1. CMB Online first
 Wang, Jianfei; Zhang, Danli

Extension operators for biholomorphic mappings
Suppose that $D\subset\mathbb{C}$ is a simply connected subdomain
containing the origin and $f(z_1)$ is a normalized convex (resp.,
starlike) function on $D$. Let
$$
\Omega_{N}(D)=\{(z_1,w_1,\ldots,w_k)\in \mathbb{C}\times{\mathbb{C}}^{n_1}\times\cdots\times{\mathbb{C}}^{n_k}:
\w_1\_{p_1}^{p_1}+\cdots+\w_k\_{p_k}^{p_k}<\frac{1}{\lambda_{D}(z_1)}\},$$
where $p_j\geq 1$, $N=1+n_1+\cdots+n_k,\,w_1\in{\mathbb{C}}^{n_1},\ldots,w_k\in{\mathbb{C}}^{n_k}$
and $\lambda_{D}$ is the density of the hyperbolic metric on
$D$. In this paper, we prove that
\begin{equation*}
\Phi_{N,{1/p_{1}},\cdots,{1}/{p_{k}}}(f)(z_1,w_1,\ldots,w_k)=\big(f(z_{1}),
(f'(z_{1}))^{1/p_{1}}w_1,\cdots,(f'(z_{1}))^{1/p_{k}}w_k\big)
\end{equation*}
is a normalized convex (resp., starlike) mapping on $\Omega_{N}(D)$.
If $D$ is the unit disk, then our result reduces to Gong and
Liu \cite{GL2} via a new method. Moreover, we give a new operator
for
convex mapping construction on an unbounded domain in ${\mathbb{C}}^{2}$.
By using a geometric approach, we prove that $\Phi_{N,{1/p_{1}},\cdots,{1}/{p_{k}}}(f)$
is a spirallike mapping of type $\alpha$ when $f$ is
a spirallike function of type $\alpha$ on the unit disk.
Keywords:biholomorphic mapping, $\varepsilon$starlike mapping, spirallike mapping Categories:32H02, 30C45 

2. CMB Online first
 Alam, Ihab Al; Lefèvre, Pascal

Embeddings of MÃ¼ntz Spaces in $L^\infty(\mu)$
In this paper, we discuss the properties of the embedding
operator $i^\Lambda_\mu : M_\Lambda^\infty\hookrightarrow L^\infty(\mu),$
where $\mu$ is a positive Borel measure on $[0,1]$ and $M_{\Lambda}^{\infty}$
is a MÃ¼ntz space. In particular, we compute the essential norm
of this embedding. As a consequence, we recover some results
of
the first author.
We also study the compactness (resp. weak compactness)
and compute the essential norm (resp. generalized essential norm)
of the embedding $i_{\mu_1,\,\mu_2} : L^\infty(\mu_1)\hookrightarrow
L^\infty(\mu_2)$, where $\mu_1$, $\mu_2$ are two positive Borel
measures on $[0,1]$ with $\mu_2$ absolutely continuous with respect
to $\mu_1$.
Keywords:MÃ¼ntz space, embedding, essential norm, compact operator Categories:32C22, 47B33, 30B10 

3. CMB Online first
4. CMB Online first
 Purbhoo, Kevin

Total nonnegativity and stable polynomials
We consider homogeneous multiaffine polynomials whose coefficients
are the PlÃ¼cker coordinates of a point $V$ of the Grassmannian.
We show that such a polynomial is stable (with respect to the
upper half plane) if and only if $V$ is in the totally nonnegative
part of the Grassmannian. To prove this, we consider an action
of
matrices on multiaffine polynomials. We show that
a matrix $A$ preserves stability of polynomials if and only if
$A$ is totally nonnegative. The proofs are applications of classical
theory of totally nonnegative matrices, and the generalized
PÃ³lyaSchur theory of Borcea and BrÃ¤ndÃ©n.
Keywords:stable polynomial, zeros of a complex polynomial, total nonnegative Grassmannian, totally nonnegative matrix Categories:32A60, 14M15, 14P10, 15B48 

5. CMB Online first
6. CMB Online first
 Pascoe, J. E.

The wedgeoftheedge theorem: edgeofthewedge type phenomenon within the common real boundary
The edgeofthewedge theorem in several complex variables gives
the analytic continuation of functions defined on the poly upper
half plane and the poly lower half plane, the set of points in
$\mathbb{C}^{n}$ with all coordinates in the upper and lower
half planes respectively, through a set in real space, $\mathbb{R}^{n}$.
The geometry of the set in the real space can force the function
to analytically continue within the boundary itself, which is
qualified in our wedgeoftheedge theorem. For example, if a
function extends to the union of two cubes in $\mathbb{R}^{n}$
that are positively oriented with some small overlap, the
functions must analytically continue to a neighborhood of that
overlap of a fixed size not depending of the size of the overlap.
Keywords:edgeofthewedge theorem, several complex variables, analytic continuation Category:32A40 

7. CMB Online first
 Li, Liangliang; Tian, Jing; Chen, Goong

Chaotic vibration of a twodimensional nonstrictly hyperbolic equation
The study of chaotic vibration for multidimensional PDEs due
to nonlinear boundary conditions is challenging. In this paper,
we mainly investigate the chaotic oscillation of a twodimensional
nonstrictly hyperbolic equation due to an energyinjecting
boundary condition and a distributed selfregulating boundary
condition. By using the method of characteristics, we give a
rigorous proof of the onset of the chaotic vibration phenomenon
of the 2D nonstrictly hyperbolic equation. We have also found
a regime of the parameters when the chaotic vibration phenomenon
occurs. Numerical simulations are also provided.
Keywords:chaotic vibration, reflection boundary condition, perioddoubling bifurcation, method of characteristics Categories:32H50, 34C28, 37K50, 54H20, 58J45 

8. CMB Online first
 Zhang, Zheng

On motivic realizations of the canonical Hermitian variations of Hodge structure of CalabiYau type over type $D^{\mathbb H}$ domains
Let $\mathcal{D}$ be the irreducible Hermitian symmetric domain
of type $D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian
variation of real Hodge structure $\mathcal{V}_{\mathbb{R}}$
of CalabiYau type over $\mathcal{D}$. This short note concerns
the problem of giving motivic realizations for $\mathcal{V}_{\mathbb{R}}$.
Namely, we specify a descent of $\mathcal{V}_{\mathbb{R}}$ from
$\mathbb{R}$ to $\mathbb{Q}$ and ask whether the $\mathbb{Q}$descent
of $\mathcal{V}_{\mathbb{R}}$ can be realized as subvariation
of rational Hodge structure of those coming from families of
algebraic varieties. When $n=2$, we give a motivic realization
for $\mathcal{V}_{\mathbb{R}}$. When $n \geq 3$, we show that
the unique irreducible factor of CalabiYau type in $\mathrm{Sym}^2
\mathcal{V}_{\mathbb{R}}$ can be realized motivically.
Keywords:variations of Hodge structure, Hermitian symmetric domain Categories:14D07, 32G20, 32M15 

9. CMB 2017 (vol 61 pp. 659)
10. CMB 2017 (vol 61 pp. 637)
11. CMB 2017 (vol 61 pp. 628)
 Marković, Marijan

Differentialfree characterisation of smooth mappings with controlled growth
In this paper we give some generalizations and
improvements of the PavloviÄ result on the
HollandWalsh type characterization of the Bloch space of
continuously differentiable (smooth) functions in
the unit ball in $\mathbf{R}^m$.
Keywords:Bloch type space, Lipschitz type space, HollandWalsh characterisation, hyperbolic distance, analytic function, Mobius transform Categories:32A18, 30D45 

12. CMB 2017 (vol 60 pp. 705)
13. CMB 2017 (vol 61 pp. 376)
 Sebbar, Abdellah; AlShbeil, Isra

Elliptic Zeta Functions and Equivariant Functions
In this paper we establish a close connection between three
notions attached to a modular subgroup. Namely the set of weight
two meromorphic modular forms, the set of equivariant functions
on the upper halfplane commuting with the action of the modular
subgroup and the set of elliptic zeta functions generalizing
the Weierstrass zeta functions. In particular, we show that the
equivariant functions can be parameterized by modular objects
as well as by elliptic objects.
Keywords:modular form, equivariant function, elliptic zeta function Categories:11F12, 35Q15, 32L10 

14. CMB 2017 (vol 61 pp. 289)
 Gupta, Purvi

A Realanalytic Nonpolynomially Convex Isotropic Torus with No Attached Discs
We show by means of an example in $\mathbb C^3$ that Gromov's
theorem on the presence of attached holomorphic discs for compact
Lagrangian manifolds is not true in the subcritical
realanalytic case, even in the absence of an obvious obstruction,
i.e, polynomial convexity.
Keywords:polynomial hull, isotropic submanifold, holomorphic disc Categories:32V40, 32E20, 53D12 

15. CMB 2017 (vol 60 pp. 462)
16. CMB 2017 (vol 61 pp. 85)
 Ding, Fan; Geiges, Hansjörg; Zhang, Guangjian

On subcritically Stein fillable 5manifolds
We make some elementary observations concerning subcritically
Stein
fillable contact structures on $5$manifolds.
Specifically, we determine the diffeomorphism type of such
contact manifolds in the case the fundamental group is finite
cyclic,
and we show that on the $5$sphere the standard contact structure
is the unique subcritically fillable one. More generally,
it is shown that subcritically fillable contact structures
on simply connected $5$manifolds are determined by their
underlying almost contact structure. Along the way, we discuss
the
homotopy classification of almost contact structures.
Keywords:subcritically Stein fillable, 5manifold, almost contact structure, thickening Categories:53D35, 32Q28, 57M20, 57Q10, 57R17 

17. CMB 2017 (vol 60 pp. 736)
 Gilligan, Bruce

Levi's Problem for Pseudoconvex Homogeneous Manifolds
Suppose $G$ is a connected complex Lie group and $H$ is a closed
complex subgroup.
Then there exists a closed complex subgroup $J$ of $G$ containing
$H$ such that
the fibration $\pi:G/H \to G/J$ is the holomorphic reduction
of $G/H$, i.e., $G/J$ is holomorphically
separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$.
In this paper we prove that if $G/H$ is pseudoconvex, i.e.,
if
$G/H$ admits a continuous plurisubharmonic exhaustion function,
then $G/J$ is Stein and $J/H$ has no nonconstant holomorphic
functions.
Keywords:complex homogeneous manifold, plurisubharmonic exhaustion function, holomorphic reduction, Stein manifold, Remmert reduction, Hirschowitz annihilator Categories:32M10, 32U10, 32A10, 32Q28 

18. CMB 2017 (vol 61 pp. 142)
 Li, Bao Qin

An Equivalent Form of Picard's Theorem and Beyond
This paper gives an equivalent form of Picard's
theorem via entire solutions of the functional equation $f^2+g^2=1$,
and then its improvements and applications to certain nonlinear
(ordinary and partial) differential equations.
Keywords:entire function, Picard's Theorem, functional equation, partial differential equation Categories:30D20, 32A15, 35F20 

19. CMB 2017 (vol 60 pp. 381)
 Rousseau, C.

The Bifurcation Diagram of Cubic Polynomial Vector Fields on $\mathbb{C}\mathbb{P}^1$
In this paper we give the bifurcation diagram
of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$
for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of
$\epsilon_1,\epsilon_0\in\mathbb{C}$.
The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure
allows describing it for parameter values in $\mathbb{S}^3$. There are
two open simply connected regions of structurally stable vector
fields separated by surfaces corresponding to bifurcations of
homoclinic connections between two separatrices of the pole at
infinity. These branch from the codimension 2 curve of double
singular points. We also explain the bifurcation of homoclinic
connection in terms of the description of Douady and Sentenac
of polynomial vector fields.
Keywords:complex polynomial vector field, bifurcation diagram, DouadySentenac invariant Categories:34M45, 32G34 

20. CMB 2017 (vol 61 pp. 166)
 MirandaNeto, Cleto B.

A moduletheoretic characterization of algebraic hypersurfaces
In this note we prove the following surprising characterization:
if
$X\subset {\mathbb A}^n$ is an (embedded, nonempty, proper)
algebraic variety defined over a
field $k$ of characteristic zero, then $X$ is a hypersurface
if and only if the module $T_{{\mathcal O}_{{\mathbb
A}^n}/k}(X)$ of logarithmic vector fields of
$X$ is a reflexive ${\mathcal
O}_{{\mathbb A}^n}$module. As a consequence of this result,
we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a
free ${\mathcal
O}_{{\mathbb A}^n}$module, which is shown to be equivalent
to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb
A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily
$X$ is a Saito free divisor.
Keywords:hypersurface, logarithmic vector field, logarithmic derivation, free divisor Categories:14J70, 13N15, 32S22, 13C05, 13C10, 14N20, , , , , 14C20, 32M25 

21. CMB 2016 (vol 59 pp. 449)
 Abdallah, Nancy

On Hodge Theory of Singular Plane Curves
The dimensions of the graded quotients of the
cohomology of a plane curve complement $U=\mathbb P^2 \setminus C$
with respect to the Hodge filtration are described in terms of
simple geometrical invariants. The case of curves with ordinary
singularities is discussed in detail. We also give a precise
numerical estimate for the difference between the Hodge filtration
and the pole order filtration on $H^2(U,\mathbb C)$.
Keywords:plane curves, Hodge and pole order filtrations Categories:32S35, 32S22, 14H50 

22. CMB 2016 (vol 59 pp. 279)
23. CMB 2016 (vol 59 pp. 346)
 Krantz, Steven

On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains
We study and generalize a classical theorem of L. Bers that classifies
domains up to biholomorphic equivalence in terms of the algebras
of
holomorphic functions on those domains. Then we develop applications
of these results to the study of domains with noncompact automorphism
group.
Keywords:Bers's theorem, algebras of holomorphic functions, noncompact automorphism group, biholomorphic equivalence Categories:32A38, 30H50, 32A10, 32M99 

24. CMB 2015 (vol 59 pp. 182)
 Naylor, Geoff; Rolfsen, Dale

Generalized Torsion in Knot Groups
In a group, a nonidentity element is called
a generalized torsion element if some product of its conjugates
equals the identity. We show that for many classical knots one
can find generalized torsion in the fundamental group of its
complement, commonly called the knot group. It follows that
such a group is not biorderable. Examples include all torus
knots, the (hyperbolic) knot $5_2$ and algebraic knots in the
sense of Milnor.
Keywords:knot group, generalized torsion, ordered group Categories:57M27, 32S55, 29F60 

25. CMB 2015 (vol 58 pp. 281)
 Kalus, Matthias

On the Relation of Real and Complex Lie Supergroups
A complex Lie supergroup can be described as a real Lie supergroup
with integrable almost complex structure. The necessary and
sufficient conditions on an almost complex structure on a real
Lie supergroup for defining a complex Lie supergroup are deduced.
The classification of real Lie supergroups with such almost
complex
structures yields a new approach to the known classification
of complex Lie supergroups by complex HarishChandra superpairs.
A universal complexification of a real Lie supergroup is
constructed.
Keywords:Lie supergroup, almost complex structure, HarishChandra pair, universal complexification Categories:32C11, 58A50 
