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51. CMB 2002 (vol 45 pp. 3)

Azagra, D.; Dobrowolski, T.
 Real-Analytic Negligibility of Points and Subspaces in Banach Spaces, with Applications We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an infinite-dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$-torus on certain Banach spaces. Categories:46B20, 58B99

52. CMB 2001 (vol 44 pp. 376)

Zhang, Xi
 A Note on $p$-Harmonic $1$-Forms on Complete Manifolds In this paper we prove that there is no nontrivial $L^{q}$-integrably $p$-harmonic $1$-form on a complete manifold with nonnegatively Ricci curvature $(0 Keywords:$p$-harmonic,$1$-form, complete manifold, Sobolev inequalityCategories:58E20, 53C21 53. CMB 2001 (vol 44 pp. 323) Schuman, Bertrand  Une classe d'hamiltoniens polynomiaux isochrones Soit$H_0 = \frac{x^2+y^2}{2}$un hamiltonien isochrone du plan$\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones qui sont des perturbations polynomiales de$H_0$. On obtient alors une condition n\'ecessaire d'isochronisme, et un crit\ere de choix pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant une g\'en\'eralisation du caract\ere isochrone des perturbations hamiltoniennes homog\enes consid\'er\'ees dans [L], [P], [S]. Let$H_0 = \frac{x^2+y^2}{2}$be an isochronous Hamiltonian of the plane$\Rset^2$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian$H_0$considered in [L], [P], [S]. Keywords:Hamiltonian system, normal forms, resonance, linearizationCategories:34C20, 58F05, 58F22, 58F30 54. CMB 2001 (vol 44 pp. 210) Leung, Man Chun  Growth Estimates on Positive Solutions of the Equation$\Delta u+K u^{\frac{n+2}{n-2}}=0$in$\R^n$We construct unbounded positive$C^2$-solutions of the equation$\Delta u + K u^{(n + 2)/(n - 2)} = 0$in$\R^n$(equipped with Euclidean metric$g_o$) such that$K$is bounded between two positive numbers in$\R^n$, the conformal metric$g=u^{4/(n-2)}g_o$is complete, and the volume growth of$g$can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on$u$, we obtain growth estimate on the$L^{2n/(n-2)}$-norm of the solution and show that it has slow decay. Keywords:positive solution, conformal scalar curvature equation, growth estimateCategories:35J60, 58G03 55. CMB 2001 (vol 44 pp. 160) Langlands, Robert P.  The Trace Formula and Its Applications: An Introduction to the Work of James Arthur James Arthur was awarded the Canada Gold Medal of the National Science and Engineering Research Council in 1999. This introduction to his work is an attempt to explain his methods and his goals to the mathematical community at large. Categories:11F70, 11F72, 58G25 56. CMB 2001 (vol 44 pp. 140) Gotay, Mark J.; Grabowski, Janusz  On Quantizing Nilpotent and Solvable Basic Algebras We prove an algebraic `no-go theorem'' to the effect that a nontrivial \pa\ cannot be realized as an associative algebra with the commu\-ta\-tor bracket. Using it, we show that there is an obstruction to quantizing the \pa\ of polynomials generated by a nilpotent \ba\ on a \sm. This result generalizes \gr 's famous theorem on the impossibility of quantizing the Poisson algebra of polynomials on$\r^{2n}$. Finally, we explicitly construct a polynomial quantization of a \sm\ with a solvable \ba, thereby showing that the obstruction in the nilpotent case does not extend to the solvable case. Categories:81S99, 58F06 57. CMB 2001 (vol 44 pp. 129) Currás-Bosch, Carlos  LinÃ©arisation symplectique en dimension 2 In this paper the germ of neighborhood of a compact leaf in a Lagrangian foliation is symplectically classified when the compact leaf is$\bT^2$, the affine structure induced by the Lagrangian foliation on the leaf is complete, and the holonomy of$\bT^2$in the foliation linearizes. The germ of neighborhood is classified by a function, depending on one transverse coordinate, this function is related to the affine structure of the nearly compact leaves. Keywords:symplectic manifold, Lagrangian foliation, affine connectionCategories:53C12, 58F05 58. CMB 2001 (vol 44 pp. 105) Pilipović, Stevan  Convolution Equation in$\mathcal{S}^{\prime\ast}$---Propagation of Singularities The singular spectrum of$u$in a convolution equation$\mu * u = f$, where$\mu$and$f$are tempered ultradistributions of Beurling or Roumieau type is estimated by $$SS u \subset (\mathbf{R}^n \times \Char \mu) \cup SS f.$$ The same is done for$SS_{*}u$. Categories:32A40, 46F15, 58G07 59. CMB 2000 (vol 43 pp. 427) Ivey, Thomas A.  Helices, Hasimoto Surfaces and BÃ¤cklund Transformations Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in$\R^3$that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the B\"acklund transformation for constant torsion curves in$\R^3$, previously derived from the well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in$H^3$or$S^3$leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces. Keywords:surfaces, filament flow, BÃ¤cklund transformationsCategories:53A05, 58F37, 52C42, 58A15 60. CMB 2000 (vol 43 pp. 183) Ionesei, Gheorghe  A Gauge Theoretic Proof of the Abel-Jacobi Theorem We present a new, simple proof of the classical Abel-Jacobi theorem using some elementary gauge theoretic arguments. Keywords:Abel-Jacobi theorem, abelian gauge theoryCategories:58D27, 30F99 61. CMB 2000 (vol 43 pp. 51) Edward, Julian  Eigenfunction Decay For the Neumann Laplacian on Horn-Like Domains The growth properties at infinity for eigenfunctions corresponding to embedded eigenvalues of the Neumann Laplacian on horn-like domains are studied. For domains that pinch at polynomial rate, it is shown that the eigenfunctions vanish at infinity faster than the reciprocal of any polynomial. For a class of domains that pinch at an exponential rate, weaker,$L^2$bounds are proven. A corollary is that eigenvalues can accumulate only at zero or infinity. Keywords:Neumann Laplacian, horn-like domain, spectrumCategories:35P25, 58G25 62. CMB 2000 (vol 43 pp. 25) Bounkhel, M.; Thibault, L.  Subdifferential Regularity of Directionally Lipschitzian Functions Formulas for the Clarke subdifferential are always expressed in the form of inclusion. The equality form in these formulas generally requires the functions to be directionally regular. This paper studies the directional regularity of the general class of extended-real-valued functions that are directionally Lipschitzian. Connections with the concept of subdifferential regularity are also established. Keywords:subdifferential regularity, directional regularity, directionally Lipschitzian functionsCategories:49J52, 58C20, 49J50, 90C26 63. CMB 1999 (vol 42 pp. 478) Pruss, Alexander R.  A Remark On the Moser-Aubin Inequality For Axially Symmetric Functions On the Sphere Let$\scr S_r$be the collection of all axially symmetric functions$f$in the Sobolev space$H^1(\Sph^2)$such that$\int_{\Sph^2} x_ie^{2f(\mathbf{x})} \, d\omega(\mathbf{x})$vanishes for$i=1,2,3\$. We prove that $$\inf_{f\in \scr S_r} \frac12 \int_{\Sph^2} |\nabla f|^2 \, d\omega + 2\int_{\Sph^2} f \, d\omega- \log \int_{\Sph^2} e^{2f} \, d\omega > -\oo,$$ and that this infimum is attained. This complements recent work of Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang concerning the Moser-Aubin inequality. Keywords:Moser inequality, borderline Sobolev inequalities, axially symmetric functionsCategories:26D15, 58G30
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