26. CMB 2015 (vol 58 pp. 381)
 Tang, Xiaomin; Liu, Taishun

The Schwarz Lemma at the Boundary of the Egg Domain $B_{p_1, p_2}$ in $\mathbb{C}^n$
Let $B_{p_1, p_2}=\{z\in\mathbb{C}^n:
z_1^{p_1}+z_2^{p_2}+\cdots+z_n^{p_2}\lt 1\}$
be an egg domain in $\mathbb{C}^n$. In this paper, we first
characterize the Kobayashi metric on $B_{p_1, p_2}\,(p_1\geq
1, p_2\geq 1)$,
and then establish a new type of the classical boundary Schwarz
lemma at $z_0\in\partial{B_{p_1, p_2}}$ for holomorphic selfmappings
of $B_{p_1, p_2}(p_1\geq 1, p_2\gt 1)$, where $z_0=(e^{i\theta},
0, \dots, 0)'$ and $\theta\in \mathbb{R}$.
Keywords:holomorphic mapping, Schwarz lemma, Kobayashi metric, egg domain Categories:32H02, 30C80, 32A30 

27. CMB 2014 (vol 57 pp. 697)
28. CMB 2014 (vol 57 pp. 658)
 Thang, Nguyen Tat

Admissibility of Local Systems for some Classes of Line Arrangements
Let $\mathcal{A}$ be a line arrangement in the complex
projective plane $\mathbb{P}^2$ and let $M$ be its complement. A rank one
local system $\mathcal{L}$ on $M$ is admissible if roughly speaking
the cohomology groups
$H^m(M,\mathcal{L})$ can be computed directly from the cohomology
algebra $H^{*}(M,\mathbb{C})$. In this work, we give a sufficient
condition for the admissibility of all rank one local systems on
$M$. As a result, we obtain some properties of the characteristic
variety $\mathcal{V}_1(M)$ and the Resonance variety $\mathcal{R}_1(M)$.
Keywords:admissible local system, line arrangement, characteristic variety, multinet, resonance variety Categories:14F99, 32S22, 52C35, 05A18, 05C40, 14H50 

29. CMB 2014 (vol 57 pp. 673)
 Ahmadi, S. Ruhallah; Gilligan, Bruce

Complexifying Lie Group Actions on Homogeneous Manifolds of Noncompact Dimension Two
If $X$ is a connected complex manifold with $d_X = 2$ that admits a (connected) Lie group $G$
acting transitively as a group of holomorphic transformations, then the action extends to an action of the
complexification $\widehat{G}$ of $G$ on $X$ except when
either the unit disk in the complex plane
or a strictly pseudoconcave homogeneous complex manifold is
the base or fiber of some homogeneous fibration of $X$.
Keywords:homogeneous complex manifold, noncompact dimension two, complexification Category:32M10 

30. CMB 2013 (vol 57 pp. 870)
 Parlier, Hugo

A Short Note on Short Pants
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and SeppÃ¤lÃ¤. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.
Keywords:hyperbolic surfaces, geodesics, pants decompositions Categories:30F10, 32G15, 53C22 

31. CMB 2013 (vol 57 pp. 794)
 Fang, ZhongShan; Zhou, ZeHua

New Characterizations of the Weighted Composition Operators Between Bloch Type Spaces in the Polydisk
We give some new characterizations for compactness of weighted
composition operators $uC_\varphi$ acting on Blochtype spaces in
terms of the power of the components of $\varphi,$ where $\varphi$
is a holomorphic selfmap of the polydisk $\mathbb{D}^n,$ thus
generalizing the results obtained by HyvÃ¤rinen and
LindstrÃ¶m in 2012.
Keywords:weighted composition operator, compactness, Bloch type spaces, polydisk, several complex variables Categories:47B38, 47B33, 32A37, 45P05, 47G10 

32. CMB 2012 (vol 57 pp. 12)
33. CMB 2011 (vol 56 pp. 593)
34. 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 

35. CMB 2011 (vol 56 pp. 44)
36. CMB 2011 (vol 55 pp. 108)
37. 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 

38. 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 

39. 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 

40. 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 

41. 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 

42. 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 

43. 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 

44. CMB 2010 (vol 54 pp. 370)
45. 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 

46. 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 

47. 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 

48. 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 

49. 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 

50. CMB 2009 (vol 40 pp. 129)