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3  Quantum Ergodicity of Boundary Values of Eigenfunctions: A Control Theory Approach Burq, N.
Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a
Riemanian manifold with piecewise smooth boundary and suppose that the
billiard associated to the geodesic flow reflecting on the boundary
according to the laws of geometric optics is ergodic.
We prove that the boundary value of the eigenfunctions of the Laplace
operator with reasonable boundary conditions are asymptotically
equidistributed in the boundary, extending previous results by
G\'erard and Leichtnam as well as Hassell and Zelditch,
obtained under the additional assumption of the convexity of~$M$.


16  On the Surjectivity of the Galois Representations Associated to NonCM Elliptic Curves Cojocaru, Alina Carmen
Let $ E $ be an elliptic curve defined over
$\Q,$ of conductor $N$ and without complex multiplication. For any
positive integer $l$, let $\phi_l$ be the Galois representation
associated to the $l$division points of~$E$. From a celebrated
1972 result of Serre we know that $\phi_l$ is surjective for any
sufficiently large prime $l$. In this paper we find conditional
and unconditional upper bounds in terms of $N$ for the primes $l$
for which $\phi_l$ is {not} surjective.


32  NonLeftOrderable 3Manifold Groups Dąbkowski, Mieczysław K.; Przytycki, Józef H.; Togha, Amir A.
We show that several torsion free 3manifold groups
are not leftorderable.
Our examples are groups of cyclic branched coverings of $S^3$
branched along links.
The figure eight knot provides simple
nontrivial examples. The groups arising in these examples are known
as Fibonacci groups which we show not to be leftorderable.
Many other examples of nonorderable groups are obtained by taking
3fold branched covers of $S^3$ branched along various hyperbolic
2bridge knots.
%with various hyperbolic 2bridge knots as branched sets.
The manifold obtained in such a way from the $5_2$ knot
is of special interest as it is conjectured to be the hyperbolic
3manifold with the smallest volume.


41  Degree Homogeneous Subgroups Dixon, John D.; Barghi, A. Rahnamai
Let $G$ be a finite group and $H$ be a subgroup. We say that $H$
is degree homogeneous if, for each $\chi\in \Irr(G)$, all
the irreducible constituents of the restriction $\chi_{H}$ have
the same degree. Subgroups which are either normal or abelian are
obvious examples of degree homogeneous subgroups. Following a
question by E.~M. Zhmud', we investigate general properties of
such subgroups. It appears unlikely that degree homogeneous
subgroups can be characterized entirely by abstract group
properties, but we provide mixed criteria (involving both group
structure and character properties) which are both necessary and
sufficient. For example, $H$ is degree homogeneous in $G$ if and
only if the derived subgroup $H^{\prime}$ is normal in $G$ and,
for every pair $\alpha,\beta$ of irreducible $G$conjugate
characters of $H^{\prime}$, all irreducible constituents of
$\alpha^{H}$ and $\beta^{H}$ have the same degree.


50  Injectivity of the Connecting Maps in AH Inductive Limit Systems Elliott, George A.; Gong, Guihua; Li, Liangqing
Let $A$ be the inductive limit of a system
$$A_{1}\xrightarrow{\phi_{1,2}}A_{2}
\xrightarrow{\phi_{2,3}} A_{3}\longrightarrow \cd
$$
with $A_n =
\bigoplus_{i=1}^{t_n} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where
$~X_{n,i}$ is a finite simplicial complex, and $P_{n,i}$ is a
projection in $M_{[n,i]}(C(X_{n,i}))$. In this paper, we will
prove that $A$ can be written as another inductive limit
$$B_1\xrightarrow{\psi_{1,2}} B_2
\xrightarrow{\psi_{2,3}} B_3\longrightarrow \cd $$
with $B_n =
\bigoplus_{i=1}^{s_n} Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$,
where $Y_{n,i}$ is a finite simplicial complex, and $Q_{n,i}$ is a
projection in $M_{\{n,i\}}(C(Y_{n,i}))$, with the extra condition
that all the maps $\psi_{n,n+1}$ are injective. (The
result is trivial if one allows the spaces $Y_{n,i}$ to be
arbitrary compact metrizable spaces.) This result is important
for the classification of simple AH algebras (see
\cite{G5,G6,EGL}. The special case that the spaces $X_{n,i}$ are
graphs is due to the third named author \cite{Li1}.


69  Biorthogonal Systems in Weakly Lindelöf Spaces Fabian, M.; Montesinos, V.; Zizler, V.
We study countable splitting of Markushevich bases in weakly Lindel\"of
Banach spaces in connection with the geometry of these spaces.


80  Trivial Units for Group Rings with $G$adapted Coefficient Rings Herman, Allen; Li, Yuanlin; Parmenter, M. M.
For each finite group $G$ for which the integral group ring
$\mathbb{Z}G$ has only trivial units, we give ringtheoretic
conditions for a commutative ring $R$ under which the group ring
$RG$ has nontrivial units. Several examples of rings satisfying
the conditions and rings not satisfying the conditions are given.
In addition, we extend a wellknown result for fields by showing
that if $R$ is a ring of finite characteristic and $RG$ has only
trivial units, then $G$ has order at most 3.


90  Products of Conjugacy Classes in $SU(2)$ Jeffrey, Lisa C.; Mare, AugustinLiviu
We obtain a complete description of the conjugacy classes
$C_1,\dots,C_n$ in $SU(2)$ with the property that $C_1\cdots
C_n=SU(2)$. The basic instrument is a characterization of the
conjugacy classes $C_1,\dots,C_{n+1}$ in $SU(2)$ with $C_1\cdots
C_{n+1}\ni I$, which generalizes a result of \cite{JeWe}.


97  On the Ranges of Bimodule Projections Katavolos, Aristides; Paulsen, Vern I.
We develop a symbol calculus for normal bimodule maps over a masa
that is the natural analogue of the Schur product theory. Using
this calculus we are easily able to give a complete description of
the ranges of contractive normal bimodule idempotents that avoids
the theory of J*algebras.
We prove that if $P$ is a normal
bimodule idempotent and $\P\ < 2/\sqrt{3}$ then $P$ is a
contraction. We finish with some attempts at extending the symbol
calculus to nonnormal maps.


112  On Negatively Curved Finsler Manifolds of Scalar Curvature Mo, Xiaohuan; Shen, Zhongmin
In this paper, we prove a global rigidity theorem for negatively
curved Finsler metrics on a compact manifold of dimension $n \geq 3$.
We show that for such a Finsler manifold, if the flag curvature is a
scalar function on the tangent bundle, then the Finsler metric is of
Randers type. We also study the case when the Finsler metric is
locally projectively flat


121  Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ Mollin, R. A.
We look at the simple continued fraction expansion of $\sqrt{D}$
for any $D=2^hc $ where $c>1$ is odd with a goal of
determining necessary and
sufficient conditions for the central norm (as determined by
the infrastructure of the underlying real quadratic order therein) to be
$2^h$. At the end of the paper, we also address the case where $D=c$
is odd and the central norm of $\sqrt{D}$ is equal to $2$.


133  Estimates of HenstockKurzweil Poisson Integrals Talvila, Erik
If $f$ is a realvalued function on $[\pi,\pi]$ that
is HenstockKurzweil integrable, let $u_r(\theta)$ be its Poisson
integral. It is shown that $\u_r\_p=o(1/(1r))$ as $r\to 1$
and this estimate is sharp for $1\leq p\leq\infty$.
If $\mu$ is a finite Borel measure and $u_r(\theta)$ is its Poisson
integral then for each $1\leq p\leq \infty$ the estimate
$\u_r\_p=O((1r)^{1/p1})$ as $r\to 1$ is sharp.
The Alexiewicz
norm estimates $\u_r\\leq\f\$ ($0\leq r<1$) and $\u_rf\\to 0$
($r\to 1$) hold. These estimates lead to two uniqueness theorems for
the Dirichlet problem
in the unit disc with HenstockKurzweil integrable boundary data.
There are similar growth estimates when $u$ is in the harmonic Hardy
space associated with the Alexiewicz
norm and when $f$ is of bounded variation.


147  BakerType Estimates for Linear Forms in the Values of $q$Series Väänänen, Keijo; Zudilin, Wadim
We obtain lower estimates for the absolute values
of linear forms of the values of generalized Heine
series at nonzero points of an imaginary quadratic field~$\II$,
in particular of the values of $q$exponential function.
These estimates depend on the individual coefficients,
not only on the maximum of their absolute values.
The proof uses a variant of classical Siegel's method
applied to a system of functional Poincar\'etype equations
and the connection between the solutions of these functional
equations and the generalized Heine series.


161  Hankel Convolution Operators on Spaces of Entire Functions of Finite Order Betancor, Jorge J.
In this paper we study Hankel transforms and Hankel convolution
operators on spaces of entire functions of finite order and their
duals.


175  Weighted Convolution Operators on $\ell_p$ Borwein, David; Kratz, Werner
The main results deal with conditions for the validity of the weighted
convolution inequality $\sum_{n\in\mathbb Z}\leftb_n\sum_{k\in\mathbb Z}
a_{nk}x_k\right^p\le C^p\sum_{k\in\mathbb Z} x_k^p$ when $p\ge1$.


180  Geometry and Arithmetic of Certain Double Octic CalabiYau Manifolds Cynk, Sławomir; Meyer, Christian
We study CalabiYau manifolds constructed as double coverings of
$\mathbb{P}^3$ branched along an octic surface. We give a list of 87
examples corresponding to arrangements of eight planes defined over
$\mathbb{Q}$. The Hodge numbers are computed for all examples. There are
10 rigid CalabiYau manifolds and 14 families with $h^{1,2}=1$. The
modularity conjecture is verified for all the rigid examples.


195  On Suslinian Continua Daniel, D.; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.
A continuum is said to be Suslinian if it does not contain uncountably
many mutually exclusive nondegenerate subcontinua. We prove that
Suslinian continua are perfectly normal and rimmetrizable. Locally
connected Suslinian continua have weight at most $\omega_1$ and under
appropriate settheoretic conditions are metrizable. Nonseparable
locally connected Suslinian continua are rimfinite on some open set.


203  NonCohenMacaulay Projective Monomial Curves with Positive ${h}$Vector de Quehen, Victoria E.; Roberts, Leslie G.
We find an infinite family of projective monomial
curves all of which have $h$vector with no negative values and
are not CohenMacaulay.


211  The Distribution of Totatives Germain, Jam
The integers coprime to $n$ are called the {\it totatives} \rm of $n$.
D. H. Lehmer and Paul Erd\H{o}s were interested in understanding when
the number of totatives between $in/k$ and $(i+1)n/k$ are $1/k$th of
the total number of totatives up to $n$. They provided criteria in
various cases. Here we give an ``if and only if'' criterion which
allows us to recover most of the previous results in this literature
and to go beyond, as well to reformulate the problem in terms of
combinatorial group theory. Our criterion is that the above holds if
and only if for every odd character $\chi \pmod \kappa$ (where
$\kappa:=k/\gcd(k,n/\prod_{pn} p)$) there exists a prime $p=p_\chi$
dividing $n$ for which $\chi(p)=1$.


221  An Elementary Proof of Suslin Reciprocity Kerr, Matt
We state and prove an important special case of Suslin reciprocity
that has found significant use in the study of algebraic cycles. An
introductory account is provided of the regulator and norm maps on Milnor
$K_2$groups (for function fields) employed in the proof.


237  Indecomposable Higher Chow Cycles Kimura, Kenichiro
Let $X$ be a projective smooth variety over a field $k$.
In the first part we show that
an indecomposable element in $CH^2(X,1)$ can be lifted
to an indecomposable element in $CH^3(X_K,2)$ where $K$ is the function
field of 1 variable over $k$. We also show that if $X$ is the selfproduct
of an elliptic curve over $\Q$ then the $\Q$vector space of
indecomposable cycles
$CH^3_{ind}(X_\C,2)_\Q$ is infinite dimensional.


244  Counting Multiple Cyclic Choices Without Adjacencies McLeod, Alice; Moser, William
We give a particularly elementary solution to the following
wellknown problem. What is the number of $k$subsets $X \subseteq
I_n=\{1,2,3,\dots,n\}$ satisfying ``no two elements of $X$ are adjacent
in the circular display of $I_n$''? Then we investigate a new
generalization (multiple cyclic choices without adjacencies) and
apply it to enumerating a class of 3line latin rectangles.


251  The Index Theory Associated to a NonFinite Trace on a $C^\ast$Algebra Murphy, G. J.
The index theory considered in this paper, a
generalisation of the classical Fredholm index theory, is obtained
in terms of a nonfinite trace on a unital $C^\ast$algebra. We relate
it to the index theory of M.~Breuer, which is developed in a
von~Neumann algebra setting, by means of a representation theorem.
We show how our new index theory can be used to obtain an index
theorem for Toeplitz operators on the compact group $\mathbf{U}(2)$,
where the classical index theory does not give any interesting result.


260  A Restriction Theorem for a \\$k$Surface in $\mathbb R ^n$ Oberlin, Daniel M.
We establish a sharp Fourier restriction estimate
for a measure on a $k$surface in $\mathbb R ^n$, where $n=k(k+3)/2$.


267  Continuous Adjacency Preserving Maps on Real Matrices Rodman, Leiba; Šemrl, Peter; Sourour, Ahmed R.
It is proved that every adjacency preserving continuous map
on the vector space of real matrices of fixed size, is either a
bijective affine tranformation
of the form $ A \mapsto PAQ+R$, possibly followed by the transposition if
the matrices are of square size, or its range is contained
in a linear subspace consisting of matrices of rank at most one
translated by some matrix $R$. The result
extends previously known
theorems where the map was assumed to be also injective.


275  Krull Dimension of Injective Modules Over Commutative Noetherian Rings Smith, Patrick F.
Let $R$ be a commutative Noetherian
integral domain with field of fractions $Q$. Generalizing a
fortyyearold theorem of E. Matlis, we prove that the $R$module
$Q/R$ (or $Q$) has Krull dimension if and only if $R$ is semilocal
and onedimensional. Moreover, if $X$ is an injective module over
a commutative Noetherian ring such that $X$ has Krull dimension,
then the Krull dimension of $X$ is at most $1$.


283  Enlarged Inclusion of Subdifferentials Thibault, Lionel; Zagrodny, Dariusz
This paper studies the integration of inclusion of subdifferentials. Under
various verifiable conditions, we obtain that if two proper lower
semicontinuous functions $f$ and $g$ have the subdifferential of $f$
included in the $\gamma$enlargement of the subdifferential of $g$, then
the difference of those functions is $ \gamma$Lipschitz over their
effective domain.


302  Discrete Sets and Associated Dynamical\\ Systems in a NonCommutative Setting Yokonuma, Takeo
We define a uniform structure on the set of discrete sets of a locally
compact topological space on which a locally compact topological group
acts continuously. Then we investigate the completeness of these
uniform spaces and study these spaces by means of topological
dynamical systems.


317  On PseudoFrobenius Rings Yousif, Mohamed F.; Zhou, Yiqiang; Zeyada, Nasr
It is proved here that a ring $R$ is right pseudoFrobenius
if and only if $R $ is a right Kasch ring such that the second
right singular ideal is injective.


321  On NonVanishing of Convolution of Dirichlet Series Akbary, Amir; Shahabi, Shahab
We study the nonvanishing on the line $Re(s)=1$ of the
convolution series associated to
two Dirichlet series in a certain class of Dirichlet series.
The nonvanishing of various $L$functions on the line $Re(s)=1$
will be simple corollaries of our general theorems.


333  Monotonicity Properties of the Hurwitz Zeta Function Alzer, Horst
Let
$$
\zeta(s,x)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^s} \quad{(s>1,\, x>0)}
$$
be the Hurwitz zeta function and let
$$
Q(x)=Q(x;\alpha,\beta;a,b)=\frac{(\zeta(\alpha,x))^a}{(\zeta(\beta,x))^b},
$$
where $\alpha, \beta>1$
and $a,b>0$ are real numbers. We prove:
(i) The function $Q$ is decreasing on $(0,\infty)$ if{}f $\alpha a\beta b\geq \max(ab,0)$.
(ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a\beta b\leq
\min(ab,0)$.
An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s1)\zeta(s,x)]^{1/(s1)}$ is decreasing on $(1,\infty)$. This settles
a conjecture of Bastien and Rogalski.


340  Short Geodesics of Unitaries in the $L^2$ Metric Andruchow, Esteban
Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$,
and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group
$U_\M$ as a subset of $\l2$ and characterize the shortest smooth
curves joining two fixed unitaries in the $L^2$ metric. As a
consequence of this we obtain that $U_\M$, though a complete (metric)
topological group, is not an embedded riemannian submanifold of $\l2$


355  On Maps Preserving Products Chebotar, M. A.; Ke, W.F.; Lee, P.H.; Shiao, L.S.
Maps preserving certain algebraic properties of elements
are often studied in Functional Analysis and Linear Algebra. The
goal of this paper is to discuss the relationships among these
problems from the ringtheoretic point of view.


370  Trigonometric Multipliers on $H_{2\pi}$ Daly, J. E.; Fridli, S.
In this paper we consider multipliers on the real Hardy space
$H_{2\pi}$. It is known that the Marcinkiewicz and the
H\"ormanderMihlin conditions are sufficient for the corresponding
trigonometric multiplier to be bounded on $L_{2\pi}^p$, $1<p<\infty$.
We show among others that the H\"ormanderMihlin
condition extends to $H_{2\pi}$ but the Marcinkiewicz condition
does not.


382  Uniform Estimates of Ultraspherical Polynomials of Large Order De Carli, Laura
In this paper we prove the sharp inequality
$$ P_n^{(s)}(x)\leq
P_n^{(s)}(1)\bigl(x^n +\frac{n1}{2 s+1}(1x^n)\bigr),$$
where
$P_n^{(s)}(x)$ is the classical ultraspherical polynomial of
degree $n$ and order $s\ge n\frac{1+\sqrt 5}{4}$. This inequality
can be refined in $[0,z_n^s]$ and $[z_n^s,1]$, where $z_n^s$
denotes the largest zero of $P_n^{(s)}(x)$.


394  Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices Đoković, D. Ž.; Szechtman, F.; Zhao, K.
Let $n=2m$ be even and denote by $\Sp_n(F)$ the symplectic group
of rank $m$ over an infinite field $F$ of characteristic different
from $2$. We show that any $n\times n$ symmetric matrix $A$ is
equivalent under symplectic congruence transformations to the
direct sum of $m\times m$ matrices $B$ and $C$, with $B$ diagonal
and $C$ tridiagonal. Since the $\Sp_n(F)$module of symmetric
$n\times n$ matrices over $F$ is isomorphic to the adjoint module
$\sp_n(F)$, we infer that any adjoint orbit of $\Sp_n(F)$ in
$\sp_n(F)$ has a representative in the sum of $3m1$ root spaces,
which we explicitly determine.


405  Liouville's Theorem in the Radially Symmetric Case Froese, Richard
We present a very short proof of Liouville's theorem for solutions
to a nonuniformly elliptic radially symmetric equation. The proof uses
the Ricatti equation satisfied by the Dirichlet to Neumann map.


409  The Existence of Universal Inner Functions on the Unit Ball of $\mathbb{C}^n$ Gauthier, P. M.; Xiao, J.
It is shown that there exists an inner function
$I$ defined on the unit ball ${\bf B}^n$ of ${\mathbb C}^n$
such that each function holomorphic on ${\bf B}^n$ and
bounded by $1$ can be approximated by
``nonEuclidean translates" of $I$.


414  Vector Fields and the Cohomology Ring of Toric Varieties Kaveh, Kiumars
Let $X$ be a smooth complex
projective variety with a holomorphic vector field with isolated
zero set $Z$. From the results of Carrell and Lieberman
there exists a filtration
$F_0 \subset F_1 \subset \cdots$ of $A(Z)$, the ring of
$\c$valued functions on $Z$, such that $\Gr A(Z) \cong H^*(X,
\c)$ as graded algebras. In this note, for a smooth projective
toric variety and a vector field generated by the action of a
$1$parameter subgroup of the torus, we work out this filtration.
Our main result is an explicit connection between this filtration
and the polytope algebra of $X$.


428  Reduction of Elliptic Curves in Equal Characteristic~3 (and~2) Miyamoto, Roland; Top, Jaap
and fibre type for elliptic curves
over discrete valued fields of equal characteristic~3.
Along the same lines, partial results are obtained
in equal characteristic~2.


445  On the Garsia Lie Idempotent Patras, Frédéric; Reutenauer, Christophe; Schocker, Manfred
The orthogonal projection of the free associative algebra onto the
free Lie algebra is afforded by an idempotent in the rational group
algebra of the symmetric group $S_n$, in each homogenous degree
$n$. We give various characterizations of this Lie idempotent and show
that it is uniquely determined by a certain unit in the group algebra
of $S_{n1}$. The inverse of this unit, or, equivalently, the Gram
matrix of the orthogonal projection, is described explicitly. We also
show that the Garsia Lie idempotent is not constant on descent classes
(in fact, not even on coplactic classes) in $S_n$.


455  On Gâteaux Differentiability of Convex Functions in WCG Spaces Rychtář, Jan
It is shown, using the BorweinPreiss variational principle
that for every continuous convex function $f$ on
a weakly compactly generated space $X$,
every $x_0\in X$ and every weakly compact convex symmetric set $K$ such that
$\cspan K=X$,
there is a point of G\^ateaux differentiability of $f$ in $x_0+K$.
This extends a Klee's result for separable spaces.


460  $B$Stable Ideals in the Nilradical of a Borel Subalgebra Sommers, Eric N.
We count the number of strictly positive $B$stable ideals in the
nilradical of a Borel subalgebra and prove that
the minimal roots of any $B$stable ideal are conjugate
by an element of the Weyl group to a subset of the simple roots.
We also count the number of ideals whose minimal roots are conjugate
to a fixed subset of simple roots.


473  Logarithms and the Topology of the Complement of a Hypersurface Zeron, E. S.
This paper is devoted to analysing the relation between the
logarithm of a nonconstant holomorphic polynomial $Q(z)$ and
the topology of the complement of the hypersurface defined by
$Q(z)=0$.


481  Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces Azagra, D.; Fabian, M.; JiménezSevilla, M.
We establish sufficient conditions on the shape of a set $A$
included in the space $\mathcal L _s^n(X,Y)$ of the $n$linear
symmetric mappings between Banach spaces $X$ and $Y$, to ensure
the existence of a $C^n$\nobreakdashsmooth
mapping $f\colon X \rightarrow Y$,
with bounded support, and such that $f^{(n)}(X)=A$, provided that
$X$ admits a $C^{n}$smooth bump with bounded $n$th derivative
and $\dens X=\dens \mathcal L ^n(X,Y)$. For instance, when $X$ is
infinitedimensional, every bounded connected and open set $U$
containing the origin is the range of the $n$th derivative of
such a mapping. The same holds true for the closure of $U$,
provided that every point in the boundary of $U$ is the end
point of a path within $U$. In the finitedimensional case, more
restrictive conditions are required. We also study the Fr\'echet
smooth case for mappings from $\mathbb R^n$ to a separable
infinitedimensional Banach space and the G\^ateaux smooth case
for mappings defined on a separable infinitedimensional Banach
space and with values in a separable Banach space.


500  Extension of Holomorphic Functions From One Side of a Hypersurface Baracco, Luca
We give a new proof of former results by G. Zampieri and the
author on extension of holomorphic
functions from one side $\Omega$ of a real hypersurface
$M$ of $\mathbb{C}^n$ in the presence of an
analytic disc tangent to $M$, attached to $\bar\Omega$
but not to $M$. Our method enables
us to weaken the regularity assumptions both
for the hypersurface and the disc.


505  On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group Bouikhalene, Belaid
Let $G$ be a compact group. Let $\sigma$ be a continuous involution
of $G$. In this paper, we are
concerned by the following functional equation
$$\int_{G}f(xtyt^{1})\,dt+\int_{G}f(xt\sigma(y)t^{1})\,dt=2g(x)h(y), \quad
x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be
determined, are complex continuous functions on $G$ such that $f$ is
central. This equation generalizes d'Alembert's and Wilson's
functional equations. We show that the solutions are expressed by
means of characters of irreducible, continuous and unitary
representations of the group $G$.


523  Angle Measures and Bisectors in Minkowski Planes Düvelmeyer, Nico
\begin{abstract}
We prove that a Minkowski plane is Euclidean if and only if Busemann's or
Glogovskij's definitions
of angular bisectors coincide
with a bisector defined by an angular measure in the sense of Brass.
In addition, bisectors defined by the area measure coincide with bisectors
defined by the circumference (arc length) measure
if and only if the unit circle is an
equiframed curve.


535  On the Error Term in Duke's Estimate for the Average Special Value of $L$Functions Ellenberg, Jordan S.
Let $\FF$ be an orthonormal basis for weight $2$
cusp forms of level $N$. We show that various weighted averages of
special values $L(f \tensor \chi, 1)$ over $f \in \FF$ are equal to $4
\pi c + O(N^{1 + \epsilon})$, where $c$ is an explicit nonzero constant. A previous result of Duke gives an error
term of $O(N^{1/2}\log N)$.


547  Degeneracy of 2Forms and 3Forms Fehér, L. M.; Némethi, A.; Rimányi, R.
We study some global aspects of differential complex 2forms and 3forms
on complex manifolds.
We compute the cohomology classes represented by the sets of points
on a manifold where such a form degenerates in various senses,
together with other similar cohomological obstructions.
Based on these results and a formula for projective
representations, we calculate the degree of the projectivization
of certain orbits of the representation $\Lambda^k\C^n$.


561  A Note on Lagrangian Loci of Quotients Foth, Philip
We study Hamiltonian actions of compact groups in the presence of
compatible involutions. We show that the Lagrangian fixed point set
on the symplectically reduced space is isomorphic to the disjoint
union of the involutively reduced spaces corresponding to
involutions on the group strongly inner to the given one.
Our techniques imply that the solution to the eigenvalues of a sum problem
for a given real form can be reduced to the quasisplit real form in the
same inner class. We also consider invariant quotients with respect to
the corresponding real form of the complexified group.


576  On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II Ichimura, Humio
Let $m=p^e$ be a power of a prime number $p$.
We say that a number field $F$ satisfies the property $(H_m')$
when for any $a \in F^{\times}$, the cyclic extension
$F(\z_m, a^{1/m})/F(\z_m)$ has a normal $p$integral basis.
We prove that $F$ satisfies $(H_m')$
if and only if the natural homomorphism $Cl_F' \to Cl_K'$ is trivial.
Here $K=F(\zeta_m)$, and $Cl_F'$ denotes the ideal class group of $F$
with respect to the $p$integer ring of $F$.


580  Exceptional Sets in Hartogs Domains Kot, Piotr
Assume that $\Omega$ is a Hartogs domain in $\mathbb{C}^{1+n}$,
defined as $\Omega=\{(z,w)\in\mathbb{C}^{1+n}:z<\mu(w),w\in H\}$, where $H$ is an open set in
$\mathbb{C}^{n}$ and $\mu$ is a continuous function with positive values in $H$ such that $\ln\mu$
is a strongly plurisubharmonic function in $H$. Let $\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$.
For a given set $E$ contained in $H$ of the type $G_{\delta}$ we construct a holomorphic function
$f\in\mathbb{O}(\Omega)$ such that
\[
E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}f(\cdot\,,w)^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}.
\]


587  Separation of Variables for $U_{q}(\mathfrak{sl}_{n+1})^{+}$ Lopes, Samuel A.
Let $U_{q}(\SL)^{+}$ be the positive part of the quantized enveloping
algebra $U_{q}(\SL)$. Using results of AlevDumas and Caldero related
to the center of $U_{q}(\SL)^{+}$, we show that this algebra is free
over its center. This is reminiscent of Kostant's separation of
variables for the enveloping algebra $U(\g)$ of a complex semisimple
Lie algebra $\g$, and also of an analogous result of JosephLetzter
for the quantum algebra $\Check{U}_{q}(\g)$. Of greater importance to
its representation theory is the fact that $\U{+}$ is free over a
larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$
to $\U{+}$ provides infinitedimensional modules with good properties,
including a grading that is inherited by submodules.


601  On the Regularity of the $s$Differential Metric Mashreghi, Javad; Pouryayevali, Mohamad R.
We show that the injective KobayashiRoyden differential metric, as
defined by Hahn, is upper semicontinous.


607  Toeplitz Algebras and Extensions of\\Irrational Rotation Algebras Park, Efton
For a given irrational number $\theta$, we define Toeplitz operators with
symbols in the irrational rotation algebra ${\mathcal A}_\theta$,
and we show that the $C^*$algebra $\mathcal T({\mathcal
A}_\theta)$ generated by these Toeplitz operators is an extension
of ${\mathcal A}_\theta$ by the algebra of compact operators. We
then use these extensions to explicitly exhibit generators of the
group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an
index theorem for $\mathcal T({\mathcal A}_\theta)$ that
generalizes the standard index theorem for Toeplitz operators on
the circle.


614  On FinitetoOne Maps Tuncali, H. Murat; Valov, Vesko
Let $f\colon X\to Y$ be a $\sigma$perfect $k$dimensional surjective
map of metrizable spaces such that $\dim Y\leq m$. It is shown that
for every positive integer $p$ with $ p\leq m+k+1$ there exists a
dense $G_{\delta}$subset ${\mathcal H}(k,m,p)$ of $C(X,\uin^{k+p})$
with the source limitation topology such that each fiber of
$f\triangle g$, $g\in{\mathcal H}(k,m,p)$, contains at most
$\max\{k+mp+2,1\}$ points. This result
provides a proof the following conjectures of
S. Bogatyi, V. Fedorchuk and J. van Mill.
Let $f\colon X\to Y$ be a $k$dimensional map between compact
metric spaces with $\dim Y\leq m$. Then:
\begin{inparaenum}[\rm(1)]
\item there exists a map
$h\colon X\to\uin^{m+2k}$ such that $f\triangle h\colon X\to
Y\times\uin^{m+2k}$ is 2toone provided $k\geq 1$;
\item there exists a
map $h\colon X\to\uin^{m+k+1}$ such that $f\triangle h\colon X\to
Y\times\uin^{m+k+1}$ is $(k+1)$toone.
\end{inparaenum}


622  Hyperplanes of the Form ${f_1(x,y)z_1+\dots+f_k(x,y)z_k+g(x,y)}$ Are Variables Vénéreau, Stéphane
The AbhyankarSathaye Embedded Hyperplane Problem asks whe\ther any
hypersurface of $\C^n$ isomorphic to $\C^{n1}$ is rectifiable, {\em
i.e.,}
equivalent to a linear hyperplane up to an automorphism of $\C^n$.
Generalizing the approach adopted by Kaliman, V\'en\'ereau, and
Zaidenberg which
consists in using almost nothing but the acyclicity of $\C^{n1}$, we solve
this problem for hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$
as in the title.


636  Correction to: On the Diophantine Equation $n(n+d)\cdots(n+(k1)d)=by^l$ Győry, K.; Hajdu, L.; Saradha, N.
In the article under consideration
(Canad. Math. Bull. \textbf{47} (2004), pp.~373388),
Lemma 6 is not true in the form presented there.
Lemma 6 is used only in the proof of part (i) of Theorem 9.
We note, however, that part (i) of Theorem 9 in question is a special
case of a theorem by Bennet, Bruin, Gy\H{o}ry and Hajdu.


637  Author Index  Index des auteurs 2005, for 2005  pour
No abstract.
