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401. 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

402. CMB 2000 (vol 43 pp. 3)

Adin, Ron; Blanc, David
 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

403. 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

404. 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

405. 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

406. 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

407. 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

408. 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

409. 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

410. 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

411. 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

412. 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

413. 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

414. 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

415. 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

416. CMB 1999 (vol 42 pp. 125)

Smith, Larry
 Modular Vector Invariants of Cyclic Permutation Representations Vector invariants of finite groups (see the introduction for an explanation of the terminology) have often been used to illustrate the difficulties of invariant theory in the modular case: see, \eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB}, \cite{poly} and \cite{survey}. It is therefore all the more surprising that the {\it unpleasant} properties of these invariants may be derived from two unexpected, and remarkable, {\it nice} properties: namely for vector permutation invariants of the cyclic group $\mathbb{Z}/p$ of prime order in characteristic $p$ the image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon \mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal, and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im (\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern classes of the action. Keywords:polynomial invariants of finite groupsCategory:13A50

417. CMB 1999 (vol 42 pp. 118)

Rao, T. S. S. R. K.
 Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$ For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak$^\ast$-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property. Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions, $M$-idealsCategories:46B20, 46E40

418. CMB 1999 (vol 42 pp. 97)

Kwon, E. G.
 On Analytic Functions of Bergman $\BMO$ in the Ball Let $B = B_n$ be the open unit ball of $\bbd C^n$ with volume measure $\nu$, $U = B_1$ and ${\cal B}$ be the Bloch space on $U$. ${\cal A}^{2, \alpha} (B)$, $1 \leq \alpha < \infty$, is defined as the set of holomorphic $f\colon B \rightarrow \bbd C$ for which $$\int_B \vert f(z) \vert^2 \left( \frac 1{\vert z\vert} \log \frac 1{1 - \vert z\vert } \right)^{-\alpha} \frac {d\nu (z)}{1-\vert z\vert} < \infty$$ if $0 < \alpha <\infty$ and ${\cal A}^{2, 1} (B) = H^2(B)$, the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic $f\colon B \rightarrow U$ for which the composition operator $C_f \colon {\cal B} \rightarrow {\cal A}^{2, \alpha}(B)$ defined by $C_f (g) = g\circ f$, $g \in {\cal B}$, is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric. Keywords:Bergman distance, \BMOA$, Hardy space, Bloch functionCategory:32A37 419. CMB 1999 (vol 42 pp. 13) Brendle, Jörg  Dow's Principle and$Q$-Sets A$Q$-set is a set of reals every subset of which is a relative$G_\delta$. We investigate the combinatorics of$Q$-sets and discuss a question of Miller and Zhou on the size$\qq$of the smallest set of reals which is not a$Q$-set. We show in particular that various natural lower bounds for$\qq$are consistently strictly smaller than$\qq$. Keywords:$Q$-set, cardinal invariants of the continuum, pseudointersection number,$\MA$($\sigma$-centered), Dow's principle, almost disjoint family, almost disjointness principle, iterated forcingCategories:03E05, 03E35, 54A35 420. CMB 1998 (vol 41 pp. 497) Borwein, J. M.; Girgensohn, R.; Wang, Xianfu  On the construction of HÃ¶lder and Proximal Subderivatives We construct Lipschitz functions such that for all$s>0$they are$s$-H\"older, and so proximally, subdifferentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz functions with prescribed H\"older and approximate subderivatives. Keywords:Lipschitz functions, HÃ¶lder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, HÃ¶lder smooth, dyadic rationalsCategories:49J52, 26A16, 26A24 421. CMB 1998 (vol 41 pp. 348) Tymchatyn, E. D.; Yang, Chang-Cheng  Characterizing continua by disconnection properties We study Hausdorff continua in which every set of certain cardinality contains a subset which disconnects the space. We show that such continua are rim-finite. We give characterizations of this class among metric continua. As an application of our methods, we show that continua in which each countably infinite set disconnects are generalized graphs. This extends a result of Nadler for metric continua. Keywords:disconnection properties, rim-finite continua, graphsCategories:54D05, 54F20, 54F50 422. CMB 1998 (vol 41 pp. 267) Fukuma, Yoshiaki  On the nonemptiness of the adjoint linear system of polarized manifold Let$(X,L)$be a polarized manifold over the complex number field with$\dim X=n$. In this paper, we consider a conjecture of M.~C.~Beltrametti and A.~J.~Sommese and we obtain that this conjecture is true if$n=3$and$h^{0}(L)\geq 2$, or$\dim \Bs |L|\leq 0$for any$n\geq 3$. Moreover we can generalize the result of Sommese. Keywords:Polarized manifold, adjoint bundleCategories:14C20, 14J99 423. CMB 1998 (vol 41 pp. 129) Lee, Young Joo  Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the$n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized. Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.Categories:47B38, 32A37 424. CMB 1998 (vol 41 pp. 207) Philos, Ch. G.; Sficas, Y. G.  An oscillation criterion for first order linear delay differential equations A new oscillation criterion is given for the delay differential equation$x'(t)+p(t)x \left(t-\tau(t)\right)=0$, where$p$,$\tau \in \C \left([0,\infty),[0,\infty)\right)$and the function$T$defined by$T(t)=t-\tau(t)$,$t\ge 0$is increasing and such that$\lim_{t\to\infty}T(t)=\infty$. This criterion concerns the case where$\liminf_{t\to\infty} \int_{T(t)}^{t}p(s)\,ds\le \frac{1}{e}$. Keywords:Delay differential equation, oscillationCategory:34K15 425. CMB 1998 (vol 41 pp. 41) Giner, E.  On the Clarke subdifferential of an integral functional on$L_p$,$1\leq p < \infty$Given an integral functional defined on$L_p$,$1 \leq p <\infty\$, under a growth condition we give an upper bound of the Clarke directional derivative and we obtain a nice inclusion between the Clarke subdifferential of the integral functional and the set of selections of the subdifferential of the integrand. Keywords:Integral functional, integrand, epi-derivativeCategories:28A25, 49J52, 46E30
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