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26. CMB 2005 (vol 48 pp. 340)

Andruchow, Esteban
 Short Geodesics of Unitaries in the $L^2$ Metric Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$ Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.Categories:46L51, 58B10, 58B25

27. CMB 2005 (vol 48 pp. 283)

Thibault, Lionel; Zagrodny, Dariusz
 Enlarged Inclusion of Subdifferentials This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma$-enlargement of the subdifferential of $g$, then the difference of those functions is $\gamma$-Lipschitz over their effective domain. Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable functionCategories:49J52, 46N10, 58C20

28. CMB 2004 (vol 47 pp. 624)

Zhang, Xi
 A Compactness Theorem for Yang-Mills Connections In this paper, we consider Yang-Mills connections on a vector bundle $E$ over a compact Riemannian manifold $M$ of dimension $m> 4$, and we show that any set of Yang-Mills connections with the uniformly bounded $L^{\frac{m}{2}}$-norm of curvature is compact in $C^{\infty}$ topology. Keywords:Yang-Mills connection, vector bundle, gauge transformationCategories:58E20, 53C21

29. CMB 2004 (vol 47 pp. 607)

Plamenevskaya, Olga
 A Residue Formula for $\SU(2)$-Valued Moment Maps Jeffrey and Kirwan gave expressions for intersection pairings on the reduced space $M_0=\mu^{-1}(0)/G$ of a Hamiltonian $G$-space $M$ in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian $\SU(2)$-space. The definition of quasi-Hamiltonian $G$-spaces was recently introduced in . Category:58F05

30. CMB 2004 (vol 47 pp. 515)

Frigon, M.
 Remarques sur l'enlacement en thÃ©orie des points critiques pour des fonctionnelles continues Dans cet article, \a partir de la notion d'enlacement introduite dans ~\cite{F} entre des paires d'ensembles $(B,A)$ et $(Q,P)$, nous \'etablissons l'existence d'un point critique d'une fonctionnelle continue sur un espace m\'etrique lorsqu'une de ces paires enlace l'autre. Des renseignements sur la localisation du point critique sont aussi obtenus. Ces r\'esultats conduisent \a une g\'en\'eralisation du th\'eor\eme des trois points critiques. Finalement, des applications \a des probl\emes aux limites pour une \'equation quasi-lin\'eaire elliptique sont pr\'esent\'ees. Categories:58E05, 35J20

31. CMB 2003 (vol 46 pp. 481)

Bachir, M.; Lancien, G.
 On the Composition of Differentiable Functions We prove that a Banach space $X$ has the Schur property if and only if every $X$-valued weakly differentiable function is Fr\'echet differentiable. We give a general result on the Fr\'echet differentiability of $f\circ T$, where $f$ is a Lipschitz function and $T$ is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm $\Vert \ \Vert_{\lip}$ on various spaces of Lipschitz functions. Categories:58C20, 46B20

32. CMB 2002 (vol 45 pp. 378)

Fernández-López, Manuel; García-Río, Eduardo; Kupeli, Demir N.
 The Local MÃ¶bius Equation and Decomposition Theorems in Riemannian Geometry A partial differential equation, the local M\"obius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local M\"obius equation and an additional partial differential equation. Keywords:submersion, MÃ¶bius equation, twisted product, warped product, product Riemannian manifoldsCategories:53C12, 58J99

33. CMB 2002 (vol 45 pp. 97)

Haas, Andrew
 Invariant Measures and Natural Extensions We study ergodic properties of a family of interval maps that are given as the fractional parts of certain real M\"obius transformations. Included are the maps that are exactly $n$-to-$1$, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures. Keywords:Continued fractions, interval maps, invariant measuresCategories:11J70, 58F11, 58F03

34. 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

35. 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 36. 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 37. 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 38. 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 39. 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 40. 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 41. 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 42. 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 43. 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 44. 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 45. 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 46. 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 47. CMB 1997 (vol 40 pp. 271) Bergweiler, Walter  Non-real periodic points of entire functions It is shown that if$f$is an entire transcendental function,$l$a straight line in the complex plane, and$n\geq 2$, then$f$has infinitely many repelling periodic points of period$n$that do not lie on$l$. Categories:30D05, 58F23 48. CMB 1997 (vol 40 pp. 285) Crawford, T. Arleigh  The space of harmonic maps from the$2$-sphere to the complex projective plane In this paper we study the topology of the space of harmonic maps from$S^2$to$\CP 2$. We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to$\CP n$for$n\geq 2$. We show that the components of maps to$\CP 2$are complex manifolds. Categories:58E20, 58D27 49. CMB 1997 (vol 40 pp. 204) Meyerhoff, Robert; Ouyang, Mingqing  The$\eta$-invariants of cusped hyperbolic$3$-manifolds In this paper, we define the$\eta$-invariant for a cusped hyperbolic$3\$-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements. Categories:57M50, 53C30, 58G25
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