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$, Hardy space, Bloch functionCategory:32A37 377. CMB 1999 (vol 42 pp. 13) Brendle, Jörg  Dow's Principle and$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$-centered), Dow's principle, almost disjoint family, almost disjointness principle, iterated forcingCategories:03E05, 03E35, 54A35 378. CMB 1999 (vol 42 pp. 125) Smith, Larry  Modular Vector Invariants of Cyclic Permutation Representations Vector invariants of finite groups (see the introduction for an explanation of the terminology) have often been used to illustrate the difficulties of invariant theory in the modular case: see, \eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB}, \cite{poly} and \cite{survey}. It is therefore all the more surprising that the {\it unpleasant} properties of these invariants may be derived from two unexpected, and remarkable, {\it nice} properties: namely for vector permutation invariants of the cyclic group$\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})$is a polynomial algebra on the top Chern classes of the action. Keywords:polynomial invariants of finite groupsCategory:13A50 379. CMB 1999 (vol 42 pp. 118) Rao, T. S. S. R. K.  Points of Weak$^\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$-idealsCategories:46B20, 46E40 380. CMB 1998 (vol 41 pp. 497) Borwein, J. M.; Girgensohn, R.; Wang, Xianfu  On the construction of HÃ¶lder and Proximal Subderivatives We construct Lipschitz functions such that for all$s>0$they are$s$-H\"older, and so proximally, subdifferentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz functions with prescribed H\"older and approximate subderivatives. Keywords:Lipschitz functions, HÃ¶lder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, HÃ¶lder smooth, dyadic rationalsCategories:49J52, 26A16, 26A24 381. CMB 1998 (vol 41 pp. 348) Tymchatyn, E. D.; Yang, Chang-Cheng  Characterizing continua by disconnection properties We study Hausdorff continua in which every set of certain cardinality contains a subset which disconnects the space. We show that such continua are rim-finite. We give characterizations of this class among metric continua. As an application of our methods, we show that continua in which each countably infinite set disconnects are generalized graphs. This extends a result of Nadler for metric continua. Keywords:disconnection properties, rim-finite continua, graphsCategories:54D05, 54F20, 54F50 382. CMB 1998 (vol 41 pp. 267) Fukuma, Yoshiaki  On the nonemptiness of the adjoint linear system of polarized manifold Let$(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$. Moreover we can generalize the result of Sommese. Keywords:Polarized manifold, adjoint bundleCategories:14C20, 14J99 383. CMB 1998 (vol 41 pp. 207) Philos, Ch. G.; Sficas, Y. G.  An oscillation criterion for first order linear delay differential equations A new oscillation criterion is given for the delay differential equation$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}$. Keywords:Delay differential equation, oscillationCategory:34K15 384. CMB 1998 (vol 41 pp. 129) Lee, Young Joo  Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the$n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized. Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.Categories:47B38, 32A37 385. CMB 1998 (vol 41 pp. 49) Harrison, K. J.; Ward, J. A.; Eaton, L-J.  Stability of weighted darma filters We study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case. Keywords:Difference equations, adaptive$\DARMA$filters, weighted shifts,, stability and boundedness, automatic continuityCategories:47A62, 47B37, 93D25, 42A85, 47N70 386. CMB 1998 (vol 41 pp. 71) Hurrelbrink, Jurgen; Rehmann, Ulf  Splitting patterns and trace forms The splitting pattern of a quadratic form$q$over a field$k$consists of all distinct Witt indices that occur for$q$over extension fields of$k$. In small dimensions, the complete list of splitting patterns of quadratic forms is known. We show that {\it all\/} splitting patterns of quadratic forms of dimension at most nine can be realized by trace forms. Keywords:Quadratic forms, Witt indices, generic splitting.Category:11E04 387. CMB 1998 (vol 41 pp. 41) Giner, E.  On the Clarke subdifferential of an integral functional on$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 388. CMB 1997 (vol 40 pp. 488) Maouche, Abdelaziz  CaractÃ©risations spectrales du radical et du socle d'une paire de jordan-banach If$f$and$g$are two analytic functions from a domain$D$of the complex plane into respectively the Banach spaces$V^+$and$V^-$, we prove that$\lambda\mapsto \Sp\bigl(f(\lambda),g(\lambda)\bigr)$is an analytic multivalued function. From this derives the subharmonicity of the functions$\lambda\mapsto \rho_V\bigl(f(\lambda),g(\lambda)\bigr)$and$\lambda\mapsto \log\rho_V\bigl(f(\lambda),g(\lambda)\bigr)$where$\rho$denotes the spectral radius. We apply these results to obtain nice caracterizations of the radical and the socle of a Banach Jordan pair, and finally we get an algebraic structural theorem. Keywords:Spectre, rayon spectral, multifonction analytique, quasi-inverse,, paire de Jordan-Banach, radical de Jacobson, socle.Categories:46H70, (17A15) 389. CMB 1997 (vol 40 pp. 169) Cruz-Uribe, David  The class$A^{+}_{\infty}(\lowercase{g})$and the one-sided reverse HÃ¶lder inequality We give a direct proof that$w$is an$A^{+}_{\infty}(g)$weight if and only if$w$satisfies a one-sided, weighted reverse H\"older inequality. Keywords:one-sided maximal operator, one-sided$(A_\infty)$, one-sided, reverse HÃ¶lder inequalityCategory:42B25 390. CMB 1997 (vol 40 pp. 54) Kechagias, Nondas E.  A note on$U_n\times U_m$modular invariants We consider the rings of invariants$R^G$, where$R$is the symmetric algebra of a tensor product between two vector spaces over the field$F_p$and$G=U_n\times U_m$. A polynomial algebra is constructed and these invariants provide Chern classes for the modular cohomology of$U_{n+m}$. Keywords:Invariant theory, cohomology of the unipotent groupCategory:13F20 391. CMB 1997 (vol 40 pp. 60) Khavinson, Dmitry  Cauchy's problem for harmonic functions with entire data on a sphere We give an elementary potential-theoretic proof of a theorem of G.~Johnsson: all solutions of Cauchy's problems for the Laplace equations with an entire data on a sphere extend harmonically to the whole space${\bf R}^N$except, perhaps, for the center of the sphere. Keywords:harmonic functions, Cauchy's problem, homogeneous harmonicsCategories:35B60, 31B20 392. CMB 1997 (vol 40 pp. 47) Hartl, Manfred  A universal coefficient decomposition for subgroups induced by submodules of group algebras Dimension subgroups and Lie dimension subgroups are known to satisfy a universal coefficient decomposition', {\it i.e.} their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the universal' coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary$N\$-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods. Keywords:induced subgroups, group algebras, Fox subgroups, relative dimension, subgroups, polynomial idealsCategories:20C07, 16A27