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51. 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

52. 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

53. 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

54. CMB 2000 (vol 43 pp. 427)

Ivey, Thomas A.
Helices, Hasimoto Surfaces and Bäcklund 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, Bäcklund transformations
Categories:53A05, 58F37, 52C42, 58A15

55. 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 theory
Categories:58D27, 30F99

56. 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 functions
Categories:49J52, 58C20, 49J50, 90C26

57. 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, spectrum
Categories:35P25, 58G25

58. 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 functions
Categories:26D15, 58G30

59. 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

60. 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

61. 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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