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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 Müntz 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 Müntz 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:Müntz space, embedding, essential norm, compact operator
Categories:32C22, 47B33, 30B10

3. CMB Online first

Kocel-Cynk, Beata; Pawłucki, Wiesław; Valette, Anna
$\mathcal{C}^p$-parametrization in o-minimal structures
We give a geometric and elementary proof of the uniform $\mathcal C^p$-parametrization theorem of Yomdin-Gromov in arbitrary o-minimal structures.

Keywords:o-minimal structure, $C^p$-parametrization
Categories:03C64, 14P15, 32B20

4. CMB Online first

Purbhoo, Kevin
Total nonnegativity and stable polynomials
We consider homogeneous multiaffine polynomials whose coefficients are the Plücker 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 Pólya-Schur theory of Borcea and Brändén.

Keywords:stable polynomial, zeros of a complex polynomial, total nonnegative Grassmannian, totally nonnegative matrix
Categories:32A60, 14M15, 14P10, 15B48

5. CMB Online first

Jiang, Cao; Dong, Xing-Tang; Zhou, Ze-Hua
Commuting and semi-commuting monomial-type Toeplitz operators on some weakly pseudoconvex domains
In this paper, we completely characterize the finite rank commutator and semi-commutator of two monomial-type Toeplitz operators on the Bergman space of certain weakly pseudoconvex domains. Somewhat surprisingly, there are not only plenty of commuting monomial-type Toeplitz operators but also non-trivial semi-commuting monomial-type Toeplitz operators. Our results are new even for the unit ball.

Keywords:Toeplitz operator, Bergman space, monomial-type symbol, weakly pseudoconvex domain
Categories:47B35, 32A36

6. CMB Online first

Pascoe, J. E.
The wedge-of-the-edge theorem: edge-of-the-wedge type phenomenon within the common real boundary
The edge-of-the-wedge 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 wedge-of-the-edge 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:edge-of-the-wedge theorem, several complex variables, analytic continuation

7. CMB Online first

Li, Liangliang; Tian, Jing; Chen, Goong
Chaotic vibration of a two-dimensional non-strictly 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 two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the 2D non-strictly 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, period-doubling 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 Calabi-Yau 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 Calabi-Yau 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 sub-variation 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 Calabi-Yau 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)

Wang, Zhenjian
On Deformations of Nodal Hypersurfaces
We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.

Keywords:nodal hypersurface, deformation, Torelli theorem
Categories:32S35, 14C30, 14D07, 32S25

10. CMB 2017 (vol 61 pp. 637)

Nemirovski, Stefan; Shafikov, Rasul Gazimovich
Uniformization and Steinness
It is shown that the unit ball in $\mathbb{C}^n$ is the only complex manifold that can universally cover both Stein and non-Stein strictly pseudoconvex domains.

Keywords:Stein manifold, covering, spherical domain
Categories:32T15, 32Q30

11. CMB 2017 (vol 61 pp. 628)

Marković, Marijan
Differential-free characterisation of smooth mappings with controlled growth
In this paper we give some generalizations and improvements of the Pavlović result on the Holland-Walsh 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, Holland-Walsh characterisation, hyperbolic distance, analytic function, Mobius transform
Categories:32A18, 30D45

12. CMB 2017 (vol 60 pp. 705)

Benelkourchi, Slimane
Envelope Approach to Degenerate Complex Monge-Ampère Equations on Compact Kähler Manifolds
We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.

Keywords:degenerate complex Monge-Ampère equation, compact Kähler manifold, big cohomology, plurisubharmonic function
Categories:32W20, 32Q25, 32U05

13. CMB 2017 (vol 61 pp. 376)

Sebbar, Abdellah; Al-Shbeil, 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 half-plane 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 Real-analytic 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 real-analytic 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)

Bayart, Frédéric; Gauthier, Paul M
Functions Universal for All Translation Operators in Several Complex Variables
We prove the existence of a (in fact many) holomorphic function $f$ in $\mathbb{C}^d$ such that, for any $a\neq 0$, its translations $f(\cdot+na)$ are dense in $H(\mathbb{C}^d)$.

Keywords:hypercyclic operator, translation operator
Categories:47A16, 32E20

16. CMB 2017 (vol 61 pp. 85)

Ding, Fan; Geiges, Hansjörg; Zhang, Guangjian
On subcritically Stein fillable 5-manifolds
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, 5-manifold, 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 non--constant 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, Douady-Sentenac invariant
Categories:34M45, 32G34

20. CMB 2017 (vol 61 pp. 166)

Miranda-Neto, Cleto B.
A module-theoretic characterization of algebraic hypersurfaces
In this note we prove the following surprising characterization: if $X\subset {\mathbb A}^n$ is an (embedded, non-empty, 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)

Dimca, Alexandru
The Poincaré-Deligne Polynomial of Milnor Fibers of Triple Point Line Arrangements is Combinatorially Determined
Using a recent result by S. Papadima and A. Suciu, we show that the equivariant Poincaré-Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined.

Keywords:line arrangement, Milnor fiber, monodromy, mixed Hodge structures
Categories:32S22, 32S35, 32S25, 32S55

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 bi-orderable. 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 Harish-Chandra superpairs. A universal complexification of a real Lie supergroup is constructed.

Keywords:Lie supergroup, almost complex structure, Harish-Chandra pair, universal complexification
Categories:32C11, 58A50
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