26. CMB 2011 (vol 56 pp. 31)
 Ayuso, Fortuny P.

Derivations and Valuation Rings
A complete characterization of valuation rings closed for a
holomorphic derivation is given, following an idea of Seidenberg,
in dimension $2$.
Keywords:singular holomorphic foliation, derivation, valuation, valuation ring Categories:32S65, 13F30, 13A18 

27. CMB 2011 (vol 56 pp. 44)
28. CMB 2011 (vol 55 pp. 108)
29. CMB 2011 (vol 55 pp. 329)
 Kamiya, Shigeyasu; Parker, John R.; Thompson, James M.

NonDiscrete Complex Hyperbolic Triangle Groups of Type $(n,n, \infty;k)$
A complex hyperbolic triangle group is a group
generated by three involutions fixing complex lines in complex
hyperbolic space. Our purpose in this paper is to improve a previous result
and to discuss discreteness of complex hyperbolic
triangle groups of type $(n,n,\infty;k)$.
Keywords:complex hyperbolic triangle group Categories:51M10, 32M15, 53C55, 53C35 

30. CMB 2011 (vol 55 pp. 249)
 Chang, DerChen; Li, Bao Qin

Description of Entire Solutions of Eiconal Type Equations
The paper describes entire solutions to the eiconal type nonlinear partial differential
equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases,
where
$X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$,
$X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$
are linearly independent operators with $p_j$ being arbitrary
polynomials in $\mathbf{C}^2$.
Keywords:entire solution, eiconal equation, polynomial, transcendental function Categories:32A15, 35F20 

31. CMB 2011 (vol 55 pp. 242)
 Cegrell, Urban

Convergence in Capacity
In this note we study the convergence of sequences of MongeAmpÃ¨re measures $\{(dd^cu_s)^n\}$,
where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.
Keywords:complex MongeAmpÃ¨re operator, convergence in capacity, plurisubharmonic function Categories:32U20, 31C15 

32. CMB 2011 (vol 55 pp. 441)
 Zorboska, Nina

Univalently Induced, Closed Range, Composition Operators on the Blochtype Spaces
While there is a large variety of univalently induced closed range
composition operators on the Bloch space,
we show that the only univalently induced, closed range, composition
operators on the Blochtype spaces $B^{\alpha}$ with $\alpha \ne 1$
are the ones induced by a disc automorphism.
Keywords:composition operators, Blochtype spaces, closed range, univalent Categories:47B35, 32A18 

33. CMB 2011 (vol 55 pp. 146)
 Li, Songxiao; Wulan, Hasi; Zhu, Kehe

A Characterization of Bergman Spaces on the Unit Ball of ${\mathbb C}^n$. II
It has been shown that a holomorphic function $f$ in the unit ball
$\mathbb{B}_n$ of ${\mathbb C}_n$ belongs to the weighted Bergman space $A^p_\alpha$,
$p>n+1+\alpha$, if and only if the function
$f(z)f(w)/1\langle z,w\rangle$ is in $L^p(\mathbb{B}_n\times\mathbb{B}_n,dv_\beta
\times dv_\beta)$, where $\beta=(p+\alphan1)/2$ and $dv_\beta(z)=
(1z^2)^\beta\,dv(z)$. In this paper
we consider the range $0 n+1+\alpha$ is
particularly interesting.
Keywords:Bergman spaces, unit ball, volume measure Category:32A36 

34. CMB 2011 (vol 54 pp. 230)
 Clouâtre, Raphaël

Universal Power Series in $\mathbb{C}^N$
We establish the existence of power series in $\mathbb{C}^N$ with the property
that the subsequences of the sequence of partial sums uniformly
approach any holomorphic function on any well chosen compact subset
outside the set of convergence of the series. We also show that, in a
certain sense, most series enjoy this property.
Categories:32A05, 32E30 

35. CMB 2011 (vol 54 pp. 338)
 Nakazi, Takahiko

SzegÃ¶'s Theorem and Uniform Algebras
We study SzegÃ¶'s theorem for a uniform algebra.
In particular, we do it for the disc algebra or the bidisc algebra.
Keywords:SzegÃ¶'s theorem, uniform algebras, disc algebra, weighted Bergman space Categories:32A35, 46J15, 60G25 

36. CMB 2010 (vol 54 pp. 370)
37. CMB 2010 (vol 54 pp. 56)
 Dinh, Thi Anh Thu

Characteristic Varieties for a Class of Line Arrangements
Let $\mathcal{A}$ be a line arrangement in the complex projective plane
$\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two
lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the
nonlocal irreducible components of the first resonance variety
$\mathcal{R}_1(\mathcal{A})$ are 2dimensional and correspond to parallelograms $\mathcal{P}$ in
$\mathbb{C}^2=\mathbb{P}^2 \setminus H_{\infty}$ whose sides are in $\mathcal{A}$ and for
which $H_0$ is a diagonal.
Keywords:local system, line arrangement, characteristic variety, resonance variety Categories:14C21, 14F99, 32S22, 14E05, 14H50 

38. CMB 2010 (vol 53 pp. 311)
 Jasiczak, Michał

Remark on Zero Sets of Holomorphic Functions in Convex Domains of Finite Type
We prove that if the $(1,1)$current of integration on an analytic subvariety $V\subset D$ satisfies the uniform Blaschke condition, then $V$ is the zero set of a holomorphic function $f$ such that $\log f$ is a function of bounded mean oscillation in $bD$. The domain $D$ is assumed to be smoothly bounded and of finite d'Angelo type. The proof amounts to nonisotropic estimates for a solution to the $\overline{\partial}$equation for Carleson measures.
Categories:32A60, 32A35, 32F18 

39. CMB 2009 (vol 53 pp. 23)
 Chen, Huaihui; Zhang, Minzhu

Boundedness From Below of Multiplication Operators Between $\alpha$Bloch Spaces
In this paper, the boundedness from below of multiplication
operators between $\alpha$Bloch spaces $\mathcal B^\alpha$, $\alpha\gt 0$, on the
unit disk $D$ is studied completely. For a bounded multiplication
operator $M_u\colon \mathcal B^\alpha\to\mathcal B^\beta$, defined by $M_uf=uf$ for
$f\in\mathcal B^\alpha$, we prove the following result:
(i) If $0\lt \beta\lt \alpha$, or
$0\lt \alpha\le1$ and $\alpha\lt \beta$, $M_u$ is not bounded below;
(ii) if $0\lt \alpha=\beta\le1$, $M_u$ is bounded below if and only if
$\liminf_{z\to\partial D}u(z)\gt 0$;
(iii) if $1\lt \alpha\le\beta$, $M_u$ is
bounded below if and only if there exist a $\delta\gt 0$ and a positive
$r\lt 1$ such that for every point $z\in D$ there is a point $z'\in
D$ with the property $d(z',z)\lt r$ and
$(1z'^2)^{\beta\alpha}u(z')\ge\delta$, where $d(\cdot,\cdot)$ denotes
the pseudodistance on $D$.
Keywords:$\alpha$Bloch function, multiplication operator Categories:32A18, 30H05 

40. CMB 2009 (vol 53 pp. 11)
 Burke, Maxim R.

Approximation and Interpolation by Entire Functions of Several Variables
Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon
\mathbb R^n\to\mathbb R$ be positive
and continuous. For any unbounded nondecreasing sequence $\{c_k\}$
of nonnegative real numbers and for any sequence without
accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire
function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that
\begin{align*}
&g^{(\alpha)}(x)f^{(\alpha)}(x)\lt h(x), \quad x\ge c_k, \alpha\le k,
k=0,1,2,\dots,
\\
&g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad x_m\ge c_k, \alpha\le k,
m,k=0,1,2,\dots.
\end{align*}
This is a version for functions of several variables of the
case $n=1$ due to L. Hoischen.
Keywords:entire function, complex approximation, interpolation, several complex variables Category:32A15 

41. CMB 2009 (vol 52 pp. 613)
 Wulan, Hasi; Zhu, Kehe

Lipschitz Type Characterizations for Bergman Spaces
We obtain new characterizations for Bergman spaces with standard
weights in terms of Lipschitz type conditions in the Euclidean,
hyperbolic, and pseudohyperbolic metrics. As a consequence, we
prove optimal embedding theorems when an analytic function
on the unit disk is symmetrically lifted to the bidisk.
Keywords:Bergman spaces, hyperbolic metric, Lipschitz condition Category:32A36 

42. CMB 2009 (vol 52 pp. 285)
43. CMB 2009 (vol 52 pp. 175)
 Biswas, Indranil

Connections on a Parabolic Principal Bundle, II
In \emph{Connections on a parabolic principal bundle over a curve, I}
we defined connections on a parabolic
principal bundle. While connections on usual principal bundles are
defined as splittings of the Atiyah exact sequence, it was noted in
the above article that the Atiyah exact sequence does not generalize to
the parabolic principal bundles.
Here we show that a twisted version
of the Atiyah exact sequence generalizes to the context of
parabolic principal bundles. For usual principal bundles, giving a
splitting of this twisted Atiyah exact sequence is equivalent
to giving a splitting of the Atiyah exact sequence. Connections on
a parabolic principal bundle can be defined using the
generalization of the twisted Atiyah exact sequence.
Keywords:Parabolic bundle, Atiyah exact sequence, connection Categories:32L05, 14F05 

44. CMB 2009 (vol 52 pp. 154)
45. CMB 2009 (vol 52 pp. 84)
46. CMB 2008 (vol 51 pp. 481)
 Bayart, Frédéric

Universal Inner Functions on the Ball
It is shown that given any sequence of automorphisms $(\phi_k)_k$ of the
unit ball $\bn$ of $\cn$ such that $\\phi_k(0)\$ tends to $1$,
there exists an inner function
$I$ such that the family of ``nonEuclidean translates"
$(I\circ\phi_k)_k$ is locally uniformly dense in the unit ball of
$H^\infty(\bn)$.
Keywords:inner functions, automorphisms of the ball, universality Categories:32A35, 30D50, 47B38 

47. CMB 2008 (vol 51 pp. 618)
 Valmorin, V.

Vanishing Theorems in Colombeau Algebras of Generalized Functions
Using a canonical linear embedding of the algebra
${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of
$\overline{\C}$valued $\C$linear maps on the space
${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing
conditions for functions and linear integral operators of class
${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic
generalized functions in dimension greater than one.
Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functions Categories:32A60, 45P05, 46F30 

48. CMB 2008 (vol 51 pp. 195)
49. CMB 2008 (vol 51 pp. 21)
 Baracco, Luca

A Remark on Extensions of CR Functions from Hyperplanes
In the characterization of the range of the Radon transform, one
encounters the problem of the holomorphic extension of functions
defined on $\R^2\setminus\Delta_\R$ (where $\Delta_\R$ is the diagonal
in $\R^2$) and which extend as ``separately holomorphic" functions of
their two arguments. In particular, these functions extend in fact to $\C^2\setminus
\Delta_\C$ where $\Delta_\C$ is the complexification of
$\Delta_\R$. We take this theorem from the integral geometry and put
it in the more natural context of the CR geometry where it accepts an
easier proof and a more general statement. In this new setting it
becomes a variant of the celebrated ``edge of the wedge" theorem of
Ajrapetyan and Henkin.
Categories:32D10, 32V25 

50. CMB 2007 (vol 50 pp. 243)
 Langlands, Robert P.

Un nouveau point de repÃ¨re dans la thÃ©orie des formes automorphes
Dans le papier Beyond Endoscopy une id\'ee pour entamer la
fonctorialit\'e en utilisant la formule des traces a \'et\'e
introduite. Maints probl\`emes, l'existence d'une limite convenable
de la formule des traces, est eqquiss\'ee dans cette note
informelle mais seulement pour $GL(2)$ et les corps des fonctions
rationelles sur un corps fini et en ne pas resolvant
bon nombre de questions.
Categories:32N10, 14xx 
