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351. 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

352. CMB 2001 (vol 44 pp. 126)

Zeron, E. Santillan
 Each Copy of the Real Line in $\C^2$ is Removable Around 1995, Professors Lupacciolu, Chirka and Stout showed that a closed subset of $\C^N$ ($N\geq 2$) is removable for holomorphic functions, if its topological dimension is less than or equal to $N-2$. Besides, they asked whether closed subsets of $\C^2$ homeomorphic to the real line (the simplest 1-dimensional sets) are removable for holomorphic functions. In this paper we propose a positive answer to that question. Keywords:holomorphic function, removable setCategory:32D20

353. CMB 2001 (vol 44 pp. 97)

Ou, Zhiming M.; Williams, Kenneth S.
 On the Density of Cyclic Quartic Fields An asymptotic formula is obtained for the number of cyclic quartic fields over $Q$ with discriminant $\leq x$. Keywords:cyclic quartic fields, density, discriminantCategories:11R16, 11R29

354. 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

355. CMB 2000 (vol 43 pp. 496)

Xu, Yuan
 Harmonic Polynomials Associated With Reflection Groups We extend Maxwell's representation of harmonic polynomials to $h$-harmonics associated to a reflection invariant weight function $h_k$. Let $\CD_i$, $1\le i \le d$, be Dunkl's operators associated with a reflection group. For any homogeneous polynomial $P$ of degree $n$, we prove the polynomial $|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$ is a $h$-harmonic polynomial of degree $n$, where $\gamma = \sum k_i$ and $\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for $h$-harmonics. We also discuss self-adjoint operators acting on the space of $h$-harmonics. Keywords:$h$-harmonics, reflection group, Dunkl's operatorsCategories:33C50, 33C45

356. CMB 2000 (vol 43 pp. 440)

Koufogiorgos, Themis; Tsichlias, Charalambos
 On the Existence of a New Class of Contact Metric Manifolds A new class of 3-dimensional contact metric manifolds is found. Moreover it is proved that there are no such manifolds in dimensions greater than 3. Keywords:contact metric manifolds, generalized $(\kappa,\mu)$-contact metric manifoldsCategories:53C25, 53C15

357. CMB 2000 (vol 43 pp. 418)

Gong, Guihua; Jiang, Xinhui; Su, Hongbing
 Obstructions to $\mathcal{Z}$-Stability for Unital Simple $C^*$-Algebras Let $\cZ$ be the unital simple nuclear infinite dimensional $C^*$-algebra which has the same Elliott invariant as $\bbC$, introduced in \cite{JS}. A $C^*$-algebra is called $\cZ$-stable if $A \cong A \otimes \cZ$. In this note we give some necessary conditions for a unital simple $C^*$-algebra to be $\cZ$-stable. Keywords:simple $C^*$-algebra, $\mathcal{Z}$-stability, weak (un)perforation in $K_0$ group, property $\Gamma$, finitenessCategory:46L05

358. CMB 2000 (vol 43 pp. 294)

Bracci, Filippo
 Fixed Points of Commuting Holomorphic Maps Without Boundary Regularity We identify a class of domains of $\C^n$ in which any two commuting holomorphic self-maps have a common fixed point. Keywords:Holomorphic self-maps, commuting functions, fixed points, Wolff point, Julia's LemmaCategories:32A10, 32A40, 32H15, 32A30

359. CMB 2000 (vol 43 pp. 362)

Kim, Hwankoo
 Examples of Half-Factorial Domains In this paper, we determine some sufficient conditions for an $A + XB[X]$ domain to be an HFD. As a consequence we give new examples of HFDs of the type $A + XB[X]$. Keywords:atomic domain, HFDCategories:13A05, 13B30, 13F15, 13G05

360. CMB 2000 (vol 43 pp. 330)

Hare, Kathryn E.
 Maximal Operators and Cantor Sets We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on $L^2$ if the Cantor set has positive Hausdorff dimension. Keywords:maximal functions, Cantor set, lacunary setCategories:42B25, 43A46

361. CMB 2000 (vol 43 pp. 268)

Bogley, W. A.; Gilbert, N. D.; Howie, James
 Cockcroft Properties of Thompson's Group In a study of the word problem for groups, R.~J.~Thompson considered a certain group $F$ of self-homeomorphisms of the Cantor set and showed, among other things, that $F$ is finitely presented. Using results of K.~S.~Brown and R.~Geoghegan, M.~N.~Dyer showed that $F$ is the fundamental group of a finite two-complex $Z^2$ having Euler characteristic one and which is {\em Cockcroft}, in the sense that each map of the two-sphere into $Z^2$ is homologically trivial. We show that no proper covering complex of $Z^2$ is Cockcroft. A general result on Cockcroft properties implies that no proper regular covering complex of any finite two-complex with fundamental group $F$ is Cockcroft. Keywords:two-complex, covering space, Cockcroft two-complex, Thompson's groupCategories:57M20, 20F38, 57M10, 20F34

362. CMB 2000 (vol 43 pp. 21)

Barnes, Bruce A.
 The Commutant of an Abstract Backward Shift A bounded linear operator $T$ on a Banach space $X$ is an abstract backward shift if the nullspace of $T$ is one dimensional, and the union of the null spaces of $T^k$ for all $k \geq 1$ is dense in $X$. In this paper it is shown that the commutant of an abstract backward shift is an integral domain. This result is used to derive properties of operators in the commutant. Keywords:backward shift, commutantCategory:47A99

363. CMB 2000 (vol 43 pp. 60)

Farkas, Daniel R.; Linnell, Peter A.
 Trivial Units in Group Rings Let $G$ be an arbitrary group and let $U$ be a subgroup of the normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$ as a subgroup of finite index, then $U = G$. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group. Keywords:units, trace, finite conjugate subgroupCategories:16S34, 16U60

364. 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

365. CMB 2000 (vol 43 pp. 3)

 Resolutions of Associative and Lie Algebras Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees. Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedronCategories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50

366. 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

367. CMB 1999 (vol 42 pp. 427)

Berndt, Bruce C.; Chan, Heng Huat
 Ramanujan and the Modular $j$-Invariant A new infinite product $t_n$ was introduced by S.~Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan's assertions about $t_n$ by establishing new connections between the modular $j$-invariant and Ramanujan's cubic theory of elliptic functions to alternative bases. We also show that for certain integers $n$, $t_n$ generates the Hilbert class field of $\mathbb{Q} (\sqrt{-n})$. This shows that $t_n$ is a new class invariant according to H.~Weber's definition of class invariants. Keywords:modular functions, the Borweins' cubic theta-functions, Hilbert class fieldsCategories:33C05, 33E05, 11R20, 11R29

368. CMB 1999 (vol 42 pp. 335)

Kim, Goansu; Tang, C. Y.
 Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups We derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of HNN-extensions of nilpotent groups with cyclic associated subgroups. Keywords:HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finiteCategories:20E26, 20E06, 20F10

369. CMB 1999 (vol 42 pp. 321)

Kikuchi, Masato
 Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section~2 the equivalence between Shimogaki's theorem and some martingale inequalities will be established, and in Section~3 the equivalence between Boyd's theorem and martingale inequalities with change of probability measure will be established. Keywords:martingale inequalities, rearrangement invariant function spacesCategories:60G44, 60G46, 46E30

370. CMB 1999 (vol 42 pp. 285)

Deng, Peiming
 On Kloosterman Sums with Oscillating Coefficients In this paper we prove: for any positive integers $a$ and $q$ with $(a,q) =1$, we have uniformly $$\sum_{\substack{n \leq N \\ (n,q) = 1, \,n\on \equiv 1 (\mod q)}} \mu (n) e \left( \frac{a\on}{q} \right) \ll Nd (q) \left\{ \frac{\log^{\frac52} N}{q^{\frac12}} + \frac{q^{\frac15} \log^{\frac{13}5} N}{N^{\frac15}} \right\}.$$ This improves the previous bound obtained by D.~Hajela, A.~Pollington and B.~Smith~\cite{5}. Keywords:Kloosterman sums, oscillating coefficients, estimateCategory:10G10

371. CMB 1999 (vol 42 pp. 274)

Dădărlat, Marius; Eilers, Søren
 The Bockstein Map is Necessary We construct two non-isomorphic nuclear, stably finite, real rank zero $C^\ast$-algebras $E$ and $E'$ for which there is an isomorphism of ordered groups $\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to \bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible with all the coefficient transformations. The $C^\ast$-algebras $E$ and $E'$ are not isomorphic since there is no $\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair of non-isomorphic, real rank zero, purely infinite $C^\ast$-algebras with similar properties. Keywords:$K$-theory, torsion coefficients, natural transformations, Bockstein maps, $C^\ast$-algebras, real rank zero, purely infinite, classificationCategories:46L35, 46L80, 19K14

372. CMB 1999 (vol 42 pp. 190)

Gilmer, Patrick M.
 Topological Quantum Field Theory and Strong Shift Equivalence Given a TQFT in dimension $d+1,$ and an infinite cyclic covering of a closed $(d+1)$-dimensional manifold $M$, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R.~Williams' work in symbolic dynamics. The Turaev-Viro module associated to a TQFT and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of $M$ has an $S^1$ factor and the infinite cyclic cover of the boundary is standard. We define a variant of a TQFT associated to a finite group $G$ which has been studied by Quinn. In this way, we recover a link invariant due to D.~Silver and S.~Williams. We also obtain a variation on the Silver-Williams invariant, by using the TQFT associated to $G$ in its unmodified form. Keywords:knot, link, TQFT, symbolic dynamics, shift equivalenceCategories:57R99, 57M99, 54H20

373. CMB 1999 (vol 42 pp. 139)

Bonet, José; Domański, Paweł; Lindström, Mikael
 Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions Every weakly compact composition operator between weighted Banach spaces $H_v^{\infty}$ of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space $H_v^{\infty}$ are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces. Keywords:weighted Banach spaces of holomorphic functions, composition operator, compact operator, weakly compact operatorCategories:47B38, 30D55, 46E15

374. CMB 1999 (vol 42 pp. 198)

Guadalupe, José J.; Pérez, Mario; Varona, Juan L.
 Commutators and Analytic Dependence of Fourier-Bessel Series on $(0,\infty)$ In this paper we study the boundedness of the commutators $[b, S_n]$ where $b$ is a $\BMO$ function and $S_n$ denotes the $n$-th partial sum of the Fourier-Bessel series on $(0,\infty)$. Perturbing the measure by $\exp(2b)$ we obtain that certain operators related to $S_n$ depend analytically on the functional parameter $b$. Keywords:Fourier-Bessel series, commutators, BMO, $A_p$ weightsCategory:42C10

375. CMB 1999 (vol 42 pp. 104)

Nikolskaia, Ludmila
 InstabilitÃ© de vecteurs propres d'opÃ©rateurs linÃ©aires We consider some geometric properties of eigenvectors of linear operators on infinite dimensional Hilbert space. It is proved that the property of a family of vectors $(x_n)$ to be eigenvectors $Tx_n= \lambda_n x_n$ ($\lambda_n \noteq \lambda_k$ for $n\noteq k$) of a bounded operator $T$ (admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1)~for the sequence $(x_n+\epsilon_n v_n)_{n\geq k(\epsilon)}$ to be admissible for every admissible $(x_n)$ and for a suitable choice of small numbers $\epsilon_n\noteq 0$ it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist $\gamma_n\in \C$ such that $v_n= \gamma_n v_{k}$ for $n\geq k$ (Theorem~2); (2)~for a bounded operator $A$ to transform admissible families $(x_n)$ into admissible families $(Ax_n)$ it is necessary and sufficient that $A$ be left invertible (Theorem~4). Keywords:eigenvectors, minimal families, reproducing kernelsCategories:47A10, 46B15
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