and has moderate growth. We prove that the groups DiffB [M ] (R n ), DiffW [M ],p (R n ), DiffS [M ] [L] (R n ), and DiffD [M ] (R n ) of C [M ] -diffeomorphisms on R n which differ from the identity by a mapping in B [M ] (global Denjoy-Carleman), W [M ],p (Sobolev-Denjoy-Carleman), S [M ] [L] (Gelfand-Shilov), or D [M ] (Denjoy-Carleman with compact support) are C [M ] -regular Lie groups. As an application we use the R-transform to show that the Hunter-Saxton PDE on the real line is well-posed in any of the classes W [M ],1 , S [M ]
We improve the main results in the paper from the title using a recent refinement of Bronshtein&#... more We improve the main results in the paper from the title using a recent refinement of Bronshtein's theorem due to Colombini, Orr\'u, and Pernazza. They are then in general best possible both in the hypothesis and in the outcome. As a consequence we obtain a result on lifting smooth mappings in several variables.
Let P (x)(z) = z n + P n j=1 (−1) j a j (x)z n−j be a family of polynomials of fixed degree n who... more Let P (x)(z) = z n + P n j=1 (−1) j a j (x)z n−j be a family of polynomials of fixed degree n whose coefficients a j are germs at 0 of smooth (C ∞ ) complex valued functions defined near 0 ∈ R q . We show: If P is generic there exists a finite collection T of transformations Ψ : R q , 0 → R q , 0 such that S {im(Ψ) : Ψ ∈ T } is a neighborhood of 0 and, for each Ψ ∈ T , the family P • Ψ allows smooth parameterizations of its roots near 0. Any Ψ ∈ T is a finite composition of linear coordinate changes and transforma-
For quasianalytic Denjoy-Carleman differentiable function classes C Q where the weight sequence Q... more For quasianalytic Denjoy-Carleman differentiable function classes C Q where the weight sequence Q = (Q k ) is log-convex, stable under derivations, of moderate growth and also an L-intersection (see (1.6)), we prove the following: The category of C Q -mappings is cartesian closed in the sense that C Q (E, C Q (F, G)) ∼ = C Q (E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C Q -diffeomorphisms is a regular C Q -Lie group but not better.
If u 7→ A(u) is a C1,�-mapping having as values unbounded self- adjoint operators with compact re... more If u 7→ A(u) is a C1,�-mapping having as values unbounded self- adjoint operators with compact resolvents and common domain of definition, parametrized by u in an (even infinite dimensional) space then any continuous arrangement of the eigenvalues u 7→ �i(u) is C0,1 in u. If u 7→ A(u) is C0,1, then the eigenvalues may be chosen C0,1/N (even C0,1 if N = 2), locally in u, where N is locally the maximal multiplicity of the eigenvalues.
We prove in a uniform way that all Denjoy-Carleman differentiable function classes of Beurling ty... more We prove in a uniform way that all Denjoy-Carleman differentiable function classes of Beurling type C (M ) and of Roumieu type C {M } , admit a convenient setting if the weight sequence M = (M k
Let C [M ] be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight seque... more Let C [M ] be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence M = (M k ) is log-convex and has moderate growth. We prove that the groups DiffB [M ] (R n ), DiffW [M ],p (R n ), DiffS [M ] [L] (R n ), and DiffD [M ] (R n ) of C [M ] -diffeomorphisms on R n which differ from the identity by a mapping in B [M ] (global Denjoy-Carleman), W [M ],p (Sobolev-Denjoy-Carleman), S [M ] [L] (Gelfand-Shilov), or D [M ] (Denjoy-Carleman with compact support) are C [M ] -regular Lie groups. As an application we use the R-transform to show that the Hunter-Saxton PDE on the real line is well-posed in any of the classes W [M ],1 , S [M ]
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