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401. 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 equivalence
Categories:57R99, 57M99, 54H20

402. 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, $A_p$ weights
Category:42C10

403. 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 operator
Categories:47B38, 30D55, 46E15

404. 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 forcing
Categories:03E05, 03E35, 54A35

405. 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 groups
Category:13A50

406. 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, $M$-ideals
Categories:46B20, 46E40

407. CMB 1999 (vol 42 pp. 104)

Nikolskaia, Ludmila
Instabilité de vecteurs propres d'opérateurs linéaires
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 kernels
Categories:47A10, 46B15

408. 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 function
Category:32A37

409. CMB 1998 (vol 41 pp. 497)

Borwein, J. M.; Girgensohn, R.; Wang, Xianfu
On the construction of Hölder 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, Hölder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, Hölder smooth, dyadic rationals
Categories:49J52, 26A16, 26A24

410. 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, graphs
Categories:54D05, 54F20, 54F50

411. 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 bundle
Categories:14C20, 14J99

412. 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, oscillation
Category:34K15

413. 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

414. 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 continuity
Categories:47A62, 47B37, 93D25, 42A85, 47N70

415. 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

416. 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-derivative
Categories:28A25, 49J52, 46E30

417. CMB 1997 (vol 40 pp. 488)

Maouche, Abdelaziz
Caractérisations 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)

418. CMB 1997 (vol 40 pp. 169)

Cruz-Uribe, David
The class $A^{+}_{\infty}(\lowercase{g})$ and the one-sided reverse Hölder 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 Hölder inequality
Category:42B25

419. 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 harmonics
Categories:35B60, 31B20

420. 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 group
Category:13F20

421. 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 ideals
Categories:20C07, 16A27
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