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3  La formule de Cauchy sur la longueur d'une courbe Ayari, S.; Dubuc, S.
Pour toute courbe rectifiable du plan, nous d\'emontrons la formule
de Cauchy relative \`a sa longueur. La formule est donn\'ee sous deux
formes: comme int\'egrale de la variation totale des projections de la
courbe dans les diverses directions et comme int\'egrale double du
nombre de rencontres de la courbe avec une droite quelconque du plan.
We give a general proof of the Cauchy formula about the length of a
plane curve. The formula is given in two ways: as the integral of the
variation of orthogonal projections of the curve, and as a double
integral of the number of intersections of the curve with an arbitrary
line of the plane.


10  Convex functions on Banach spaces not containing $\ell_1$ Borwein, Jon; Vanderwerff, Jon
There is a sizeable class of results precisely
relating boundedness, convergence and differentiability properties
of continuous convex functions on Banach spaces to whether or
not the space contains an isomorphic copy of $\ell_1$. In this
note, we provide constructions showing that the main such
results do not extend to natural broader classes of functions.


19  Lattice trees and superBrownian motion Derbez, Eric; Slade, Gordon
This article discusses our recent proof that above eight dimensions
the scaling limit of sufficiently spreadout lattice trees is the variant
of superBrownian motion called {\it integrated superBrownian excursion\/}
($\ISE$), as conjectured by Aldous. The same is true for nearestneighbour
lattice trees in sufficiently high dimensions. The proof, whose details will
appear elsewhere, uses the lace expansion. Here, a related but simpler
analysis is applied to show that the scaling limit of a meanfield theory
is $\ISE$, in all dimensions. A connection is drawn between $\ISE$ and
certain generating functions and critical exponents, which may be useful
for the study of highdimensional percolation models at the critical point.


39  On projective $Z$frames Zhao, Dongsheng
This paper deals with the projective objects in the category of all
$Z$frames, where the latter is a common generalization of
different types of frames. The main result obtained here is that a
$Z$frame is ${\bf E}$projective if and only if it is stably
$Z$continuous, for a naturally arising collection ${\bf E}$ of morphisms.


47  A universal coefficient decomposition for subgroups induced by submodules of group algebras Hartl, Manfred
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.


54  A note on $U_n\times U_m$ modular invariants Kechagias, Nondas E.
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 $U_{n+m}$.


60  Cauchy's problem for harmonic functions with entire data on a sphere Khavinson, Dmitry
We give an elementary potentialtheoretic 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.


67  On a Brownian motion problem of T. Salisbury Knight, Frank B.
Let $B$ be a Brownian motion on $R$, $B(0)=0$, and let
$f(t,x)$ be continuous. T.~Salisbury conjectured that if the total variation
of $f(t,B(t))$, $0\leq t\leq 1$, is finite $P$a.s., then $f$ does not
depend on $x$. Here we prove that this is true if the expected total
variation is finite.


72  Generalized Siegel modular forms and cohomology of locally symmetric varieties Lee, Min Ho
We generalize Siegel modular forms and construct an exact sequence
for the cohomology of locally symmetric varieties which plays the
role of the EichlerShimura isomorphism for such generalized Siegel
modular forms.


81  Une caractérisation des corps satisfaisant le théorème de l'axe principal Movahhedi, A.; Salinier, A.
Resum\'e. On caract\'erise les corps $K$ satisfaisant le th\'eor\`eme
de l'axe principal \`a l'aide de propri\'et\'es des formes
carac\t\'erisation de ces m\^emes corps due \`a Waterhouse,
on retrouve \`a partir de l\`a, de fa\c{c}on \'el\'ementaire,
un r\'esultat de Becker selon lequel un pro$2$groupe qui se
r\'ealise comme groupe de Galois absolu d'un tel corps $K$ est
engendr\'e par des involutions.
ABSTRACT. We characterize general fields $K$, satisfying the
Principal Axis Theorem, by means of properties of trace forms of
the finite extensions of $K$. From this and Waterhouse's
characterization of the same fields, we rediscover, in quite an
elementary way, a result of Becker according to which a
pro$2$group which occurs as the absolute Galois group of such
a field $K$, is generated by


88  The multidirectional mean value theorem in Banach spaces Radulescu, M. L.; Clarke, F. H.
Recently, F.~H.~Clarke and Y.~Ledyaev established a
multidirectional mean value theorem applicable to lower
semicontinuous functions on Hilbert spaces, a result which
turns out to be useful in many applications. We develop a
variant of the result applicable to locally Lipschitz functions
on certain Banach spaces, namely those that admit a
${\cal C}^1$Lipschitz continuous bump function.


103  The transfer of a commutator law from a nilring to its adjoint group Riley, David M.; Tasić, Vladimir
For every field $F$ of characteristic $p\geq 0$,
we construct an example of a finite dimensional nilpotent
$F$algebra $R$ whose adjoint group $A(R)$ is not
centrebymetabelian, in spite of the fact that $R$ is Lie
centrebymetabelian
and satisfies the identities $x^{2p}=0$ when $p>2$ and
$x^8=0$ when $p=2$. The
existence of such algebras answers a question raised by
A.~E.~Zalesskii, and is in contrast to
positive results obtained by Krasilnikov, Sharma and Srivastava
for Lie metabelian rings
and by Smirnov for the class Lie centrebymetabelian nilalgebras
of exponent 4 over a field of characteristic 2 of cardinality at least 4.


108  Continuous Selfmaps of the Circle Schaer, J.
Given a continuous map $\delta$ from the circle $S$ to itself we
want to find all selfmaps $\sigma\colon S\to S$ for which
$\delta\circ\sigma = \delta$. If the degree $r$ of $\delta$ is not
zero, the transformations $\sigma$ form a subgroup of the cyclic
group $C_r$. If $r=0$, all such invertible transformations form a
group isomorphic either to a cyclic group $C_n$ or to a dihedral
group $D_n$ depending on whether all such transformations are
orientation preserving or not. Applied to the tangent image of
planar closed curves, this generalizes a result of Bisztriczky and
Rival [1]. The proof rests on the theorem: {\it Let
$\Delta\colon\bbd R\to\bbd R$ be continuous, nowhere constant, and
$\lim_{x\to \infty}\Delta(x)=\infty$, $ \lim_{x\to+\infty}\Delta
(x)=+\infty$; then the only continuous map $\Sigma\colon\bbd R\to\bbd
R$ such that $\Delta\circ\Sigma=\Delta$ is the identity
$\Sigma=\id_{\bbd R}$.


117  Un lemme de Schwarz pour les boulesunités ouvertes Vigué, JeanPierre
Let $B_1$ and $B_2$ be the open unit balls of ${\bbd C}^{n_1}$ and
${\bbd C}^{n_2}$ for the norms $\Vert\,{.}\,\Vert_1$ and $\Vert\,{.}\,
\Vert_2$. Let $f \colon B_1 \rightarrow B_2$ be a holomorphic
mapping such that $f(0)=0$. It is well known that, for every $z \in B_1$,
$\Vert f(z)\Vert_2 \leq \Vert z \Vert_1$, and $\Vert f'(0)\Vert \leq
1$.
In this paper, I prove the converse of this result. Let $f \colon B_1
\rightarrow B_2$ be a holomorphic mapping such that $f'(0)$ is an
isometry. If $B_2$ is strictly convex, I prove that $f(0) =0$ and
that $f$ is linear. I also define the rank of a point $x$ belonging to the
boundary of $B_1$ or $B_2$. Under some hypotheses on the ranks, I
prove that a holomorphic mapping such that $f(0) = 0$ and that $f'(0)$ is
an isometry is linear.


129  Sur les caractères d'une algèbre de Banach Badea, Catalin
A new proof for the GleasonKahane\.Zelazko theorem concerning the
characters of a Banach algebra is given. A theorem due to P\'olya and
Saxer is used instead of the Hadamard factorization theorem.


133  Derivations from totally ordered semigroup algebras into their duals Blackmore, T. D.
For a wellbehaved measure $\mu$, on a locally compact
totally ordered set $X$, with continuous part $\mu_c$, we make
$L^p(X,\mu_c)$
into a commutative Banach bimodule over the totally ordered
semigroup algebra
$L^p(X,\mu)$, in such a way that the natural surjection from the algebra
to the module is a bounded derivation. This gives rise to bounded
derivations from $L^p(X,\mu)$
into its dual module and in particular shows that if $\mu_c$ is not
identically zero then $L^p(X,\mu)$ is not weakly
amenable. We show that all bounded derivations from $L^1(X,\mu)$
into its dual module arise in this way and also describe all bounded
derivations from
$L^p(X,\mu)$ into its dual for $1<p<\infty$ in the case that $X$ is
compact and $\mu$ continuous.


143  Quantum deformations of simple Lie algebras Bremner, Murray
It is shown that every simple complex Lie algebra $\fg$ admits a
1parameter family $\fg_q$ of deformations outside the category of
Lie algebras.
These deformations are derived from a tensor product decomposition for
$U_q(\fg)$modules;
here $U_q(\fg)$ is the quantized enveloping algebra of $\fg$.
From this it follows that the multiplication on $\fg_q$ is
$U_q(\fg)$invariant.
In the special case $\fg = {\ss}(2)$, the structure constants for
the deformation ${\ss}(2)_q$ are obtained from the quantum
ClebschGordan
formula applied to $V(2)_q \otimes V(2)_q$;
here $V(2)_q$ is the simple 3dimensional
$U_q\bigl({\ss}(2)\bigr)$module of
highest weight $q^2$.


149  Monochromatic homothetic copies\\ of $\{1,1+s,1+s+t\}$ Brown, Tom C.; Landman, Bruce M.; Mishna, Marni
For positive integers $s$ and $t$, let $f(s, t)$ denote the smallest positive
integer $N$ such that every $2$colouring of $[1,N]=\{1,2, \ldots , N\}$ has
a monochromatic homothetic copy of $\{1, 1+s, 1+s+t\}$.
We show that $f(s, t) = 4(s+t) + 1$ whenever $s/g$ and $t/g$ are not
congruent to $0$ (modulo $4$), where $g=\gcd(s,t)$. This can be viewed as
a generalization of part of van~der~Waerden's theorem on
arithmetic progressions, since the $3$term arithmetic progressions are the
homothetic copies of $\{1, 1+1, 1+1+1\}$. We also show that $f(s, t) = 4(s+t)
+ 1$ in many other cases (for example, whenever $s > 2t > 2$ and $t$ does not
divide $s$), and that $f(s, t) \le 4(s+t) + 1$ for all $s$, $t$.
Thus the set of homothetic copies of $\{1, 1+s, 1+s+t\}$ is a set of
triples with a particularly simple Ramsey function (at least for the case
of two colours), and one wonders what other ``natural'' sets of triples,
quadruples, {\it etc.}, have simple (or easily estimated) Ramsey functions.


158  The trigonometry of hyperbolic tessellations Coxeter, H. S. M.
For positive integers $p$ and $q$ with $(p2)(q2) >
4$ there is, in the hyperbolic plane, a group $[p,q]$
generated by reflections in the three sides of a triangle
$ABC$ with angles $\pi /p$, $\pi/q$, $\pi/2$. Hyperbolic
trigonometry shows that the side $AC$ has length $\psi$,
where $\cosh \psi = c/s$, $c = \cos \pi/q$, $s = \sin\pi/p$.
For a conformal drawing inside the unit circle with centre
$A$, we may take the sides $AB$ and $AC$ to run straight
along radii while $BC$ appears as an arc of a circle
orthogonal to the unit circle. The circle containing this
arc is found to have radius $1/\sinh \psi = s/z$, where $z
= \sqrt{c^2s^2}$, while its centre is at distance $1/\tanh
\psi = c/z$ from $A$. In the hyperbolic triangle $ABC$,
the altitude from $AB$ to the rightangled vertex $C$ is
$\zeta$, where $\sinh\zeta = z$.


169  The class $A^{+}_{\infty}(\lowercase{g})$ and the onesided reverse Hölder inequality CruzUribe, David
We give a direct proof that $w$ is an $A^{+}_{\infty}(g)$ weight if and only
if $w$ satisfies a onesided, weighted reverse H\"older inequality.


174  Nonuniqueness for the $p$harmonic flow Hungerbühler, Norbert
If $f_0\colon\Omega\subset \R^m\to S^n$ is a weakly $p$harmonic map from
a bounded smooth domain $\Omega$ in $\R^m$ (with $2<p<m$) into a sphere
and if $f_0$ is not stationary $p$harmonic, then there exist infinitely
many weak solutions of the $p$harmonic flow with initial and boundary
data $f_0$, {\it i.e.,} there are infinitely many global weak solutions
$f\colon\Omega\times \R_+\to S^n$ of
\begin{gather*}
\partial_tf\rmdiv(\nabla f^{p2}\nabla f)=
f = f_0\quad \mbox{on the parabolic boundary of $\Omega\times \R_+$.}
\end{gather*}
We also show that there exist nonstationary weakly $(m1)$harmonic
maps $f_0\colon B^m\to S^{m1}$.


183  The range of group algebra homomorphisms Kepert, Andrew G.
A characterisation of the range of a homomorphism between two
commutative group algebras is presented which implies, among other
things, that this range is closed. The work relies mainly on the
characterisation of such homomorphisms achieved by P.~J.~Cohen.


193  Finite rank operators and functional calculus on Hilbert modules over abelian $C^{\ast}$algebras Kucerovsky, Dan
We consider the problem: If $K$ is a compact normal operator on a Hilbert
module $E$, and $f\in C_0(\Sp K)$ is a function which is zero in a
neighbourhood of the origin, is $f(K)$ of finite rank? We show that
this is the case if the underlying $C^{\ast}$algebra is abelian, and that
the range of $f(K)$ is contained in a finitely generated projective
submodule of $E$.


198  The ${\cal J}_0$radical of a matrix nearring can be intermediate Meldrum, J. D. P.; Meyer, J. H.
An example is constructed to show that the ${\cal J}_0$radical of a matrix
nearring can be an intermediate ideal. This solves a conjecture put forward
in [1].


204  The $\eta$invariants of cusped hyperbolic $3$manifolds Meyerhoff, Robert; Ouyang, Mingqing
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.


214  Polynomials of quadratic type producing strings of primes Mollin, R. A.; Goddard, B.; Coupland, S.
The primary purpose of this paper is to provide necessary and
sufficient conditions for certain quadratic polynomials of negative
discriminant (which we call EulerRabinowitsch type), to produce
consecutive prime values for an initial range of input values less than
a Minkowski bound. This not only generalizes the classical work of
Frobenius, the later developments by Hendy, and the generalizations by
others, but also concludes the line of reasoning by providing a
complete list of all such primeproducing polynomials, under the
assumption of the generalized Riemann hypothesis ($\GRH$). We demonstrate
how this primeproduction phenomenon is related to the exponent of the
class group of the underlying complex quadratic field. Numerous
examples, and a remaining conjecture, are also given.


221  On semiregular rings whose finitely generated modules embed in free modules Rada, Juan; Saorín, Manuel
We consider rings as in the title and find the precise obstacle for them not
to be QuasiFrobenius, thus shedding new light on an old open question in
Ring Theory. We also find several partial affirmative answers for that
question.


231  Asymptotic theory and the foundations of statistics Reid, N.
Statistics in the 20th century
has been enlivened by a passionate, occasionally bitter, and still vibrant
debate on the foundations of statistics and in particular on
Bayesian vs. frequentist approaches to inference.
In 1975 D.~V.~Lindley predicted a Bayesian 21st century for statistics.
This prediction has often been discussed since,
but there is still no consensus on the probability
of its correctness.
Recent developments in the asymptotic theory of statistics are,
surprisingly, shedding new light on this debate, and may have the
potential to provide a common middle ground.


244  Nonexistence results of positive entire solutions for quasilinear elliptic inequalities Naito, Yūki; Usami, Hiroyuki
This paper treats the quasilinear elliptic inequality
$$
\div (Du^{m2}Du) \geq p(x)u^{\sigma},
\quad x \in \Rs^N,
$$
where $N \geq 2$, $m > 1$, $ \sigma > m  1$, and $p \colon \Rs^N
\rightarrow (0, \infty)$ is continuous. Sufficient conditions are
given for this inequality to have no positive entire solutions. When
$p$ has radial symmetry, the existence of positive entire solutions can
be characterized by our results and some known results.


254  Subdiagonal algebras for subfactors II (finite dimensional case) Saito, KichiSuke; Watatani, Yasuo
We show that finite dimensional subfactors do not have subdiagonal
algebras unless the Jones index is one.


257  A characterization of real hypersurfaces in complex space forms in terms of the Ricci tensor Baikoussis, Christos
We study real hypersurfaces of a complex space form $M_n(c)$,
$c\ne 0$ under certain conditions of the Ricci tensor on the orthogonal
distribution $T_o$.


266  Finite groups with large automizers for their Abelian subgroups Bechtell, H.; Deaconescu, M.; Silberberg, Gh.
This note contains the classification of the finite groups $G$
satisfying the condition $N_{G}(H)/C_{G}(H)\cong \Aut(H)$ for every abelian
subgroup $H$ of $G$.


271  Nonreal periodic points of entire functions Bergweiler, Walter
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 $l$.


276  Fonctions elliptiques et équations différentielles ordinaires Chouikha, Raouf
In this paper, we detail some results of a previous note concerning
a trigonometric expansion of the Weierstrass elliptic function
$\{\wp(z);\, 2\omega, 2\omega'\}$. In particular, this implies its
classical Fourier expansion. We use a direct integration method of
the ODE $$(E)\left\{\matrix{{d^2u \over dt^2} = P(u, \lambda)\hfill \cr
u(0) = \sigma\hfill \cr {du \over dt}(0) = \tau\hfill \cr}\right.$$
where $P(u)$ is a polynomial of degree $n = 2$ or $3$. In this case,
the bifurcations of $(E)$ depend on one parameter only. Moreover, this
global method seems not to apply to the cases $n > 3$.


285  The space of harmonic maps from the $2$sphere to the complex projective plane Crawford, T. Arleigh
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.


296  A general approach to LittlewoodPaley theorems for orthogonal families Hare, Kathryn E.
A general lacunary LittlewoodPaley type theorem is proved, which applies in a
variety of settings including Jacobi polynomials in $[0, 1]$, $\su$, and the
usual classical trigonometric series in $[0, 2 \pi)$. The theorem is used to
derive new results for $\LP$ multipliers on $\su$ and Jacobi $\LP$ multipliers.


309  On the homology of finite abelian coverings of links Hillman, J. A.; Sakuma, M.
Let $A$ be a finite abelian group and $M$ be a
branched cover of an homology $3$sphere, branched over a link $L$,
with covering group $A$. We show that $H_1(M;Z[1/A])$ is determined
as a $Z[1/A][A]$module by the Alexander ideals of $L$ and certain
ideal class invariants.


316  On geometric properties of OrliczLorentz spaces Hudzik, H.; Kamińska, A.; Mastyło, M.
Criteria for local uniform rotundity and midpoint local uniform
rotundity in OrliczLorentz spaces with the Luxemburg norm are given.
Strict $K$monotonicity and KadecKlee property are also discussed.


330  Amalgamated products and the Howson property Kapovich, Ilya
We show that if $A$ is a torsionfree word hyperbolic group
which belongs to class $(Q)$, that is all finitely generated subgroups of $A$
are quasiconvex in $A$, then any maximal cyclic subgroup $U$ of $A$ is a Burns
subgroup of $A$. This, in particular, implies that if $B$ is a Howson group
(that is the intersection of any two finitely generated subgroups is finitely
generated) then $A\ast_U B$, $\langle A,t \mid U^t=V\rangle$ are also Howson
groups. Finitely generated free groups, fundamental groups of closed
hyperbolic surfaces and some interesting $3$manifold groups are known to
belong to class $(Q)$ and our theorem applies to them. We also describe a
large class of word hyperbolic groups which are not Howson.


341  The stable and unstable types of classifying spaces Lee, HyangSook
The main purpose of this paper is to study groups $G_1$, $G_2$ such that
$H^\ast(BG_1,{\bf Z}/p)$ is isomorphic to $H^\ast(BG_2,{\bf Z}/p)$
in ${\cal U}$, the category of unstable modules over the Steenrod algebra
${\cal A}$, but not isomorphic as graded algebras over ${\bf Z}/p$.


352  A New Proof of a Theorem of Magnus Liriano, Sal
Using naive algebraic geometric methods a new proof of the
following celebrated theorem of Magnus is given:
Let $G$ be a group with a presentation having $n$ generators and $m$
relations. If $G$ also has a presentation on $nm$ generators, then
$G$ is free of rank $nm$.


356  Principe du maximum et lemme de Schwarz, a valeurs vectorielles Mazet, Pierre
Nous {\'e}tablissons un
th{\'e}or{\`e}me pour les fonctions holomorphes {\`a} valeurs dans une
partie convexe ferm{\'e}e. Ce th{\'e}or{\`e}me pr{\'e}cise
la position des coefficients de Taylor de telles fonctions et peut
{\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des
in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce
th{\'e}or{\`e}me permet de retrouver des versions connues du principe
du maximum et d'obtenir de nouveaux r{\'e}sultats sur les
applications holomorphes {\`a} valeurs vectorielles.


364  On the nonvanishing of a certain class of Dirichlet series Narayanan, Sridhar
In this paper,
we consider Dirichlet series with Euler products of the form
$F(s) = \prod_{p}{\bigl(1 + {a_p\over{p^s}}\bigr)}$ in $\Re(s) > 1$,
and which are regular in $\Re(s) \geq 1$ except for a pole of
order $m$ at $s = 1$.
We establish criteria for such a Dirichlet series to be nonvanishing
on the line of convergence. We also show that our results
can be applied to yield nonvanishing results for a subclass of the
Selberg class and the SatoTate conjecture.


370  Which $3$manifolds embed in $\Triod \times I \times I$? Rolfsen, Dale; Zhongmou, Li
We classify the compact $3$manifolds whose boundary is a union of
$2$spheres, and which embed in $T \times I \times I$, where $T$ is a
triod and $I$ the unit interval. This class is described explicitly as
the set of punctured handlebodies. We also show that any $3$manifold
in $T \times I \times I$ embeds in a punctured handlebody.


376  The dual pair $PGL_3 \times G_2$ Gross, Benedict H.; Savin, Gordan
Let $H$ be the split, adjoint group of type $E_6$ over a $p$adic field.
In this paper we study the restriction of the minimal representation of
$H$ to the closed subgroup $PGL_3 \times G_2$.


385  Elliptic units and class fields of global function fields Bae, Sunghan; Kang, PyungLyun
Elliptic units of global function fields were first studied by
D.~Hayes in the case that $\deg\infty$ is assumed to be $1$, and he
obtained some class number formulas using elliptic units. We
generalize Hayes' results to the case that $\deg\infty$ is arbitrary.


395  $D$spaces and resolution Boudhraa, Zineddine
A space $X$ is a $D$space if, for every neighborhood assignment $f$
there is a closed discrete set $D$ such that $\bigcup{f(D)}=X$. In this
paper we give some necessary conditions and some sufficient conditions
for a resolution of a topological space to be a $D$space. In particular,
if a space $X$ is resolved at each $x\in X$ into a $D$space $Y_x$ by
continuous mappings $f_x\colon X\{{x}\} \rightarrow Y_x$, then the
resolution is a $D$space if and only if $\bigcup{\{{x}\}}\times \Bd(Y_x)$
is a $D$space.


402  On the Preservation of Root Numbers and the Behavior of Weil Characters Under Reciprocity Equivalence Carpenter, Jenna P.
This paper studies how the local root numbers and the Weil additive
characters of the Witt ring of a number field behave under
reciprocity equivalence. Given a reciprocity equivalence between
two fields, at each place we define a local square class which
vanishes if and only if the local root numbers are preserved. Thus
this local square class serves as a local obstruction to the
preservation of local root numbers. We establish a set of
necessary and sufficient conditions for a selection of local square
classes (one at each place) to represent a global square class.
Then, given a reciprocity equivalence that has a finite wild set,
we use these conditions to show that the local square classes
combine to give a global square class which serves as a global
obstruction to the preservation of all root numbers. Lastly, we
use these results to study the behavior of Weil characters under
reciprocity equivalence.


416  On the singular behaviour of the TitchmarshWeyl $m$function for the perturbed Hill's equation on the line Clemence, Dominic P. 

422  On compact separable radial spaces Dow, Alan
If ${\cal A} $ and ${\cal B}$ are disjoint ideals on $\omega$, there is
a {\it tower preserving\/} $\sigma$centered forcing which introduces a
subset of $\omega$ which meets every infinite member of ${\cal A}$ in
an infinite set and is almost disjoint from every member of ${\cal B}$.
We can then produce a model in which all compact separable radial
spaces are Fr\'echet, thus answering a question of P.~Nyikos. The
question of the existence of compact ccc radial spaces which are not
Fr\'echet was first asked by Chertanov (see \cite{Ar78}).


433  A uniform $L^{\infty}$ estimate of the smoothing operators related to plane curves Guo, Kanghui
In dealing with the spectral synthesis property for a plane curve with
nonzero curvature, a key step is to have a uniform $L^{\infty}$ estimate
for some smoothing operators related to the curve. In this paper, we will
show that the same $L^{\infty}$ estimate holds true for a plane curve
that may have zero curvature.


443  Reflective Representations and Banach C*Modules Hadwin, Don; Orhon, Mehmet
Suppose ${\cal A}$ is a unital $C$*algebra and $m\colon{\cal A}\to B(X)$


448  Stable index pairs for discrete dynamical systems Kaczynski, Tomasz; Mrozek, Marian
A new shorter proof of the existence of index pairs for discrete
dynamical systems is given. Moreover, the index pairs defined in
that proof are stable with respect to small perturbations of the
generating map. The existence of stable index pairs was previously
known in the case of diffeomorphisms and flows generated by smooth
vector fields but it was an open question in the general discrete
case.


456  Approximation of smooth maps by real algebraic morphisms Kucharz, Wojciech; Rusek, Kamil
Let $\Bbb G_{p,q}(\Bbb F)$ be the Grassmann space of all
$q$dimensional $\Bbb F$vector subspaces of $\Bbb F^{p}$, where $\Bbb F$
stands for $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the quaternions). Here
$\Bbb G_{p,q}(\Bbb F)$ is regarded as a real algebraic variety. The paper
investigates which ${\cal C}^\infty$ maps from a nonsingular real algebraic
variety $X$ into $\Bbb G_{p,q}(\Bbb F)$ can be approximated, in the
${\cal C}^\infty$ compactopen topology, by real algebraic morphisms.


464  On the solvability of a Neumann boundary value problem at resonance Kuo, ChungCheng
We study the existence of solutions of the semilinear equations (1)
$\triangle u + g(x,u)=h$, ${\partial u \over \partial n} = 0$ on
$\partial \Omega$ in which the nonlinearity $g$ may grow
superlinearly in $u$ in one of directions $u \to \infty$ and $u \to
\infty$, and (2) $\triangle u + g(x,u)=h$, ${\partial u \over
\partial n} = 0$ on $\partial \Omega$ in which the nonlinear term $g$
may grow superlinearly in $u$ as $u \to \infty$. The purpose of this
paper is to obtain solvability theorems for (1) and (2) when the
LandesmanLazer condition does not hold. More precisely, we require
that $h$ may satisfy $\int g^\delta_ (x) \, dx < \int h(x) \, dx = 0<
\int g^\gamma_+ (x)\,dx$, where $\gamma, \delta$ are arbitrarily
nonnegative constants, $g^\gamma_+ (x) = \lim_{u \to \infty} \inf
g(x,u) u^\gamma$ and $g^\delta_ (x)=\lim_{u \to \infty} \sup
g(x,u)u^\delta$. The proofs are based upon degree theoretic arguments.


471  A short proof of Euler's relation for convex polytopes* Lawrence, Jim
The purposen of this paper is to present a short, selfcontained
proof of Euler's relation. The ingredients of this proof are (i) the
principle of inclusion and exclusion of combinatorics and (ii) the
Euler characteristic; a development of the Euler characteristic is included.


475  Coefficient multipliers of Bergman spaces $A^p$, II Lou, Zengjian
We show that the multiplier space $(A^1,X)=\{g:M_\infty(r,g'')
=O(1r)^{1}\}$, where $X$ is $\BMOA$, $\VMOA$, $B$, $B_0$ or disk algebra $A$.
We give the multipliers from $A^1$ to $A^q(H^q)(1\le q\le \infty)$, we
also give the multipliers from $l^p(1\le p\le 2), C_0, \BMOA$, and
$H^p(2\le p<\infty)$ into $A^q(1\le q\le 2)$.


488  Caractérisations spectrales du radical et du socle d'une paire de jordanbanach Maouche, Abdelaziz
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.


498  Matrix transformations based on Dirichlet convolution Selvaraj, Chikkanna; Selvaraj, Suguna
This paper is a study of summability methods that are based
on Dirichlet convolution. If $f(n)$ is a function on positive integers
and $x$ is a sequence such that $\lim_{n\to \infty} \sum_{k\le n}
{1\over k}(f\ast x)(k) =L$, then $x$ is said to be {\it $A_f$summable\/}
to $L$. The necessary and sufficient condition for the matrix $A_f$ to
preserve bounded variation of sequences is established. Also, the
matrix $A_f$ is investigated as $\ell  \ell$ and $GG$ mappings. The
strength of the $A_f$matrix is also discussed.


508  Author Index  Index des auteurs 1997, for 1997  pour
No abstract.

